Curriculum Note, Study Session 13, Reading 33

Pricing and Valuation of Forward Commitments 1. INTRODUCTION TO PRICING AND VALUATION OF FORWARD COMMITMENTS Forward commitments: A forward is an agreement between two parties to buy or sell an asset at a pre-determined future time for a certain price. • Forward price for a forward contract is defined as the delivery price, which make the value of the contract at initiation be zero. The buyer of a forward contract has a “long position” in the asset/commodity. • • Principles of Arbitrage-Free Pricing And Valuation of Forward Commitments 1.1 Forward commitment pricing: Forward commitment pricing involves determining the appropriate forward commitment price or rate at which the forward commitment contract is initiated. Forward commitment valuation: Forward commitment valuation involves determining the appropriate value of the forward commitment once it has been initiated. Forward value refers to the monetary value of an existing forward or futures contract. Key assumptions made in pricing and valuation of contracts: i. ii. iii. iv. Replicating instruments are identifiable and investable; There are no market frictions; Short selling is allowed with full use of proceeds; Borrowing and lending are available at a known risk-free rate. Note: Cash inflows to the arbitrageur have a positive sign and outflows are negative. Carry arbitrage models used for forward commitment pricing and valuation are based on the no-arbitrage approach. • • Arbitrage occurs when equivalent assets or combination of assets sell for two different prices. The law of one price states that two identical goods must sell for the same current price in the absence of transaction costs. According to law of one price, arbitrage will drive prices of 1.2 equivalent assets to a single price so that no riskless profits can be earned. The law of one price is based on the value additivity principle, according to which the value of a portfolio is simply the sum of the values of each instrument held in the portfolio. Arbitrage opportunities should disappear quickly in an efficient and frictionless market. Pricing and Valuing Generic Forward And Futures Contracts 1.2.1) Forwards and Futures § § § § § • • • 0 = today, T = expiration, underlying asset = S0(or t or T), forward = F(0,T) S0 denotes the underlying price at the time of forward contract initiation ST denotes the underlying price when the forward contract expires. F0(T) denote the forward price established at the initiation date, 0, and expiring at date T, where T represents a period of time later. Uppercase “F” denotes the forward price, whereas lowercase “f” denotes the futures price. Similarly, uppercase “V” denotes the forward value, whereas lowercase “v” denotes the futures value. Forward contracts are traded over-the-counter, no money changes hand initially and during the life time of the contract. Hence, the contract value at the initiation of the contract is ZERO. The forward contract value when initiated is expressed as V0(T) = v0(T) = 0. The contract price is set such that the value of the contract is Zero, that is, Present value of contract price = Prevailing spot price of the underlying Subsequent to the initiation date, the value can be significantly positive or negative. At Market Contract: The forward contracts having value of zero at contract initiation are referred to as at market. Property of Convergence: According to property of convergence, at Time T (expiration), both the forward price and the futures price are equivalent to the spot price, that is, FT(T) = fT(T) = ST Important to Remember: • The market value of a long position in a forward contract value at maturity is VT(T) = ST – F0(T). –––––––––––––––––––––––––––––––––––––– Copyright © FinQuiz.com. All rights reserved. –––––––––––––––––––––––––––––––––––––– FinQuiz Notes – 2 0 2 2 Reading 33 READING 37 • • • • • Pricing and Valuation of Forward Commitments The market value of a short position in a forward contract value at maturity is VT(T) = F0(T) – ST. The market value of a long position in a futures contract value before marking to market is vt(T) = ft(T) – ft–(T). The market value of a short position in a futures contract value before marking to market is vt(T) = ft–(T) – ft(T). The futures contract value after daily settlement is vt(T) = 0. If value of underlying > initial forward price, a long position in a forward contract will have a positive value. 2. 2.1 2) If value of underlying < initial forward price, a short position in a forward contract will have a positive value at expiration. Note: The forward value and the futures value will be different because futures contracts are marked to market while forward contracts are not being marked to market. CARRY ARBITRAGE Carry Arbitrage Model When There Are No Underlying Cash Carry arbitrage model is based on following two rules: 1) • FinQuiz.com Do not use your own money, i.e. borrow money to buy the underlying. Do not take any price risk (here refers to market risk); i.e. invest the proceeds from short selling transactions at risk-free rate or in other words, lend the money by selling the underlying). Cash Flows related to Carrying the Underlying through Calendar Time: If an arbitrageur enters a forward contract to sell an underlying instrument for delivery at Time T, then this exposure can be hedged by buying the underlying instrument at Time 0 with borrowed funds and carry it to the forward expiration date so it can be sold under the terms of the forward contract. The table below shows Cash Flows Related to Carrying the Underlying through Calendar Time. 1Note that all four transactions are done simultaneously not sequentially. 2Note that if the forward contract is priced correctly, there will be no arbitrage profit and, hence, no Step 4. • • • • • The above figure shows that arbitrageur borrows the money to buy the asset, so at Time T, he will pay back FV(S0), based on the risk-free rate. When ST FV(S0), then the forward contract is sold and the underlying is purchased. • If F0(T) < FV(S0), then the forward contract is purchased and the underlying is sold short. • If the forward contract price is equal to its equilibrium price, there will be no arbitrage profit and thus no Step 4. • The quoted forward price does not directly reflect expectations of future underlying prices. Relationship between Forward price and interest rate: Forward price is directly related to interest rates – i.e., when interest rate falls (rises), forward price decreases (increases). This relationship between forward prices and interest rates will generally hold except for interest rates forward contracts. Practice: Example 1, Reading 33, Curriculum. Cash Flows for the Valuation of a Long Forward Position: 1) Reverse Carry Arbitrage: Suppose forward price of F0(T) = 104, which is less than the forward price determined by the carry arbitrage model (105). In this case, the opposite strategy – named “Reverse Carry Arbitrage” is followed. It involves the following steps: 1) 2) 3) 4) 3 Buy a forward contract, and the value at T is ST – F0(T). Sell short the underlying instrument. Lend the short sale proceeds. Borrow the arbitrage profit. Present value is calculated over the remaining life of the contract. • • “Value at Time t” represents the value of the forward contracts. Ft(T) = FVt,T(St) The value observed at Time t of the original forward contract initiated at Time 0 and expiring at Time T is simply the present value3 of the difference in the forward prices, as stated below. Alternatively, Vt(T) = St – PVt,T[F0(T)] READING 37 Pricing and Valuation of Forward Commitments Practice: Example 2, Reading 33, Curriculum. 1) 2) 2.1 Carry Arbitrage Model When Underlying Has Cash Flows In Carry arbitrage, we are required to pay the interest cost, whereas in reverse carry arbitrage, we receive the interest benefit. 3) 4) 5) FinQuiz.com Purchase the underlying at Time 0, receive the dividend at Time t = 0.5 and sell the underlying at Time T. Reinvest the dividend received at Time t = 0.5 at the risk-free interest rate until Time T. Borrow the initial cost of the underlying. Sell a forward contract at Time 0 and the underlying will be delivered at Time T. Borrow the arbitrage profit. Cash flows are reflected in the table: Let γ denote the carry benefits (for example, dividends, foreign interest, and bond coupon payments that would arise from certain underlyings). Future value of underlying carry benefits = γT = FV0,T(γ0) Present value of underlying carry benefits = γ0 = PV0,T(γT) • Let θ denote the carry costs. These refer to additional costs to hold the commodities, like storage, insurance, deterioration, etc. These can be considered as negative dividends. Carry costs are zero for financial instruments but holding these assets does involve opportunity cost of interest. Future value of underlying costs = θT = FV0,T(θ0) Present value of underlying costs = θ0 = PV0,T(θT) • Forward price is the future value of the underlying adjusted for carry cash flows. Forward pricing equation is stated as below: Ø Ø Carry costs (e.g. interest rate) are added to forward price because they increase the cost of carrying the underlying instrument through time. Carry benefits are subtracted from forward price because they decrease the cost of carrying the underlying instrument through time. Example: Suppose, S0 = 100, r = 5%, T = 1, and ST = 90.Assuming the underlying will distribute 2.9277 at Time t = 0.5: γt = 2.9277. The time until the distribution of 2.9277 is t, and hence, the present value is γ0 = 2.9277/(1 + 0.05)0.5 = 2.8571 The time between the distribution and the forward expiration is T – t = 0.5, and thus, the Future value = γT = 2.9277(1 + 0.05)0.5 = 3 Cash Flows for Financed Position in the Underlying with Forward: The steps involved in this strategy are as below: The value of the cash flow at Time 0 is zero, or V0(T) +PV[F0(T) + γT – FV(S0)] = 0 and V0(T) = –PV[F0(T) + γT – FV(S0)]. If theForward contract has zero value, then Forward Price = F0(T) = Future value of underlying – Future value of carry benefits= FV(S0) – γT Initial forward price = Future value of the underlying Value of any ownership benefits at expiration or F0(T) = FV0,T(S0 – γ0) Forward value for a long position is estimated using the following: Where, Ft(T) = FVt,T(St + θt – γt) Annual compounding and continuous compounding: The equivalence between annual compounding and continuous compounding can be expressed as follows: READING 37 Pricing and Valuation of Forward Commitments (1 + r)T = erct or rc = ln[(1 + r)T]/T = ln(1 + r); If the quoted interest rate is 5% based on annual compounding, then the implied interest rate based on continuous compounding is rc = ln(1 + r) = ln(1 + 0.05) = 0.0488, or 4.88% The future value of the underlying adjusted for carry, i.e., the dividend payments, is F0(T) = Ø Ø Ø Ø This implies that a cash flow compounded at 5% annually is equivalent to being compounded at 4.88% continuously. Continuous compounding results in a lower quoted rate. FinQuiz.com If a dividend payment is announced between the forward’s valuation and expiration dates, assuming the news announcement does not change the current underlying price, the forward value will most likely decrease. If a new dividend is imposed, the new forward price will decrease and consequently, the value of the old forward contract will be lower. Practice: Example 4, Reading 33, Curriculum. Carry arbitrage model with continuous compounding: The carry arbitrage model with continuous compounding is expressed as 3 3.1 PRICING EQUITY FORWARD AND FUTURES Equity Forward and Futures Contracts Since, futures contracts are marked to market daily, the equity futures value is zero each day after settlement has occurred. Practice: Example 5, 6 & 7 Reading 33, Curriculum. 3. Interest Rate Forward and Futures Contracts Libor, which stands for London Interbank Offered Rate, is a widely used interest rate that serves as the underlying for many derivative instruments. It represents the rate at which London banks can borrow from other London banks. Ø Ø Ø When these loans are in dollars, they are known as Eurodollar time deposits, with the rate referred to as dollar Libor. Average Libor rates are derived and posted each day at 11:30 a.m. London time. Libor is stated on an actual over 360-day count basis (often denoted ACT/360) with interest paid on an add-on basis. Let, Li(m) = Libor on an m-day deposit observed on day i NA = notional amount, quantity of funds initially deposited NTD = number of total days in a year, used for interest calculations (always 360 in the Libor market) tm = accrual period, fraction of year for m-day deposit— tm = m/NTD TA = terminal amount, quantity of funds repaid when the Libor deposit is withdrawn Example: Suppose day i is designated as Time 0, and we are considering a 90-day Eurodollar deposit (m = 90). Dollar Libor is quoted at 2%; thus, Li(m) = L0(90) = 0.02. $50,000 is initially deposited, i.e. NA = $50,000. Hence, tm = 90/360 = 0.25 TA = NA [1 + L0(m)tm] = $50,000[1 + 0.02(90/360)] = $50,250 Interest paid = TA – NA = $50,250 – $50,000 = $250 Forward market for Libor: A forward rate agreement (FRA) is an over the counter (OTC) forward contract in which the underlying is an interest rate on a deposit. An FRA involves two counterparties: the fixed receiver (short) and the floating receiver (long). Ø Ø Ø Being long the FRA means that we gain when Libor rises. The fixed receiver counterparty receives an interest payment based on a fixed rate and makes an interest payment based on a floating rate. The floating receiver counterparty receives an interest payment based on a floating rate and READING 37 Ø Ø Ø Pricing and Valuation of Forward Commitments makes an interest payment based on a fixed rate. FRA price is the fixed interest rate such that the FRA value is zero on the initiation date. The underlying of an FRA is an interest payment. It is also important to understand that the parties to an FRA do not necessarily engage in a Libor deposit in the spot market. Rather, a Libor spot market is simply the benchmark from which the payoff of the FRA is determined. A 3 × 9 FRA is pronounced as “3 by 9.” It implies that FRA expires in three months and the payoff of the FRA is 6months Libor (i.e. 9 -3) when the FRA expires in 3 months. Ø Ø A short (long) FRA will effectively replicate going short (long) a nine-month Libor deposit and long (short) a three-month FRA deposit. FRA value is the market value on the evaluation date and reflects the fair value of the original position. Example: A 30-day FRA on 90-day Libor would have h = 30, m = 90, and h + m = 120. If we want to value the FRA prior to expiration, g could be any day between 0 and 30. Ø Ø Ø Ø FRA (0, h, m) denotes the fixed forward rate set at Time 0 that expires at Time h wherein the underlying Libor deposit has m days to maturity at expiration of the FRA. Thus, the rate set at initiation of a contract expiring in 30 days in which the underlying is 90day Libor is denoted FRA (0, 30, 90). Like all standard forward contracts, at initiation, no money changes hands, implying value is zero. We can estimate price of FRA by determining the fixed rate [FRA(0,30,90)] such that the value is zero on the initiation date. How to settle interest rate derivative at expiration: There are two ways to settle an interest rate derivative when it expires: 1) Advanced set, settled in arrears:Advanced set implies that the reference interest rate is set at the time the money is deposited. The term settled in arrears means that the interest payment is made at Time h + m, (i.e. at the maturity of the underlying instrument). Swaps and interest rate options are normally based on advanced set, settled in arrears. 2) FinQuiz.com Advanced set, advanced settled: FRAs are typically settled based on advanced set, advanced settled. In an FRA, the term “advanced” refers to the fact that the interest rate is set at Time h, the FRA expiration date, which is the time when the underlying deposit starts. Here, advanced settled means the settlement is made at Time h. Libor spot deposits are settled in arrears, whereas FRA payoffs are settled in advance. The settlement amounts for advanced set, advanced settled are determined in the following manner: • Settlement amount at h for receive-floating: NA{[ (m) Lh − FRA(0,h,m)]tm}/[1 + Dh(m)tm] • Settlement amount at h for receive-fixed: NA{[FRA(0,h,m) − Lh(m)]tm}/[1 + Dh(m)tm] Where, 1 + Dh(m)tmis a discount factor applied to the FRA payoff.It reflects that the rate on which the payoff is determined, Lh(m), is obtained on day h from the Libor spot market, which uses settled in arrears, that is, interest to be paid on day h + m. Example: In 30 days, a UK company expects to make a bank deposit of £10,000,000 for a period of 90 days at 90-day Libor set 30 days from today. The company is concerned about a possible decrease in interest rates. The company enters into a £10,000,000 notional amount 1 × 4 receive-fixed FRA that is advanced set, advanced settled. This implies that an instrument that expires in 30 days and is based on 90-day (4 – 1) Libor. The discount rate for the FRA settlement cash flows is 0.40%. After 30 days, 90-day Libor in British pounds is 0.55%. TA = 10,000,000[1 + 0.0055(0.25)] = £10,013,750. Interest paid at maturity = TA – NA = £10,013,750 £10,000,000 = £13,750. • If the FRA was initially priced at 0.60%, the payment received to settle it will be closest to: m = 90 (number of days in the deposit) tm = 90/360 h = 30 (number of days initially in the FRA) The settlement amount of the 1 × 4 FRA at h for receive-fixed = [10,000,000(0.0060 – 0.0055)(0.25)]/[1 + 0.0040(0.25)] = £1,248.75 • If the FRA was initially priced at 0.50%, the payment received to settle it will be closest to as follows: Settlement amount of the 1 × 4 FRA at h for receivefixed = [10,000,000(0.0050 – 0.0055)(0.25)]/[1 + 0.0040(0.25)] = –£1,248.75 • In pay floating FRA, the long benefits when interest rate declines. Practice: Example 8, Reading 33, Curriculum. READING 37 Pricing and Valuation of Forward Commitments FRA pricing: Steps are as follows: Step 1:Deposit funds for h + m days: Ø At Time 0:deposit an amount = 1/[1 + L0(h)th], the present value of 1 maturing in h days, in a bank for h+ m days at an agreed upon rate of L0(h + m). Ø After h + m days,withdraw an amount = [1 + L0(h + m)th+m]/[1 + L0(h)th] Step 2: Borrow funds for h days: Ø At Time 0: Borrow {1/[1 + L0(h)th]}, for h days so that the net cash flow at Time 0 is zero. Ø In h days, this borrowing will be worth 1. Step 3:At Time h, roll over the maturing loan in Step 2 by borrowing funds for m days at the rate Lh(m). At the end of m days, we will owe [1 + Lh(m)tm]. In order to mitigate the risk of increase in interest rate, we would enter into a receive-floating FRA on m-day Libor that expires at Time h and has the rate set at FRA(0,h,m) as defined in step 4. Step 4:Enter a receive-floating FRA and roll the payoff at h to h + m at the rate Lh(m). The payoff at Time h will be ([Lh(m) – FRA(0,h,m)]tm)/(1 + Lh(m)tm). There will be no cash flow from this FRA at Time h because this amount will be rolled forward at the rate Lh(m)tm. Therefore, the value realized at Time h + m will be [Lh(m) – FRA(0,h,m)]tm. Cash Flow Table for Deposit and Lending Strategy with FRA FinQuiz.com Practice: Example 9, Reading 33, Curriculum. Valuing an existing FRA:If we are long the old FRA, we will receive the rate Lh(m) at h. We will go short a new FRA that will force us to pay Lh(m) at h. Suppose that we initiate an FRA that expires in 90 days and is based on 90day Libor. The fixed rate at initiation is 2.49%. Thus, tm = 90/360, and FRA (0,h,m) = FRA(0,90,90) = 2.49%. Ø Ø When the FRA expires and makes its payoff, assume that we roll it forward by lending it (if a gain) or borrowing it (if a loss) from period h to period h + m at the rate Lh(m). We then collect or pay the rolled forward value at h + m. Thus, there is no cash realized at Time h. Assume 30 days later, the rate on an FRA based on 90-day Libor that expires in 60 days is 2.59%. Thus, FRA (g, h – g, m) = FRA(30,60,90) = 2.59%. We go short this FRA, and as with the long FRA, we roll forward its payoff from Time h to h + m. Therefore, there is no cash realized from this FRA at Time h. Value of the offset position = (2.59% – 2.49%) = 10 bps times 90/360 paid at Time h + m To determine the fair value of the original FRA at Time g, we need the present value of this Time h + m cash flow at Time g. Value of the old FRA = Present value of the difference in the new FRA rate and the old FRA rate Hence, the value is Ø The terminal cash flows as expressed in the table can be used to solve for the FRA fixed rate. Because the transaction starts off with no initial investment or receipt of cash, the net cash flows at Time h + m should equal zero; thus, +[1 + L0( h+m )th+m]/[1 + L0(h)th] − [1 + Lh(m)tm] + [Lh(m) − FRA(0,h,m)]tm = 0 FRA fixed rate: FRA(0,h,m) = {[1 + L0( h+m )th+m]/[1 + L0(h)th] − 1}/ tm E.g. FRA(0,90,90) = {[1 + L0(180)t180]/[1 + L0(90)t90] – 1}/t90 Where, Vg(0,h,m) is the value of the FRA at Time g that was initiated at Time 0, expires at Time h, and is based on m-day Libor. Dg(h + m – g) is the discount rate. Traditionally, it is assumed that the discount rate, Dg(h + m – g), is equal to the underlying floating rate, Lg(h + m – g), but that is not necessary. Example: Suppose a 60- day rate of 3% on day g. Thus, Lg(h – g) = L30(60) = 3%. Then the value of the FRA would be Vg(0,h,m) = V60(0,90,90) = 0.00025/[1 + 0.03(60/360)] = 0.000249 READING 37 Pricing and Valuation of Forward Commitments FinQuiz.com Cash Flows for FRA Valuation are as following: Practice: Example 10, Reading 33, Curriculum. 4 PRICING FIXED-INCOME FORWARD AND FUTURES Accrued interest = Accrual period × Periodic coupon amount or AI = (NAD/NTD) × (C/n) Where NAD denotes the number of accrued days since the last coupon payment, NTD denotes the number of total days during the coupon payment period, n denotes the number of coupon payments per year, and C is the stated annual coupon amount. Example: After two months (60 days), a 3% semiannual coupon bond with par of 1,000 would have accrued interest of AI = (60/180) × (30/2) = 5. Important to remember: • The accrued interest is expressed in currency (not percent) and the number of total days (NTD) depends on the coupon payment frequency (semi-annual on 30/360 day count convention would be 180). We know that Forward price is equal to Future value of underlying adjusted for carry cash flows, as stated below: = FV0,T(S0 + θ0 – γ0) • For the fixed-income bond, let T + Y denote the underlying instrument’s current time to maturity. Therefore, Y is the time to maturity of the underlying bond at Time T, when the contract expires. • • • • • Let B0(T + Y) denote the quoted price observed at Time 0 of a fixed-rate bond that matures at Time T + Y and pays a fixed coupon rate. For bonds quoted without accrued interest, let AI0 denote the accrued interest at Time 0. The carry benefits are the bond’s fixed coupon payments, γ0 = present value of all coupon interest paid over the forward contract horizon from Time 0 to Time T = PVCI0,T. Future value of these coupons is γT = FVCI0,T. Assuming no carry costs, θ0 = 0. S0 = Quoted bond price + Accrued interest = B0(T + Y) + AI0 (1) Fixed-income futures contracts: Fixed-income futures contracts often have more than one bond that can be delivered by the seller. These bonds are usually traded at different prices based on maturity and stated coupon, therefore, an adjustment known as the conversion factor is used to make prices of all deliverable bonds equal (roughly, not exactly). In Fixed-income futures contracts markets, the futures price, F0(T), is defined as Quoted futures price × conversion factor= QF0(T) × CF(T) In general, the futures contract are settled against the quoted bond price without accrued interest. Thus, the total profit or loss on a long futures position = BT(T + Y) – F0(T). Based on above equation (1), this profit or loss can be expressed as follows: (ST – AIT) – F0(T) READING 37 Pricing and Valuation of Forward Commitments Adjusted Price of fixed-income forward or futures price including the conversion factor can be expressed as F0(T) = QF0(T) CF(T) =Future value of underlying adjusted for carry cash flows = FV0,T[S0 − PVCI0,T] = Future value (Quoted bond price + accrued interest - coupon payments made during the life of the contract) = FV0,T[B0( T+Y ) + AI0 − PVCI0,T] Steps of Carry arbitrage in the bond market: Step 1: Buy the underlying bond, requiring S0 cash flow. Step 2: Borrow an amount equivalent to the cost of the underlying bond, S0. Step 3: Sell the futures contract at F0(T). Step 4: Borrow the arbitrage profit. FinQuiz.com QF0(T) = Conversion factor adjusted future value of underlying adjusted for carry = [1/CF(T)]{FV0,T[B0(T + Y) + AI0] – AIT – FVCI0,T} Practice: Example 11, Reading 33, Curriculum. Cash Flows for Offsetting a Long Forward Position: Practice: Example 12, Reading 33, Curriculum. 4.1 Ø Ø Ø The value of the Time 0 cash flows should be zero or else there is an arbitrage opportunity. If the value in the Time 0 column for net cash flows is positive, then we buy bond, borrow, and sell futures. If the Time 0 column is negative, then we conduct the reverse carry arbitrage strategy, i.e. short sell bond, lend, and buy futures. In equilibrium, to eliminate an arbitrage opportunity, PV0,T[F0(T) – FV0,T(S0) + AIT + FVCI0,T] = 0 or F0(T) = FV0,T(S0) – AIT – FVCI0,T 5. Comparing Forward and Futures Contracts Forward pricing: F0 (T) = FV0,T(S0 + θ0 – γ0) Note that the price of a forward commitment is a function of the price of the underlying instrument, financing costs, and other carry costs and benefits. Forward valuation: Vt(T) = PVt,T [Ft(T) – F0(T)] Futures prices are generally found using the same model, but unlike forwards, futures values are zero at the end of each day because daily market to market settlement. PRICING AND VALUING SWAP CONTRACTS Swap contracts can be synthetically created by either a portfolio of underlying instruments or a portfolio of forward contracts. Thus, swaps can be viewed as a portfolio of futures contracts. A swap can also be viewed as a portfolio of option because a single forward contract can be viewed as a portfolio of a call and a put option. READING 37 Pricing and Valuation of Forward Commitments Generic Swap Cash Flows: Receive-Floating, Pay-Fixed 5.1 FinQuiz.com Interest Rate Swap Contracts Interest rate swaps have two legs, typically a floating leg (FLT) and a fixed leg (FIX). The floating leg cash flow (denoted Si) can be expressed as follows: • A receive-floating, pay-fixed swap is equivalent to being long a floating-rate bond and short a fixed-rate bond. If both bonds are purchased at par, the initial cash flows are zero and the par payments at the end offset each other. Also, note that the coupon dates on the bonds match the settlement dates on the swap and the maturity date matches the expiration date of the swap. Receive-Floating, Pay-Fixed as a Portfolio of Bonds The fixed leg cash flow (denoted FS) can be expressed as follows: Where, o CFi represents cash flows o APi denotes the accrual period o r denotes the observed floating rate appropriate for Time i o NADi denotes the number of accrued days during the payment period o NTDi denotes the total number of days during the year applicable to cash flow i o rFIX denotes the fixed swap rate. Types of day count methods:The two most popular day count methods are known as 30/360 and ACT/ACT. • • Uses of Swaps: Swaps can be used to manage interest rate risk. E.g. we can create a synthetic floating-rate bond by entering a receive-fixed, pay-floating interest rate swap. This swap can be used to hedge exposure to fixed rate loan. The two fixed rate payments (i.e. on loan and swap) cancel each other, leaving on net the floating-rate payments. There are also currency swaps and equity swaps. Currency swaps can be used to manage both interest rate and currency exposures. Equity swaps can be used to manage equity exposure. Like OTC products, swaps can be designed with an infinite number of variations. A swap can have both semi-annual payments and quarterly payments, as well as actual day counts and day counts based on 30 days per month. Also, the notional amount can vary across the maturities. Due to differences in payment frequency and day count methods as well as identifying the appropriate discount rate to apply to the future cash flows, the pricing and valuation of swaps is a bit tricky. As the name suggests, 30/360 treats each month as having 30 days, and thus a year has 360 days. ACT/ACT treats the accrual period as having the actual number of days divided by the actual number of days in the year (365 or 366). In swap market, the floating interest rate is assumed to be advanced set and settled in arrears; thus, rFLT,i is set at the beginning of period and paid at the end. If we assume constant accrual periods, the receive-fixed, pay-floating net cash flow can be expressed as follows: FS−Si = AP(rFIX − rFLT,i) And the receive-floating, pay-fixed net cash flow can be expressed as follows: Si − FS = AP(rFLT,i− rFIX) Example: Suppose, a fixed rate is 5%, the floating rate is 5.2%, and the accrual period is 30 days based on a 360 day year, the payment of a receive-fixed, pay-floating swap is calculated as (30/360)(0.05 – 0.052) = –0.000167 per notional of 1. Because the floating rate > fixed rate, the party that pays floating (and receives fixed)would pay this amount to the party that receives floating (and pays fixed). Swap pricing: Swap pricing involves determining the equilibrium fixed swap rate. The fixed swap rate is simply READING 37 Pricing and Valuation of Forward Commitments one minus the final present value term divided by the sum of present values (as discussed in detail below). Suppose the arbitrageur enters a receive-fixed, payfloating interest rate swap with some initial value V. Please see the cash flows for receive-fixed swap hedge with bonds as stated below: Cash Flows for Receive-Fixed Swap Hedge with Bonds Ø Ø FinQuiz.com adjusted for the accrual period, i.e. FSi = APFIX,irFIX. The annualized fixed swap rate = fixed swap leg cash flow / fixed rate accrual period, or rFIX,i = FS/APFIX The fixed swap payment will vary if the accrual period varies across the swap payments. Practice: Example 13, Reading 33, Curriculum. In equilibrium, we must have –V – VB + FB = 0 or else there is an arbitrage opportunity. For a receive fixed and pay floating swap, the value of the swap is V = Value of fixed bond – Value of floating bond = FB – VB Ø Ø Ø The value of a receive-fixed, pay-floating interest rate swap is simply the value of buying a fixedrate bond and issuing a floating-rate bond. The value of a floating-rate bond, assuming we are on a reset date and the interest payment matches the discount rate, is par, assumed to be 1 here. The value of a fixed bond is as follows: Where, C denotes the coupon amount for the fixed-rate bond and PV0,ti (1) is the appropriate present value factor for the ith fixed cash flow. Ø Ø Positive (negative) value of FS0 represents value to the party receiving (paying) fixed. Please refer below to the Cash Flows for Receive-fixed Swap Valued at Time t: Practice: Example 14, Reading 33, Curriculum. The fixed swap leg cash flow for a unit of notional amount is simply the fixed swap rate 6. PRICING AND VALUING CURRENCY SWAP CONTRACTS A currency swap is a contract in which two counterparties agree to exchange future interest payments in different currencies. There are four major types of currency swaps: 1. 2. 3. Interest rate swap valuation: The value of a fixed rate swap at some future point in Time t is simply the sum of the present value of the difference in fixed swap rates times the stated notional amount (denoted NA), or Fixed-for-fixed Floating for- fixed Fixed-for-floating 4. Floating-for-floating Important to Remember: • Currency swaps often (but do not always) involve an exchange of notional amounts at both the initiation of the swap and at the expiration of the swap. READING 37 • • Pricing and Valuation of Forward Commitments The payment on each leg of the swap is in a different currency unit, such as euros and dollars, and the payments are not netted. Each leg of the swap can be either fixed or floating. Currency swap pricing has three key variables: These include two fixed interest rates and one notional amount. The value of a fixed-rate bond in Currency k can be expressed as FinQuiz.com In order to determine the fixed rates of the swap such that the current swap value is zero, we have FBa(C0,a,Para) = S0FBb(C0,b,Parb) The equilibrium fixed swap rate equations for each currency: and Where k = a or b, Ck denotes the periodic fixed coupon amount in Currency k, denotes the present value from Time 0 to Time ti discounting at the Currency k risk-free rate, and Park denotes the k currency unit par value. The fixed swap rate in each currency is simply one minus the final present value term divided by the sum of present values. Numerical Example of Currency Swap Hedged with Bonds Here, par is not assumed to be equal to 1 because the notional amounts are typically different in each currency within the currency swap. Please refer to table below for cash flows for currency swaps hedged with Bonds. Cash Flows for Currency Swap Hedged with Bonds Ø Based on this table, in equilibrium we must have–Va + FBa – S0FBb = 0 Fixed-for-fixed currency swap value is Va = FBa – S0FBb or else there is an arbitrage opportunity. Note that the exchange rate S0 is the number of Currency a units for one unit of Currency b at Time 0; thus, S0FBb is expressed in Currency a units. Swap value after initiation = Va= FBa – S0FBb In equilibrium, the notional amounts of the two legs of the currency swap are NAb = Parb and NAa = Para = S0Parb. Ø If the initial swap value is positive, then we would follow the set of transactions stated in the table above. If the initial swap value is negative, then the opposite set of transactions would be implemented, that is, we would enter into a payUS dollar, receive-euro swap, buy Currency a bonds, and short sell Currency b bonds. Practice: Example 15, Reading 33 Curriculum. Fixed-for-floating currency swap: A fixed-for-floating currency swap is simply a fixed-for-fixed currency swap paired with a floating-for-fixed interest rate swap. Cash Flows for Currency Swap Hedged with Bonds READING 37 Pricing and Valuation of Forward Commitments FinQuiz.com The currency swap valuation equation can be expressed as Value of a fixed-for-fixed currency swap = Va = FBa – S0FBb 7. PRICING AND VALUING EQUITY SWAP CONTRACTS An equity swap is an OTC derivative contract in which two parties agree to exchange a series of cash flows whereby one party pays variable cash flows based on an equity and the other party pays either (1) a variable cash flows based on a different equity or rate or (2) a fixed cash flow. Equity swaps are widely used in equity portfolio investment management to modify returns and risks. Three common types of equity swaps are i. Receive-equity return ii. Pay-fixed; receive equity return iii. Pay-floating; and receive-equity return, payanother equity return. It can be viewed simply as a receive-equity a, pay-fixed swap combined with a pay-equity b, receive-fixed swap. The fixed payments cancel out each other – remaining with equity portion. Important to Remember: • The underlying reference instrument for the equity leg of an equity swap can be an individual stock, a published stock index, or a custom portfolio. • The equity leg cash flow can include or exclude dividends. • Like interest rate swaps, equity swaps have a fixed or floating interest rate leg. The equity swap cash flows can be expressed as follows: 1. 2. Practice: Example 16, Reading 33, Curriculum. For receive-equity, pay-fixed = NA (Equity return – Fixed rate) For receive-equity, pay-floating = NA(Equity return – Floating rate) 3. For receive-equity, pay-equity = NA(Equity returna – Equity returnb) Where, a and b denote different equities. Ø Ø When the equity leg of the swap is negative, then the receive-equity counterparty must pay both the equity return as well as the fixed rate. Equity swaps may cause liquidity problems because if the equity return is negative, then the receive-equity return, pay-floating or pay-fixed swap may result in a large negative cash flow. Practice: Example 17, Reading 33, Curriculum. The cash flows for the equity leg of an equity swap can be expressed as Si = NAEREi Where, NAE denotes the notional amount and REi denotes the periodic return of the equity either with or without dividends as specified in the swap contract. The cash flows for the fixed interest rate leg of the equity swap can be expressed as FS = NAEAPFIXrFIX Where APFIX denotes the accrual period for the fixed leg for which we assume the accrual period is constant and rFIXdenotes the fixed rate on the equity swap. READING 37 Pricing and Valuation of Forward Commitments Please refer to the table below for Cash Flows for Receive-Fixed Equity Swap Hedged with Equity and Bond. Equity swap value is V = –NAE + FB – PV (Par – NAE) The fixed swap rate can be expressed as Ø Ø Ø In a pay-floating swap, there is no need to calculate price of the swap because the floating side effectively prices itself at par automatically at the start. If the swap involves paying one equity return against another, there would be no need to price the swap because this arrangement can be viewed as paying equity a and receiving a fixed rate as specified above and receiving equity b and paying the same fixed rate. The fixed rates would cancel each other. Valuing an equity swap after the swap is initiated (Vt) is similar to valuing an interest rate swap except that rather than adjust the floatingrate bond for the last floating rate observed (remember, advanced set), the value of the notional amount of equity is adjusted as below. Where, o FBt(C0) denotes the Time t value of a fixed-rate bond initiated with coupon C0 at Time 0, o St denotes the current equity price, o St– denotes the equity price observed at the last reset date, and o PV() denotes the present value function from Time t to the swap maturity time. Practice: Example 18 & 19, Reading 33, Curriculum. Practice: Questions from End of Chapter Curriculum and FinQuiz Question-bank FinQuiz.com