Forward Contracts And Forward Rates - New York University

Debt Instruments and MarketsProfessor CarpenterForward Contracts andForward RatesOutline and Readings Outline––––Forward ContractsForward PricesForward RatesInformation in ForwardRates Buzzwords- settlement date, delivery,underlying asset- spot rate, spot price, spotmarket- forward purchase, forwardsale, forward loan, forwardlending, forward borrowing,synthetic forward- expectations theory, termpremium Reading– Veronesi, Chapters 5 and 7– Tuckman, Chapters 2 and 16Forward Contracts and Forward Rates1

Debt Instruments and MarketsProfessor CarpenterForward Contracts Aforward contract is an agreement to buy anasset at a future settlement date at a forwardprice specified today.– No money changes hands today.– The pre-specified forward price isexchanged for the asset at settlement date. Bycontrast, an ordinary transaction that settlesimmediately is called a spot or cash transaction,and the price is called the spot price or cashprice.Motivation Supposetoday, time 0, you know you will needto do a transaction at a future date, time t. Onething you can do is wait until time t and thendo the transaction at prevailing market prices- i.e., do a spot transaction in the future. Alternatively,you can try to lock in the terms ofthe transaction today- i.e., arrange a forward transaction today.Forward Contracts and Forward Rates2

Debt Instruments and MarketsProfessor CarpenterWhat is the fair forward price? Insome cases, the forward contract can besynthesized with transaction in the current spotmarket. Inthat case, no arbitrage will require that thecontractual forward price must be the same as theforward price that could be synthesized.Synthetic Forward Price Forexample, if the underlying asset doesn’tdepreciate, make any payments, or entail any storagecosts or convenience yield, the synthetic forwardprice of the asset is SpotPrice Interest to settlement date Howto synthesize?– Buy the asset now for the spot price.– Borrow the amount of the spot price, with repaymenton the settlement date– You pay nothing now, and you pay the spot priceplus interest at the settlement date.Forward Contracts and Forward Rates3

Debt Instruments and MarketsProfessor CarpenterSynthetic Forward Contract on a ZeroSuppose r0.5 5.54%, d0.5 0.9730, r1 5.45%, and d1 0.9476.Synthesize a forward contract to buy 1 par of the zeromaturing at time 1 by1) buying 1 par of the 1-year zero and2) borrowing the money from time 0.5 to pay for it:1)-0.9476 12) 0.9476?Net:0-F ? 1 ------------------------------ -------------------------- 00.51Class Problem: What is the no-arbitrage forward price F?Arbitrage Argument ClassProblem: Suppose a bank quoted aforward price of 0.98. How could you makearbitrage profit?Forward Contracts and Forward Rates4

Debt Instruments and MarketsProfessor CarpenterSynthetic Forward Price for a Zero In general, suppose the underlying asset is 1 parof a zero maturing at time T. In the forward contract, you agree to buy this zeroat time t. The forward price you could synthesize is spotprice plus interest to time t:FtT dT (1 rt /2) 2t If the quoted contractual forward price differs,there is an arbitrage opportunity. Class ProblemSuppose the spot price of 1 par of the 1.5-year zerois 0.9222.What is the no arbitrage forward price of this zerofor settlement at time 1, F11.5 ?Forward Contracts and Forward Rates5

Debt Instruments and MarketsProfessor CarpenterForward Contract as a Portfolio of ZeroesHere’s another way to view the contract: You agree today (t 0) to pay at t the sum F to get 1worth of par at This contract is a portfolio of cash flows: 0- F 1 ----------------- ----------------- 0tT What is the PV of this contract? It is a portfolio:Long 1 par of T-year zerosShort F par of t-year zeros So its present value is V -F x dt 1 x dT Zero Cost Forward Price Att 0 the contract “costs” zero. The forward price is negotiated to make that true. What is the forward price that makes the contractworth zero? 0- F 1 ----------------- ----------------- 0tTV -F x dt 1 x dT 0 F dT / dtwhich is equivalent to F dT (1 rt /2)2t FtTForward Contracts and Forward Rates6

Debt Instruments and MarketsProfessor CarpenterExamplesRecall the spot prices of 1 par of the 0.5-, 1-, and 1.5year zeroes are 0.9730, 0.9476, and 0.9222.The no-arbitrage forward price of the 1-year zero forsettlement at time 0.5 isF0.51 d1/d0.5 0.9476/0.9730 0.9739The no-arbitrage forward price of the 1.5-year zero forsettlement at time 1 isF11.5 d1.5/d1 0.9222/0.9476 0.9732Class Problem Suppose a firm has an old forward contract on itsbooks. The contract commits the firm to buy, at time t 0.5, 1000 par of the zero maturing at time T 1.5 for aprice of 950. At inception, the contract was worth zero, but nowmarkets have moved. What is the value of thiscontract to the firm now?Forward Contracts and Forward Rates7

Debt Instruments and MarketsProfessor CarpenterForward Contract on a Zero as aForward Loan Justas we can think of the spot purchase of a zero aslending money, we can think of a forward purchase ofa zero as a forward loan.forward lender agrees today to lend FtT on thesettlement date t and get back 1 on the date T. Thethe forward rate, ftT, as the interest rate earnedfrom lending FtT for T-t years and getting back 1: Define Thisis the same transaction, just described in terms oflending or borrowing at rate instead of buying orselling at a price.Class Problem Recall that the no-arbitrage forward price of the 1.5year zero for settlement at time 1 is What is the implied forward rate f11.5 that you couldlock in today for lending from time 1 to time 1.5?Forward Contracts and Forward Rates8

Debt Instruments and MarketsProfessor CarpenterArbitrage Argument in Terms of Rates:New Riskless Lending Possibilities Consider the lending possibilities when a forwardcontract for lending from time t to time T is available. Now there are two ways to lend risklessly from time 0to time T:1) Lend at the current spot rate rT (i.e., buy a T-yearzero). A dollar invested at time 0 would growrisklessly to (1 rT /2)2T.2) Lend risklessly to time t (i.e., buy a t-year zero)and roll the time t payoff into the forward contractto time T. A dollar invested at time 0 would growrisklessly to (1 rt /2)2t x(1 ftT/2)2(T-t) .No Arbitrage Forward RateIn the absence of arbitrage, the two ways of lendingrisklessly to time T must be equivalent:0tTExample: The forward rate from time t 0.5 to timeT 1 must satisfyForward Contracts and Forward Rates9

Debt Instruments and MarketsProfessor CarpenterNo Arbitrage Forward Rate Class Problem:The 1.5-year zero rate is r3 5.47%. What is theforward rate from time t 0.5 to time T 1.5?Connection Between Forward Pricesand Forward RatesOf course, this is the same as the no arbitrage equationswe saw before:Example: The implied forward rate for a loan from time 0.5to time 1 is 5.36%. This gives a discount factor of 0.9739,which we showed before is the synthetic forward price topay at time 0.5 for the zero maturing at time 1.Forward Contracts and Forward Rates10

Debt Instruments and MarketsProfessor CarpenterSummary: One No Arbitrage Equation,Three Economic Interpretations:(1) Forward price Spot price Interest(2) Present value of forward contract cash flows at inception 0:(3) Lending short Rolling into forward loan Lending long:Using the relations between prices and rates,andorwe can verify that these equations are all the same. Other arrangements:Spot Rates as Averages of Forward Rates Rolling money through a series of short-term forwardcontracts is a way to lock in a long term rate and thereforesynthesizes an investment in a long zero. Here are twoways to lock in a rate from time 0 to time t: The growth factor (1 rt/2) is the geometric average of the(1 f/2)’s and so the interest rate rt is approximately theaverage of the forward rates. Recall the example– The spot 6-month rate is 5.54% and the forward 6-monthrate is 5.36%.– Their average is equal to the 1-year rate of 5.45%.Forward Contracts and Forward Rates11

Debt Instruments and MarketsProfessor CarpenterZero rates are averages of the one-period forward rates up to theirmaturity, so while the zero curve is rising, the marginal forward ratemust be above the zero rate, and while the zero curve is falling, themarginal forward rate must be below the zero rate.Forward Rates vs. Future Spot Rates The forward rate is the rate you can fix todayfor a loan that starts at some future date. By contrast, you could wait around until thatfuture date and transact at whatever is theprevailing spot rate. Is the forward rate related to the random futurespot rate? For example, is the forward rate equal topeople’s expectation of the future spot rate?Forward Contracts and Forward Rates12

Debt Instruments and MarketsProfessor CarpenterThe Pure Expectations Hypothesis The “Pure Expectations Hypothesis” says that theforward rate is equal to the expected future spot rate. It turns out that’s roughly equivalent to the hypothesisthat expected returns on all bonds over a given horizonare the same, as if people were risk-neutral. For example, if the forward rate from time 0.5 to time 1equals the expected future spot rate over that time, thenthe expected one-year rate of return from rolling two sixmonth zeroes is equal to the one-year rate of return fromholding a one-year zero:Example in which the Pure ExpectationsHypothesis Holds: Upward-Sloping Yield CurveTime 0Time 0.50.5-yr horizonROR onROR on0.5-yr z.1-yr z.5.00%4.008%1-yr horizonROR onROR on0.5-yr z.1-yr z.5.749%5.25%d .25%Zero rateu0.5r1 6.50%0r0.5 5.00%0r1 5.25%0.5r1Expected:Forward ratef0.51 5.50%If the pure expectations hypothesis holds, then an upward-slopingyield curve indicates rates are expected to rise.Forward Contracts and Forward Rates13

Debt Instruments and MarketsProfessor CarpenterExample in which the Pure ExpectationsHypothesis Holds: Downward-Sloping Yield CurveProblem with the Pure ExpectationsHypothesis: Expected Rates of ReturnMay Differ Across Bonds Different bonds may have different expected rates ofreturn because their returns have different riskproperties (variance, covariance with other risks, etc.) In that case, the pure expectations hypothesis cannothold. For example, the yield curve is typically upwardsloping.– If the pure expectations hypothesis were true, thatwould mean people generally expect rates to rise.– An alternative explanation is that investors generallyrequire a higher expected return to be willing to holdlonger bonds.Forward Contracts and Forward Rates14

Debt Instruments and MarketsProfessor CarpenterExample in which Longer Bonds Have HigherExpected ReturnsTime 0Time 0.50.5-yr horizonROR onROR on0.5-yr z.1-yr z.5.00%4.503%1-yr horizonROR onROR on0.5-yr z.1-yr z.5.499%5.25%d .25%Zero rateu0.5r1 6.00%0r0.5 5.00%0r1 5.25%0.5r1Expected:Forward ratef0.51 5.50%Here, the yield curve is upward-sloping, not because rates are expected torise, but because longer bonds are priced to offer a higher expected return.Term Premiums in Forward Rates Empirically, forward rates tend to be higher than thespot rate that ultimately prevails for that investmenthorizon, or equivalently, longer bonds appear to havehigher average returns. The “term premium” is defined roughly byForward rate Expected future spot rate TermPremium A more general version of expectations hypothesissays that term premiums are roughly constant. If that’s true, then changes in forward rates reflectchanges in expectations about future rates. On the other hand it could be because risk premiumshave changed.Forward Contracts and Forward Rates15