Learning Module 1 - Pricing and Valuation of Forward Commitments (EOC)

64 Learning Module 1 Pricing and Valuation of Forward Commitments PRACTICE PROBLEMS The following information relates to questions 1-6 Tim Doyle is a portfolio manager at BestFutures Group, a hedge fund that frequently enters into derivative contracts either to hedge the risk of investments it holds or to speculate outside of those investments. Doyle works alongside Diane Kemper, a junior analyst at the hedge fund. They meet to evaluate new investment ideas and to review several of the firm’s existing investments. Carry Arbitrage Model Doyle and Kemper discuss the carry arbitrage model and how they can take advantage of mispricing in bond markets. Specifically, they would like to execute an arbitrage transaction on a Eurodollar futures contract in which the underlying Eurodollar bond is expected to make an interest payment in two months. Doyle makes the following statements: Statement 1 If the Eurodollar futures price is less than the price suggested by the carry arbitrage model, the futures contract should be purchased. Statement 2 Based on the cost of carry model, the futures price would be higher if the underlying Eurodollar bond’s upcoming interest payment was expected in five months instead of two. Three-Year Treasury Note Futures Contract Kemper then presents two investment ideas to Doyle. Kemper’s first investment idea is to purchase a three-year Treasury note futures contract. The underlying 1.5%, semi-annual three-year Treasury note is quoted at a clean price of 101. It has been 60 days since the three-year Treasury note’s last coupon payment, and the next coupon payment is payable in 120 days. Doyle asks Kemper to calculate the full spot price of the underlying three-year Treasury note. 10-Year Treasury Note Futures Contract Kemper’s second investment idea is to purchase a 10-year Treasury note futures contract. The underlying 2%, semi-annual 10-year Treasury note has a dirty price of 104.17. It has been 30 days since the 10-year Treasury note’s last coupon payment. The futures contract expires in 90 days. The quoted futures contract price is 129. The current annualized three-month risk-free rate is 1.65%. The conversion factor is 0.7025. Doyle asks Kemper to calculate the equilibrium quoted futures contract price based on the carry arbitrage model. Japanese Government Bonds After discussing Kemper’s new investment ideas, Doyle and Kemper evaluate one of their existing forward contract positions. Three months ago, BestFutures took a long position in eight 10-year Japanese government bond (JGB) forward contracts, with each contract having a contract notional value of 100 million yen. The contracts had a price of JPY153 (quoted as a percentage of par) when the contracts were purchased. Now, the contracts have six months left to expiration and have a price of JPY155. The annualized six-month interest rate is 0.12%. Doyle asks Kemper to value the JGB forward position. Interest Rate Swaps Additionally, Doyle asks Kemper to price a one-year plain vanilla swap. The spot Practice Problems 65 rates and days to maturity at each payment date are presented in Exhibit 1. Exhibit 1: Selected US Spot Rate Data Days to Maturity Spot Interest Rates (%) 90 1.90 180 2.00 270 2.10 360 2.20 Finally, Doyle and Kemper review one of BestFutures’s pay-fixed interest rate swap positions. Two years ago, the firm entered into a JPY5 billion five-year interest rate swap, paying the fixed rate. The fixed rate when BestFutures entered into the swap two years ago was 0.10%. The current term structure of interest rates for JPY cash flows, which are relevant to the interest rate swap position, is presented in Exhibit 2. Exhibit 2: Selected Japanese Interest Rate Data Yen Spot Interest Rates (%) Present Value Factors 1 0.03 0.9997 2 0.06 0.9988 3 0.08 0.9976 Maturity (Years) Sum 2.9961 Doyle asks Kemper to calculate the value of the pay-fixed interest rate swap. 1. Which of Doyle’s statements regarding the Eurodollar futures contract price is correct? A. Only Statement 1 B. Only Statement 2 C. Both Statement 1 and Statement 2 2. The full spot price of the three-year Treasury note is: A. 101.00. B. 101.25. C. 101.50. 3. The equilibrium 10-year Treasury note quoted futures contract price is closest to: A. 147.94. B. 148.89. C. 149.78. 66 Learning Module 1 Pricing and Valuation of Forward Commitments 4. The value of the JGB long forward position is closest to: A. JPY15,980,823. B. JPY15,990,409. C. JPY16,000,000. 5. Based on Exhibit 1, the fixed rate of the one-year plain vanilla swap is closest to: A. 0.12%. B. 0.55%. C. 0.72%. 6. Based on Exhibit 2, the value of the pay-fixed interest rate swap is closest to: A. –JPY6,491,550. B. –JPY2,980,500. C. –JPY994,793. The following information relates to questions 7-11 Donald Troubadour is a derivatives trader for Southern Shores Investments. The firm seeks arbitrage opportunities in the forward and futures markets using the carry arbitrage model. Troubadour identifies an arbitrage opportunity relating to a fixed-income futures contract and its underlying bond. Current data on the futures contract and underlying bond are presented in Exhibit 1. The current annual compounded risk-free rate is 0.30%. Exhibit 1: Current Data for Futures and Underlying Bond Futures Contract Underlying Bond Quoted futures price 125.00 Conversion factor 0.90 Time remaining to contract expiration Three months Accrued interest over life of futures contract Quoted bond price 112.00 Accrued interest since last coupon payment 0.08 Accrued interest at futures contract expiration 0.20 0.00 Troubadour next gathers information on a Japanese equity index futures contract, the Nikkei 225 Futures Contract: Troubadour holds a long position in a Nikkei 225 futures contract that has a remaining maturity of three months.The continuously compounded dividend yield on the Nikkei 225 Stock Index is 1.1%, and the current stock index level is 16,080. The continuously compounded annual interest rate is 0.2996%. Troubadour next considers an equity forward contract for Texas Steel, Inc. (TSI). Information regarding TSI common shares and a TSI equity forward contract is Practice Problems presented in Exhibit 2. Exhibit 2: Selected Information for TSI ■ The price per share of TSI’s common shares is $250. ■ The forward price per share for a nine-month TSI equity forward contract is $250.562289. ■ Assume annual compounding. Troubadour takes a short position in the TSI equity forward contract. His supervisor asks, “Under which scenario would our position experience a loss?” Three months after contract initiation, Troubadour gathers information on TSI and the risk-free rate, which is presented in Exhibit 3. Exhibit 3: Selected Data on TSI and the Risk-Free Rate (Three Months Later) ■ The price per share of TSI’s common shares is $245. ■ The risk-free rate is 0.325% (quoted on an annual compounding basis). ■ TSI recently announced its regular semiannual dividend of $1.50 per share that will be paid exactly three months before contract expiration. ■ The market price of the TSI equity forward contract is equal to the no-arbitrage forward price. 7. Based on Exhibit 1 and assuming annual compounding, the arbitrage profit on the bond futures contract is closest to: A. 0.4158. B. 0.5356. C. 0.6195. 8. The current no-arbitrage futures price of the Nikkei 225 futures contract is closest to: A. 15,951.81. B. 16,047.86. C. 16,112.21. 9. Based on Exhibit 2, Troubadour should find that an arbitrage opportunity relating to TSI shares is A. not available. B. available based on carry arbitrage. C. available based on reverse carry arbitrage. 10. The most appropriate response to Troubadour’s supervisor’s question regarding the TSI forward contract is: A. a decrease in TSI’s share price, all else equal. B. an increase in the risk-free rate, all else equal 67 68 Learning Module 1 Pricing and Valuation of Forward Commitments C. a decrease in the market price of the forward contract, all else equal. 11. Based on Exhibits 2 and 3, and assuming annual compounding, the per share value of Troubadour’s short position in the TSI forward contract three months after contract initiation is closest to: A. $1.6549. B. $5.1561. C. $6.6549. The following information relates to questions 12-20 Sonal Johnson is a risk manager for a bank. She manages the bank’s risks using a combination of swaps and forward rate agreements (FRAs). Johnson prices a three-year MRR-based interest rate swap with annual resets using the present value factors presented in Exhibit 1. Exhibit 1: Present Value Factors Maturity (years) Present Value Factors 1 0.990099 2 0.977876 3 0.965136 Johnson also uses the present value factors in Exhibit 1 to value an interest rate swap that the bank entered into one year ago as the pay-fixed (receive-floating) party. Selected data for the swap are presented in Exhibit 2. Johnson notes that the current equilibrium two-year fixed swap rate is 1.12%. Exhibit 2: Selected Data on Fixed for Floating Interest Rate Swap Swap notional amount $50,000,000 Original swap term Three years, with annual resets Fixed swap rate (since initiation) 3.00% One of the bank’s investments is exposed to movements in the Japanese yen, and Johnson desires to hedge the currency exposure. She prices a one-year fixed-for-fixed currency swap involving yen and US dollars, with a quarterly reset. Johnson uses the interest rate data presented in Exhibit 3 to price the currency swap. Practice Problems 69 Exhibit 3: Selected Japanese and US Interest Rate Data Days to Maturity Yen Spot Interest Rates US Dollar Spot Interest Rates 90 0.05% 0.20% 180 0.10% 0.40% 270 0.15% 0.55% 360 0.25% 0.70% Johnson next reviews an equity swap with an annual reset that the bank entered into six months ago as the receive-fixed, pay-equity party. Selected data regarding the equity swap, which is linked to an equity index, are presented in Exhibit 4. At the time of initiation, the underlying equity index was trading at 100.00. Exhibit 4: Selected Data on Equity Swap Swap notional amount $20,000,000 Original swap term Five years, with annual resets Fixed swap rate 2.00% The equity index is currently trading at 103.00, and relevant US spot rates, along with their associated present value factors, are presented in Exhibit 5. Exhibit 5: Selected US Spot Rates and Present Value Factors Maturity (years) Spot Rate Present Value Factors 0.5 0.40% 0.998004 1.5 1.00% 0.985222 2.5 1.20% 0.970874 3.5 2.00% 0.934579 4.5 2.60% 0.895255 Johnson reviews a 6 × 9 FRA that the bank entered into 90 days ago as the pay-fixed/receive-floating party. Selected data for the FRA are presented in Exhibit 6, and current MRR data are presented in Exhibit 7. Based on her interest rate forecast, Johnson also considers whether the bank should enter into new positions in 1 × 4 and 2 × 5 FRAs. Exhibit 6: 6 × 9 FRA Data FRA term 6×9 FRA rate 0.70% FRA notional amount US$20,000,000 FRA settlement terms Advanced set, advanced settle 70 Learning Module 1 Pricing and Valuation of Forward Commitments Exhibit 7: Current MRR (Market Reference Rate) 30-day MRR 0.75% 60-day MRR 0.82% 90-day MRR 0.90% 120-day MRR 0.92% 150-day MRR 0.94% 180-day MRR 0.95% 210-day MRR 0.97% 270-day MRR 1.00% Three months later, the 6 × 9 FRA in Exhibit 6 reaches expiration, at which time the three-month US dollar MRR is 1.10% and the six-month US dollar MRR is 1.20%. Johnson determines that the appropriate discount rate for the FRA settlement cash flows is 1.10%. 12. Based on Exhibit 1, Johnson should price the three-year MRR-based interest rate swap at a fixed rate closest to: A. 0.34%. B. 1.16%. C. 1.19%. 13. From the bank’s perspective, using data from Exhibit 1, the current value of the swap described in Exhibit 2 is closest to: A. –$2,951,963. B. –$1,849,897. C. –$1,943,000. 14. Based on Exhibit 3, Johnson should determine that the annualized equilibrium fixed swap rate for Japanese yen is closest to: A. 0.0624%. B. 0.1375%. C. 0.2496%. 15. From the bank’s perspective, using data from Exhibits 4 and 5, the fair value of the equity swap is closest to: A. –$1,139,425. B. –$781,322. C. –$181,323. 16. Based on Exhibit 5, the current value of the equity swap described in Exhibit 4 would be zero if the equity index was currently trading the closest to: A. 97.30. B. 99.09. Practice Problems C. 100.00. 17. From the bank’s perspective, based on Exhibits 6 and 7, the value of the 6 × 9 FRA 90 days after inception is closest to: A. $14,820. B. $19,647. C. $29,635. 18. Based on Exhibit 7, the no-arbitrage fixed rate on a new 1 × 4 FRA is closest to: A. 0.65%. B. 0.73%. C. 0.98%. 19. Based on Exhibit 7, the fixed rate on a new 2 × 5 FRA is closest to: A. 0.61%. B. 1.02%. C. 1.71%. 20. Based on Exhibit 6 and the three-month US dollar MRR at expiration, the payment amount that the bank will receive to settle the 6 × 9 FRA is closest to: A. $19,945. B. $24,925. C. $39,781. 71 72 Learning Module 1 Pricing and Valuation of Forward Commitments SOLUTIONS 1. C is correct. Doyle’s first statement is correct. Unless the Eurodollar futures contract’s quoted price is equal to the no-arbitrage futures price, there is an arbitrage opportunity. Moreover, if the quoted futures price is less than the no-arbitrage futures price, then to take advantage of the arbitrage opportunity, the Eurodollar futures contract should be purchased and the underlying Eurodollar bond should be sold short. Doyle would then lend the short sale proceeds at the risk-free rate. The strategy that comprises those transactions is known as reverse carry arbitrage. Doyle’s second statement is also correct. Based on the cost of carry model, the futures price is calculated as the future value of the sum of the underlying plus the underlying carry costs minus the future value of any ownership benefits. If the Eurodollar bond’s interest payment was expected in five months instead of two, the benefit of the cash flow would occur three months later, so the future value of the benefits term would be slightly lower. Therefore, the Eurodollar futures contract price would be slightly higher if the Eurodollar bond’s interest payment was expected in five months instead of two months. A is incorrect because Doyle’s Statement 2 is correct (not incorrect). Based on the cost of carry model, the futures price would be higher if the underlying Eurodollar bond’s interest payment took place in five months instead of two months. B is incorrect because Doyle’s Statement 1 is correct (not incorrect). If the Eurodollar’s futures contract price is less than the price suggested by the carry arbitrage model, the futures contract should be purchased. 2. B is correct. The full spot price of the three-year Treasury note is calculated as S0 = Quoted bond price + Accrued interest = B0 + AI0. ​Accrued interest ​​(​AI​)​​ = Accrural period × Periodic coupon amount ​NTD ​)​​× ​( ​_ ​n ) ​ ​​. = ​( ​_ NAD C AI = (60/180) × (0.015/2) = 0.25. S0 = 101 + 0.25 = 101.25. A is incorrect because 101 is the quoted clean (not the full spot) price of the three-year Treasury note. The clean price excludes accrued interest; the full price, also referred to as the dirty price, includes accrued interest. C is incorrect because the number of days until the next coupon payment (instead of the accrual period) is incorrectly used to compute accrued interest: AI = (120/180) × (0.015/2) = 0.50. S0 = 101 + 0.50 = 101.50. 3. A is correct. The equilibrium 10-year quoted futures contract price based on the carry arbitrage model is calculated as Q0 = (1/CF) × [FV(B0 + AI0) − AIT− FVCI]. CF = 0.7025. B0 = 104.00. AI0 = 0.17. Solutions 73 AIT = (120/180 × 0.02/2) = 0.67. FVCI = 0. ​​Q0​ ​ = ​(​ ​1 / 0.7025​)​​× ​[​ ​(​1 + 0.0165​)​3/12​​(​104.17​)​​− 0.67 − 0​]​​ = 147.94. B is incorrect because accrued interest at expiration is not subtracted in the equilibrium quoted futures contract price formula: ​​Q0​ ​ = ​(​ ​1 / 0.7025​)​​× ​[​ (​ ​1 + 0.0165​)​3/12​(​ ​104.17​)​​− 0​]​​ = 148.89. C is incorrect because the future value is incorrectly calculated (the exponent of 3/12 is omitted): ​​Q0​ ​ = ​(​ ​1 / 0.7025​)​​× ​[​ ​(​ ​1 + 0.0165​)​​(​ ​104.17​)​​− 0.67 − 0​]​​ = 149.78.​ 4. B is correct. The value of the JGB forward position is calculated as ​ 0​ ​]​​ = ​(​ ​155 − 153​)​​/ ​(​1 + 0.0012​)​​12 ​​ = 1.9988. ​​Vt​​ = PV​​[​Ft​​− F 6 _ Therefore, the value of the long forward position is 1.9988 per JPY100 par value. For the long position in eight contracts with each contract having a par value of 100 million yen, the value of the position is calculated as 0.019988 × (JPY100,000,000) × 8 = JPY15,990,409. A is incorrect because the present value of the difference between the price when the contracts were purchased and the current price of the contracts was incorrectly computed (the exponent of 6/12 is omitted): Vt = Ft − F0 = (155 − 153)/(1 + 0.0012) = 1.9980. 0.019980 × (JPY100,000,000) × 8 = JPY15,980,823. C is incorrect because the absolute difference (not the present value of the difference) between the price when the contracts were purchased and the current price of the contracts was computed: Vt = Ft − F0 = (155 − 153) = 2. 0.02 × (JPY100,000,000) × 8 = JPY16,000,000. 5. B is correct. The swap’s fixed rate is calculated as n ​​r​FIX​ = ​[​ ​1 − ​PV​n​​(​1​)​​]​​/ ​∑​​​PV​i​​(​1​)​​. i=1 PVi (1) = 1/[1 + Rspoti (NADi/NTD)]. 90 − day PV factor = 1/[1 + 0.019 × (90/360)] = 0.9953. 180 − day PV factor = 1/[1 + 0.020 × (180/360)] = 0.9901. 270 − day PV factor = 1/[1 + 0.021 × (270/360)] = 0.9845. 360 − day PV factor = 1/[1 + 0.022 × (360/360)] = 0.9785. 74 Learning Module 1 Pricing and Valuation of Forward Commitments 4 ​​∑​​​PV​i​(​ ​1)​ ​​ = 0.9953 + 0.9901 + 0.9845 + 0.9785 = 3.9483.​ i=1 rFIX = (1 − 0.9785)/3.9483 = 0.0055 = 0.55%. A is incorrect because the 90-day PV factor is incorrectly used in the numerator of the swap pricing equation instead of the final present value term: n ​​r​FIX​ = ​[​ ​1 − ​PV​n​​(​1​)​​]​​/ ​∑​​​PV​i​​(​1​)​​.​ i=1 rFIX = (1 − 0.9953)/3.9483 = 0.0012 = 0.12%. C is incorrect because the sum of the present value terms excludes the final present value term: 3 ​​∑​​​PV​i​(​ ​1)​ ​​ = 0.9953 + 0.9901 + 0.9845 = 2.9699.​ i=1 n r ​​ ​FIX​ = ​[​ ​1 − ​PV​n​(​ ​1)​ ​​]​​/ ​∑​​​PV​i​(​ ​1)​ ​​.​ i=1 rFIX = (1 − 0.9785)/2.9699 = 0.0072 = 0.72%. 6. B is correct. The value of the pay-fixed interest rate swap is calculated as ​− ​VSWAP,t ​ ​ = NA×​​(​FS​t​− ​FS​0​)​​× ​∑​ni=1​​PV​i​. 3 ​​FS​t​ = r​ ​FIX​ = ​[​ ​1 − ​PV​n​​(​1​)​​]​​/ ​∑​​​PV​i​​(​1​)​​ = ​(​ ​1 − 0.9976​)​​/ 2.9961 = 0.000801 i=1 = 0.08%.​ ​ = NA×​​(​FS​t​− ​FS​0​)​​× ​∑​3i=1​​PV​i −​VSWAP,t ​           ​ = ​​ JPY5billion × ​​(​0.000801 − 0.001​)​​× 2.9961 = − JPY2, 980, 500. Given that rates have declined since the inception of the swap, the value of the pay-fixed, receive-floating position is currently a loss of JPY2,980,500. A is incorrect because the arithmetic average of the yen spot rates (instead of the current fixed swap rate) was incorrectly used to calculate the value of the pay-fixed swap: Arithmetic average of yen spot rates = (0.0003 + 0.0006 + 0.0008)/3 = 0.0006. ​ = NA × ​​(​FS​t​− ​FS​0​)​​× ​∑​3i=1​P ​ V​i −​VSWAP,t ​           ​​ JPY5billion × ​​(​0.0006 − 0.001​)​​× 2.9961 = = − JPY6, 491, 550. C is incorrect because the product of the notional amount and the difference between the initial swap fixed rate and the current swap fixed rate was not multiplied by the sum of the present values: ​− ​VSWAP,t ​ ​ = NA×​​(​FS​t​− ​FS​0​)​​ = JPY5billion × ​(​ ​0.0008 − 0.001​) = − JPY994, 793. Solutions 7. B is correct. The no-arbitrage futures price is equal to the following: F0 = FV[B0 + AI0 – PVCI] F0 = (1 + 0.003)0.25(112.00 + 0.08 – 0) = 112.1640. The adjusted price of the futures contract is equal to the conversion factor multiplied by the quoted futures price: F0 = CF × Q0 F0 = (0.90)(125) = 112.50. Adding the accrued interest of 0.20 in three months (futures contract expiration) to the adjusted price of the futures contract gives a total price of 112.70. This difference means that the futures contract is overpriced by 112.70 – 112.1640 = 0.5360. The available arbitrage profit is the present value of this difference: 0.5360/(1.003)0.25 = 0.5356. 8. B is correct. The no-arbitrage futures price is ​​F​0​ = ​S0​ ​e​ xp​​​​(​rc​ ​+CC−CB​)​​T F0 = 16,080exp(0.002996 + 0 – 0.011)(3/12) = 16,047.86. 9. A is correct. The carry arbitrage model price of the forward contract is FV(S0) = S0(1 + r)T = $250(1 + 0.003)0.75 = $250.562289. The market price of the TSI forward contract is $250.562289. A carry or reverse carry arbitrage opportunity does not exist because the market price of the forward contract is equal to the carry arbitrage model price. 10. B is correct. From the perspective of the long position, the forward value is equal to the present value of the difference in forward prices: Vt = PV[Ft – F0], where Ft = FV(St +CCt –CBt). All else equal, an increase in the risk-free rate before contract expiration would cause the forward price, Ft, to increase. This increase in the forward price would cause the value of the TSI forward contract, from the perspective of the short, to decrease. Therefore, an increase in the risk-free rate would lead to a loss on the short position in the TSI forward contract. 11. C is correct. The no-arbitrage price of the forward contract, three months after 75 76 Learning Module 1 Pricing and Valuation of Forward Commitments contract initiation, is F0.25 = FV0.25(S0.25 + CC0.25 – CB0.25) F0.25 = [$245 + 0 – $1.50/(1 + 0.00325)(0.5 – 0.25)](1 + 0.00325)(0.75 – 0.25) = $243.8966. Therefore, from the perspective of the long, the value of the TSI forward contract is V0.25 = PV0.25[F0.25 – F0] V0.25 = ($243.8966 – $250.562289)/(1 + 0.00325)0.75 – 0.25 = –$6.6549. Because Troubadour is short the TSI forward contract, the value of his position is a gain of $6.6549. 12. C is correct. The swap pricing equation is 1 − P ​Vn​ ​​(​1)​ ​​ ​​∑​n ​P ​V​​(​ ​1​)​​​​. ​​r​FIX​ = _ i=1 i That is, the fixed swap rate is equal to 1 minus the final present value factor (in this case, Year 3) divided by the sum of the present values (in this case, the sum of Years 1, 2, and 3). The sum of present values for Years 1, 2, and 3 is calculated as n ​∑​​P V ​ i​​(​ ​1)​ ​​ = 0.990099 + 0.977876 + 0.965136 = 2.933111​. i=1 Thus, the fixed-swap rate is calculated as 1 − 0.965136 ​ 2.933111 ​ = 0.01189 or 1.19%​. ​​r​FIX​ = _ 13. B is correct. The value of a swap from the perspective of the receive-fixed (pay-floating) party is calculated as ​V = NA×​(​ ​FS​0​− ​FS​t​)​​× ​∑​ni=1​P ​ V​i​​. The swap has two years remaining until expiration. The sum of the present values for Years 1 and 2 is ​​∑​ni=1​​PV​i​ = 0.990099 + 0.977876 = 1.967975​. Given the current equilibrium two-year swap rate of 1.12% and the fixed swap rate at initiation of 3.00%, the swap value per dollar notional is calculated as V = 1×(0.03 – 0.0112) x 1.967975 = 0.036998. The current value of the swap, from the perspective of the receive-fixed party, is $50,000,000 × 0.036998 = $1,849,897. From the perspective of the bank, as the pay-fixed party, the value of the swap is –$1,849,897. 14. C is correct. The equilibrium swap fixed rate for yen is calculated as 1 − P ​Vn,JPY ​ ​(​ ​1)​ ​​       ​4 .​​ ​​r​JPY​ = ____________ ( ) ​∑​i=1​P ​Vi,JPY ​ ​​ ​1​ ​​ The yen present value factors are calculated as 1 ​   ​PV ​(​1​)​i,JPY​ = _________________ NA D ​ i​​ ​​, 1 + Rspo ​t​i,JPY​​(_ ​NTD ​)​​ Solutions 77 where 90-day PV factor = 1/[1 + 0.0005(90/360)] = 0.999875 180-day PV factor = 1/[1 + 0.0010(180/360)] = 0.999500 270-day PV factor = 1/[1 + 0.0015(270/360)] = 0.998876 360-day PV factor = 1/[1 + 0.0025(360/360)] = 0.997506 Sum of present value factors = 3.995757 Therefore, the yen periodic rate is calculated as 1 − ​PV​n​​(​1​)​​ 1 − 0.997506 ​4 ​= _ ​ 3.995757 ​ = 0.000624 = 0.0624%. ​​r​JPY​ = _ ( ) ​∑​i=1​​PV​i​​ ​1​ ​​ The annualized rate is (360/90) times the periodic rate of 0.0624%, or 0.2496%. 15. B is correct. The value of an equity swap at time t is calculated as VEQ,t = VFIX(C0) – (St/St–1)NAE – PV(Par – NAE). The swap was initiated six months ago, so the first reset has not yet passed; thus, there are five remaining cash flows for this equity swap. The fair value of the swap is determined by comparing the present value of the implied fixed-rate bond with the return on the equity index. The fixed swap rate of 2.00%, the swap notional amount of $20,000,000, and the present value factors in Exhibit 5 result in a present value of the implied fixed-rate bond’s cash flows of $19,818,678: PV Factors Fixed Cash Flow PV (fixed cash flow) 0.5 0.998004 or 1/[1 + 0.0040(180/360)] $400,000 $399,202 1.5 0.985222 or 1/[1 + 0.0100(540/360)] $400,000 $394,089 2.5 0.970874 or 1/[1 + 0.0120(900/360)] $400,000 $388,350 3.5 0.934579 or 1/[1 + 0.0200(1,260/360)] $400,000 $373,832 4.5 0.895255 or 1/[1 + 0.0260(1,620/360)] $20,400,000 $18,263,205 Date (in years) Total $19,818,678 The value of the equity leg of the swap is calculated as (103/100)($20,000,000) = $20,600,000. Note the swap’s notional amount and the implied fixed-rate bond’s par value are both $20,000,000; therefore, the term – PV(Par – NAE) reduces to zero. The swap was designed to profit if rates fell or equities declined. Neither happened, so the swap value will be negative for the bank. The fair value of the equity swap, from the perspective of the bank (receive-fixed, pay-equity party) is calculated as VEQ = $19,818,678 - $20,600,000 = –$781,322. 16. B is correct. The equity index level at which the swap’s fair value would be zero can be calculated by setting the swap valuation formula equal to zero and solving for St: 78 Learning Module 1 Pricing and Valuation of Forward Commitments VEQ,t = VFIX(C0) – (St/St–1)NAE = 0. The value of the fixed leg of the swap has a present value of $19,818,678, or 99.0934% of par value: Date (years) PV Factors Fixed Cash Flow PV (fixed cash flow) 0.998004 $400,000 $399,202 0.5 1.5 0.985222 $400,000 $394,089 2.5 0.970874 $400,000 $388,350 3.5 0.934579 $400,000 $373,832 4.5 0.895255 $20,400,000 $18,263,205 Total $19,818,678 Treating the swap notional value as par value and substituting the present value of the fixed leg and S0 into the equation yields ​St​​ ​0 = 99.0934 − ( ​_ ​100 ​)​​100​ Solving for St yields St = 99.0934. 17. A is correct. The current value of the 6 × 9 FRA is calculated as Vg= NA × {[FRAg – FRA0]tm}/[1 + D(T–g) t(T–g)]. The 6 × 9 FRA expires six months after initiation. The bank entered into the FRA 90 days ago; thus, the FRA will expire in 90 days. To value the FRA, the first step is to compute the new FRA rate, which is the rate on Day 90 of an FRA that expires in 90 days in which the underlying is the 90-day MRR: FRAg = {[1 + LTtT]/[1+Lhth] – 1}/tm FRAg = {[1 + L180(180/360)]/[1 + L90 (90/360)] – 1}/(90/360) Exhibit 7 indicates that L90 = 0.90% and L180 = 0.95%, so FRAg = {[1 + 0.0095(180/360)]/[1 + 0.0090(90/360)] – 1}/(90/360) FRAg= {[1.00475/1.00225] – 1]} × 4 = 0.009978, or 0.9978%. Therefore, given the FRA rate at initiation of 0.70% and notional principal of $20 million from Exhibit 1, the current value of the forward contract is calculated as Vg = $20,000,000 × [(0.009978 – 0.0070)(90/360)]/[1 + 0.0095(180/360)]. = $14,890.00/1.00475 = $14,819.61. 18. C is correct. The no-arbitrage fixed rate on the 1 × 4 FRA is calculated as FRA0 = {[1 + LTtT]/[1 + Lhth] – 1}/tm . For a 1 × 4 FRA, the two rates needed to compute the no-arbitrage FRA fixed rate are L30 = 0.75% and L120 = 0.92%. Therefore, the no-arbitrage fixed rate on the 1 × 4 FRA rate is calculated as FRA0= {[1 + 0.0092(120/360)]/[1 + 0.0075(30/360)] – 1}/(90/360). FRA0= {[1.003066/1.000625] – 1} × 4 = 0.009761, or 0.98% rounded. Solutions 19. B is correct. The fixed rate on the 2 × 5 FRA is calculated as FRA0 = {[1 + LTtT]/[1 + Lhth] – 1}/tm. For a 2 × 5 FRA, the two rates needed to compute the no-arbitrage FRA fixed rate are L60= 0.82% and L150 = 0.94%. Therefore, the no-arbitrage fixed rate on the 2 × 5 FRA rate is calculated as FRA0= {[1 + 0.0094(150/360)]/[1 + 0.0082(60/360)] – 1}/(90/360) FRA0 = {[(1.003917/1.001367) – 1} × 4 = 0.010186, or 1.02% rounded. 20. A is correct. Given a three-month US dollar MRR of 1.10% at expiration, the settlement amount for the bank as the pay-fixed (receive-floating) party is calculated as Settlement amount pay-fixed (receive floating) = NA × {[Lm – FRA0]tm}/[1 + Dmtm]}. Settlement amount pay-fixed (receive floating) = $20,000,000 × {[0.011 – 0.0070] × (90/360)]/[1 + 0.011(90/360)]}. Settlement amount pay-fixed (receive floating) = $20,000,000 × (0.001)/1.00275 = $19,945.15. Therefore, the bank will receive $19,945 (rounded) as the receive-floating party. 79 146 Learning Module 2 Valuation of Contingent Claims PRACTICE PROBLEMS The following information relates to questions 1-9 Bruno Sousa has been hired recently to work with senior analyst Camila Rocha. Rocha gives him three option valuation tasks. Alpha Company Sousa’s first task is to illustrate how to value a call option on Alpha Company with a one-period binomial option pricing model. It is a non-dividend-paying stock, and the inputs are as follows. ■ The current stock price is 50, and the call option exercise price is 50. ■ In one period, the stock price will either rise to 56 or decline to 46. ■ The risk-free rate of return is 5% per period. Based on the model, Rocha asks Sousa to estimate the hedge ratio, the risk-neutral probability of an up move, and the price of the call option. In the illustration, Sousa is also asked to describe related arbitrage positions to use if the call option is overpriced relative to the model. Beta Company Next, Sousa uses the two-period binomial model to estimate the value of a European-style call option on Beta Company’s common shares. The inputs are as follows. ■ The current stock price is 38, and the call option exercise price is 40. ■ The up factor (u) is 1.300, and the down factor (d) is 0.800. ■ The risk-free rate of return is 3% per period. Sousa then analyzes a put option on the same stock. All of the inputs, including the exercise price, are the same as for the call option. He estimates that the value of a European-style put option is 4.53. Exhibit 1 summarizes his analysis. Sousa next must determine whether an American-style put option would have the same value. Practice Problems 147 Exhibit 1: Two-Period Binomial European-Style Put Option on Beta Company Item Value Underlying 38 Put 4.5346 Hedge Ratio –0.4307 Time = 0 Item Value Underlying 49.4 Put 0.2517 Hedge Ratio –0.01943 Item Value Underlying 30.4 Put 8.4350 Hedge Ratio –1 Time = 1 Item Value Underlying 64.22 Put 0 Item Value Underlying 39.52 Put 0.48 Item Value Underlying 24.32 Put 15.68 Time = 2 Sousa makes two statements with regard to the valuation of a European-style option under the expectations approach. Statement 1 The calculation involves discounting at the risk-free rate. Statement 2 The calculation uses risk-neutral probabilities instead of true probabilities. Rocha asks Sousa whether it is ever profitable to exercise American options prior to maturity. Sousa answers, “I can think of two possible cases. The first case is the early exercise of an American call option on a dividend-paying stock. The second case is the early exercise of an American put option.” Interest Rate Option The final option valuation task involves an interest rate option. Sousa must value a two-year, European-style call option on a one-year spot rate. The notional value of the option is 1 million, and the exercise rate is 2.75%. The risk-neutral probability of an up move is 0.50. The current and expected one-year interest rates are shown in Exhibit 2, along with the values of a one-year zero-coupon bond of 1 notional value for each interest rate. 148 Learning Module 2 Valuation of Contingent Claims Exhibit 2: Two-Year Interest Rate Lattice for an Interest Rate Option Maturity Value Rate 1 0.961538 4% Maturity Value Rate 1 0.952381 5% Maturity Value Rate Maturity Value Rate 1 0.970874 3% 1 0.970874 3% Maturity Value Rate 1 0.990099 1% Time = 0 Maturity Value Rate 1 0.980392 2% Time = 1 Time = 2 Rocha asks Sousa why the value of a similar in-the-money interest rate call option decreases if the exercise price is higher. Sousa provides two reasons. Reason 1 The exercise value of the call option is lower. Reason 2 The risk-neutral probabilities are changed. 1. The optimal hedge ratio for the Alpha Company call option using the one-period binomial model is closest to: A. 0.60. B. 0.67. C. 1.67. 2. The risk-neutral probability of the up move for the Alpha Company stock is closest to: A. 0.06. B. 0.40. C. 0.65. 3. The value of the Alpha Company call option is closest to: A. 3.71. B. 5.71. C. 6.19. 4. For the Alpha Company option, the positions to take advantage of the arbitrage opportunity are to write the call and: A. short shares of Alpha stock and lend. Practice Problems B. buy shares of Alpha stock and borrow. C. short shares of Alpha stock and borrow. 5. The value of the European-style call option on Beta Company shares is closest to: A. 4.83. B. 5.12. C. 7.61. 6. The value of the American-style put option on Beta Company shares is closest to: A. 4.53. B. 5.15. C. 9.32. 7. Which of Sousa’s statements about binomial models is correct? A. Statement 1 only B. Statement 2 only C. Both Statement 1 and Statement 2 8. Based on Exhibit 2 and the parameters used by Sousa, the value of the interest rate option is closest to: A. 5,251. B. 6,236. C. 6,429. 9. Which of Sousa’s reasons for the decrease in the value of the interest rate option is correct? A. Reason 1 only B. Reason 2 only C. Both Reason 1 and Reason 2 The following information relates to questions 10-17 Trident Advisory Group manages assets for high-net-worth individuals and family trusts. Alice Lee, chief investment officer, is meeting with a client, Noah Solomon, to discuss risk management strategies for his portfolio. Solomon is concerned about recent volatility and has asked Lee to explain options valuation and the use of options in risk management. Options on Stock Lee uses the BSM model to price TCB, which is one of Solomon’s holdings. 149 150 Learning Module 2 Valuation of Contingent Claims Exhibit 1 provides the current stock price (S), exercise price (X), risk-free interest rate (r), volatility (σ), and time to expiration (T) in years as well as selected outputs from the BSM model. TCB does not pay a dividend. Exhibit 1: BSM Model for European Options on TCB BSM Inputs S X r σ T $57.03 55 0.22% 32% 0.25 d1 N(d1) d2 N(d2) BSM Call Price BSM Put Price 0.3100 0.6217 0.1500 0.5596 $4.695 $2.634 BSM Outputs Options on Futures The Black model valuation and selected outputs for options on another of Solomon’s holdings, the GPX 500 Index (GPX), are shown in Exhibit 2. The spot index level for the GPX is 187.95, and the index is assumed to pay a continuous dividend at a rate of 2.2% (δ) over the life of the options being valued, which expire in 0.36 years. A futures contract on the GPX also expiring in 0.36 years is currently priced at 186.73. Exhibit 2: Black Model for European Options on the GPX Index Black Model Inputs GPX Index 187.95 Black Model Call Value $14.2089 X r σ T δ Yield 180 0.39% 24% 0.36 2.2% Black Model Put Value Market Call Price Market Put Price $7.4890 $14.26 $7.20 Option Greeks Delta (call) 0.6232 Delta (put) Gamma (call or put) Theta (call) daily Rho (call) per % Vega per % (call or put) –0.3689 0.0139 –0.0327 0.3705 0.4231 After reviewing Exhibit 2, Solomon asks Lee which option Greek letter best describes the changes in an option’s value as time to expiration declines. Solomon observes that the market price of the put option in Exhibit 2 is $7.20. Lee responds that she used the historical volatility of the GPX of 24% as an input to the BSM model, and she explains the implications for the implied volatility for the GPX. Options on Interest Rates Solomon forecasts the three-month MRR will exceed 0.85% in six months and is considering using options to reduce the risk of rising rates. He asks Lee to value an interest rate call with a strike price of 0.85%. The current three-month MRR Practice Problems is 0.60%, and an FRA for a three-month MRR loan beginning in six months is currently 0.75%. Hedging Strategy for the Equity Index Solomon’s portfolio currently holds 10,000 shares of an exchange-traded fund (ETF) that tracks the GPX. He is worried the index will decline. He remarks to Lee, “You have told me how the BSM model can provide useful information for reducing the risk of my GPX position.” Lee suggests a delta hedge as a strategy to protect against small moves in the GPX Index. Lee also indicates that a long position in puts could be used to hedge larger moves in the GPX. She notes that although hedging with either puts or calls can result in a delta-neutral position, they would need to consider the resulting gamma. 10. Based on Exhibit 1 and the BSM valuation approach, the initial portfolio required to replicate the long call option payoff is: A. long 0.3100 shares of TCB stock and short 0.5596 shares of a zero-coupon bond. B. long 0.6217 shares of TCB stock and short 0.1500 shares of a zero-coupon bond. C. long 0.6217 shares of TCB stock and short 0.5596 shares of a zero-coupon bond. 11. To determine the long put option value on TCB stock in Exhibit 1, the correct BSM valuation approach is to compute: A. 0.4404 times the present value of the exercise price minus 0.6217 times the price of TCB stock. B. 0.4404 times the present value of the exercise price minus 0.3783 times the price of TCB stock. C. 0.5596 times the present value of the exercise price minus 0.6217 times the price of TCB stock. 12. What are the correct spot value (S) and the risk-free rate (r) that Lee should use as inputs for the Black model? A. 186.73 and 0.39%, respectively B. 186.73 and 2.20%, respectively C. 187.95 and 2.20%, respectively 13. Which of the following is the correct answer to Solomon’s question regarding the option Greek letter? A. Vega B. Theta C. Gamma 14. Based on Solomon’s observation about the model price and market price for the put option in Exhibit 2, the implied volatility for the GPX is most likely: A. less than the historical volatility. 151 152 Learning Module 2 Valuation of Contingent Claims B. equal to the historical volatility. C. greater than the historical volatility. 15. The valuation inputs used by Lee to price a call reflecting Solomon’s interest rate views should include an underlying FRA rate of: A. 0.60% with six months to expiration. B. 0.75% with nine months to expiration. C. 0.75% with six months to expiration. 16. The strategy suggested by Lee for hedging small moves in Solomon’s ETF position would most likely involve: A. selling put options. B. selling call options. C. buying call options. 17. Lee’s put-based hedge strategy for Solomon’s ETF position would most likely result in a portfolio gamma that is: A. negative. B. neutral. C. positive. Solutions SOLUTIONS 1. A is correct. The hedge ratio requires the underlying stock and call option values for the up move and down move. S+ = 56, and S– = 46. c+ = Max(0,S+ – X) = Max(0,56 – 50) = 6, and c– = Max(0,S– – X) = Max(0,46 – 50) = 0. The hedge ratio is ​c​+​− ​c​−​ 6−0 6 ​= _ ​56 − 46 ​ = _ ​10 ​ = 0.60 ​h = _ ​ ​S​+​− ​S​−​ 2. C is correct. For this approach, the risk-free rate is r = 0.05, the up factor is u = S+/S = 56/50 = 1.12, and the down factor is d = S–/S = 46/50 = 0.92. The risk-neutral probability of an up move is π = [FV(1) – d]/(u – d) = (1 + r – d]/(u – d) π = (1 + 0.05 – 0.92)/(1.12 – 0.92) = 0.13/0.20 = 0.65 3. A is correct. The call option can be estimated using the no-arbitrage approach or the expectations approach. With the no-arbitrage approach, the value of the call option is c = hS + PV(–hS– + c–). h = (c+ – c–)/(S+ – S–) = (6 – 0)/(56 – 46) = 0.60. c = (0.60 × 50) + (1/1.05) × [(–0.60 × 46) + 0]. c = 30 – [(1/1.05) × 27.6] = 30 – 26.286 = 3.714. Using the expectations approach, the risk-free rate is r = 0.05, the up factor is u = S+/S = 56/50 = 1.12, and the down factor is d = S–/S = 46/50 = 0.92. The value of the call option is c = PV × [πc+ + (1 – π)c–]. π = [FV(1) – d]/(u – d) = (1.05 – 0.92)/(1.12 – 0.92) = 0.65. c = (1/1.05) × [0.65(6) + (1 – 0.65)(0)] = (1/1.05)(3.9) = 3.714. Both approaches are logically consistent and yield identical values. 4. B is correct. You should sell (write) the overpriced call option and then go long (buy) the replicating portfolio for a call option. The replicating portfolio for a call option is to buy h shares of the stock and borrow the present value of (hS– – c–). c = hS + PV(–hS– + c–). h = (c+ – c–)/(S+ – S–) = (6 – 0)/(56 – 46) = 0.60. For the example in this case, the value of the call option is 3.714. If the option is overpriced at, say, 4.50, you short the option and have a cash flow at Time 0 of +4.50. You buy the replicating portfolio of 0.60 shares at 50 per share (giving you a cash flow of –30) and borrow (1/1.05) × [(0.60 × 46) – 0] = (1/1.05) × 27.6 = 26.287. Your cash flow for buying the replicating portfolio is –30 + 26.287 = –3.713. Your net cash flow at Time 0 is + 4.50 – 3.713 = 0.787. Your net cash flow at Time 1 for either the up move or down move is zero. You have made an arbitrage profit of 0.787. In tabular form, the cash flows are as follows: 153 154 Learning Module 2 Transaction Sell the call option Buy h shares Borrow –PV(–hS– + c–) Net cash flow Valuation of Contingent Claims Time Step 0 Time Step 1 Down Occurs Time Step 1 Up Occurs 4.50 0 –6.00 –0.6 × 50 = –30 0.6 × 46 = 27.6 0.6 × 56 = 33.6 –(1/1.05) × [(–0.6 × 46) + 0] = 26.287 –0.6 × 46 = –27.6 –0.6 × 46 = –27.6 0.787 0 0 5. A is correct. Using the expectations approach, the risk-neutral probability of an up move is π = [FV(1) – d]/(u – d) = (1.03 – 0.800)/(1.300 – 0.800) = 0.46. The terminal value calculations for the exercise values at Time Step 2 are c++ = Max(0,u2S – X) = Max[0,1.302(38) – 40] = Max(0,24.22) = 24.22. c–+ = Max(0,udS – X) = Max[0,1.30(0.80)(38) – 40] = Max(0,–0.48) = 0. c– – = Max(0,d2S – X) = Max[0,0.802(38) – 40] = Max(0,–15.68) = 0. Discounting back for two years, the value of the call option at Time Step 0 is c = PV[π2c++ + 2π(1 – π)c–+ + (1 – π)2c– –]. c = [1/(1.03)]2[0.462(24.22) + 2(0.46)(0.54)(0) + 0.542(0)]. c = [1/(1.03)]2[5.1250] = 4.8308. 6. A is correct. Using the expectations approach, the risk-neutral probability of an up move is π = [FV(1) – d]/(u – d) = (1.03 – 0.800)/(1.300 – 0.800) = 0.46. An American-style put can be exercised early. At Time Step 1, for the up move, p+ is 0.2517 and the put is out of the money and should not be exercised early (X < S, 40 < 49.4). However, at Time Step 1, p– is 8.4350 and the put is in the money by 9.60 (X – S = 40 – 30.40). So, the put is exercised early, and the value of early exercise (9.60) replaces the value of not exercising early (8.4350) in the binomial tree. The value of the put at Time Step 0 is now p = PV[πp+ + (1 – π)p–] = [1/(1.03)][0.46(0.2517) + 0.54(8.4530)] = 4.54. Following is a supplementary note regarding Exhibit 1. The values in Exhibit 1 are calculated as follows. Solutions At Time Step 2: p++ = Max(0,X – u2S) = Max[0,40 – 1.3002(38)] = Max(0,40 – 64.22) = 0. p–+ = Max(0,X – udS) = Max[0,40 – 1.300(0.800)(38)] = Max(0,40 – 39.52) = 0.48. p– – = Max(0,X – d2S) = Max[0,40 – 0.8002(38)] = Max(0,40 – 24.32) = 15.68. At Time Step 1: p+ = PV[πp++ + (1 – π)p–+] = [1/(1.03)][0.46(0) + 0.54(0.48)] = 0.2517. p– = PV[πp–+ + (1 – π)p– –] = [1/(1.03)][0.46(0.48) + 0.54(15.68)] = 8.4350. At Time Step 0: p = PV[πp+ + (1 – π)p–] = [1/(1.03)][0.46(0.2517) + 0.54(8.4350)] = 4.5346. 7. C is correct. Both statements are correct. The expected future payoff is calculated using risk-neutral probabilities, and the expected payoff is discounted at the risk-free rate. 8. C is correct. Using the expectations approach, per 1 of notional value, the values of the call option at Time Step 2 are c++ = Max(0,S++ – X) = Max(0,0.050 – 0.0275) = 0.0225. c+– = Max(0,S+– – X) = Max(0,0.030 – 0.0275) = 0.0025. c– – = Max(0,S– – – X) = Max(0,0.010 – 0.0275) = 0. At Time Step 1, the call values are c+ = PV[πc++ + (1 – π)c+–] . c+ = 0.961538[0.50(0.0225) + (1 – 0.50)(0.0025)] = 0.012019. c– = PV[πc+– + (1 – π)c– –]. c– = 0.980392[0.50(0.0025) + (1 – 0.50)(0)] = 0.001225. At Time Step 0, the call option value is c = PV[πc+ + (1 – π)c–]. c = 0.970874[0.50(0.012019) + (1 – 0.50)(0.001225)] = 0.006429. The value of the call option is this amount multiplied by the notional value, or 0.006429 × 1,000,000 = 6,429. 9. A is correct. Reason 1 is correct: A higher exercise price does lower the exercise value (payoff ) at Time 2. Reason 2 is not correct because the risk-neutral probabilities are based on the paths that interest rates take, which are determined by the market and not the details of a particular option contract. 10. C is correct. The no-arbitrage approach to creating a call option involves buying Delta = N(d1) = 0.6217 shares of the underlying stock and financing with –N(d2) = –0.5596 shares of a risk-free bond priced at exp(–rt)(X) = exp(–0.0022 × 0.25) (55) = $54.97 per bond. Note that the value of this replicating portfolio is nSS + nBB = 0.6217(57.03) – 0.5596(54.97) = $4.6943 (the value of the call option with slight rounding error). 155 156 Learning Module 2 Valuation of Contingent Claims 11. B is correct. The formula for the BSM price of a put option is p = e–rtXN(–d2) – SN(–d1). N(–d1) = 1 – N(d1) = 1 – 0.6217 = 0.3783, and N(–d2) = 1 – N(d2) = 1 – 0.5596 = 0.4404. Note that the BSM model can be represented as a portfolio of the stock (nSS) and zero-coupon bonds (nBB). For a put, the number of shares is nS = –N(–d1) < 0 and the number of bonds is nB = –N(d2) > 0. The value of the replicating portfolio is nSS + nBB = –0.3783(57.03) + 0.4404(54.97) = $2.6343 (the value of the put option with slight rounding error). B is a risk-free bond priced at exp(–rt)(X) = exp(–0.0022 × 0.25)(55) = $54.97. 12. A is correct. Black’s model to value a call option on a futures contract is c = e– rT[F (T)N(d ) – XN(d )]. The underlying F is the futures price (186.73). The 0 1 2 0 correct discount rate is the risk-free rate, r = 0.39%. 13. B is correct. Lee is pointing out the option price’s sensitivity to small changes in time. In the BSM approach, option price sensitivity to changes in time is given by the option Greek theta. 14. A is correct. The put is priced at $7.4890 by the BSM model when using the historical volatility input of 24%. The market price is $7.20. The BSM model overpricing suggests the implied volatility of the put must be lower than 24%. 15. C is correct. Solomon’s forecast is for the three-month MRR to exceed 0.85% in six months. The correct option valuation inputs use the six-month FRA rate as the underlying, which currently has a rate of 0.75%. 16. B is correct because selling call options creates a short position in the ETF that would hedge his current long position in the ETF. Exhibit 2 could also be used to answer the question. Solomon owns 10,000 shares of the GPX, each with a delta of +1; by definition, his portfolio delta is +10,000. A delta hedge could be implemented by selling enough calls to make the portfolio delta neutral: Portfolio delta + 10, 000 ​ ​Delta​ ​ ​ = − _ ​+ 0.6232 ​ = − 16, 046 calls. ​​NH ​ ​ = − ___________ H 17. C is correct. Because the gamma of the stock position is 0 and the put gamma is always non-negative, adding a long position in put options would most likely result in a positive portfolio gamma. Gamma is the change in delta from a small change in the stock’s value. A stock position always has a delta of +1. Because the delta does not change, gamma equals 0. The gamma of a call equals the gamma of a similar put, which can be proven using put–call parity. Alternative Investments 216 Learning Module 2 Investments in Real Estate through Private Vehicles PRACTICE PROBLEMS The following information relates to questions 1-5 Jane Lee is the new chief investment officer of the Eastland University Endowment Fund (the “Fund”). Historically, the Fund has invested based on a 60% equity/40% fixed-income allocation. However, Lee seeks to change the Fund’s investment strategy by shifting a large amount of assets from traditional investments to alternative investments, including real estate, to increase portfolio diversification and the Fund’s risk-adjusted return. Lee recently hired Jon Tarrow as the Fund’s new head of real estate. During a recent meeting, Tarrow and Lee concluded that the initial real estate investment strategy would be to acquire stabilized office buildings located in primary markets. Tarrow has been working on the potential acquisition of a 100% occupied, Class A office building located in New York City. Tarrow and his junior analyst will conduct an independent valuation analysis of the office building; however, Tarrow has also hired an appraiser to value the building and provide valuation support. The appraiser will use three different approaches to estimate the building’s value but will place emphasis on both the direct capitalization and the discounted cash flow methods. During an internal meeting, the junior analyst makes the following three statements about the appraiser’s two primary valuation methods: Statement 1: The direct capitalization method requires an estimation of a terminal value. Statement 2: Properties expected to generate faster growth will require a capitalization rate that results in a higher value. Statement 3: The direct capitalization and the discounted cash flow methods both consider growth and property quality. The appraiser has concluded that net operating income (NOI) for the New York City office building is expected to be $15,600,000 for the first year, and after that, NOI is expected to increase at 0.50% annually for the foreseeable future. The building’s value is also expected to increase by 0.50% annually, and investors expect to achieve a 5.75% internal rate of return given the level of risk. Lee then tells Tarrow that the Fund can procure a loan with an interest-only 5.25% coupon provided that the loan-to-value (LTV) ratio does not exceed 65% and the debt service coverage ratio (DSCR) is at least 1.65x. The balance of the loan would be due after 10 years. 1. Which of the following three valuation approaches best represents the one the appraiser will place the most emphasis on? A. Cost approach B. Income approach C. Sales comparison approach 2. Which of the junior analyst’s three statements regarding the appraiser’s two primary valuation methods is incorrect? A. Statement 1 Practice Problems B. Statement 2 C. Statement 3 3. Which of the following will lead to a decrease in the appraiser’s value of the New York City office building? A. Increase in building NOI B. Decrease in discount rate C. Decrease in NOI and building value growth rates 4. Based on the direct capitalization method, the appraiser’s current value of the New York City office building is closest to: A. $249,600,000. B. $271,300,000. C. $297,100,000. 5. The maximum loan amount that the Fund can obtain on the New York City office building is closest to: A. $117,100,000. B. $180,100,000. C. $193,100,000. 217 218 Learning Module 2 Investments in Real Estate through Private Vehicles SOLUTIONS 1. B is correct. The direct capitalization method and the discounted cash flow method are income approaches used to appraise or estimate the value of a commercial (income-producing) property. A is incorrect because the cost approach considers what reproducing or replicating a property would cost and deducts depreciation and other factors that reduce the property’s value. Replacement cost includes the expense of buying the land and constructing a new property on the site that has the same utility or functionality as the property being appraised. C is incorrect because the sales comparison approach considers what similar or comparable properties (comparables) were transacted for in the current market. It is also referred to as the market approach. 2. A is incorrect. The discounted cash flow method (not direct capitalization method) requires a terminal value by capitalizing NOI at some future date. B is correct because properties expected to generate faster growth will use a capitalization rate that results in a higher value, and properties with slower growth will use a capitalization rate that results in a lower value. C is correct because both income approaches consider growth and property quality. 3. C is correct. The value of a property = NOI/(r − g), where r = the discount rate (required return) and g = the growth rate for income (given constant growth in income, value will grow at the same rate). Therefore, a decrease in the NOI and building value growth rates will lead to a decrease in the property’s value. A is incorrect because an increase in building NOI will lead to a higher property value. B is incorrect because a decrease in the discount rate will lead to a higher property value. 4. C is correct. The value of a property = NOI/(r − g), where r = the discount rate (required return) and g = the growth rate for income (given constant growth in income, value will grow at the same rate). Therefore, the building’s value is calculated as follows: $15,600,000/(5.75% – 0.50%) = $297,142,857 ($297,100,000 rounded). A is incorrect because a property value of $249,600,000 is incorrectly calculated as follows: $15,600,000/(5.75% + 0.50%) = $249,600,000. B is incorrect because a property value of $271,300,000 is incorrectly calculated as follows: $15,600,000/5.75% = $271,300,000. 5. B is correct. Based on the LTV ratio, the maximum loan amount would be 65% of $297,100,000, or $193,115,000 ($193,100,000 rounded). With a minimum DSCR of 1.65x, the maximum debt service would be $15,600,000/1.65 = $9,454,545. This amount is the mortgage payment that would result in a 1.65x DSCR for an interest-only loan. If the loan is interest only, then we can obtain the loan amount by simply dividing the mortgage payment by the interest rate. Therefore, the loan amount would be $9,454,545/0.0525 = $180,086,580 ($180,100,000 rounded). Because the loan amount based on the minimum DSCR results in a lower loan amount, that would be the maximum amount that could be borrowed. A is incorrect because $117,100,000 is incorrectly calculated as follows: Solutions $15,600,000/0.0525/1.65 = $180,086,580. $180,086,580 × 65% = $117,056,277 ($117,100,000 rounded). C is incorrect because $193,100,000 is the rounded loan amount based on the maximum LTV of 65%. Because the loan amount based on the minimum DSCR results in a lower loan amount, $180,086,580 ($180,100,000 rounded) would be the maximum loan amount that could be borrowed. 219 252 Learning Module 3 Investments in Real Estate Through Publicly Traded Securities PRACTICE PROBLEMS The following information relates to questions 1-4 Maitha Smith is the CIO of the Westland Pension Fund (the “Fund”). Smith and her junior analyst are analyzing Bay Realty Corp. (“Bay”), a publicly traded REIT based in San Francisco, for a potential investment. Bay currently owns and operates 40 office buildings totaling 8 million square feet. These properties exhibit an average LTV (loan-to-value) ratio of 40%. Bay owns no other real estate–related assets. Bay’s senior executives are company employees who report to the board of directors, whose members are elected by shareholders. Smith first instructs her junior analyst to conduct a NAVPS (net asset value per share) analysis on Bay. The junior analyst makes the following three statements: Statement 1: NAVPS should not include investors’ assessments of the value of any non-asset-based income streams, the value of non–real estate assets, or the value added by management. Statement 2: REITs whose shares trade below NAVPS or have high leverage might have a more difficult time raising new capital to fund acquisitions and development, which could limit long-term growth. Statement 3: Shares priced at discounts to NAVPS are interpreted as indications of potential overvaluation. To complement the NAVPS, Smith instructs her junior analyst to also calculate FFO and AFFO measures for Bay. The junior analyst then makes the following three statements as part of the ongoing discussion: Statement 4: AFFO better approximates a company’s sustainable dividend-paying capacity than FFO. Statement 5: FFO and AFFO are based on net income available to equity and thus represent levered income. Statement 6: FFO is superior to AFFO as a measure of economic income and thus economic return because it takes into account the capital expenditures necessary to maintain the economic income of a property portfolio. After the discussion, the junior analyst obtains selected information on Bay, which is shown in Exhibit 1: Exhibit 1 Non-cash (straight-line) rent $215,000 Recurring maintenance–type capital expenditures and leasing commissions $700,000 Adjusted funds from operations $4,000,000 AFFO per share $5.00 Current stock price $80.00 1. Which of the following best describes Bay? A. Mortgage REIT Practice Problems B. Internally managed REIT C. Real estate operating company (REOC) 2. Which of the junior analyst’s three statements regarding NAVPS is correct? A. Statement 1 B. Statement 2 C. Statement 3 3. Which of the junior analyst’s three statements regarding FFO and AFFO is incorrect? A. Statement 4 B. Statement 5 C. Statement 6 4. Based on Exhibit 1, Bay’s P/FFO is closest to: A. 14.3x B. 13.0x. C. 20.7x. 253 254 Learning Module 3 Investments in Real Estate Through Publicly Traded Securities SOLUTIONS 1. B is correct. Bay is internally managed, or self-managed. Bay’s senior executives are company employees who report to the board of directors, whose members are elected by shareholders. Fully integrated REITs, such as Bay, generally have fewer conflicts than REITs that are externally advised or externally managed. A is incorrect because Bay is an equity REIT, not a mortgage REIT. Bay currently owns and operates 40 office buildings totaling 8 million square feet. C is incorrect because REOCs are ordinary taxable real estate ownership companies, which are different from REITs. Businesses are organized as REOCs, as opposed to REITs, if they are located in countries that do not have a tax-advantaged REIT regime in place, if they engage to a large extent in the development of for-sale real estate properties, or if they offer other non-qualifying services, such as brokerage and third-party property management. 2. B is correct. REITs whose shares trade below NAVPS or have high leverage might have a more difficult time raising new capital to fund acquisitions and development, which in turn could limit long-term growth, in contrast to REITs that trade at or above NAVPS. A is incorrect because NAVPS should include investors’ assessments of the value of any non-asset-based income streams, the value of non–real estate assets, and the value added by management. C is incorrect because shares priced at discounts to NAVPS are interpreted as indications of potential undervaluation. 3. C is incorrect. AFFO is superior to FFO (not the other way around) as a measure of economic income and thus economic return because it takes into account the capital expenditures necessary to maintain the economic income of a property portfolio. A is correct because AFFO better approximates a company’s sustainable dividend-paying capacity than FFO. B is correct because both FFO and AFFO are based on net income available to equity and thus represent levered income. 4. B is correct. FFO = AFFO + Non-cash (straight-line) rent + Recurring maintenance–type capital expenditures and leasing commissions = $4,000,000 + $215,000 + $700,000 = $4,915,000. The number of shares outstanding = 4,000,000/5.00 = 800,000. FFO per share = 4,915,000/800,000 = $6.14. Current stock price = $80.00 per share P/FFO = $80.00/$6.14 = 13.0x A is incorrect because 14.3x is incorrectly calculated as follows: $4,000,000 – $215,000 + $700,000 = $4,485,000. The number of shares outstanding = 4,000,000/5.00 = 800,000. FFO per share = 4,485,000/800,000 = $5.61. Current stock price = $80.00 per share P/FFO = $80.00/$5.61 = 14.3x C is incorrect because 20.7x is incorrectly calculated as follows: $4,000,000 – $215,000 – $700,000 = $3,085,000. The number of shares outstanding = 4,000,000/5.00 = 800,000. FFO per share = 3,085,000/800,000 = $3.86. Current stock price = $80.00 per share P/FFO = $80.00/$3.86 = 20.7x Practice Problems 291 PRACTICE PROBLEMS 1. Discuss the ways that private equity funds can create value. 2. Jean-Pierre Dupont is the chief investment officer (CIO) of a French pension fund that allocates a substantial portion of its assets to private equity. The fund’s PE portfolio comprises mainly large buyout funds and mezzanine funds with a limited allocation to a special situations fund. A decision has been made to increase allocations to European venture capital. The investment committee of the pension fund requested that Dupont present an analysis of five key investment characteristics specific to venture capital relative to buyout investing. Can you assist Dupont in this request? The following information relates to questions 3-8 Martha Brady is the CIO of the Upper Darby County (UDC) public employees’ pension system. Brady is considering an allocation of the pension system’s assets to private equity. She has asked two of her analysts, Jennifer Chau, CFA, and Matthew Hermansky, to provide more information about the workings of the private equity market. Brady recognizes that the private equity asset class covers a broad spectrum of equity investments that are not traded in public markets. She asks Chau to describe the major differences between assets within this asset class. Chau notes that private equity ranges from venture capital financing of early-stage companies to complete buyouts of large publicly traded or even privately held companies. Chau describes some of the characteristics of venture capital and buyout investments. Chau mentions that private equity firms take care to align the economic interests of the managers of the investments they control with their own. Various contractual clauses are inserted in the compensation contracts of the management team in order to reward or punish managers who meet or do not meet agreed-upon target objectives. One concern Chau highlights is the illiquidity of private equity investments over time. Some funds are returned to investors, however, over the life of the fund because a number of investment opportunities are exited early. Provisions in a fund’s prospectus describe the distribution of returns to investors, some of which favor the limited partners. One such provision is the distribution waterfall mechanism that provides distributions to limited partners (LPs) before the general partner (GP) receives the carried interest. This distribution mechanism is called the total return waterfall. Chau prepares the following data to illustrate the distribution waterfall mechanism and the funds provided to limited partners when a private equity fund with a zero hurdle rate exits from its first three projects during a three-year period. Exhibit 1: Investment Returns and Distribution Waterfalls Private equity committed capital $400 million Carried interest 20% 292 Learning Module 4 Private Equity Investments First project investment capital $20 million Second project investment capital $45 million Third project investment capital $50 million Proceeds from first project $25 million Proceeds from second project $35 million Proceeds from third project $65 million Chau cautions that investors must understand the terminology used to describe the performance of private equity funds. Interpretation of performance numbers should be made with the awareness that much of the fund assets are illiquid during a substantial part of the fund’s life. She provides the latest data in Exhibit 2 for the Alpha, Beta, and Gamma Funds, diversified high-technology venture capital funds formed five years ago, each with five years remaining to termination. Exhibit 2: Financial Performance of Alpha, Beta, and Gamma Funds Fund PIC DPI RVPI Alpha 0.30 0.10 0.65 Beta 0.85 0.10 1.25 Gamma 0.85 1.25 0.75 Chau studies the data and comments, Of the three funds, the Alpha Fund has the best chance to outperform over the remaining life. First, it’s because the management has earned such a relatively high residual value on capital and will be able to earn a high return on the remaining funds called down. At termination, the RVPI will be double the 0.65 value when the rest of the funds are called down. Second, its cash-on-cash return as measured by DPI is already as high as that of the Beta Fund. The PIC (or paid-in capital) ratio indicates the proportion of capital already called by the GP. The PIC of Alpha is relatively low relative to Beta and Gamma. Hermansky notes that a private equity fund’s ability to properly plan and execute its exit from an investment is vital for the fund’s success. Venture funds, such as Alpha, Beta, and Gamma, take special care to plan their exits. Brady then asks the analysts what procedures private equity firms would use to value investments in their portfolios as well as investments that are added later. She is concerned about buying into a fund with existing assets that do not have public market prices that can be used to ascertain value. In such cases, she worries, what if a GP overvalues the assets and new investors in the fund pay more for the fund assets than they are worth? Hermansky makes three statements regarding the valuation methods used in private equity transactions during the early stages of selling a fund to investors. Statement 1 For venture capital investment in the early stages of analysis, emphasis is placed on the discounted cash flow approach to valuation. Statement 2 For buyout investments, income-based approaches are used frequently as a primary method of valuation. Practice Problems Statement 3 293 If a comparable group of companies exist, multiples of revenues or earnings are used frequently to derive a value for venture capital investments. 3. The characteristic that is most likely common to both the venture capital and buyout private equity investment is: A. measurable and assessable risk. B. the extensive use of financial leverage. C. the strength of the individual track record and ability of members of management. 4. The contractual term enabling management of the private equity–controlled company to be rewarded with increased equity ownership as a result of meeting performance targets is called: A. a ratchet. B. the tag-along right. C. the clawback provision. 5. For the projects described in Exhibit 8, under a deal-by-deal method with a clawback provision and true-up every three years, the cumulative dollar amount the GP receives by the end of the three years is equal to: A. 1 million. B. 2 million. C. 3 million. 6. Are Chau’s two reasons for interpreting Alpha Fund as the best-performing fund over the remaining life correct? A. No B. Yes C. The first reason is correct, but the second reason is incorrect. 7. The exit route for a venture capital investment is least likely to be in the form of a(n): A. initial public offering (IPO). B. sale to other venture funds targeting the same sector. C. buyout by the management of the venture investment. 8. Which statement by Hermansky is the least valid? A. Statement 1 B. Statement 2 C. Statement 3 294 Learning Module 4 Private Equity Investments The following information relates to questions 9-14 Daniel Collin is a junior analyst at JRR Equity Partners (JRR), a private equity firm. Collin is assigned to work with Susan Tseng, a senior portfolio manager. Tseng and Collin meet to discuss existing and potential investments. Tseng starts the meeting with a discussion of LBO firms and VC firms. Collin tells Tseng, LBO firms tend to invest in companies with predictable cash flows and experienced management teams, whereas VC firms tend to invest in companies with high EBITDA or EBIT growth and where an exit is fairly predictable. Tseng and Collin next analyze a potential investment in the leveraged buyout of Stoneham Industries. Specifically, they assess the expected gain if they elect to purchase all of the preference shares and 90% of the common equity through the LBO. Details of the LBO include the following: ■ The buyout requires an initial investment of $10 million. ■ Financing for the deal includes $6 million in debt, $3.6 million in preference shares that promise a 15% annual return paid at exit, and $0.4 million in common equity. The expected exit value in six years is $15 million, with an estimated reduction in debt of $2.8 million over the six years prior to exit. Tseng and Collin next discuss JRR’s investment in Venture Holdings, a private equity fund. Selected details on the Venture Holdings fund include the following: ■ Total committed capital is $115 million. ■ The distribution waterfall follows the deal-by-deal method, and carried interest is 20%. ■ On its first exit event a few years ago, the fund generated a $10 million profit. ■ At the end of the most recent year, cumulative paid-in capital was $98 million, cumulative distributions paid out to LPs were $28 million, and the year-end NAV, before and after distributions, was $170.52 million and $131.42 million, respectively. ■ Tseng and Collin estimate that the fund’s NAV before distributions will be $242.32 million at the end of next year. Finally, Tseng and Collin evaluate two venture capital funds for potential investment: the Squire Fund and the Treble Fund. Both funds are in Year 7 of an estimated 10-year term. Selected data for the two funds are presented in Exhibit 1. Exhibit 1: Selected Data for the Squire Fund and the Treble Fund Squire Fund Treble Fund 0.11 0.55 RVPI 0.95 0.51 Gross IRR –11% 10% Net IRR –20% 8% DPI Practice Problems 295 After reviewing the performance data in Exhibit 1, Collin draws the following conclusions: Conclusion 1 The unrealized return on investment for the Squire Fund is greater than the unrealized return on investment for the Treble Fund. Conclusion 2 The TVPI for the Treble Fund is higher than the TVPI for the Squire Fund because the Treble Fund has a higher gross IRR. 9. Is Collin’s statement about LBO firms and VC firms correct? A. Yes B. No, because he is wrong with respect to VC firms C. No, because he is wrong with respect to LBO firms 10. The multiple of expected proceeds at exit to invested funds for JRR’s Stoneham LBO investment is closest to: A. 2.77. B. 2.89. C. 2.98. 11. The distribution available to the limited partners of the Venture Holdings fund from the first exit is closest to: A. $2 million. B. $8 million. C. $10 million. 12. At the end of the most recent year, the ratio of total value to paid-in capital (TVPI) for the Venture Holdings fund was closest to: A. 0.29. B. 1.34. C. 1.63. 13. Based on Tseng and Collin’s estimate of NAV next year, the estimate of carried interest next year is closest to: A. $14.36 million. B. $22 million. C. $25.46 million. 14. Which of Collin’s conclusions regarding the Squire Fund and the Treble Fund is correct? A. Only Conclusion 1 296 Learning Module 4 Private Equity Investments B. Only Conclusion 2 C. Both Conclusion 1 and Conclusion 2 15. What problems are encountered when using comparable publicly traded companies to value private acquisition targets? 16. Jo Ann Ng is a senior analyst at SING INVEST, a large regional mid-market buyout manager in Singapore. She is considering the exit possibilities for an existing investment in a mature automotive parts manufacturer that was acquired three years ago at a multiple of 7.5 times EBITDA. SING INVEST originally anticipated exiting its investment in China Auto Parts, Inc., within three to six years. Ng noted that market conditions have deteriorated and that companies operating in a similar business trade at an average multiple of 5.5 times EBITDA. She expects, however, based on analyst reports and industry knowledge, that the market will recover strongly within the next two years because of the fast-increasing demand for cars in emerging markets. Upon review of market opportunities, Ng also noted that China Gear Box, Inc., a smaller Chinese automotive parts manufacturer that presents strong potential synergies with China Auto Parts, Inc., is available for sale at an EBITDA multiple of 4.5. Exits by means of an IPO or a trade sale to a financial or strategic (company) buyer are possible in China. How would you advise Ng to enhance value upon exit of China Auto Parts? 17. Wenda Lee, CFA, is a portfolio manager at a UK-based private equity institutional investor. She is considering an investment in a mid-market European buyout fund to achieve better diversification for her firm’s private equity portfolio. She short-listed two funds that she sees as having similar risk–return profiles. Before deciding which one to invest in, she is carefully reviewing and comparing the terms of each fund. Mid-Market Fund A Mid-Market Fund B Management fees 2.5% 1.5% Transaction fees 100% to the GP 50–50% split Carried interest 15% 20% Hurdle rate 6% 9% Clawback provision No Yes Deal-by-deal Total return Distribution waterfall Based on the analysis of terms, which fund would you recommend to Lee? 18. What are the main ways that the performance of private equity limited partnerships can be measured (A) during the life of the fund and (B) once all investments have been exited? Solutions 297 SOLUTIONS 1. The main ways that private equity funds can create value include the following: ■ Operational improvements and clearly defined strategies: In the case of later-stage companies and buyouts, private equity owners can often create value by focusing the business on its most profitable opportunities and providing strategic direction for the business. In the case of venture capital deals, the private equity funds provide valuable business experience, mentor management, and offer access to their network of contacts and other portfolio companies. ■ Creating incentives for managers and aligning their goals with the investors: This is often achieved by providing significant monetary rewards to management if the private equity fund secures a profitable exit. In the case of buyouts, the free cash flow available to management is minimized by taking on significant amounts of debt financing. ■ Optimizing the financial structure of the company: In the case of buyouts, the use of debt can reduce the tax payments made by the company and also reduce the cost of capital. There may also be opportunities in certain market conditions to take advantage of any mispricing of risk by lenders, which can allow the private equity funds to take advantage of interest rates that do not fully reflect the risks being carried by the lenders. Many would point to various periods from 2015 to 2019 when government interest rates were low, debt spreads were tight, and/or lender covenants were loose as examples of such prevailing conditions. 2. Venture Capital Buyout Primarily equity funded. Use of leverage is rare and very limited. Extensive use of leverage consisting of a large proportion of senior debt and a significant layer of junior and/or mezzanine debt. Returns of investment portfolios are generally characterized by very high returns from a limited number of highly successful investments and a significant number of write-offs from low performing investments or failures. Returns of investment portfolios are generally characterized by lower variance across returns from underlying investments. Bankruptcies are rare events. Venture capital firm monitors achievement of milestones defined in business plan and growth management. Buyout firm monitors cash flow management and strategic and business planning. Expanding capital requirement if in the growth phase. Low working capital requirement. Assessment of risk is difficult because of new technologies, new markets, and lack of operating history. Risk is measurable (e.g., mature businesses, long operating history, etc.). 3. C is correct. Members of both the firm being bought out and the venture capital investment usually have strong individual management track records. Extensive financial leverage is common in buyouts but not venture capital investments, whereas measurable risk is more common in buyouts than in venture capital situations. 298 Learning Module 4 Private Equity Investments 4. A is correct. 5. B is correct. On a cumulative basis for three years, the fund earns $10 million, of which $2 million goes to the GP. The $2 million earned by the GP corresponds to 20% of the difference between total three-year proceeds and three-year invested capital, or 0.2 × [(25 + 35 + 65) − (20 + 45 + 50)]. 6. A is correct. Chau misinterprets DPI, RVPI, and PIC. The returns earned to date are for each dollar of invested capital—that which has been drawn down—not total returns. Chau mistakenly believes (assuming the same management skill) the result for Alpha Fund at termination will be on the order of 3 × 0.65 = 1.95, instead of 0.65. In both cases, Alpha Fund has underperformed relative to the other two funds. 7. C is correct. Leverage needed to finance a management buyout is not readily available to firms with limited history. 8. A is correct. Statement 1 is the least likely to be valid. 9. B is correct. LBO firms generally invest in firms with a predictable cash flow pattern (EBITDA or EBIT growth) and experienced management teams. In contrast, venture capital firms tend to invest in new firms and new technologies with high revenue growth. Also, VC investments tend to be characterized as having exits that are difficult to anticipate. 10. B is correct. The investment exit value is $15 million. The expected payoff to JRR is calculated as follows (all amounts in millions): Expected exit value: $15.00 Debt remaining at exit: ($6.0 – 2.8) 3.20 Preference shares: [$3.60 × 8.33 (1.15)6] Common equity: ($15 exit – 3.2 debt – 8.33 preference) 3.47 Initial investment: $3.6 (preference) + 0.9 × $0.4 (common) = $3.96. Proceeds at exit: $8.33 (preference) + 0.9 × $3.47 (common) = $11.45. Multiple of expected proceeds to invested funds: $11.45 exit value/$3.96 initial investment = 2.89×. 11. B is correct. The distribution waterfall for the Venture Holdings fund follows the deal-by-deal method. The investment generated a profit of $10 million, and with carried interest of 20%, the general partner would receive $2 million ($10 million × 20%), leaving $8 million for the limited partners. 12. C is correct. Total value to paid-in capital (TVPI) represents the fund’s distributed value and undistributed value as a proportion of the cumulative invested capital. TVPI is the sum of distributed to paid-in capital (DPI) and residual value to paid-in capital (RVPI): $28 million Cumulative distributions _       ​DPI = ____________________ ​Cumulative invested capital ​ = ​$98 million ​ = 0.29× NAV (after distributions) $131.42 million ____________ ​RVPI = ____________________ ​Cumulative       ​ $98 million ​ = 1.34× invested capital ​ =    Solutions Cumulative distribution + NAV (after distributions) TVPI = ____________________________________ ​          Cumulative invested capital $28 million + 131.42 million = _____________________    ​    ​ = 1.63× $98 million 13. A is correct. Provided that NAV before distribution exceeds committed capital, the general partner is entitled to carried interest, calculated as the 20% multiplied by the increase in NAV before distributions. So, the carried interest is calculated as follows: Carried interest = 20% × ($242.32 – $170.52) = $14.36 million. 14. A is correct. DPI provides an indication of a fund’s realized return, whereas RVPI provides an indication of a fund’s unrealized return. The Squire Fund has a higher RVPI (0.95) than the Treble Fund (0.51). TVPI, which is the sum of DPI and RVPI, is the same for both funds: 0.11 + 0.95 = 1.06 for the Squire Fund and 0.55 + 0.51 = 1.06 for the Treble Fund. 15. There are many complexities in using comparable companies to value private targets, including the following: ■ The lack of public comparison companies operating in the same business, facing the same risks, and at the same stage of development. It is often possible to identify “approximate” comparisons but very rare to find an exact match. It is essential, therefore, to use judgment when using comparison company information, rather than just taking the average multiples derived from a sample of disparate companies. ■ Comparison companies may have different capital structures, so estimated beta coefficients and some financial ratios should be adjusted accordingly. ■ Reported accounting numbers for earnings must be chosen carefully and adjusted for any exceptional items, atypical revenues, and costs in the reference year. Care must also be taken to decide which earnings figures to compare; the main choices are trailing earnings (the last 12 months), earnings from the last audited accounts, or prospective year-ahead earnings. 16. The exit strategies available to SING INVEST to divest their holding in China Auto Parts, Inc., will largely depend on the following two factors: ■ Time remaining until the fund’s term expires: If the time remaining is sufficiently long, the fund’s manager has more flexibility to work out an exit at more favorable market circumstances and terms. ■ Amount of undrawn commitments from LPs in the fund: If sufficient LP commitments can be drawn, the fund manager may take advantage of current investment opportunities at depressed market prices to enhance returns upon exit in a more favorable market environment. In the case of China Auto Parts, Inc., depending on an analysis of the factors discussed, Ng could offer an opinion to support the acquisition of China Gear Box, Inc., subject to an in-depth analysis of potential synergies with China Auto Parts, Inc. The objective here may thus be twofold: to benefit from short-term market conditions and to enhance the value of existing investments by reinforcing their market potential with a strategic merger. 17. Assuming that both funds have similar risk–return characteristics, a closer analysis of economic and corporate governance terms should be instrumental in 299 300 Learning Module 4 Private Equity Investments determining which fund to select. In economic terms, Mid-Market Fund B has a higher carried interest than Mid-Market Fund A, but Mid-Market Fund B has a fee structure that is better aligned with the interests of LPs. A larger proportion of Mid-Market Fund B’s fees (through the carried interest) will come from achieving successful exits, whereas Mid-Market Fund A will earn relatively larger fees on running the fund (management fees and transaction fees) without necessarily achieving high performance. In addition, the 9% hurdle rate of Mid-Market Fund B is indicative of confidence in the fund manager’s ability to achieve a minimum compounded 9% return to LPs for which no carried interest will be paid. In corporate governance terms, Mid-Market Fund B is far better aligned with the interests of LPs as a result of a clawback provision and a more favorable distribution waterfall that will allow payment of carried interest on a total return basis instead of deal by deal. The conclusion is that Mid-Market Fund B appears better aligned with the interests of LPs. 18. In the early years of a fund, all measures of return are of little relevance because fees drag down the reported returns and investments are initially valued at cost. This produces the J-curve effect. After a few years (or longer in the case of venture capital investments), performance measures become more meaningful, and the two main measures used by investors are IRR and return multiples (of the initial sum invested). During the life of the fund, it is necessary to value the non-exited investments and add them to the realized returns. The former inevitably involves an element of judgment on the part of the General Partner, especially when it is difficult to estimate the likely market value of the investment. Once all the investments have been exited, the multiples and IRR can be estimated easily, taking account of the exact timing of the cash flows into and out of the fund. The most relevant measures for investors are computed net of management fees and any carried interest earned by the general partner. Practice Problems 347 PRACTICE PROBLEMS The following information relates to questions 1-8 Raffi Musicale is the portfolio manager for a defined benefit pension plan. He meets with Jenny Brown, market strategist with Menlo Bank, to discuss possible investment opportunities. The investment committee for the pension plan has recently approved expanding the plan’s permitted asset mix to include alternative asset classes. Brown proposes the Apex Commodity Fund (Apex Fund) offered by Menlo Bank as a potentially suitable investment for the pension plan. The Apex Fund attempts to produce trading profits by capitalizing on the mispricing between the spot and futures prices of commodities. The fund has access to storage facilities, allowing it to take delivery of commodities when necessary. The Apex Fund’s current asset allocation is presented in Exhibit 1. Exhibit 1: Apex Fund’s Asset Allocation Commodity Sector Allocation (%) Energy 31.9 Livestock 12.6 Softs 21.7 Precious metals 33.8 Brown explains that the Apex Fund has had historically low correlations with stocks and bonds, resulting in diversification benefits. Musicale asks Brown, “Can you identify a factor that affects the valuation of financial assets like stocks and bonds but does not affect the valuation of commodities?” Brown shares selected futures contract data for three markets in which the Apex Fund invests. The futures data are presented in Exhibit 2. Exhibit 2: Selected Commodity Futures Data* Month Gold Price Coffee Price Gasoline Price July 1,301.2 0.9600 2.2701 September 1,301.2 0.9795 2.2076 December 1,301.2 1.0055 2.0307 * Gold: US$/troy ounce; coffee: US$/pound; gasoline: US$/gallon. Menlo Bank recently released a report on the coffee market. Brown shares the key conclusion from the report with Musicale: “The coffee market had a global harvest that was greater than expected. Despite the large harvest, coffee futures trading activity is balanced between producers and consumers. This balanced condition is not expected to change over the next year.” 348 Learning Module 5 Introduction to Commodities and Commodity Derivatives Brown shows Musicale the total return of a recent trade executed by the Apex Fund. Brown explains that the Apex Fund took a fully collateralized long futures position in nearby soybean futures contracts at the quoted futures price of 865.0 (US cents/bushel). Three months later, the entire futures position was rolled when the near-term futures price was 877.0 and the farther-term futures price was 883.0. During the three-month period between the time that the initial long position was taken and the rolling of the contract, the collateral earned an annualized rate of 0.60%. Brown tells Musicale that the pension fund could alternatively gain long exposure to commodities using the swap market. Brown and Musicale analyze the performance of a long position in an S&P GSCI total return swap having monthly resets and a notional amount of $25 million. Selected data on the S&P GSCI are presented in Exhibit 3. Exhibit 3: Selected S&P GSCI Data Reference Date Index Level April (swap initiation) 2,542.35 May 2,582.23 June 2,525.21 1. The Apex Fund is most likely to be characterized as: A. a hedger. B. a speculator. C. an arbitrageur. 2. Which factor would most likely affect the supply or demand of all four sectors of the Apex Fund? A. Weather B. Spoilage C. Government actions 3. The most appropriate response to Musicale’s question regarding the valuation factor is: A. storage costs. B. transportation costs. C. expected future cash flows. 4. Which futures market in Exhibit 2 is in backwardation? A. Gold B. Coffee C. Gasoline 5. Based on the key conclusion from the Menlo Bank coffee market report, the Practice Problems 349 shape of the coffee futures curve in Exhibit 2 is most consistent with the: A. insurance theory. B. theory of storage. C. hedging pressure hypothesis. 6. Based on Exhibit 2, which commodity’s roll returns will most likely be positive? A. Gold B. Coffee C. Gasoline 7. The Apex Fund’s three-month total return on the soybean futures trade is closest to: A. 0.85%. B. 1.30%. C. 2.22%. 8. Based on Exhibit 3, on the June settlement date, the party that is long the S&P GSCI total return swap will: A. owe a payment of $552,042.23. B. receive a payment of $1,502,621.33. C. receive a payment of $1,971,173.60. The following information relates to questions 9-15 Mary McNeil is the corporate treasurer at Farmhouse, which owns and operates several farms and ethanol production plants in the United States. McNeil’s primary responsibility is risk management. Katrina Falk, a recently hired junior analyst at Farmhouse, works for McNeil in managing the risk of the firm’s commodity price exposures. Farmhouse’s risk management policy requires the use of futures to protect revenue from price volatility, regardless of forecasts of future prices, and prohibits risk managers from taking speculative positions. McNeil meets with Falk to discuss recent developments in two of Farmhouse’s commodity markets, grains and livestock. McNeil asks Falk about key characteristics of the two markets that affect revenues and costs. Falk tells McNeil the following: Statement 1 The life cycle for livestock depends on the product and varies widely by product. Statement 2 Grains have uniform, well-defined seasons and growth cycles specific to geographic regions. A material portion of Farmhouse’s revenue comes from livestock exports, and a major input cost is the cost of grains imported from outside the United States. Falk and McNeil next discuss three conclusions that Falk reached in an analysis 350 Learning Module 5 Introduction to Commodities and Commodity Derivatives of the grains and livestock markets: Conclusion 1 Assuming demand for grains remains constant, extreme heat in the regions from which we import our grains will result in a benefit to us in the form of lower grain prices. Conclusion 2 New tariffs on cattle introduced in our primary export markets will likely result in higher prices for our livestock products in our local market. Conclusion 3 Major improvements in freezing technology allowing for longer storage will let us better manage the volatility in the prices of our livestock products. McNeil asks Falk to gather spot and futures price data on live cattle, wheat, and soybeans, which are presented in Exhibit 1. Additionally, she observes that (1) the convenience yield of soybeans exceeds the costs of its direct storage and (2) commodity producers as a group are less interested in hedging in the forward market than commodity consumers are. Exhibit 1: Selected Commodity Price Data* Market Live Cattle Price Wheat Price Soybeans Price Spot 109 407 846 Futures 108 407 850 * Live cattle: US cents per pound; wheat and soybeans: US cents per bushel. A key input cost for Farmhouse in producing ethanol is natural gas. McNeil uses positions in natural gas (NG) futures contracts to manage the risk of natural gas price volatility. Three months ago, she entered into a long position in natural gas futures at a futures price of $2.93 per million British thermal units (MMBtu). The current price of the same contract is $2.99. Exhibit 2 presents additional data about the three-month futures position. Exhibit 2: Selected Information—Natural Gas Futures Three-Month Position* Prices Commodity Natural Gas (NG) Total Current $ Exposure 5,860,000 Position Near-Term Futures (Current Price) Farther-Term Futures Long 2.99 3.03 * NG: $ per MMBtu; 1 contract = 10,000 MMBtu. The futures position is fully collateralized earning a 3% rate. McNeil decides to roll forward her current exposure in the natural gas position. Each month, McNeil reports the performance of the energy futures positions, including details on price returns, roll returns, and collateral returns, to the firm’s executive committee. A new committee member is concerned about the negative roll returns on some of the positions. In a memo to McNeil, the committee member asks her to explain why she is not avoiding positions with negative roll Practice Problems returns. 9. With respect to its risk management policy, Farmhouse can be best described as: A. a trader. B. a hedger. C. an arbitrageur. 10. Which of Falk’s statements regarding the characteristics of the grains and livestock markets is correct? A. Only Statement 1 B. Only Statement 2 C. Both Statement 1 and Statement 2 11. Which of Falk’s conclusions regarding commodity markets is correct? A. Conclusion 1 B. Conclusion 2 C. Conclusion 3 12. Which commodity market in Exhibit 1 is currently in a state of contango? A. Wheat B. Soybeans C. Live cattle 13. Based on Exhibit 1 and McNeil’s two observations, the futures price of soybeans is most consistent with the: A. insurance theory. B. theory of storage. C. hedging pressure hypothesis. 14. Based on Exhibit 2, the total return from the long position in natural gas futures is closest to: A. 1.46%. B. 3.71%. C. 4.14%. 15. The most appropriate response to the new committee member’s question is that: A. roll returns are negatively correlated with price returns. B. such roll returns are the result of futures markets in backwardation. C. such positions may outperform other positions that have positive roll returns. 351 352 Learning Module 5 Introduction to Commodities and Commodity Derivatives The following information relates to questions 16-22 Jamal Nabli is a portfolio manager at NextWave Commodities (NWC), a commodity-based hedge fund located in the United States. NWC’s strategy uses a fixed-weighting scheme to allocate exposure among 12 commodities, and it is benchmarked against the Thomson Reuters/CoreCommodity CRB Index (TR/ CC CRB). Nabli manages the energy and livestock sectors with the help of Sota Yamata, a junior analyst. Nabli and Yamata meet to discuss a variety of factors that affect commodity values in the two sectors they manage. Yamata tells Nabli the following: Statement 1 Storage costs are negatively related to futures prices. Statement 2 In contrast to stocks and bonds, most commodity investments are made by using derivatives. Statement 3 Commodities generate future cash flows beyond what can be realized through their purchase and sale. Nabli and Yamata then discuss potential new investments in the energy sector. They review Brent crude oil futures data, which are presented in Exhibit 1. Exhibit 1: Selected Data on Brent Crude Oil Futures Spot Price 77.56 Near-Term Futures Price Longer-Term Futures Price 73.64 73.59 Yamata presents his research related to the energy sector, which has the following conclusions: ■ Consumers have been more concerned about prices than producers have. ■ Energy is consumed on a real-time basis and requires minimal storage. After concluding the discussion of the energy sector, Nabli reviews the performance of NWC’s long position in lean hog futures contracts. Nabli notes that the portfolio earned a −12% price return on the lean hog futures position last year and a −24% roll return after the contracts were rolled forward. The position was held with collateral equal to 100% of the position at a risk-free rate of 1.2% per year. Yamata asks Nabli to clarify how the state of the futures market affects roll returns. Nabli responds as follows: Statement 1 Roll returns are generally negative when a futures market is in contango. Statement 2 Roll returns are generally positive when a futures market is in backwardation. As part of their expansion into new markets, NWC is considering changing its benchmark index. Nabli investigates two indexes as a possible replacement. These indexes both use similar weighting and rebalancing schemes. Index A includes contracts of commodities typically in contango, whereas Index B includes contracts of commodities typically in backwardation. Nabli asks Yamata how the Practice Problems two indexes perform relative to each other in a market that is trending upward. Because of a substantial decline in drilling activity in the North Sea, Nabli believes the price of Brent crude oil will increase more than that of heavy crude oil. The actual price volatility of Brent crude oil has been lower than its expected volatility, and Nabli expects this trend to continue. Nabli also expects the level of the ICE Brent Index to increase from its current level. Nabli and Yamata discuss how to use swaps to take advantage of Nabli’s expectations. The possible positions are (1) a basis swap long on Brent crude oil and short on heavy crude oil, (2) a long volatility swap on Brent crude oil, and (3) a short position in an excess return swap that is based on a fixed level (i.e., the current level) of the ICE Brent Index. 16. Which of Nabli’s statements regarding the valuation and storage of commodities is correct? A. Statement 1 B. Statement 2 C. Statement 3 17. Based on Exhibit 1, Yamata should conclude that the: A. calendar spread for Brent crude oil is $3.97. B. Brent crude oil futures market is in backwardation. C. basis for the near-term Brent crude oil futures contract is $0.05 per barrel. 18. Based on Exhibit 1 and Yamata’s research on the energy sector, the shape of the futures price curve for Brent crude oil is most consistent with the: A. insurance theory. B. theory of storage. C. hedging pressure hypothesis. 19. The total return (annualized excluding leverage) on the lean hog futures contract is: A. −37.2%. B. −36.0%. C. −34.8%. 20. Which of Nabli’s statements about roll returns is correct? A. Only Statement 4 B. Only Statement 5 C. Both Statement 4 and Statement 5 21. The best response to Nabli’s question about the relative performance of the two indexes is that Index B is most likely to exhibit returns that are: A. lower than those of Index A. B. the same as those of Index A. 353 354 Learning Module 5 Introduction to Commodities and Commodity Derivatives C. higher than those of index A. 22. Given Nabli’s expectations for crude oil, the most appropriate swap position is the: A. basis swap. B. volatility swap. C. excess return swap. Solutions SOLUTIONS 1. C is correct. Commodity arbitrage involves an ability to inventory physical commodities and the attempt to capitalize on mispricing between the commodity (along with related storage and financing costs) and the futures price. The Apex Fund has access to storage facilities and uses these facilities in the attempt to capitalize on mispricing opportunities. 2. C is correct. Government actions can affect the supply or demand of all four sectors of the Apex Fund. With respect to energy, environmental mandates imposed by governments have tightened pollution standards, which have led to increasing processing costs that negatively affect demand. The supply of livestock, such as hogs and cattle, is affected by government-permitted use of drugs and growth hormones. Softs, or cash crops, can be affected by government actions, such as the attempt to maintain strategic stockpiles to control domestic prices. The level of demand and relative value of a precious metal, such as gold, is directly linked to government actions associated with managing to inflation targets. 3. C is correct. Expected future cash flows affect the valuation of financial assets, such as stocks and bonds, but do not affect the valuation of commodities. Financial assets (stocks and bonds) are valued based on expected future cash flows. In contrast, the valuation of a commodity is based on a discounted forecast of a future commodity price, which incorporates storage and transportation costs. 4. C is correct. When the near-term (i.e., closer to expiration) futures contract price is higher than the longer-term futures contract price, the futures market for the commodity is in backwardation. Because gasoline is the only one of the three futures markets in Exhibit 2 in which the near-term futures contract price ($2.2701) is higher than the longer-term contract price ($2.0307), the gasoline futures market is the only one in backwardation. 5. B is correct. The theory of storage focuses on the level of commodity inventories and the state of supply and demand. A commodity that is regularly stored should have a higher price in the future (contango) to account for those storage costs. Because coffee is a commodity that requires storage, its higher future price is consistent with the theory of storage. 6. C is correct. Roll returns are generally positive (negative) when the futures market is in backwardation (contango) and zero when the futures market is flat. Because the gasoline market is in backwardation, its roll returns will most likely be positive. 7. A is correct. The total return on the trade represents the sum of three components: price return, roll return, and collateral return. Price return = (Current price − Previous price)/Previous price = (877.0 − 865.0)/865.0 = 1.387%. Roll return = [(Near-term futures contract closing price − Farther-term futures contract closing price)/Near-term futures contract closing price] × Percentage of the position in the futures contract being rolled. Because the entire position is being rolled, the percentage of the position in the futures contract being rolled is equal to 100%. So: 355 356 Learning Module 5 Introduction to Commodities and Commodity Derivatives Roll return = [(877.0 − 883.0)/877.0] × 100% = −0.684%. Collateral return = [3 months/12 months] × 0.60% = 0.15%. Total return = 1.387% − 0.684% + 0.15% = 0.853%. 8. A is correct. The total return swap involves a monthly cash settlement (reset) based on the performance of the underlying reference asset (S&P GSCI) given a notional amount of $25 million. If the level of the index increases between the two valuation dates (in this case, May and June), the long position (the swap buyer) receives payment. If the level of the index decreases between the two valuation dates, the swap seller receives payment. The return on the reference index for the month of June is [(2,525.21 − 2,582.23)/2,582.23], which is equivalent to −2.2082%. Therefore, the swap buyer (long position) must pay the swap seller a cash settlement for the month of June. The June payment calculation is equal to $25,000,000 × −2.2082%, or −$552,042.23. 9. B is correct. Hedgers trade in the futures markets to hedge their exposures related to the commodity, as stated in Farmhouse’s risk management policy. 10. C is correct. The life cycle of livestock does vary widely by product. Grains have uniform, well-defined seasons and growth cycles specific to geographic regions. Therefore, both statements are correct. 11. C is correct. Commodity prices are affected by supply and demand, and improvements in freezing technology can improve the firm’s ability to store its products for longer periods and manage the volatility of supply and demand. For example, during times of excess supply, a livestock producer, such as Farmhouse, can freeze its products and offer them during better market supply conditions. 12. B is correct. The futures market for soybeans is in a state of contango because the spot price is lower than the futures price. 13. C is correct. In Exhibit 1, the spot price of soybeans is less than the futures price. This observation can be explained only by the hedging pressure hypothesis. According to this hypothesis, hedging pressure occurs when both producers and consumers seek to protect themselves from commodity market price volatility by entering into price hedges to stabilize their projected profits and cash flows. If consumers are more interested in hedging than producers are, the futures price will exceed the spot price. In contrast, the insurance theory predicts that the futures price has to be lower than the current spot price as a form of payment or remuneration to the speculator who takes on the price risk and provides price insurance to the commodity seller. Similarly, the theory of storage also predicts that when a commodity’s convenience yield is greater than its direct storage costs, the futures price will be lower than the spot price. 14. A is correct. The total return for a fully collateralized position is the sum of the price return, the roll return, and the collateral return: Price return = (Current price − Previous price)/Previous price = (2.99 − 2.93)/2.93 = 2.05%. Roll return Solutions = (Near-term futures closing price − Farther-term futures closing price)/Near-term futures closing price × Percentage of position in futures contract being rolled = [(2.99 − 3.03)/2.99] × 100% = −1.34%. Collateral return = Annual rate × Period length as a fraction of the year = 3% × 0.25 = 0.75%. Therefore, the total return for three months = 2.05% − 1.34% + 0.75% = 1.46%. 15. C is correct. Investment positions are evaluated on the basis of total return, and the roll return is part of the total return. Even though negative roll return negatively affects the total return, this effect could be more than offset by positive price and collateral returns. Therefore, it is possible that positions with negative roll returns outperform positions with positive roll returns, depending on the price and collateral returns. 16. B is correct. The most common way to invest in commodities is via derivatives, and commodities do not generate future cash flows beyond what can be realized through their purchase and sale. Also, storage costs are positively related to futures prices. Physical assets have to be stored, and storage incurs costs (rent, insurance, spoilage, etc.). Therefore, a commodity that is regularly stored should have a higher price in the future to account for those storage costs. 17. B is correct. The Brent crude oil futures market is in a state of backwardation. Commodity futures markets are in a state of backwardation when the spot price is greater than the price of near-term (i.e., nearest-to-expiration) futures contracts and, correspondingly, the price of near-term futures contracts is greater than that of longer-term contracts. The calendar spread is the difference between the near-term futures contract price and the longer-term futures contract price, which is $73.64 − $73.59 = $0.05. The basis for the near-term Brent crude oil futures contract is the difference between the spot price and the near-term futures price: $77.56 − $73.64 = $3.92. 18. B is correct. The Brent crude oil futures market is in a state of backwardation: The spot price is greater than the price of near-term (i.e., nearest-to-expiration) futures contracts. Commodities (in this case, Brent crude oil) are physical assets, not virtual assets, such as stocks and bonds. Physical assets have to be stored, and storage incurs costs (rent, insurance, inspections, spoilage, etc.). According to the theory of storage, a commodity that is consumed along a value chain that allows for just-in-time delivery and use (i.e., minimal inventories and storage) can avoid these costs. Yamata’s research concluded that energy is consumed on a real-time basis and requires minimal storage. In this situation, demand dominates supply, and current prices are higher than futures prices (state of backwardation). 19. C is correct. The contract was held for one year, so the price return of −12% is an annualized figure. Additionally, the −24% roll return is also annualized. Nabli’s collateral return equals 1.2% per year × 100% initial collateral investment = 1.2%. Therefore, the total return (annualized) is calculated as follows: Total return = Price return + Roll return + Collateral return. Total return = −12% + (−24%) + 1.2% = −34.8%. 357 358 Learning Module 5 Introduction to Commodities and Commodity Derivatives 20. C is correct. Roll returns are generally negative (positive) when the futures market is in contango (backwardation) and zero when the futures market is flat. 21. C is correct. Index B is likely to have higher performance than Index A in a market that is trending upward. Indexes that (perhaps inadvertently) contain contracts that more commonly trade in backwardation may improve forward-looking performance because this generates a positive roll return. Similarly, indexes that contain contracts that more commonly trade in contango may hurt performance for the same reason (i.e., negative roll return). 22. A is correct. Nabli expects the price of Brent crude oil to increase more than that of heavy crude oil, and Nabli can take advantage of this prediction by entering into a basis swap that is long Brent crude oil and short heavy crude oil. Nabli should take a short (not long) position in a volatility swap to take advantage of his prediction that Brent crude oil’s price volatility will be lower than its expected volatility. Nabli should take a long (not short) position in an excess return swap to take advantage of his expectation that the level of the ICE Brent Index will increase faster than leading oil benchmarks. Portfolio Management 398 Learning Module 1 Exchange-Traded Funds: Mechanics and Applications PRACTICE PROBLEMS 1. Which of the following statements regarding exchange-traded funds (ETFs) is correct? ETFs: A. disclose their holdings on a quarterly basis. B. trade in both primary and secondary markets. C. offer a creation/redemption mechanism that allows any investor to create or redeem shares. 2. The list of securities that a particular ETF wants to own, which is disclosed daily by all ETFs, is referred to as the: A. creation unit. B. creation basket. C. redemption basket. 3. When an authorized participant transacts to create or redeem ETF shares, the related costs are ultimately borne: A. solely by the ETF sponsor. B. solely by the AP. C. proportionally by all existing ETF shareholders. 4. Assuming arbitrage costs are minimal, which of the following is most likely to occur when the share price of an ETF is trading at a premium to its intraday NAV? A. New ETF shares will be created by the ETF sponsor. B. Redemption baskets will be received by APs from the ETF sponsor. C. Retail investors will exchange baskets of securities that the ETF tracks for creation units. The following information relates to questions 5-10 Howie Rutledge is a senior portfolio strategist for an endowment fund. Rutledge meets with recently hired junior analyst Larry Stosur to review the fund’s holdings. Rutledge asks Stosur about the mechanics of exchange-traded funds (ETFs). Stosur responds by making the following statements: Statement 1 Unlike mutual fund shares that can be shorted, ETF shares cannot be shorted. Statement 2 In the ETF creation/redemption process, the authorized participants (APs) absorb the costs of transacting securities for the ETF’s portfolio. Practice Problems Statement 3 399 If ETF shares are trading at a discount to NAV and arbitrage costs are sufficiently low, APs will buy the securities in the creation basket and exchange them for ETF shares from the ETF sponsor. Rutledge notes that one holding, ETF 1, is trading at a premium to its intraday NAV. He reviews the ETF’s pricing and notes that the premium to the intraday NAV is greater than the expected arbitrage costs. Stosur is evaluating three ETFs for potential investment. He notes that the ETFs have different portfolio characteristics that are likely to affect each ETF’s tracking error. A summary of the characteristics for the ETFs is presented in Exhibit 1. Exhibit 1: ETF Characteristics Affecting Tracking Error ETF 2 ETF 3 ETF 4 Portfolio Construction Approach Full Replication Representative Sampling Full Replication Type of Foreign Holdings Local shares ADRs* ADRs* Yes Yes No Engagement in Securities Lending *ADRs are American Depositary Receipts. Rutledge and Stosur discuss the factors that influence ETF bid–ask spreads. Stosur tells Rutledge that quoted bid–ask spreads for a particular transaction size are (1) negatively related to the amount of the ongoing order flow in the ETF, (2) positively related to the costs and risks for the ETF liquidity provider, and (3) positively related to the amount of competition among market makers for the ETF. As ETF shares may trade at prices that are different from the NAV, Rutledge examines selected data in Exhibit 2 for three ETFs that might have this problem. Exhibit 2: Selected Data on ETFs ETF 5 ETF 6 ETF 7 Percentage of Foreign Holdings 10% 50% 90% Trading Frequency High Low Low Rutledge considers a new ETF investment for the fund. He plans to own the ETF for nine months. The ETF has the following trading costs and management fees: ■ Annual management fee of 0.32% ■ Round-trip trading commissions of 0.20% ■ Bid–offer spread of 0.10% on purchase and sale Rutledge asks Stosur to compute the expected total holding period cost for investing in the ETF. 5. Which of Stosur’s statements regarding ETF mechanics is correct? A. Statement 1 B. Statement 2 C. Statement 3 400 Learning Module 1 Exchange-Traded Funds: Mechanics and Applications 6. Given the current pricing of ETF 1, the most likely transaction to occur is that: A. new ETF shares will be created by the APs. B. redemption baskets will be received by APs from the ETF sponsor. C. retail investors will exchange baskets of securities that the ETF tracks for creation units. 7. Which ETF in Exhibit 1 is most likely to have the lowest tracking error? A. ETF 2 B. ETF 3 C. ETF 4 8. Stosur’s statement about quoted bid–ask spreads is incorrect with respect to the: A. amount of the ongoing order flow in the ETF. B. costs and risks for the ETF liquidity providers. C. amount of competition among market makers for the ETF. 9. Which ETF in Exhibit 2 is most likely to trade at the largest premium or discount relative to NAV? A. ETF 5 B. ETF 6 C. ETF 7 10. Excluding the compounding effect, the expected total holding period cost for investing in the ETF over a nine-month holding period is closest to: A. 0.54%. B. 0.62%. C. 0.64%. 11. An ETF’s reported tracking error is typically measured as the: A. standard deviation of the difference in daily returns between an ETF and its benchmark. B. difference in annual return between an ETF and its benchmark over the past 12 months. C. annualized standard deviation of the difference in daily returns between an ETF and its benchmark. 12. To best assess an ETF’s performance, which reflects the impact of portfolio rebalancing expenses and other fees, an investor should: A. review daily return differences between the ETF and its benchmark. B. perform a rolling return assessment between the ETF and its benchmark. Practice Problems C. compare the ETF’s annual expense ratio with that of other ETFs in its asset class category. 13. An ETF’s tracking error, as traditionally reported, indicates to investors: A. whether the ETF is underperforming or outperforming its underlying index. B. the magnitude by which an ETF’s returns deviate from its benchmark over time. C. the distribution of differences in daily returns between the ETF and its benchmark. 14. For a typical ETF, which of the following sources of tracking error is most likely to be the smallest contributor to tracking error? A. Representative sampling B. Fees and expenses incurred by the ETF C. Changes to the underlying index securities 15. Which of the following statements relating to capital gains in ETFs and mutual funds is correct? A. ETFs tend to distribute less in capital gains than mutual funds do. B. Mutual funds may elect not to distribute all realized capital gains in a given year. C. The selling of ETF shares by some investors may create capital gains that affect the remaining ETF investors in terms of taxes. 16. Which of the following statements regarding distributions made by ETFs is correct? A. Return-of-capital (ROC) distributions are generally not taxable. B. ETFs generally reinvest any dividends received back into the ETF’s holdings. C. A dividend distribution is a distribution paid to investors in excess of an ETF’s earnings. 17. Investors buying ETFs: A. incur management fees that decrease with the length of the holding period. B. are assured of paying a price equal to the NAV if they purchase shares at the market close. C. incur trading costs in the form of commissions and bid–ask spreads at the time of purchase. 18. Consider an ETF with the following trading costs and management fees: ■ Annual management fee of 0.40% ■ Round-trip trading commissions of 0.55% 401 402 Learning Module 1 Exchange-Traded Funds: Mechanics and Applications ■ Bid–offer spread of 0.20% on purchase and sale Excluding compound effects, the expected total holding-period cost for investing in the ETF over a nine-month holding period is closest to: A. 1.05%. B. 1.15%. C. 1.25%. 19. The bid–ask spread for very liquid, high-volume ETFs will be least influenced by the: A. market maker’s desired profit spread. B. creation/redemption fees and other direct costs. C. likelihood of receiving an offsetting ETF order in a short time frame. 20. Such factors as regulations, competition, and corporate actions relate to: A. fund-closure risk. B. counterparty risk. C. expectation-related risk. 21. John Smith has invested in an inverse ETF. Smith is a novice investor who is not familiar with inverse ETFs, and therefore, he is unsure how the ETF will perform because of a lack of understanding of the ETF’s risk and return characteristics. This risk is best described as: A. counterparty risk. B. holdings-based risk. C. expectation-related risk. 22. Factor (smart beta) strategy ETFs are least likely to be used by investors: A. to modify portfolio risk. B. for tactical trading purposes. C. to seek outperformance versus a benchmark. 23. Which of the following statements regarding applications of ETFs in portfolio management is correct? A. Equity ETFs tend to be more active than fixed-income ETFs. B. The range of risk exposures available in the futures market is more diverse than that available in the ETF space. C. ETFs that have the highest trading volumes in their asset class category are generally preferred for tactical trading applications. Solutions SOLUTIONS 1. B is correct. ETFs trade in both primary and secondary markets. The primary market for ETF trading is that which exists on an over-the-counter basis between authorized participants (APs), a special group of institutional investors, and the ETF issuer or sponsor. This process is referred to as creation/redemption, and it is only through these primary market transactions that shares of the ETF can be created or destroyed. ETFs also trade in the secondary market on exchanges. Secondary market trading happens between any pair of market participants— individual or institutional investors, market makers, and so on. 2. B is correct. Each day, ETF managers publicly disclose a list of securities that they want to own, which is referred to as the creation basket. This basket also serves as the portfolio for determining the intrinsic net asset value (NAV) of the ETF on the basis of prices during the trading day. 3. B is correct. The AP generally absorbs all the costs associated with buying or selling the securities in the baskets or the ETF shares and pays an additional fee to the ETF provider to cover processing fees associated with creation/redemption activities. APs pass these costs to investors in the ETF’s bid–ask spread, which is incurred by investors entering (ETF share buyers) and exiting (ETF share sellers) the fund. 4. A is correct. When the share price of an ETF is trading at a premium to its intraday NAV and assuming arbitrage costs are minimal, APs will step in and take advantage of the arbitrage. Specifically, APs will step in and buy the basket of securities that the ETF tracks (the creation basket) and exchange it with the ETF provider for new ETF shares (a creation unit). These new shares received by APs can then be sold on the open market to realize arbitrage profits. 5. B is correct. Statement 2 is correct. A significant advantage of the ETF creation/ redemption process is that the AP absorbs all costs of transacting the securities for the fund’s portfolio. APs pass these costs to investors in the ETF’s bid–ask spread, incurred by ETF buyers and sellers. Thus, non-transacting shareholders of an ETF are shielded from the negative impact of transaction costs caused by other investors entering and exiting the fund. In contrast, when investors enter or exit a traditional mutual fund, the mutual fund manager incurs costs to buy or sell investments arising from this activity, which affects all fund shareholders. This makes the ETF structure inherently fairer: Frequent ETF traders bear the cost of their activity, while buy-and-hold ETF shareholders are shielded from those costs. Investors cannot short mutual fund shares, but they can short ETF shares. Also, if ETF shares are trading at a discount to NAV and arbitrage costs are sufficiently low, APs will buy ETF shares and exchange them for the securities in the redemption basket. Statement 3 describes the scenario that would occur if the ETF shares are trading at a premium to NAV. A is incorrect because Statement 1 is incorrect. Investors cannot short mutual fund shares, but they can short ETF shares. C is incorrect because Statement 3 is incorrect. If ETF shares are trading at a discount to NAV and arbitrage costs are sufficiently low, APs will buy ETF shares and exchange them for the securities in the redemption basket. Statement 3 describes the scenario that would occur if ETF shares are trading at a premium to NAV. 6. A is correct. When the share price of an ETF is trading at a premium to its intraday NAV and arbitrage costs are minimal, APs will step in and take advantage 403 404 Learning Module 1 Exchange-Traded Funds: Mechanics and Applications of the arbitrage. Specifically, APs will buy the basket of securities that the ETF tracks (the creation basket) and exchange it with the ETF sponsor for new ETF shares (a creation unit). These new ETF shares received by APs can then be sold on the open market to realize arbitrage profits. B is incorrect because in the case of an ETF trading at a premium to NAV, the APs will not receive redemption baskets of securities. Instead, the APs will deliver creation baskets to the ETF sponsor and receive new ETF shares. C is incorrect because only APs can deliver creation baskets or receive redemption baskets from the ETF sponsors. Retail investors can buy and sell ETF shares on the open market. 7. A is correct. Compared with a full replication approach, ETF portfolios managed using a representative sampling/optimization approach are likely to have greater tracking error. Also, differences in trading hours for depositary receipts and local constituent shares create discrepancies between the portfolio and index values. These discrepancies can lead to greater tracking error for portfolios holding ADRs in lieu of the underlying local shares. Further, ETF sponsors that engage in securities lending can generate additional portfolio income to help offset fund expenses, thereby lowering tracking error. ETF 2 uses a full replication approach, holds only local foreign shares, and engages in securities lending. Therefore, ETF 2 will likely have the lowest tracking error out of the ETFs in Exhibit 1. ETF 3 will likely have greater tracking error than ETF 2 because it is managed using a representative sampling approach and is invested in depositary receipts in lieu of local shares. ETF 4 will likely have greater tracking error than ETF 2 because it is invested in depositary receipts in lieu of local shares and does not engage in securities lending. 8. C is correct. Several factors determine the width of an ETF’s quoted bid–ask spread. First, the amount of ongoing order flow in the ETF is negatively related to the bid–ask spread (more flow means lower spreads). Second, the actual costs and risks for the liquidity provider are positively related to spreads (more costs and risks mean higher spreads); the spread is compensation to the liquidity provider for incurring these costs and risks. Finally, the amount of competition among market makers for that ETF is negatively related to the bid–ask spread (more competition means lower spreads). A is incorrect because Stosur is correct in stating that the quoted bid–ask spread for a particular transaction size is negatively related to the amount of the ongoing order flow in the ETF (more flow means lower spreads). B is incorrect because Stosur is correct in stating that the quoted bid–ask spread for a particular transaction size is positively related to the costs and risks for the ETF liquidity provider (more costs and risks mean higher spreads). The bid–ask spread represents the market maker’s price for taking the other side of the ETF transaction, which includes the costs and risks to carry the position on its books and/or to hedge the position using underlying securities or closely related ETFs or derivatives. 9. C is correct. ETFs that trade infrequently may have large premiums or discounts to NAV, because the ETF may not have traded in the hours leading up to the market close and NAV may have significantly risen or fallen during that time because of market movement. Furthermore, NAV is often a poor fair value indicator for ETFs holding foreign securities because of differences in exchange closing times between the underlying (e.g., foreign stocks, bonds, or commodities) and the exchange where the ETF trades. Therefore, ETF 7 is most likely to have the largest discount or premium because it has a low trading frequency and has the highest Solutions percentage of foreign holdings among the three ETFs. A is incorrect because ETF 5 has the lowest percentage of foreign holdings among the three ETFs and is the one ETF with a high trading frequency. Therefore, relative to ETF 7, with its low trading frequency and high foreign holdings, ETF 5 is likely to trade at smaller premiums or discounts. B is incorrect because ETF 6 has a lower percentage of foreign holdings than ETF 7. Even though both ETF 6 and ETF 7 have the same low trading frequency, the lower percentage of foreign holdings for ETF 6 is likely to result in it trading at smaller premiums or discounts. 10. A is correct. The expected total holding period cost for investing in the ETF over the nine-month holding period is calculated as follows: Total expected holding period cost = Annual management fee + Round-trip trading commissions + Bid–offer spread on purchase/sale. Total expected holding period cost = (9/12) × (0.32%) + 0.20% + 0.10% = 0.54%. 11. C is correct. An ETF’s tracking error is typically reported as the annualized standard deviation of the daily differential returns of the ETF and its benchmark. 12. B is correct. A rolling return assessment, referred to in the ETF industry as the “tracking difference,” provides a more informative picture of the investment outcome for an investor in an ETF. Such an analysis allows investors to see the cumulative effect of portfolio management and expenses over an extended period. It also allows for comparison with other annual metrics such as a fund’s expense ratio. Tracking error, as a statistic, reveals only ETF tracking variability; it does not reveal to investors whether the fund is over- or underperforming its index or whether that tracking error is concentrated over a few days or is more consistently experienced. An ETF’s expense ratio does not fully reflect the investor experience. That is, the expense ratio does not reflect the cost of portfolio rebalancing or other fees, making it an inferior assessment measure relative to a rolling return assessment. 13. B is correct. An ETF’s tracking error is typically reported as the annualized standard deviation of the daily differential returns of the ETF and its benchmark. Therefore, an ETF’s reported tracking error indicates to investors the magnitude by which an ETF’s returns deviate from those of its benchmark over time. 14. C is correct. Although additions and deletions of securities from the underlying benchmark index may occur and result in tracking error, such index changes generally occur infrequently (often quarterly). In addition, ETF portfolio managers may work with APs for index rebalance trades to ensure market-on-close pricing to minimize this source of tracking error. Therefore, the resulting tracking error caused by index changes will not likely be as large as the tracking error caused by representative sampling or by fees and expenses incurred by the ETF. 15. A is correct. ETFs tend to distribute far less in capital gains relative to mutual funds. This is mostly due to the fact that ETFs have historically had significantly lower turnover than mutual funds have had. 16. A is correct. Return-of-capital distributions are amounts paid out in excess of an ETF’s earnings and serve to reduce an investor’s cost basis by the amount of the distribution. These distributions are generally not taxable. 17. C is correct. ETF trading costs in the form of commissions and bid–ask spreads 405 406 Learning Module 1 Exchange-Traded Funds: Mechanics and Applications are paid by investors buying or selling ETF shares on an exchange. These trading costs are influenced by the bid–ask spread of the ETF, the size of the trade relative to the normal trading activity of the ETF, and the ease of hedging the ETF by the market-making community. Even the closing price of the ETF on the exchange includes a premium or discount to the NAV, driven by supply and demand factors on the exchange and the market impact costs of executing an exchange transaction. The purchase and sale trading costs of an ETF are paid regardless of holding period, whereas other costs, such as management fees, increase as the holding period lengthens. 18. A is correct. The expected total holding-period cost for investing in the ETF over a nine-month holding period is calculated as follows: Total holding-period cost = Annual management fee + Round-trip trading commissions + Bid–offer spread on purchase/sale. Total holding-period cost = (9/12) × (0.40%) + 0.55% + 0.20% = 1.05%. 19. B is correct. ETF bid–ask spreads are generally less than or equal to the combination of the following: ■ ± Creation/redemption fees and other direct costs, such as brokerage and exchange fees ■ + Bid–ask spread of the underlying securities held by the ETF ■ + Compensation for the risk of hedging or carrying positions by liquidity providers (market makers) for the remainder of the trading day ■ + Market maker’s desired profit spread ■ − Discount related to the likelihood of receiving an offsetting ETF order in a short time frame For very liquid and high-volume ETFs, buyers and sellers are active throughout the trading day. Therefore, because most of these ETF trades are matched extremely quickly and never involve the creation/redemption process, the first three factors listed do not contribute heavily to their bid–ask spreads. So, creation/redemption fees and other direct costs are not likely to have much influence on these ETFs’ bid–ask spreads. 20. A is correct. Fund-closure risk is the risk that an ETF may shut down. The reasons that lead to an ETF closing down often have to do with changes in regulations, increased competition, and corporate activity (merger and acquisition activity within the ETF industry). 21. C is correct. Expectation-related risk is the risk that some ETF investors may not fully understand how more complex ETFs will perform because of a lack of understanding of sophisticated assets classes and strategies. 22. B is correct. Factor strategy ETFs are usually benchmarked to an index created with predefined rules for screening and/or weighting stock holdings and are considered longer-term, buy-and-hold investment options rather than tactical trading instruments. The strategy index rules are structured around return drivers or factors, such as value, dividend yield, earnings or dividend growth, quality, stock volatility, or momentum. Investors using factor-based investing seek outperformance versus a benchmark or portfolio risk modification. 23. C is correct. ETFs that have the highest trading volumes in their asset class category are generally preferred for tactical trading applications. 444 Learning Module 2 Using Multifactor Models PRACTICE PROBLEMS 1. Compare the assumptions of the arbitrage pricing theory (APT) with those of the capital asset pricing model (CAPM). 2. Assume that the following one-factor model describes the expected return for portfolios: E(Rp) = 0.10 + 0.12βp,1 Also assume that all investors agree on the expected returns and factor sensitivity of the three highly diversified Portfolios A, B, and C given in the following table: Portfolio Expected Return Factor Sensitivity A 0.196 0.80 B 0.156 1.00 C 0.244 1.20 Assuming the one-factor model is correct and based on the data provided for Portfolios A, B, and C, determine if an arbitrage opportunity exists and explain how it might be exploited. The following information relates to questions 3-8 Carlos Altuve is a manager-of-managers at an investment company that uses quantitative models extensively. Altuve seeks to construct a multi-manager portfolio using some of the funds managed by portfolio managers within the firm. Maya Zapata is assisting him. Altuve uses arbitrage pricing theory (APT) as a basis for evaluating strategies and managing risks. From his earlier analysis, Zapata knows that Funds A and B in Exhibit 1 are well diversified. He has not previously worked with Fund C and is puzzled by the data because it is inconsistent with APT. He asks Zapata gather additional information on Fund C’s holdings and to determine if an arbitrage opportunity exists among these three investment alternatives. Her analysis, using the data in Exhibit 1, confirms that an arbitrage opportunity does exist. Exhibit 1: Expected Returns and Factor Sensitivities (One-Factor Model) Fund Expected Return Factor Sensitivity A 0.02 0.5 B 0.04 1.5 C 0.03 0.9 Using a two-factor model, Zapata now estimates the three funds’ sensitivity to inflation and GDP growth. That information is presented in Exhibit 2. Zapata assumes a zero value for the error terms when working with the selected two-factor model. Practice Problems 445 Exhibit 2: Expected Returns and Factor Sensitivities (Two-Factor Model) Factor Sensitivity Expected Return Inflation GDP Growth A 0.02 0.5 1.0 B 0.04 1.6 0.0 C 0.03 1.0 1.1 Fund Altuve asks Zapata to calculate the return for Portfolio AC, composed of a 60% allocation to Fund A and 40% allocation to Fund C, using the surprises in inflation and GDP growth in Exhibit 3. Exhibit 3: Selected Data on Factors Research Staff Forecast Actual Value Inflation 2.0% 2.2% GDP Growth 1.5% 1.0% Factor Finally, Altuve asks Zapata about the return sensitivities of Portfolios A, B, and C given the information provided in Exhibit 3. 3. Which of the following is not a key assumption of APT, which is used by Altuve to evaluate strategies and manage risks? A. A factor model describes asset returns. B. Asset-specific risk can be eliminated through diversification. C. Arbitrage opportunities exist among well-diversified portfolios. 4. The arbitrage opportunity identified by Zapata can be exploited with: A. Strategy 1: Buy $50,000 Fund A and $50,000 Fund B; sell short $100,000 Fund C. B. Strategy 2: Buy $60,000 Fund A and $40,000 Fund B; sell short $100,000 Fund C. C. Strategy 3: Sell short $60,000 of Fund A and $40,000 of Fund B; buy $100,000 Fund C. 5. The two-factor model Zapata uses is a: A. statistical factor model. B. fundamental factor model. C. macroeconomic factor model. 446 Learning Module 2 Using Multifactor Models 6. Based on the data in Exhibits 2 and 3, the return for Portfolio AC, given the surprises in inflation and GDP growth, is closest to: A. 2.02%. B. 2.40%. C. 4.98%. 7. The surprise in which of the following had the greatest effect on fund returns? A. Inflation on Fund B B. GDP growth on Fund A C. GDP growth on Fund C 8. Based on the data in Exhibit 2, which fund is most sensitive to the combined surprises in inflation and GDP growth in Exhibit 3? A. Fund A B. Fund B C. Fund C The following information relates to questions 9-14 Hui Cheung, a portfolio manager, asks her assistant, Ronald Lam, to review the macroeconomic factor model currently in use and to consider a fundamental factor model as an alternative. The current macroeconomic factor model has four factors: Ri = ai + bi1FGDP + bi2FCAP + bi3FCON + bi4FUNEM + εi, Where FGDP, FCAP, FCON, and FUNEM represent unanticipated changes in four factors: gross domestic product, manufacturing capacity utilization, consumer spending, and the rate of unemployment, respectively. Lam assumes the error term is equal to zero when using this model. Lam estimates the current model using historical monthly returns for three portfolios for the most recent five years. The inputs used in and estimates derived from the macroeconomic factor model are presented in Exhibit 1. The US Treasury bond rate of 2.5% is used as a proxy for the risk-free rate of interest. Exhibit 1: Inputs for and Estimates from the Current Macroeconomic Model Factor Sensitivities and Intercept Coefficients Factor Benchmark Factor Surprise (%) 0.24 0.50 0.8 0.00 −1.45 −1.00 0.5 0.00 0.50 1.10 2.5 Portfolio 1 Portfolio 2 Portfolio 3 Intercept (%) 2.58 3.20 4.33 FGDP 0.75 1.00 −0.23 1.23 FCAP FCON Practice Problems 447 Factor Sensitivities and Intercept Coefficients Factor Portfolio 1 Portfolio 2 Portfolio 3 Benchmark Factor Surprise (%) FUNEM −0.14 0.00 −0.05 −0.10 1.0 Annual Returns, Most Recent Year Return (%) 6.00 4.00 5.00 4.50 Lam uses the macroeconomic model to calculate the tracking error and the mean active return for each portfolio. He presents these statistics in Exhibit 2. Exhibit 2: Macroeconomic Factor Model Tracking Error and Mean Active Return Portfolio Tracking Error Mean Active Return Portfolio 1 1.50% 1.50% Portfolio 2 1.30% −0.50% Portfolio 3 1.00% 0.50% Lam considers a fundamental factor model with four factors: Ri = aj + bj1FLIQ + bj2FLEV + bj3FEGR + bj4FVAR + εj, where FLIQ, FLEV, FEGR, and FVAR represent liquidity, financial leverage, earnings growth, and the variability of revenues, respectively. Lam and Cheung discuss similarities and differences between macroeconomic factor models and fundamental factor models, and Lam offers a comparison of those models to statistical factor models. Lam makes the following statements. Statement 1 The factors in fundamental factor models are based on attributes of stocks or companies, whereas the factors in macroeconomic factor models are based on surprises in economic variables. Statement 2 The factor sensitivities are generally determined first in fundamental factor models, whereas the factor sensitivities are estimated last in macroeconomic factor models. Lam also tells Cheung: An advantage of statistical factor models is that they make minimal assumptions, and therefore, statistical factor model estimation lends itself to easier interpretation than macroeconomic and fundamental factor models. Lam tells Cheung that multifactor models can be useful in active portfolio management, but not in passive management. Cheung disagrees; she tells Lam that multifactor models can be useful in both active and passive management. 9. Based on the information in Exhibit 1, the expected return for Portfolio 1 is closest to: A. 2.58%. B. 3.42%. C. 6.00%. 448 Learning Module 2 Using Multifactor Models 10. Based on Exhibit 1, the active risk for Portfolio 2 is explained by surprises in: A. GDP. B. consumer spending. C. all four model factors. 11. Based on Exhibit 2, which portfolio has the best information ratio? A. Portfolio 1 B. Portfolio 2 C. Portfolio 3 12. Which of Lam’s statements regarding macroeconomic factor models and fundamental factor models is correct? A. Only Statement 1 B. Only Statement 2 C. Both Statements 1 and 2 13. Is Lam’s comment regarding statistical factor models correct? A. Yes B. No, because he is incorrect with respect to interpretation of the models’ results C. No, because he is incorrect with respect to the models’ assumptions 14. Whose statement regarding the use of multifactor models in active and passive portfolio management is correct? A. Lam only B. Cheung only C. Both Lam and Cheung 15. Last year the return on Harry Company stock was 5 percent. The portion of the return on the stock not explained by a two-factor macroeconomic factor model was 3 percent. Using the data given below, calculate Harry Company stock’s expected return. Macroeconomic Factor Model for Harry Company Stock Actual Value (%) Expected Value (%) Stock’s Factor Sensitivity Change in interest rate 2.0 0.0 −1.5 Growth in GDP 1.0 4.0 2.0 Variable 16. Which type of factor model is most directly applicable to an analysis of the style orientation (for example, growth vs. value) of an active equity investment manag- Practice Problems er? Justify your answer. 17. Suppose an active equity manager has earned an active return of 110 basis points, of which 80 basis points is the result of security selection ability. Explain the likely source of the remaining 30 basis points of active return. The following information relates to questions 18-19 Address the following questions about the information ratio. 18. What is the information ratio of an index fund that effectively meets its investment objective? 19. What are the two types of risk an active investment manager can assume in seeking to increase his information ratio? 20. A wealthy investor has no other source of income beyond her investments and that income is expected to reliably meet all her needs. Her investment advisor recommends that she tilt her portfolio to cyclical stocks and high-yield bonds. Explain the advisor’s advice in terms of comparative advantage in bearing risk. 449 450 Learning Module 2 Using Multifactor Models SOLUTIONS 1. APT and the CAPM are both models that describe what the expected return on a risky asset should be in equilibrium given its risk. The CAPM is based on a set of assumptions including the assumption that investors’ portfolio decisions can be made considering just returns’ means, variances, and correlations. The APT makes three assumptions: 1. A factor model describes asset returns. 2. There are many assets, so investors can form well-diversified portfolios that eliminate asset-specific risk. 3. No arbitrage opportunities exist among well-diversified portfolios. 2. According to the one-factor model for expected returns, the portfolio should have these expected returns if they are correctly priced in terms of their risk: Portfolio 1 E(R A) = 0.10 + 0.12βA,1 = 0.10 + (0.12)(0.80) = 0.10 + 0.10 = 0.20 Portfolio 2 E(RB) = 0.10 + 0.12βB,1 = 0.10 + (0.12)(1.00) = 0.10 + 0.12 = 0.22 Portfolio 3 E(RC) = 0.10 + 0.12βC,1 = 0.10 + (0.12)(1.20) = 0.10 + 0.14 = 0.24 In the table below, the column for expected return shows that Portfolios A and C are correctly priced but Portfolio B offers too little expected return for its risk, 0.15 or 15%. By shorting Portfolio B (selling an overvalued portfolio) and using the proceeds to buy a portfolio 50% invested in A and 50% invested in C with a sensitivity of 1 that matches the sensitivity of B, for each monetary unit shorted (say each euro), an arbitrage profit of €0.22 − €0.15 = €0.07 is earned. Portfolio Expected Return Factor Sensitivity A 0.196 0.80 B 0.156 1.00 C 0.244 1.20 0.5A + 0.5C 0.22 1.00 3. C is correct. Arbitrage pricing theory (APT) is a framework that explains the expected return of a portfolio in equilibrium as a linear function of the risk of the portfolio with respect to a set of factors capturing systematic risk. A key assumption of APT is that, in equilibrium, there are no arbitrage opportunities. 4. C is correct. The expected return and factor sensitivities of a portfolio with a 60% weight in Fund A and a 40% weight in Fund B are calculated as weighted averages Solutions 451 of the expected returns and factor sensitivities of Funds A and B: Expected return of Portfolio 60/40 = (0.60)(0.02) + (0.40)(0.04) = 0.028, or 2.8% Factor sensitivity of Portfolio 60/40 = (0.60)(0.5) + (0.40)(1.5) = 0.9 Fund Expected Return Factor Sensitivity A 0.02 0.5 B 0.04 1.5 C 0.03 0.9 0.028 0.900 0.030 1.000 Portfolio 60/40 60%A + 40%B Portfolio 50/50 50%A + 50%B The factor sensitivity of Portfolio 60/40 is identical to that of Fund C; therefore, this strategy results in no factor risk relative to Portfolio C. However, Fund C’s expected return of 3.0% is higher than Portfolio 60/40’s expected return of 2.8%. This difference supports Strategy 3: buying Fund C and selling short Portfolio 60/40 to exploit the arbitrage opportunity. 5. C is correct. In a macroeconomic factor model, the factors are surprises in macroeconomic variables, such as inflation risk and GDP growth, that significantly explain returns. 6. A is correct. The macroeconomic two-factor model takes the following form: Ri = ai + bi1FINF + bi2FGDP + εi, where FINF and FGDP represent surprises in inflation and surprises in GDP growth, respectively, and ai represents the expected return to asset i. Using this model and the data in Exhibit 2, the returns for Fund A and Fund C are represented by the following: RA = 0.02 + 0.5FINF + 1.0FGDP + εA RC = 0.03 + 1.0FINF + 1.1FGDP + εC Surprise in a macroeconomic model is defined as actual factor minus predicted factor. The surprise in inflation is 0.2% (= 2.2% − 2.0%). The surprise in GDP growth is −0.5% (= 1.0% − 1.5%). The return for Portfolio AC, composed of a 60% allocation to Fund A and 40% allocation to Fund C, is calculated as the following: RAC = (0.6)(0.02) + (0.4)(0.03) + [(0.6)(0.5) + (0.4)(1.0)](0.002)+ [(0.6)(1.0) + (0.4)(1.1)](−0.005) + 0.6(0) + 0.4(0) = 2.02% 7. C is correct. Surprise in a macroeconomic model is defined as actual factor minus predicted factor. For inflation, the surprise factor is 2.2% − 2.0% = 0.2%; for 452 Learning Module 2 Using Multifactor Models GDP growth, the surprise factor is 1.0% − 1.5% = −0.5%. The effect on returns is the product of the surprise and the factor sensitivity. Change in Portfolio Return due to Surprise in Fund Inflation GDP Growth A 0.5 × 0.2% = 0.10% 1.0 × −0.5% = −0.50% B 1.6 × 0.2% = 0.32% 0.0 × −0.5% = 0.00% C 1.0 × 0.2% = 0.20% 1.1 × −0.5% = −0.55% The effect of the GDP growth surprise on Fund C was the largest single-factor effect on Fund returns (−0.55%). 8. A is correct. The effect of the surprises in inflation and GDP growth on the returns of the three funds is calculated as the following. Change in Portfolio Return Because of Surprise in Fund Inflation GDP Growth A 0.5 × 0.2% = 0.10% 1.0 × −0.5% = −0.50% B 1.6 × 0.2% = 0.32% 0.0 × −0.5% = 0.00% C 1.0 × 0.2% = 0.20% 1.1 × −0.5% = −0.55% The combined effects for the three funds are the following. Fund A: 0.10% + (−0.50%) = −0.40% Fund B: 0.32% + (0.00%) = 0.32% Fund C: 0.20% + (−0.55%) = −0.35% Therefore, Fund A is the most sensitive to the surprises in inflation and GDP growth in Exhibit 3. 9. A is correct. When using a macroeconomic factor model, the expected return is the intercept (when all model factors take on a value of zero). The intercept coefficient for Portfolio 1 in Exhibit 1 is 2.58. 10. C is correct. Active risk, also referred to as tracking risk or tracking error, is the sample standard deviation of the time series of active returns, where the active returns consist of the differences between the portfolio return and the benchmark return. Whereas GDP is the only portfolio non-zero sensitivity for Portfolio 2, the contribution to the portfolio’s active return is the sum of the differences between the portfolio’s and the benchmark’s sensitivities multiplied by the factor return. Because all four of the factor sensitivities of Portfolio 2 are different from the factor sensitivities of the benchmark, all four factors contribute to the portfolio’s active return and, therefore, to its active risk. 11. A is correct. Portfolio 1 has the highest information ratio, 1.0, and thus has the best mean active return per unit of active risk: _ _ ​R ​P​− ​R ​B​ _ IR = ​ ​ s​​(​RP​ ​− ​RB​ ​) 1.50% _ = ​1.50% = 1.00 This information ratio exceeds that of Portfolio 2 (−0.38) or Portfolio 3 (0.50). Solutions 12. C is correct. In a macroeconomic factor model, the factors are surprises in macroeconomic variables that significantly explain returns. Factor sensitivities are generally specified first in fundamental factor models, whereas factor sensitivities are estimated last in macroeconomic factor models. 13. B is correct. An advantage of statistical factor models is that they make minimal assumptions. However, the interpretation of statistical factors is generally more difficult than the interpretation of macroeconomic and fundamental factor models. 14. B is correct. Analysts can use multifactor models in passively managed portfolios to replicate an index fund’s factor exposures. 15. In a macroeconomic factor model, the surprise in a factor equals actual value minus expected value. For the interest rate factor, the surprise was 2 percent; for the GDP factor, the surprise was −3 percent. The intercept represents expected return in this type of model. The portion of the stock’s return not explained by the factor model is the model’s error term. 5% = Expected return − 1.5(Interest rate surprise) + 2(GDP surprise) + Error term = Expected return − 1.5(2%) + 2(−3%) + 3% = Expected return − 6% Rearranging terms, the expected return for Harry Company stock equals 5% + 6% = 11%. 16. A fundamental factor model. Such models typically include many factors related to the company (e.g., earnings) and to valuation that are commonly used indicators of a growth orientation. A macroeconomic factor model may provide relevant information as well, but typically indirectly and in less detail. 17. This remainder of 30 basis points would be attributable to the return from factor tilts. A portfolio manager’s active return is the sum of two components, factor tilts and security selection. Factor tilt is the product of the portfolio manager’s higher or lower factor sensitivities relative to the benchmark’s factor sensitivities and the factor returns. Security selection reflects the manager’s ability to overweight securities that outperform or underweight securities that underperform. 18. An index fund that effectively meets its investment objective is expected to have an information ratio of zero, because its active return should be zero. 19. The active manager may assume active factor risk and active specific risk (security selection risk) in seeking a higher information ratio. 20. This wealthy investor has a comparative advantage in bearing business cycle risk compared with the average investor who depends on income from employment. Because the average investor is sensitive to the business cycle and in particular the risk of recession, we would expect there to be a risk premium to hold recession-sensitive securities. Cyclical stocks and high-yield bonds are both very sensitive to the risk of recessions. Because the welfare of the wealthy investor is not affected by recessions, she can tilt her portfolio to include cyclical stocks and high yield bonds to attempt to capture the associated risk premiums. 453 510 Learning Module 3 Measuring and Managing Market Risk PRACTICE PROBLEMS The following information relates to questions 1-5 Randy Gorver, chief risk officer at Eastern Regional Bank, and John Abell, assistant risk officer, are currently conducting a risk assessment of several of the bank’s independent investment functions. These reviews include the bank’s fixed-income investment portfolio and an equity fund managed by the bank’s trust department. Gorver and Abell are also assessing Eastern Regional’s overall risk exposure. Eastern Regional Bank Fixed-Income Investment Portfolio The bank’s proprietary fixed-income portfolio is structured as a barbell portfolio: About half of the portfolio is invested in zero-coupon Treasuries with maturities in the 3- to 5-year range (Portfolio P1), and the remainder is invested in zero-coupon Treasuries with maturities in the 10- to 15-year range (Portfolio P2). Georges Montes, the portfolio manager, has discretion to allocate between 40% and 60% of the assets to each maturity “bucket.” He must remain fully invested at all times. Exhibit 1 shows details of this portfolio. Exhibit 1: US Treasury Barbell Portfolio Maturity Average duration Average yield to maturity Market value P1 P2 3–5 Years 10–15 Years 3.30 11.07 1.45% 2.23% $50.3 million $58.7 million Trust Department’s Equity Fund a. Use of Options: The trust department of Eastern Regional Bank manages an equity fund called the Index Plus Fund, with $325 million in assets. This fund’s objective is to track the S&P 500 Index price return while producing an income return 1.5 times that of the S&P 500. The bank’s chief investment officer (CIO) uses put and call options on S&P 500 stock index futures to adjust the risk exposure of certain client accounts that have an investment in this fund. The portfolio of a 60-year-old widow with a below-average risk tolerance has an investment in this fund, and the CIO has asked his assistant, Janet Ferrell, to propose an options strategy to bring the portfolio’s delta to 0.90. Practice Problems b. Value at Risk: The Index Plus Fund has a value at risk (VaR) of $6.5 million at 5% for one day. Gorver asks Abell to write a brief summary of the portfolio VaR for the report he is preparing on the fund’s risk position. Combined Bank Risk Exposures The bank has adopted a new risk policy, which requires forward-looking risk assessments in addition to the measures that look at historical risk characteristics. Management has also become very focused on tail risk since the subprime crisis and is evaluating the bank’s capital allocation to certain higher-risk lines of business. Gorver must determine what additional risk metrics to include in his risk reporting to address the new policy. He asks Abell to draft a section of the risk report that will address the risk measures’ adequacy for capital allocation decisions. 1. If Montes is expecting a 50 bp increase in yields at all points along the yield curve, which of the following trades is he most likely to execute to minimize his risk? A. Sell $35 million of P2 and reinvest the proceeds in three-year bonds B. Sell $15 million of P2 and reinvest the proceeds in three-year bonds C. Reduce the duration of P2 to 10 years and reduce the duration of P1 to 3 years 2. Which of the following options strategies is Ferrell most likely to recommend for the client’s portfolio? A. Long calls B. Short calls C. Short puts 3. Which of the following statements regarding the VaR of the Index Plus Fund is correct? A. The expected maximum loss for the portfolio is $6.5 million. B. Five percent of the time, the portfolio can be expected to experience a loss of at least $6.5 million. C. Ninety-five percent of the time, the portfolio can be expected to experience a one-day loss of no more than $6.5 million. 4. To comply with the new bank policy on risk assessment, which of the following is the best set of risk measures to add to the chief risk officer’s risk reporting? A. Conditional VaR, stress test, and scenario analysis B. Monte Carlo VaR, incremental VaR, and stress test C. Parametric VaR, marginal VaR, and scenario analysis 5. Which of the following statements should not be included in Abell’s report to management regarding the use of risk measures in capital allocation decisions? A. VaR measures capture the increased liquidity risk during stress periods. 511 512 Learning Module 3 Measuring and Managing Market Risk B. Stress tests and scenario analysis can be used to evaluate the effect of outlier events on each line of business. C. VaR approaches that can accommodate a non-normal distribution are critical to understand relative risk across lines of business. The following information relates to questions 6-13 Tina Ming is a senior portfolio manager at Flusk Pension Fund (Flusk). Flusk’s portfolio is composed of fixed-income instruments structured to match Flusk’s liabilities. Ming works with Shrikant McKee, Flusk’s risk analyst. Ming and McKee discuss the latest risk report. McKee calculated value at risk (VaR) for the entire portfolio using the historical method and assuming a lookback period of five years and 250 trading days per year. McKee presents VaR measures in Exhibit 1. Exhibit 1: Flusk Portfolio VaR (in $ millions) Confidence Interval 95% Daily VaR Monthly VaR 1.10 5.37 After reading McKee’s report, Ming asks why the number of daily VaR breaches over the last year is zero even though the portfolio has accumulated a substantial loss. Next, Ming requests that McKee perform the following two risk analyses on Flusk’s portfolio: Analysis 1 Use scenario analysis to evaluate the impact on risk and return of a repeat of the last financial crisis. Analysis 2 Estimate over one year, with a 95% level of confidence, how much Flusk’s assets could underperform its liabilities. Ming recommends purchasing newly issued emerging market corporate bonds that have embedded options. Prior to buying the bonds, Ming wants McKee to estimate the effect of the purchase on Flusk’s VaR. McKee suggests running a stress test using a historical period specific to emerging markets that encompassed an extreme change in credit spreads. At the conclusion of their conversation, Ming asks the following question about risk management tools: “What are the advantages of VaR compared with other risk measures?” 6. Based on Exhibit 1, Flusk’s portfolio is expected to experience: A. a minimum daily loss of $1.10 million over the next year. B. a loss over one month equal to or exceeding $5.37 million 5% of the time. C. an average daily loss of $1.10 million 5% of the time during the next 250 trading days. Practice Problems 7. The number of Flusk’s VaR breaches most likely resulted from: A. using a standard normal distribution in the VaR model. B. using a 95% confidence interval instead of a 99% confidence interval. C. lower market volatility during the last year compared with the lookback period. 8. To perform Analysis 1, McKee should use historical bond: A. prices. B. yields. C. durations. 9. The limitation of the approach requested for Analysis 1 is that it: A. omits asset correlations. B. precludes incorporating portfolio manager actions. C. assumes no deviation from historical market events. 10. The estimate requested in Analysis 2 is best described as: A. liquidity gap. B. surplus at risk. C. maximum drawdown. 11. Which measure should McKee use to estimate the effect on Flusk’s VaR from Ming’s portfolio recommendation? A. Relative VaR B. Incremental VaR C. Conditional VaR 12. When measuring the portfolio impact of the stress test suggested by McKee, which of the following is most likely to produce an accurate result? A. Marginal VaR B. Full revaluation of securities C. The use of sensitivity risk measures 13. The risk management tool referenced in Ming’s question: A. is widely accepted by regulators. B. takes into account asset liquidity. C. usually incorporates right-tail events. 513 514 Learning Module 3 Measuring and Managing Market Risk The following information relates to questions 14-20 Carol Kynnersley is the chief risk officer at Investment Management Advisers (IMA). Kynnersley meets with IMA’s portfolio management team and investment advisers to discuss the methods used to measure and manage market risk and how risk metrics are presented in client reports. The three most popular investment funds offered by IMA are the Equity Opportunities, the Diversified Fixed Income, and the Alpha Core Equity. The Equity Opportunities Fund is composed of two exchange-traded funds: a broadly diversified large-cap equity product and one devoted to energy stocks. Kynnersley makes the following statements regarding the risk management policies established for the Equity Opportunities portfolio: Statement 1 IMA’s preferred approach to model value at risk (VaR) is to estimate expected returns, volatilities, and correlations under the assumption of a normal distribution. Statement 2 In last year’s annual client performance report, IMA stated that a hypothetical $6 million Equity Opportunities Fund account had a daily 5% VaR of approximately 1.5% of portfolio value. Kynnersley informs the investment advisers that the risk management department recently updated the model for estimating the Equity Opportunities Fund VaR based on the information presented in Exhibit 1. Exhibit 1: Equity Opportunities Fund—VaR Model Input Assumptions Large-Cap ETF Energy ETF Total Portfolio Portfolio weight 65.0% 35.0% 100.0% Expected annual return 12.0% 18.0% 14.1% Standard deviation 20.0% 40.0% 26.3% Correlation between ETFs: 0.90 Number of trading days/year: 250 For clients interested in fixed-income products, IMA offers the Diversified Fixed-Income Fund. Kynnersley explains that the portfolio’s bonds are all subject to interest rate risk. To demonstrate how fixed-income exposure measures can be used to identify and manage interest rate risk, Kynnersley distributes two exhibits featuring three hypothetical Treasury coupon bonds (Exhibit 2) under three interest rate scenarios (Exhibit 3). Exhibit 2: Fixed-Income Risk Measure Hypothetical Bond Duration Bond 1 1.3 Bond 2 3.7 Bond 3 10.2 Practice Problems 515 Exhibit 3: Interest Rate Scenarios Scenario Interest Rate Environment Scenario 1 Rates increase 25 bps Scenario 2 Rates increase 10 bps Scenario 3 Rates decrease 20 bps One of the investment advisers comments that a client recently asked about the performance of the Diversified Fixed-Income Fund relative to its benchmark, a broad fixed-income index. Kynnersley informs the adviser as follows: Statement 3 The Diversified Fixed-Income Fund manager monitors the historical deviation between portfolio returns and benchmark returns. The fund prospectus stipulates a target deviation from the benchmark of no more than 5 bps. Kynnersley concludes the meeting by reviewing the constraints IMA imposes on securities included in the Alpha Core Equity Fund. The compliance department conducts daily oversight using numerous risk screens and, when indicated, notifies portfolio managers to make adjustments. Kynnersley makes the following statement: Statement 4 It is important that all clients investing in the fund be made aware of IMA’s compliance measures. The Alpha Core Equity Fund restricts the exposure of individual securities to 1.75% of the total portfolio. 14. Based on Statement 1, IMA’s VaR estimation approach is best described as the: A. parametric method. B. historical simulation method. C. Monte Carlo simulation method. 15. In Statement 2, Kynnersley implies that the portfolio: A. is at risk of losing $4,500 each trading day. B. value is expected to decline by $90,000 or more once in 20 trading days. C. has a 5% chance of falling in value by a maximum of $90,000 on a single trading day. 16. Based only on Statement 2, the risk measurement approach: A. ignores right-tail events in the return distribution. B. is similar to the Sharpe ratio because it is backward looking. C. provides a relatively accurate risk estimate in both trending and volatile regimes. 516 Learning Module 3 Measuring and Managing Market Risk 17. Based on Exhibit 1, the daily 5% VaR estimate is closest to: A. 1.61%. B. 2.42%. C. 2.69%. 18. Based only on Exhibits 2 and 3, it is most likely that under: A. Scenario 1, Bond 2 outperforms Bond 1. B. Scenario 2, Bond 1 underperforms Bond 3. C. Scenario 3, Bond 3 is the best performing security. 19. The risk measure referred to in Statement 3 is: A. active share. B. beta sensitivity C. ex post tracking error. 20. In Statement 4, Kynnersley describes a constraint associated with a: A. risk budget. B. position limit. C. stop-loss limit. The following information relates to questions 21-26 Hiram Life (Hiram), a large multinational insurer located in Canada, has received permission to increase its ownership in an India-based life insurance company, LICIA, from 26% to 49%. Before completing this transaction, Hiram wants to complete a risk assessment of LICIA’s investment portfolio. Judith Hamilton, Hiram’s chief financial officer, has been asked to brief the management committee on investment risk in its India-based insurance operations. LICIA’s portfolio, which has a market value of CAD260 million, is currently structured as shown in Exhibit 1. Despite its more than 1,000 individual holdings, the portfolio is invested predominantly in India. The Indian government bond market is highly liquid, but the country’s mortgage and infrastructure loan markets, as well as the corporate bond market, are relatively illiquid. Individual mortgage and corporate bond positions are large relative to the normal trading volumes in these securities. Given the elevated current and fiscal account deficits, Indian investments are also subject to above-average economic risk. Hamilton begins with a summary of the India-based portfolio. Exhibit 1 presents the current portfolio composition and the risk and return assumptions used to estimate value at risk (VaR). Practice Problems 517 Exhibit 1: Selected Assumptions for LICIA’s Investment Portfolio Allocation Average Daily Standard Daily Return Deviation India government securities 50% 0.015% 0.206% India mortgage/infrastructure loans 25% 0.045% 0.710% India corporate bonds 15% 0.025% 0.324% India equity 10% 0.035% 0.996% Infrastructure is a rapidly growing asset class with limited return history; the first infrastructure loans were issued just 10 years ago. Hamilton’s report to the management committee must outline her assumptions and provide support for the methods she used in her risk assessment. If needed, she will also make recommendations for rebalancing the portfolio to ensure its risk profile is aligned with that of Hiram. Hamilton develops the assumptions shown in Exhibit 2, which will be used for estimating the portfolio VaR. Exhibit 2: VaR Input Assumptions for Proposed CAD260 Million Portfolio Method Average Return Assumption Standard Deviation Assumption Monte Carlo simulation 0.026% 0.501% Parametric approach 0.026% 0.501% Historical simulation 0.023% 0.490% Hamilton elects to apply a one-day, 5% VaR limit of CAD2 million in her risk assessment of LICIA’s portfolio. This limit is consistent with the risk tolerance the committee has specified for the Hiram portfolio. The markets’ volatility during the last 12 months has been significantly higher than the historical norm, with increased frequency of large daily losses, and Hamilton expects the next 12 months to be equally volatile. She estimates the one-day 5% portfolio VaR for LICIA’s portfolio using three different approaches: Exhibit 3: VaR Results over a One-Day Period for Proposed Portfolio Method 5% VaR Monte Carlo simulation CAD2,095,565 Parametric approach CAD2,083,610 Historical simulation CAD1,938,874 The committee is likely to have questions in a number of key areas—the limitations of the VaR report, potential losses in an extreme adverse event, and the reliability of the VaR numbers if the market continues to exhibit higher-than-normal volatility. Hamilton wants to be certain that she has thoroughly evaluated the risks inherent in the LICIA portfolio and compares them with the risks in Hi- 518 Learning Module 3 Measuring and Managing Market Risk ram’s present portfolio. Hamilton believes the possibility of a ratings downgrade on Indian sovereign debt is high and not yet fully reflected in securities prices. If the rating is lowered, many of the portfolio’s holdings will no longer meet Hiram’s minimum ratings requirement. A downgrade’s effect is unlikely to be limited to the government bond portfolio. All asset classes can be expected to be affected to some degree. Hamilton plans to include a scenario analysis that reflects this possibility to ensure that management has the broadest possible view of the risk exposures in the India portfolio. 21. Given Hamilton’s expectations, which of the following models is most appropriate to use in estimating portfolio VaR? A. Parametric method B. Historical simulation method C. Monte Carlo simulation method 22. Which risk measure is Hamilton most likely to present when addressing the committee’s concerns regarding potential losses in extreme stress events? A. Relative VaR B. Incremental VaR C. Conditional VaR 23. The scenario analysis that Hamilton prepares for the committee is most likely a: A. stress test. B. historical scenario. C. hypothetical scenario. 24. The scenario analysis that Hamilton prepares for the committee is a valuable tool to supplement VaR because it: A. incorporates historical data to evaluate the risk in the tail of the VaR distribution. B. enables Hamilton to isolate the risk stemming from a single risk factor—the ratings downgrade. C. allows the committee to assess the effect of low liquidity in the event of a ratings downgrade. 25. Using the data in Exhibit 2, the portfolio’s annual 1% parametric VaR is closest to: A. CAD17 million. B. CAD31 million. C. CAD48 million. Practice Problems 26. What additional risk measures would be most appropriate to add to Hamilton’s risk assessment? A. Delta B. Duration C. Tracking error 519 520 Learning Module 3 Measuring and Managing Market Risk SOLUTIONS 1. B is correct. Duration is a measure of interest rate risk. To reduce risk in anticipation of an increase in interest rates, Montes would seek to shorten the portfolio’s duration. He is limited, however, in the amount he can shift from P2 to P1. Selling $15 million of P2 reduces that portfolio to the lower end of the permitted 40% to 60% range. By reinvesting the proceeds at the shortest maturities allowed, Montes substantially reduces the portfolio duration. 2. B is correct. An index-tracking portfolio without options has a delta of 1. To achieve a delta of 0.9, the delta of the options position must be negative. Of the three choices, only short calls have a negative delta. Long call options have deltas ranging from 0 to 1. Short calls, therefore, have deltas ranging from 0 to −1. The short call position lowers the portfolio’s overall delta as desired. 3. B is correct. VaR measures the frequency of losses of a given minimum magnitude. Here the VaR indicates that on 5% of trading days, the portfolio will experience a loss of at least $6.5 million. (Although C may appear to say the same thing as B, it actually implies that the portfolio will experience a loss on 95% of trading days.) The correct interpretation is that returns will be equal to or greater than −$6.5 million on 95% of trading days; those returns include gains as well as losses. 4. A is correct. The bank policy requires the addition of forward-looking risk assessments, and management is focused on tail risk. Conditional VaR measures tail risk, and stress tests and scenario analysis subject current portfolio holdings to historical or hypothetical stress events. 5. A is correct. VaR measures do not capture liquidity risk. “If some assets in a portfolio are relatively illiquid, VaR could be understated, even under normal market conditions. Additionally, liquidity squeezes are frequently associated with tail events and major market downturns, thereby exacerbating the risk.” 6. B is correct. VaR is the minimum loss that would be expected a certain percentage of the time over a specified period of time given the assumed market conditions. A 5% VaR is often expressed as its complement—a 95% level of confidence. Therefore, the monthly VaR in Exhibit 5 indicates that $5.37 million is the minimum loss that would be expected to occur over one month 5% of the time. Alternatively, 95% of the time, a loss of more than $5.37 million would not be expected. 7. C is correct. Flusk experienced zero daily VaR breaches over the last year yet incurred a substantial loss. A limitation of VaR is its vulnerability to different volatility regimes. A portfolio might remain under its VaR limit every day but lose an amount approaching this limit each day. If market volatility during the last year is lower than in the lookback period, the portfolio could accumulate a substantial loss without technically breaching the VaR constraint. A is incorrect because VaR was calculated using historical simulation, so the distribution used was based on actual historical changes in the key risk factors experienced during the lookback period. Thus, the distribution is not characterized using estimates of the mean return, the standard deviation, or the correlations among the risk factors in the portfolio. In contrast, the parametric method of estimating VaR generally assumes that the distribution of returns for the risk factors is normal. B is incorrect because a specification with a higher confidence level will produce a higher VaR. If a 99% confidence interval was used to calculate historical VaR, Solutions the VaR would be larger (larger expected minimum loss). During the last year, none of Flusk’s losses were substantial enough to breach the 5% VaR number (95% confidence interval); therefore, if McKee used a 1% VaR (99% confidence interval), the number of VaR breaches would not change. 8. B is correct. In order to simulate the impact of the latest financial crisis on the current bond portfolio holdings, McKee’s valuation model for bonds should use the historical yields of bonds with similar maturity. Historical yields drive the pricing of bonds more than the price history or the current duration. Historical prices for the fixed-income positions currently held in the portfolio may not exist, and even when historical prices do exist, they may not be relevant to the current characteristics (e.g., maturity) of the instrument. Even if the same bonds existed at the time of the latest financial crisis, their durations would change because of the passage of time. A is incorrect because using a bond’s past price history would mischaracterize the risk of the current portfolio holdings. For this reason, the historical yields are more important in explaining the risks. Historical prices for the fixed-income positions currently held in the portfolio may not exist, and even when historical prices do exist, they may not be relevant to the current characteristics (e.g., maturity) of the instrument. C is incorrect because historical bond durations would not capture the current characteristics of the bonds in the portfolio. Duration is a sensitivity measure and is the weighted-average time to maturity of a bond. Even if the same bonds existed at the time of the latest financial crisis, their remaining time to maturity and durations would change because of the passage of time. 9. C is correct. Ming suggested in Analysis 1 to use a historical scenario that measures the hypothetical portfolio return that would result from a repeat of a particular period of financial market history. Historical scenarios are complementary to VaR but are not going to happen in exactly the same way again, and they require additional measures to overcome the shortcomings of the VaR. 10. B is correct. Analysis 2 describes surplus at risk. Surplus at risk is an application of VaR; it estimates how much the assets might underperform the liabilities with a given confidence level, usually over a year. 11. B is correct. Incremental VaR measures the change in a portfolio’s VaR as a result of adding or removing a position from the portfolio or if a position size is changed relative to the remaining positions. 12. B is correct. McKee suggests running a stress test using a historical scenario specific to emerging markets that includes an extreme change in credit spreads. Stress tests, which apply extreme negative stress to a particular portfolio exposure, are closely related to scenario risk measures. A scenario risk measure estimates the portfolio return that would result from a hypothetical change in markets (hypothetical scenario) or a repeat of a historical event (historical scenario). When the historical simulation fully revalues securities under rate and price changes that occurred during the scenario period, the results should be highly accurate. A is incorrect because marginal VaR measures the change in portfolio VaR given a very small change in a portfolio position (e.g., change in VaR for a $1 or 1% change in the position). Therefore, marginal VaR would not allow McKee to estimate how much the value of the option-embedded bonds would change under an extreme change in credit spreads. C is incorrect because sensitivity risk measures use sensitivity exposure measures, such as first-order (delta, duration) and second-order (gamma, convexity) 521 522 Learning Module 3 Measuring and Managing Market Risk sensitivity, to assess the change in the value of a financial instrument. Although gamma and convexity can be used with delta and duration to estimate the impact of extreme market movements, they are not suited for scenario analysis related to option-embedded bonds. 13. A is correct. VaR has emerged as one of the most popular risk measures because global banking regulators require or encourage the use of it. VaR is also frequently found in annual reports of financial firms and can be used for comparisons. 14. A is correct. VaR is an estimate of the loss that is expected to be exceeded with a given level of probability over a specified time period. The parametric method typically assumes that the return distributions for the risk factors in the portfolio are normal. It then uses the expected return and standard deviation of return for each risk factor and correlations to estimate VaR. 15. B is correct. Value at risk is the minimum loss that would be expected a certain percentage of the time over a certain period of time. Statement 2 implies that there is a 5% chance the portfolio will fall in value by $90,000 (= $6,000,000 × 1.5%) or more in a single day. If VaR is measured on a daily basis and a typical month has 20–22 business days, then 5% of the days equates to about 1 day per month or once in 20 trading days. 16. A is correct. Statement 2 indicates that the Equity Opportunities Fund reported a daily VaR value. One of the limitations of VaR is that it focuses so heavily on left-tail events (the losses) that right-tail events (potential gains) are often ignored. B is incorrect because VaR is viewed as forward looking in that it uses the current portfolio holdings and measures its potential loss. The Sharpe ratio represents a backward-looking, return-based measure and is used to assess the skill of the manager. C is incorrect because VaR does not provide an accurate risk estimate in either trending or volatile regimes. A portfolio might remain under its VaR limit every day but lose an amount approaching this limit each day. Under such circumstances, the portfolio could accumulate substantial losses without technically breaching the VaR constraint. Also, during periods of low volatility, VaR will appear quite low, underestimating the losses that could occur when the environment returns to a normal level of volatility. 17. C is correct. Measuring VaR at a 5% threshold produces an estimated value at risk of 2.69%. From Exhibit 6, the expected annual portfolio return is 14.1% and the standard deviation is 26.3%. Annual values need to be adjusted to get their daily counterparts. Assuming 250 trading days in a year, the expected annual return is adjusted by dividing by 250 and the standard deviation is adjusted by dividing by the square root of 250. Thus, _ the daily expected return is 0.141/250 = 0.000564, and volatility is ​0.263 / ​ √250 ​= 0.016634. 5% daily VaR = E(Rp) − 1.65σp = 0.000564 − 1.65(0.016634) = −0.026882. The portfolio is expected to experience a potential minimum loss in percentage terms of 2.69% on 5% of trading days. 18. C is correct. The change in value of a bond is inversely related to a change in yield. Given a bond priced at B with duration D and yield change of Δy, the rate of return or percentage price change for the bond is approximately given as follows: ΔB/B ≈ −DΔy/(1 + y). Under Scenario 3, interest rates decrease by 20 bps. In an environment of decreasing interest rates, the bond with the highest dura- Solutions tion will have the greatest positive return. Bond 3 has a duration of 10.2, which is greater than that of both Bond 1 (duration = 1.3) and Bond 2 (duration = 3.7). 19. C is correct. A traditional asset manager uses ex post tracking error when analyzing backward-looking returns. The Diversified Fixed-Income Fund prospectus stipulates a target benchmark deviation of no more than 5 bps. Tracking error is a measure of the degree to which the performance of a given investment deviates from its benchmark. 20. B is correct. Position limits are limits on the market value of any given investment; they are excellent controls on overconcentration. Position limits can be expressed in currency units or as a percentage of net assets. The Alpha Core Equity Fund restricts the exposure of individual securities to 1.75% of the total portfolio. 21. C is correct. The Monte Carlo simulation method can accommodate virtually any distribution, an important factor given the increased frequency of large daily losses. This method can also more easily accommodate the large number of portfolio holdings. The Monte Carlo method allows the user to develop her own forward-looking assumptions about the portfolio’s risk and return characteristics, unlike the historical simulation method, which uses the current portfolio and re-prices it using the actual historical changes in the key factors experienced during the lookback period. Given the limited return history for infrastructure investments and Hamilton’s expectations for higher-than-normal volatility, the historical simulation method would be a suboptimal choice. 22. C is correct. Conditional VaR is a measure of tail risk that provides an estimate of the average loss that would be incurred if the VaR cutoff is exceeded. 23. C is correct. A hypothetical scenario analysis allows the risk manager to estimate the likely effect of the scenario on a range of portfolio risk factors. A sovereign ratings downgrade would affect Hiram’s India equity and corporate bond exposures as well as the government bond exposure. In addition, the assumptions used in constructing the scenario analysis can specifically address the effect of a need to sell large position sizes under decreased liquidity conditions resulting from a ratings downgrade. VaR alone does not accurately reflect the risk of large position sizes, which may be difficult to trade. 24. C is correct. A hypothetical scenario analysis allows Hamilton to estimate the direct effect of a ratings downgrade on the portfolio’s government bond holdings and the resulting need to sell a number of the portfolio’s holdings because they no longer meet the ratings guidelines. VaR alone does not accurately reflect the risk of large position sizes, which may be difficult to trade. The hypothetical scenario analysis will also highlight the effect of increased economic turmoil on all of the portfolio’s exposures, not only the government bond exposures. 523 524 Learning Module 3 Measuring and Managing Market Risk 25. B is correct. The VaR is derived as follows: VaR = {[E(Rp) − 2.33σp](−1)}(Portfolio value), where E(Rp) = Annualized daily return = (0.00026 × 250) = 0.065 250 = Number of trading days annually 2.33 = Number of standard deviations to attain 1% VaR _ σp = Annualized standard deviation = ​​(​0.00501 × √ ​ 250 )​ ​​ = 0.079215 Portfolio value = CAD260,000,000 VaR = −(0.065 − 0.184571) × CAD260,000,000 = CAD31,088,460. 26. B is correct. Given the large fixed-income exposure in the LICIA portfolio, examining the portfolio duration more closely would be prudent. Duration is the primary sensitivity exposure measure for fixed-income investments. 574 Learning Module 4 Backtesting and Simulation PRACTICE PROBLEMS The following information relates to questions 1-8 Kata Rom is an equity analyst working for Gimingham Wealth Partners (GWP), a large investment advisory company. Rom meets with Goran Galic, a Canadian private wealth client, to explain investment strategies used by GWP to generate portfolio alpha for its clients. Rom states that GWP is recognized in the Canadian investment industry as a leading factor-based value portfolio manager and describes how GWP creates relevant investment strategies and explains GWP’s backtesting process. Rom notes the following: Statement 1 Using historical data, backtesting approximates a real-life investment process to illustrate the risk–return tradeoff of a particular proposed investment strategy. Statement 2 Backtesting is used almost exclusively by quantitative investment managers and rarely by fundamental investment managers, who are more concerned with information such as forward estimates of company earnings, macroeconomic factors, and intrinsic values. Galic, who is 62 years old, decides to allocate C$2 million (representing 10% of his net worth) to an account with GWP and stipulates that portfolio assets be restricted exclusively to domestic securities. Although GWP has not backtested its strategies with such a restriction, it has backtested its strategies using a global index that includes domestic securities. Rom shows the following risk measures to Galic for three factor portfolios. Exhibit 1: Downside Risk Measures for Model Factors Risk Measure Factor 1 Factor 2 Factor 3 Value at risk (VaR) (95%) (6.49%) (0.77%) (2.40%) Conditional VaR (CVaR) (95%) (15.73%) (4.21%) (3.24%) 35.10% 38.83% 45.98% Maximum drawdown Galic asks Rom, “What happens if the future is different from the past?” Rom gives the following replies: Statement 3 Although backtesting can offer some comfort, you are correct that it does have a weakness: Backtesting generally does not capture the dynamic nature of financial markets and in particular may not capture extreme downside risk. Practice Problems Statement 4 575 As a result, we have captured extreme downside risk and the dynamic nature of financial markets by using the Value-at-Risk and Conditional Value-at-Risk measures. In an effort to make Galic fully aware of the risks inherent in GWP’s strategies, Rom describes a recent study that investigated the return distributions of value and momentum factors that GWP uses to construct portfolios. The study found that these distributions were non-normal based on their negative skewness, excess kurtosis, and tail dependence. Rom indicated that investment strategies based on this type of data are prone to significantly higher downside risk. Rom informs Galic that GWP also uses a technique commonly referred to as scenario analysis to examine how strategies perform in different structural regimes. Exhibit 2 compares the performance of two of GWP’s factor allocation strategies in different regimes: Exhibit 2: Scenario Analysis Using the Sharpe Ratio Strategy/Regime High Volatility Low Volatility Recession Non-recession Strategy I 0.88 0.64 0.20 1.00 Strategy II 1.56 1.60 1.76 1.52 Galic is surprised to see that some of the backtest results are unfavorable. He asks, “Why has GWP not considered strategies that perform better in backtesting?” Galic recently met with Fastlane Wealth Managers, who showed much better performance results. The portfolio manager at Fastlane told Galic that the company selects the top-performing strategies after performing thousands of backtests. 1. Which of Rom’s statements concerning backtesting is correct? A. Only Statement 1 B. Only Statement 2 C. Both Statement 1 and Statement 2 2. Which key parameter needs to be changed for a new backtest that includes Galic’s restrictions? A. Start and end dates B. Consideration of transaction costs C. Investment universe 3. Galic’s concern embedded in the question “What happens if the future is different from the past?” is a problem most relevant for which investment strategy evaluation technique? A. Sensitivity analysis B. Backtesting C. Monte Carlo simulation 576 Learning Module 4 Backtesting and Simulation 4. Which of the following conclusions of Exhibit 1 is least likely to be true? A. 5% of the time, losses from Factor 1 would be at least 6.49%. B. When the VaR is exceeded in Factor 1, we should expect an average loss of 15.73%. C. 5% of the time, losses from Factor 2 are likely to be worse than losses from Factor 1. 5. Based on the statistical study performed by GWP, which of the following represents a suggested course of action if GWP were to conduct Monte Carlo simulation analyses on the factor strategies? A. Inverse transformation B. Bootstrapping C. Sensitivity analysis 6. Based on Exhibit 1, which factor has the smallest downside risk as measured by the weighted average of all losses that exceed a threshold? A. Factor 1 B. Factor 2 C. Factor 3 7. The approach used by Fastlane Wealth Managers most likely incorporates: A. risk parity. B. data snooping. C. cross-validation. 8. Comparing the two strategies in Exhibit 2, the best risk-adjusted performance is demonstrated by: A. Strategy II in periods of low volatility and recession. B. Strategy I in periods of high volatility and non-recession. C. Strategy II in periods of high volatility and non-recession. The following information relates to questions 9-16 Emily Yuen is a senior analyst for a consulting firm that specializes in assessing equity strategies using backtesting and simulation techniques. She is working with an assistant, Cameron Ruckey, to develop multifactor portfolio strategies based on nine factors common to the growth style of investing. To do so, Yuen and Ruckey plan to construct nine separate factor portfolios and then use them to create factor-weighted allocation portfolios. Yuen tasks Ruckey with specifying the investment universe and determining the availability of appropriate reporting data in vendor databases. Ruckey selects Practice Problems 577 a vendor database that does not provide point-in-time data, so he adjusts the database to include point-in-time constituent stocks and a reporting lag of four months. Next, Yuen and Ruckey run initial backtests on the nine factor portfolios, calculating performance statistics and key metrics for each. For backtesting purposes, the portfolios are rebalanced monthly over a 30-year time horizon using a rolling-window procedure. Yuen and Ruckey consider a variety of metrics to assess the results of the factor portfolio backtests. Yuen asks Ruckey what can be concluded from the data for three of the factor strategies in Exhibit 1: Exhibit 1: Backtest Metrics for Factor Strategies Factor 1 Factor 2 Factor 3 (3.9%) (1.3%) (8.4%) Standard deviation of returns 2.1% 1.2% 4.6% Maximum drawdown 27.2% 8.3% 59.7% VaR (95%) Ruckey tells Yuen the following: Statement 1 We do not need to consider maximum drawdown, because standard deviation sufficiently characterizes risk. Statement 2 Factor 2 has the highest downside risk. From her professional experience Yuen knows that benchmark and risk parity factor portfolios, in which factors are equally weighted and equally risk weighted, respectively, are popular with institutional and high-net-worth clients. To gain a more complete picture of these investment strategies’ performance, Yuen and Ruckey design a Benchmark Portfolio (A) and a Risk Parity Portfolio (B), and then run two simulation methods to generate investment performance data based on the underlying factor portfolios, assuming 1,000 simulation trials for each approach: Approach 1 Historical simulation Approach 2 Monte Carlo simulation Yuen and Ruckey discuss the differences between the two approaches and then design the simulations, making key decisions at various steps. During the process, Yuen expresses a number of concerns: Concern 1: Returns from six of the nine factors are correlated. Concern 2: The distribution of Factor 1 returns exhibits excess kurtosis and negative skewness. Concern 3: The number of simulations needed for Approach 1 is larger than the size of the historical dataset. For each approach, Yuen and Ruckey run 1,000 trials to obtain 1,000 returns for Portfolios A and B. To help understand the effect of the skewness and excess kurtosis observed in the Factor 1 returns on the performance of Portfolios A and B, Ruckey suggests simulating an additional 1,000 factor returns using a multivariate skewed Student’s t-distribution, then repeating the Approach 2 simulation. 578 Learning Module 4 Backtesting and Simulation 9. Following Ruckey’s adjustments to the initial vendor database, backtested returns will most likely be subject to: A. stale data. B. data snooping C. p-hacking 10. Based on Exhibit 1, Ruckey should conclude that: A. Factor Strategy 3 has the highest portfolio turnover. B. Factor Strategy 2 has less downside risk than Strategy 3. C. Factor Strategy 2 has the highest returns. 11. Which of Ruckey’s statements about Exhibit 1 is incorrect? A. Only Statement 1 B. Only Statement 2 C. Both Statement 1 and Statement 2 12. Simulation Approach 1 (historical simulation) differs from Approach 2 (Monte Carlo simulation) in that: A. it is deterministic. B. a functional form of the statistical distribution for each decision variable needs to be specified. C. it assumes that sampling the returns from the actual data provides sufficient guidance about future asset returns. 13. To address Concern 1 when designing Approach 2, Yuen should: A. model each factor or asset on a standalone basis. B. calculate the 15 covariance matrix elements needed to calibrate the model. C. specify a multivariate distribution rather than modeling each factor or asset on a standalone basis. 14. Based on Concern 2, the Factor 1 strategy is most likely to: A. be favored by risk-averse investors. B. generate surprises in the form of negative returns. C. have return data that line up tightly around a trend line. 15. To address Concern 3 when designing Approach 1, Yuen should: A. add monthly return observations to the dataset using interpolation. B. randomly sample from the historical returns with replacement. C. choose the multivariate normal distribution as the initial functional form. Practice Problems 16. The process Ruckey suggests to better understand how the performance of Portfolios A and B using Approach 2 is affected by the distribution of Factor 1 returns is best described as: A. data snooping. B. sensitivity analysis. C. inverse transformation. 579 580 Learning Module 4 Backtesting and Simulation SOLUTIONS 1. A is correct. Statement 1 is correct because the main objective of backtesting is to understand the risk–return tradeoff of an investment strategy by approximating the real-life investment process. B is incorrect because Statement 2 is inaccurate. Although backtesting fits quantitative and systematic investment styles more naturally, it has also been heavily used by fundamental managers. C is incorrect because Statement 2 is not accurate. Backtesting, used in quantitative and systematic investment styles, is also heavily used by fundamental managers. 2. C is correct. Investment universe represents the securities in which a strategy can potentially invest. Galic’s restriction to exclusively own domestic securities means the investment universe of a backtest for a strategy for Galic’s account should use a domestic rather than global investment universe. A is incorrect. Galic’s restriction to domestic securities does not affect the start and end dates for a backtest. B is incorrect. Galic’s restriction to domestic securities does not change the inclusion of transaction costs in the study. 3. B is correct. An implicit assumption of backtesting is that past returns are a guide to future asset returns. A is incorrect. Sensitivity analysis refers to modifying assumptions such as probability distributions of key variables in a Monte Carlo simulation, which is a non-deterministic evaluation technique that does not use historical data. C is incorrect. Monte Carlo simulation is a non-deterministic evaluation technique that does not use historical data. 4. C is correct. The VaR metrics in Exhibit 1 show that 5% of the time, losses will be at least 6.49% and 0.77%, respectively, for Factor 1 and Factor 2. The CVaR metrics in Exhibit 1 show that the weighted average of all loss outcomes that exceed the VaR loss are 15.73% and 4.21% for Factor 1 and Factor 2, respectively. Thus, A is true because it correctly defines VaR, and B is true because it correctly defines CVaR, whereas C is untrue because both VaR and CVaR are lower for Factor 2 than Factor 1. 5. C is correct. Performing sensitivity analysis represents best practice given these characteristics, because the user could test different probability distributions that relax the assumptions of the normal distribution, for example. A is incorrect. Inverse transformation is a method of random observation generation, often used in simulation. B is incorrect. Bootstrapping refers to random sampling with replacement, often used in historical simulation. 6. C is correct. Exhibit 1 presents three downside risk measures: VaR, CVaR, and maximum drawdown. Conditional VaR is defined as the weighted average of all loss outcomes in the return distribution that exceed the VaR loss. Thus, CVaR is a more comprehensive measure of tail loss than VaR. Based on Exhibit 1, the factor with the smallest downside risk based on CVaR is Factor 3. 7. B is correct. The fact that the two firms’ investment performance results differ over similar time horizons using the same data and factors may be the result of selection bias. Data snooping is a type of selection bias. Fastlane Wealth Manag- Solutions ers is most likely selecting the best-performing modeling approach and publishing its results (i.e., data snooping). A is incorrect because risk parity is a portfolio construction technique that accounts for the volatility of each factor and the correlations of returns among all factors to be combined in the portfolio. It is not regarded as selection bias. C is incorrect because cross-validation is a technique used in the machine learning field, as well as in backtesting investment strategies, to partition data for model training and testing. It is not considered selection bias. 8. A is correct. Using the Sharpe ratio, the best risk-adjusted relative performance can be determined by comparing the sensitivity of the two strategies under differing macroeconomic regimes: recession versus non-recession and high volatility versus low volatility. The best risk-adjusted return will exhibit the highest Sharpe ratio. Strategy II demonstrates higher risk-adjusted returns compared with Strategy I under all four macroeconomic conditions, particularly in periods of low volatility, when the Sharpe ratio outperformance is 0.96, and recessions, when the Sharpe ratio outperformance is 1.56. 9. A is correct. A reporting lag of four months is likely to introduce stale data into the backtest because many large-capitalization companies report earnings within 30–50 days of quarter end. Although assuming four months (120 days) of reporting lag will eliminate a source of look-ahead bias, it introduces a new problem (i.e., stale data). B and C are incorrect. Data snooping and p-hacking refer to the same problem: a flawed approach to using data to make decisions. Data snooping and p-hacking are not characteristics of data, nor can they be added to a dataset by making an adjustment. 10. B is correct. Both VaR and maximum drawdown are downside risk measures, and both measures are lower for Strategy 2 than Strategy 3. A is incorrect. We cannot deduce portfolio turnover from the metrics provided in Exhibit 1. C is incorrect. We cannot deduce returns from the metrics provided in Exhibit 1. 11. C is correct. Both statements are incorrect. Statement 1 is incorrect because maximum drawdown and standard deviation are different measures. Maximum drawdown is typically used to represent downside risk, because it is the minimum cumulative return observed. Standard deviation is a measure of volatility. Although the two measures may be correlated, they are not substitutes for each other. Statement 2 is incorrect because two downside risk measures are presented: VaR and maximum drawdown. Factor Strategy 2 has the lowest reading for both measures, indicating that it has the least downside risk among the three strategies presented in Exhibit 1. 12. C is correct. Approach 1 is a historical simulation and assumes that past asset returns provide sufficient guidance about future asset returns. A is incorrect because both approaches are non-deterministic and random in nature. Approach 1 is a historical simulation, and Approach 2 is a Monte Carlo simulation. B is incorrect because Approach 1 is a historical simulation and each random variable of interest (key driver and/or decision variable) is randomly drawn from historical data. A functional form of the statistical distribution of returns for each decision variable needs to be specified for a Monte Carlo simulation, which is Approach 2. 581 582 Learning Module 4 Backtesting and Simulation 13. C is correct. Approach 2 is a Monte Carlo simulation. The returns of Portfolios A and B are driven by the returns of the nine underlying factor portfolios (based on nine common growth factors). In the case of asset or factor allocation strategies, the returns from six of the nine factors are correlated, and therefore it is necessary to specify a multivariate distribution rather than modeling each factor or asset on a standalone basis. A is incorrect because Approach 2 is a Monte Carlo simulation to generate investment performance data for the nine underlying factor portfolios. The returns of six of the nine factors are correlated, which means specifying a multivariate distribution rather than modeling each factor or asset on a standalone basis. B is incorrect because the analyst should calculate the elements of the covariance matrix for all factors, not only the correlated factors. Doing so entails calculating 36, not 15, elements of the covariance matrix. Approach 2 is a Monte Carlo simulation using the factor allocation strategies for Portfolios A and B for the nine factor portfolios, the returns of which are correlated, which means specifying a multivariate distribution. To calibrate the model, a few key parameters need to be calculated: the mean, the standard deviation, and the covariance matrix. For 9 assets, we need to estimate 9 mean returns, 9 standard deviations, 9 × ​​(​9 − 1​)​​ and _ ​ ​ = 36​elements of the covariance matrix. Assuming just the 6 cor2 6 × ​(​ ​6 − 1​)​​ related assets, the calculation is ​_ ​ = 15. 2 14. B is correct. The distribution of Factor 1 returns exhibits excess kurtosis and negative skewness (relative to the normal distribution). The excess kurtosis implies that these strategies are more likely to generate surprises, meaning extreme returns, whereas the negative skewness suggests those surprises are more likely to be negative (than positive). A is incorrect because risk-averse investors are more likely to prefer distribution properties such as positive skew (higher probability of positive returns) and lower to moderate kurtosis (lower probability of extreme negative surprises). The distribution of Factor 1 returns exhibits excess kurtosis and negative skewness. C is incorrect because the distribution of Factor 1 returns exhibits excess kurtosis and negative skewness. The joint distribution of such returns is rarely multivariate normal—so, typically the means and variances of these returns and the correlations between them are insufficient to describe the joint return distribution. In other words, the return data do not line up tightly around a trend line because of fat tails and outliers. 15. B is correct. Random sampling with replacement, also known as bootstrapping, is often used in historical simulations because the number of simulations needed is often larger than the size of the historical dataset. Because Approach 1 is a historical simulation and Concern 3 notes that the number of simulations needed is larger than the size of the historical dataset, bootstrapping should be used. A is incorrect because this approach would result in creating observations that do not exist in the historical record. Doing so would violate the assumption and procedures of historical simulation. C is incorrect because choosing the multivariate normal distribution as the initial functional form is typically done in a Monte Carlo simulation (Approach 2), not in a historical simulation (Approach 1). Historical simulation randomly samples from the historical dataset by drawing a number from a uniform distribution so that there is equal probability of being selected. Choice of distribution would not address the concern about the size of the dataset. 16. B is correct. Sensitivity analysis can be implemented to help managers understand how the target variable (portfolio returns) and risk profiles are affected by changes in input variables. Approach 2 is a Monte Carlo simulation, and the Solutions results depend on whether the multivariate normal distribution is the correct functional form or a reasonable proxy for the true distribution. Because this information is almost never known, sensitivity analysis using a multivariate skewed Student’s t-distribution helps to account for empirical properties such as the skewness and the excess kurtosis observed in the underlying factor return data. A is incorrect. Data snooping is the subconscious or conscious manipulation of data in a way that produces a statistically significant result (i.e., a p-value that is sufficiently small or a t-statistic that is sufficiently large to indicate statistically significance). C is incorrect. The inverse transformation method is the process of converting a randomly generated number into a simulated value of a random variable. 583