Determination of Forward and Futures Prices, Swaps

Chapter 5 - Determination of Forward and Futures Prices             Investment assets vs. consumption assets Short selling Assumptions and notations Forward price for an investment asset that provides no income Forward price for an investment asset that provides a known cash income Forward price for an investment asset that provides a known dividend yield Valuing forward contracts Forward prices and futures prices Stock index futures Currency futures Commodity futures Cost of carry  Investment assets vs. consumption assets An investment asset is an asset that is held mainly for investment purpose, for example, stocks, bonds, gold, and silver A consumption asset is an asset that is held primarily for consumption purpose, for example, oil, meat, and corn  Short selling Selling an asset that is not owned – Table 5.1, cash flows from short sale and purchase of shares, a review  Assumptions and notations Assumptions Perfect capital markets: transaction costs are ignored, borrowing and lending rates are the same, taxes are ignored (or subject to the same tax rate), and arbitrage profits are exploited away Arbitrage profit is the profit from a portfolio that involves 1. Zero net cost 2. No risk in terminal portfolio value 3. Positive profit Notations T: time until delivery date (years) S0: spot price of the underlying asset today F0: forward price today = delivery price K if the contract were negotiated today r: zero coupon risk-free interest rate with continuous compounding for T years to maturity 20  Forward price for an investment asset that provides no income Consider a forward contract negotiated today T = 3 months = ¼ year, S0 = $40, r = 5%, F0 = ? General solution: F0 = S0 erT (5.1) So, F0 = $40.50 If equation (5.1) does not hold, an arbitrage opportunity exists For example, if F0 = 43 > 40.50, an arbitrage profit = 43 - 40.50 = $2.50 Strategy: (Forward price is too high relative to spot price) Today: (1) Borrow $40.00 at 5% for 3 months and buy one unit of the asset (2) Sell a 3-month forward contract for one unit of the asset at $43 After 3 months: (1) Make the delivery and collect $43 (2) Reply the loan of $40.50 = 40e0.05*3/12 (3) Count for profit = 43.00 - 40.50 = $2.50 If F0 = 39 < 40.50, an arbitrage profit = 40.50 - 39 = $1.50 Show the proof by yourself as an exercise Application: stocks, bonds, and any other securities that do not pay current income during the specified period  Forward price for an investment asset that provides a known cash income Consider a forward contract negotiated today T = 3 months = ¼ year, S0 = $40, r = 5% In addition, the asset provides a known income in the future (dividends, coupon payments, etc.) with a PV of I = $4, F0 = ? General solution, F0 = (S0 -I)erT (5.2) So F0 = $36.45 If F0 = 38 > 36.45, an arbitrage profit = 38 - 36.45 = $1.55 Show the proof by yourself as an exercise 21 If F0 = 35 < 36.45, an arbitrage profit = 36.45 - 35 = $1.45 Strategy: (Forward price is too low relative to spot price) Today: (1) Short sell one unit of asset at the spot price for $40.00 (2) Deposit $40.00 at 5% for 3 months (3) Buy a 3-month forward contract for one unit of the asset at $35 After 3 months: (1) Take money out of the bank ($40.50) (2) Take the delivery by paying $35.00 and return the asset plus income ($4.05) (3) Count for profit = 40.50 - 35.00 - 4.05 = $1.45 Application: stocks, bonds, and any other securities that pay a known cash income during the specified period  Forward price for an investment asset that provides a known yield Consider a forward contract negotiated today T = 3 months = ¼ year, S0 = $40, r = 5% In addition, a constant yield which is paid continuously as the percentage of the current asset price is q = 4% per year. Then, in a smaller time interval, for example, return on one day = current asset price * 0.04/365, F0 = ? General solution: F0 = S0 e(r-q)T (5.3) So, F0 = $40.10 If F0 = 42 > 40.10, an arbitrage profit = 42 - 40.10 = $1.90 Strategy: (forward price is too high relative to spot price) Today: (1) Borrow $40 at 5% for 3 months and buy one unit of the asset at $40 (2) Sell a 3-month forward contract for one unit of the asset at $42 After 3 months: (1) Make the delivery and collect $42.00 (2) Pay off the loan in the amount of $40.50 (40e0.05*3/12) (3) Receive a known yield for three months of $0.40 (3/12 of 40*0.04) (4) Count for profit = 42 - 40.50 + 0.40 = $1.90 If F0 = 39 < 40.10, an arbitrage profit = 40.10 - 39 = $1.10 Show the proof by yourself as an exercise Application: stock indexes 22  Valuing forward contracts K: delivery price f: value of the forward contract today, f = 0 at the time when the contract is first entered into the market (F0 = K) In general: f = (F0 - K) e-rT for a long position, where F0 is the current forward price For example, you entered a long forward contract on a non-dividend-paying stock some time ago. The contract currently has 6 months to maturity. The risk-free rate is 10%, the delivery price is $24, and the current market price of the stock is $25. Using (5.1), F0 = 25e0.1*6/12 = $26.28, f = (26.28 - 24) e-0.1*6/12 = $2.17 Similarly, f = (K - F0) e-rT for a short position  Forward prices and futures prices Under the assumption that the risk-free interest rate is constant and the same for all maturities, the forward price for a contract with a certain delivery date is the same as the futures price for a contract with the same delivery date. Futures price = delivery price determined as if the contract were negotiated today The formulas for forward prices apply to futures prices after daily settlement Patterns of futures prices It increases as the time to maturity increases - normal market Futures price Normal Maturity month It decreases as the time to maturity increases - inverted market Futures price Inverted Maturity month Futures prices and expected future spot prices Keynes and Hicks: hedgers tend to hold short futures positions and speculators tend to hold long futures position, futures price < expected spot price because speculators ask for compensation for bearing the risk (or hedgers are willing to pay a premium to reduce the risk) 23 Risk and return explanation: if the return from the asset is not correlated with the market (beta is zero), k = r, F0 = E(ST); if the return from the asset is positively correlated with the market (beta is positive), k > r, F0 < E(ST); if the return from the asset is negatively correlated with the market (beta is negative) k < r, F0 > E(ST)  Stock index futures Stock index futures: futures contracts written on stock indexes Futures contracts can be written on many indices, such as DJIA: price weighted, $10 time the index Nikkei: price weighted, $5 times the index S&P 500 index: value weighted, $250 times the index NASDAQ 100 index: value weighted, $20 times the index Stock index futures prices, recall (5.3) General formula: F0 = S0 e(r-q)T, where q is the continuous dividend yield If this relationship is violated, you can arbitrage - index arbitrage Speculating with stock index futures If you bet that the general stock market is going to fall, you should short (sell) stock index futures If you bet that the general stock market is going to rise, you should long (buy) stock index futures Hedging with stock index futures Short hedging: take a short position in stock index futures to reduce downward risk in portfolio value Long hedging: take a long position in stock index futures to not miss rising stock market  Currency futures Exchange rate and exchange rate risk Direct quotes vs. indirect quotes 1 pound / $1.60 (direct) vs. 0.625 pound / $1.00 (indirect) Exchange rate risk: risk caused by fluctuation of exchange rates 24 Currency futures prices, recall (5.3) General formula: F0 = S0 e(r-rf)T, where rf is the foreign risk-free rate For example, if the 2-year risk-free interest rate in Australia and the US are 5% and 7% respectively, and the spot exchange rate is 0.6200 USD per AUD, then the 2-year forward exchange rate should be 0.6453. If the 2-year forward rate is 0.6300, arbitrage opportunity exists. To arbitrage: (1) Borrow 1,000 AUD at 5%, convert it to 620 USD and invest it in the U.S. at 7% for 2 years (713.17 USD in 2 years) (2) Enter a 2-year forward contract to buy AUD at 0.6300 (3) After 2 years, collect 713.17 USD and convert it to 1,132.02 AUD (4) Repay the loan plus interest of 1,105.17 AUD (5) Net profit of 26.85 AUD (or 16.91 USD) Speculation using foreign exchange futures If you bet that the British pound is going to depreciate against US dollar you should sell British pound futures contracts If you bet that the British pound is going to appreciate against US dollar you should buy British pound futures contracts Hedging with foreign exchange futures to reduce exchange rate risk – Table 5.4 Quotes  Commodity futures Commodities: consumption assets with no investment value, for example, wheat, corn, crude oil, etc. For commodity futures prices, recall (5.1), (5.2), and (5.3) For commodities with no storage cost: F0 = S0erT For commodities with storage cost: F0 = (S0 + U)erT, where U is the present value of all storage costs F0 = S0 e(r+u)T, where u is the storage costs per year as a percentage of the spot price Example 5.8: consider a one-year futures contract on gold. We assume no income and that it costs $2 per ounce per year to store gold, with the payment being made at the end of the year. The gold spot price is $1,600 and the risk-free rate is 5% per year for all maturities. U = 2e-rT = 2*e-0.05*1 = 1.90 25 F0 = (S0 + U)erT = (1,600 + 1.90)e0.05*1 = $1,684.03 If the futures price is too high, say $1,700, an arbitrager can (1) Borrow $160,000 at the risk-free rate of 5% for one year and buy 100 ounces of gold (2) Store gold for one year (3) Short one gold futures contract at 1,700 for delivery in one year After one year, the arbitrager can (1) Get out gold from storage and pay $200 storage fee (2) Deliver gold for $170,000 (3) Payout the loan (principal plus interest of $168,203 = 160,000e0.05*1 (4) Count for profit of 170,000 – 168,203 – 200 = $1,587 If the futures price is too low, say $1,650, an arbitrager can reverse the above steps to make risk-free profit – an exercise for students Consumption purpose: reluctant to sell commodities and buy forward contracts F0  S0 erT, with no storage cost F0  (S0 + U)erT, where U is the present value of all storage costs F0  S0 e(r+u)T, where u is the storage costs per year as a percentage of the spot price  Cost of carry It measures the storage cost plus the interest that is paid to finance the asset minus the income earned on the asset Asset Stock without dividend Stock index with dividend yield q Currency with interest rate rf Commodity with storage cost u  Futures price F0 = S0 erT F0 = S0 e(r-q)T F0 = S0 e(r-rf)T F0 = S0 e(r+u)T Assignments Quiz (required) Practice Questions: 5.9, 5.10, 5.14 and 5.15 26 Cost of carry r r-q r-rf r+u Chapter 6 - Interest Rate Futures       Day count and quotation conventions T-bond futures T-bill futures Duration Duration based hedging Speculation and hedging with interest rate futures  Day count and quotation conventions Three day counts are used in the U.S. Actual/actual: T-bonds For example, the coupon payment for a T-bond with 8% coupon rate (semiannual payments on March 1 and September 1) between March 1 and July 3 is (124/184)*4 = $2.6957 30/360: corporate and municipal bonds For example, for a corporate bond with the same coupon rate and same time span mentioned above, the coupon payment is (122/180)*4 = $2.7111 Actual/360: T-bills and other money market instruments The reference period is 360 days but the interest earned in a year is 365/360 times the quoted rate Quotes for T-bonds: dollars and thirty-seconds for a face value of $100 For example, 95-05 indicates that it is $95 5/32 ($95.15625) for $100 face value or $95,156.25 for $100,000 (contract size) Minimum tick = 1/32 Daily price limit is 3 full points (96 of 1/32, 3% of the face value, equivalent to $3,000) For T-bonds, cash price = quoted cash price + accrued interest Example 0 40 days 182 days 142 days remaining until next coupon Suppose annual coupon is $8 and the quoted cash price is 99-00 (or $99 for a face value of $100) then the cash price = 99 + (4/182)*40 = $99.8791 27 T-bills: sold at a discount with no interest payment, the denomination usually is $1,000 Price quotes are for a face value of $100 The cash price and quoted price are different For example, if y is the cash price of a T-bill that will mature in n days, then the quoted price (P) is given by P = (360/n)*(100 - y), known as the discount rate If y = 98, n = 91 days, then the quoted price = 7.91 Return on the T-bill = (2/98)*(365/91) = 8.186% (2/98 = 2.04% for 91 days)  T-bond futures T-bond futures are quoted as T-bonds and the price received for each $100 face value with a short position upon delivery is Invoice amount = quoted futures price*(CF) + AI, where CF = conversion factor and AI = accrued interest For example, if the quoted futures price = 90-00, CF = 1.38, and AI = $3.00, then Invoice amount = 90(1.38) + 3.00 = $127.20 Conversion factor: equal to the quoted price the bond would have per dollar of principal on the first day of the delivery month on the assumption that the interest rate for all maturities equals 6% per year (with semiannual compounding) For example, let’s consider a 10% coupon bond with 20 years and 2 months to maturity. For the purpose of calculating the CF, the bond is assumed to mature in 20 years (round down to the nearest 3 months). The value of the bond is 40 5 100   $146.23 . Dividing by the face value gives the CF 1.4623.  i (1  0.03) 40 i 1 (1  0.03) (Or you can use a financial calculator to figure out the CF: PMT = 5, FV = 100, N = 40, i/y = 3%, solve for PV = 146.23, CF = 146.23/100 = 1.4623) For example, consider an 8% coupon bond with 18 years and 4 months to maturity. To calculate CF, we assume that the bond has 18 years and 3 months to maturity. 36 4 100 4   $125.83 i (1  0.03) 36 i 1 (1  0.03) The interest rate for a 3-month period is (1.03)1/2 – 1 = 1.4889%. Hence, the present value of the bond is 125.83/(1+1.4889% ) = $123.99. Subtracting the accrued interest of $2.00 we get $121.99. The conversion factor is therefore 1.2199. 28 Cheapest-to-deliver issues Since the person with a short position can deliver any T-bond that has more than 15 years to maturity and that it is not callable within 15 years, which bond is the cheapest to deliver? Recall: invoice price = quoted futures price*(CF) + AI, where CF = conversion factor, and AI = accrued interest The cost of purchasing a bond is Cash price = quoted cash price + accrued interest The net cost is the difference between the cash price and the invoice price Net cost to deliver = quoted cash price - (quoted futures price*CF) (Note, AI is cancelled out) Choose the bond that minimizes the net cost to deliver Determining the quoted T-bond futures price, use F0 = (S0 - I)erT Example 6.1: choose a bond from below to deliver, assuming the most recent settlement price (quoted futures price) is 93-08 (or 93.25) Bond 1 2 3 Quoted cash price 99.50 143.50 119.75 CF 1.0382 1.5188 1.2615 Net cost to deliver 99.50 – (93.25*1.0382) = $2.69 143.50 – (93.25*1.5188) = $1.87 119.75 – (93.25*1.2615) = $2.12 The cheapest-to-deliver is bond 2  T-bill futures Call for delivery of T-bills with a face value of $1,000,000 and a time to maturity of 90 days The contract size is $1,000,000 with delivery months set in March, June, September, and December No daily price limit Minimum tick is 0.01% of discount yield (one basis point, equivalent to $25.00; or when the interest rate changes by 1 basis point, 0.01%, the interest earned on $1,000,000 face value for 3 months changes by $25.00) 29 V = $1,000,000 0 T1 r1 T2 time 90 days r2 Determining the current (present) value, use PV = V e  r2T2 Determining the futures price, use F0 = PV e r1T1 Why buy a futures contract on T-bills? To lock in a short-term interest rate Eurodollar futures are similar to T-bill futures  Duration A measure of how long on average the bondholder has to wait before receiving cash payments Suppose a bond provides cash flows ct at time t. The bond price B and bond yield y (with continuous compounding) are related by: B n c e  yt i i i 1  The duration of the bond (D) is defined as: D  t c e  yti n i 1 i i B ci e  yti ] =  ti [ B i 1 n For a small change in yield ( y ) the change in bond price is approximately: B  n dB y  y  ci t i e  yti   BDy dy i 1 Rearranging, we get B   Dy B Percentage change in bond price = – duration*the change in yield (where yield is expressed with continuous compounding) 30 For example, a 3-year bond with coupon rate of 10%, payable semiannually, sells for 94.213 and has a yield to maturity of 12% (continuous compounding). The duration for the bond is 2.653 years. If the yield increases by 0.1% the bond price will drop by 0.250 to 93.963 [-94.213*2.653*(0.001) = -0.25]. Time (years) 0.5 1.0 1.5 2.0 2.5 3.0 Total Cash flow ($) 5 5 5 5 5 105 130 Present value 4.709 4.435 4.176 3.933 3.704 73.256 94.213 Weight 0.050 0.047 0.044 0.042 0.039 0.778 1.000 Time*weight 0.025 0.047 0.066 0.083 0.098 2.333 2.653 What if y is not continuous (for example, semiannual compounding)? If y is not expressed with continuous compounding then B   BD * y , where D* is called modified duration D*  D 2.653   2.50 , where D is the duration of the bond and y is the yield 1  y / m 1  0.12 2 with a compounding frequency of m times per year For a zero-coupon bond, the duration is equal to the time to maturity  Duration based hedging Interest rate risk: interest rate price risk vs. interest rate reinvestment risk Interest rate price risk: risk that the bond value (price) falls when the market interest rates rise Reinvestment risk: risk that the interest received will be reinvested at a lower rate Relationship between duration and bond price volatility: B   BDy Define VF: contract price for the interest rate futures contract DF: duration of the asset underlying the futures contract at the maturity of the futures contract P: forward value of the portfolio being hedged at the maturity of the hedge (assumed to be the value of the portfolio today) DP: duration of the portfolio at the maturity of the hedge 31 Then we have approximations of P   PD P y and VF  VF DF y , the optimal number of contracts to hedge against an uncertain y is N*  PDP , called duration-based hedge ratio VF DF For example, on August 2, a fund manager with $10 million invested in government bonds is concerned that interest rates are expected to be volatile over the next 3 months. The manager decides to use December T-bond futures contract to hedge the portfolio. The current futures price is 93-02, or 93.0625 for $100 face value. Since the contract size is $100,000, the futures price is $93,062.50. Further suppose that the duration of the bond portfolio in 3 months is 6.80 years. The cheapest-to-deliver bond in the T-bond contract is expected to be a 20-year 12% coupon bond. The yield on this bond is currently 8.8% and the duration is 9.20 years at maturity of the futures contract. The duration-based hedge ratio is 10,000,000 6.80 N*  *  79.42 Contracts, short positions 93,062.50 9.20 Application of bond immunization: banking management, pension fund management  Speculation and hedging with interest rate futures Speculation with outright positions A speculator with a long position is betting that the interest rate is going to fall so the price of interest rate futures is going to rise A speculator with a short position is betting that the interest rate is going to rise so the price of interest rate futures is going to fall Speculation with spreads An intra-commodity T-bill spread: speculating the term structure of interest rates, for example, T-bill futures with different maturities (nearby vs. distant) An inter-commodity spread: speculating on shifting risk levels between different instruments, for example, T-bill futures vs. T-bond futures Hedging with interest rate futures A long hedge: take a long position in futures market to reduce interest rate risk A short hedge: take a short position in futures market to reduce interest rate risk  Assignments Quiz (required) Practice Questions: 6.8, 6.9, 6.10 and 6.11 32 Chapter 7 - Swaps         Swaps Interest-rate swaps Role of financial intermediary Comparative advantage Valuation of interest-rate swaps Currency swaps Valuation of currency swaps Other types of swaps  Swaps A swap is a private agreement between two companies to exchange cash flows in the future according to a prearranged formula: an extension of a forward contract. For example, Intel and Microsoft agreed a swap in interest payments on a notional principal of $100 million. Microsoft agreed to pay Intel a fixed rate of 5% per year while Intel agreed to pay Microsoft the 6-month LIBOR. 5.0% (fixed) Intel Microsoft LIBOR (floating) Cash flows to Microsoft in a $100 million 3-year interest rate swap when a fixed rate of 5% is paid and LIBOR is received: Date 3-5-2013 9-5-2013 3-5-2014 9-5-2014 3-5-2015 9-5-2015 3-5-2016 6-month LIBOR 4.20% 4.80% 5.30% 5.50% 5.60% 5.90% Floating cash flow Fixed cash flow Net cash flow +2.10 million +2.40 million +2.65 million +2.75 million +2.80 million +2.95 million -2.50 million -2.50 million -2.50 million -2.50 million -2.50 million -2.50 million -0.40 million -0.10million +0.15 million +0.25 million +0.30 million +0.45 million Cash flows to Intel will be the same in amounts but opposite in signs. The floating rate in most interest rate swaps is the London Interbank Offered Rate (LIBOR). It is the rate of interest for deposits between large international banks. We ignored the day count convention (LIBOR is quoted on an actual/360 basis while a fixed rate uses 365 days per year). Since there are 184 days between March 5, 2013 and September 5, 2013 the actual payment should be 100*0.042*184/360 = $2.1467 million 33  Interest-rate swaps Using swaps to transform a liability For example, Microsoft could transform a floating-rate loan to a fixed rate loan while Intel could transform a fixed rate loan to a floating-rate loan 5.2% 5.0% Intel Microsoft LIBOR LIBOR + 0.1% After swap, Intel pays LIBOR + 0.2% and Microsoft pays 5.1% Using swaps to transform an asset For example, Microsoft could transform an asset earning fixed-rate to an asset earning floating-rate while Intel could transform an asset that earns floating-rate to an asset that earns fixed-rate 5.0% Intel LIBOR - 0.2% 4.7% Microsoft LIBOR After swap, Intel earns 4.8% and Microsoft earns LIBOR - 0.3% We will ignore the day count convention (LIBOR uses 360 days per year while a fixed rate uses 365 days per year)  Role of financial intermediary Arranging the swaps to earn about 3-4 basis points (0.03-0.04%) Refer to the swap between Microsoft and Intel again when a financial institution is involved to earn 0.03% (shared evenly by both companies) 5.2% 4.985% Intel 5.015% Institution LIBOR Microsoft LIBOR LIBOR + 0.1% After swap, Intel pays LIBOR + 0.215%, Microsoft pays 5.115%, and the financial institution earns 0.03% 34  Comparative advantage Some companies have a comparative advantage when borrowing in fixed rate (U.S dollars) markets, whereas others have a comparative advantage in floating-rate (foreign currency) markets. Let us consider two corporations, AAACorp and BBBCorp. Both companies are going to borrow $10 million and are facing the following rates. Assume that AAACorp wants floating rate and BBBCorp wants fixed rate: AAACorp BBBCorp Fixed 4.00% 5.20% Floating LIBOR - 0.1% LIBOR + 0.6% How can a swap benefit both companies? Answer: AAACorp borrows at the fixed rate and BBBCorp borrows at the floating rate and then two companies engage in a swap Without a financial institution involved 4.00% 4.35% AAACorp BBBCorp LIBOR LIBOR + 0.6% After swap, AAACorp pays LIBOR - 0.35% and BBBCorp pays 4.95% (both benefit by 0.25%) The total benefit is equal to a - b (0.5%), where a is the difference between the interest rates in fixed rate markets (1.20%) and b is the difference between the interest rates in floating rate markets (0.7%). The total gain doesn’t have to be shared evenly. When a financial institution is involved and it earns 4 basis points (0.04%): 4.00% 4.33% AAACorp 4.37% Institution LIBOR BBBCorp LIBOR LIBOR + 0.6% After swap, AAACorp pays LIBOR - 0.33% and BBBCorp pays 4.97% (both benefit by 0.23%) while the financial institution earns 0.04%. The total gain remains at 0.5%. 35  Valuation of interest-rate swaps Vswap = Bfl - Bfix (or Vswap = Bfix - Bfl), where Bfix is the present value of fixed-rate bond underlying the swap and Bfl is the preset value of floating-rate bond underlying the swap Suppose that a financial institution has agreed to pay 6-month LIBOR and receive 8% fixed rate (semiannual compounding) on a notional principal of $100 million. The swap has a remaining life of 1.25 years. The LIBOR rates with continuous compounding for 3month, 9-month, and 15-month maturities are 10%, 10.5%, and 11%, respectively. The 6month LIBOR rate at the last payment date was 10.2%. Time Years 0.25 0.75 1.25 Bfix cash flow 4.0 4.0 104.0 Bfl cash flow 105.10 Discount factor 0.9753 0.9243 0.8715 Present value Bfix cash flow 3.901 3.697 90.640 Present value Bfl cash flow 102.505 Vswap = Bfix - Bfl = (3.901+3.697+90.640) – 102.505 = -4.267 million Note: Bfl = L (notional principal immediately after an interest payment) and therefore Bfl = 100 + 100*0.051 = 105.10 million after 3 months If the financial institution pays fixed and receives floating, the value of the swap would be +4.267 million (again a zero-sum game)  Currency swaps A currency swap is an agreement to exchange interest payments and principal in one currency for principal and interest payments in another currency. It can transform a loan in one currency into a loan in another currency. Let us consider two corporations, GE and Qantas Airway. Both companies are going to borrow money and are facing the following rates. Assume GM wants to borrow AUD and Qantas Airway wants to borrow USD. GE Qantas USD 5.00% 7.00% AUD 7.60% 8.00% How can a swap benefit both companies? Answer: GE borrows USD and Qantas borrows AUD and then two companies engage in a swap (fixed-for-fixed currency swap) Since a = 2.00% and b = 0.4%, therefore the net gain = a – b = 1.6% 36 Assuming a financial institution arranges the swap and earns 0.2% (by taking the exchange rate risk) 5.0% USD 5.0% GE USD 6.3% Institution Quntas USD AUD 6.9% AUD 8.0% AUD 8.0% Net outcome: GE borrows AUD at 6.9% (0.7% better than it would be if it went directly to AUD markets) Quntas borrows USD at 6.3% (0.7% better than it would be if it went directly to USD markets) Financial institution receives 1.3% USD and pays 1.1% AUD and has a net gain of 0.2%  Valuation of currency swaps Vswap = BD - S0BF (or Vswap = S0BF - BD), where S0 is the spot exchange rate, BF and BD are the values of the foreign-denominated bond underlying the swap and the U.S. dollar bond underlying the swap  Other types of swaps Fixed-for-floating currency swap: a floating interest rate in one currency is exchanged for a fixed interest rate in another currency Floating-for-floating currency swap: a floating interest rate in one currency is exchanged for a floating interest rate in another currency Credit default swap (CDS): allows companies to hedge credit risks in the same way that they have hedged market risks - buys insurance to hedge default risk Equity swap: an agreement to exchange the total returns (dividends and capital gains) realized on an equity index for either a fixed or floating rate of interest Commodity swaps Volatility swaps  Assignments Quiz (required) Practice Questions: 7.9, 7.10 and 7.11 37 Swap examples 1. Companies X and Y have been offered the following rates per year on a $5 million ten-year loan: Fixed-rate Floating-rate Company X 7.0% LIBOR + 0.5% Company Y 8.8% LIBOR + 1.5% Company X requires a floating-rate loan and Company Y requires a fixed-rate loan. Design a swap that will net a financial institution acting as an intermediary 0.1% per year and it will be equally attractive to X and Y. Answer: a = 1.8% (8.8% - 7.0%) and b = 1.0% [(LIBOR + 1.5%) – (LIBOR + 0.5%)], swap provides a net gain of 0.8% (1.8% - 1.0%), less 0.1% for the bank, leaving 0.7% net gain to be shared by X and Y (0.35% each) After the swap, X pays floating at LIBOR + 0.15%, Y pays fixed at 8.45%, and the institution earns 0.1% 2. Companies A and B are facing the following annual interest rates in US and UK: Sterling U.S. Dollar Company A 8.0% 7.0% Company B 7.6% 6.2% A wants to borrow dollar and B wants to borrow sterling. Design a swap that will net a financial institution acting as an intermediary 0.10% per year and it will be equally attractive to both companies. Answer: a = 0.4% (8.0% - 7.6%) and b = 0.8% (7.0 - 6.2%), a swap provides a total gain of 0.4% (0.8% - 0.4%), less 0.1% for the bank, leaving 0.3% net gain to be shared by A and B (0.15% each) After the swap, A borrows dollar at 6.85%, B borrows sterling at 7.45%, and the financial institution earns 0.1%. 38 Chapter 8 - Securitization and the Credit Crisis of 2007     Securitization The U.S. housing market What went wrong? Aftermath  Securitization Banks could not keep pace with the increasing demand for residential mortgages, especially during real estate booming periods. This led to the development of the mortgage-backed security (MBS) market. Portfolios of mortgages were created and the cash flows (interests and principal payments) generated by the portfolios were packaged as securities and sold to investors. Asset-backed security (ABS): a portfolio of income-producing assets, such as loans, is sold by the originating banks to a special purpose vehicle (SPV) and the cash flows from the assets are then allocated to tranches. A tranche is one of a number of related securities offered as part of the same transaction. Figure 8.1 Cash flows are allocated to different tranches by specifying what is known as a waterfall Figure 8.2 The extent to which the tranches get their principal back depends on losses on the underlying assets. The first 5% of losses are borne by the equity tranche. If losses exceed 5% the equity tranche loses its entire principal and some losses are borne by the principal of the mezzanine tranche. If losses exceed 20%, the mezzanine tranche loses its entire principal and some losses are borne by the principal of the senior tranche. Usually, the senior tranche is rated AAA. The mezzanine is rated BBB. The equity tranche is not rated. Finding investors to buy the senior tranche (AAA-rated) was not difficult since it offered attractive return with relatively low risk. Equity tranche usually was sold to hedge funds (high risk high returns) It is difficult to find investors to buy mezzanine tranche. Figure 8.3 39  The U.S. housing market The relaxation of lending standards Subprime mortgage securitization The S&P/Case-Shiller Composite-10 index of U.S. real estate prices Figure 8.4 The bubble burst in 2007: many mortgage holders cannot afford their payments, hosing price drops, leading to many foreclosures. The borrowers hold a free American-style put option since when there is a default, the lender is able to take the possession of the house. The losses: far greater than the losses in house prices  What went wrong? Irrational exuberance Mortgage lending standards Mortgage-backed products Rating agencies Default risk Incentives (agency problems)  Aftermath More regulations Higher standards for mortgage lending  Assignments Quiz (required) Practice Questions: 8.9, 8.10. 8.11 and 8.12 40