FIN 472 Fixed-Income Securities Forward Rates

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1 FIN 472 Fixed-Income Securities Forward Rates Professor Robert B.H. Hauswald Kogod School of Business, AU Interest-Rate Forwards Review of yield curve analysis Forwards yet another use of yield curve forward deposits, loan commitments Introduction to synthetic FI securities FRAs, FRNs pricing and valuing: marked-to-market Preview of topics to come 2

2 Term Structure of Interest Rates r zero Yield curve Time to maturity 0 3m 6m 1yr 3yr 5yr 10yr 30yr slide 3 Yield Curve = Term Structure of IR r Flat Normal Inverted maturity slide 4

3 Determinants of the Yield Curve Federal Reserve Board sets a target level for the Fed Funds rate implementation? the rate at which depository institutions make uncollaterized overnight loans to one another. what do banks need Fed Funds for? so, the Fed controls what? Long-term rates reflect expectations of future rates: influenced by the monetary policy general economic outlook slide 5 Yield Curve Calculus Yield curves: term structure of interest rates time value of money representation pricing and arbitrage: spot rates Simple yield curve analysis: price - yield yield to maturity, discount yields, yield curves SO approximation of price changes due to parallel shifts: duration and convexity What about more realistic shifts? 6

4 Forward Contracts Definition: a contract to exchange a good at some future date for a price determined now prespecified price, prespecified date no cash/good exchange at contracting date Time line: now: price, quantity and future transaction date later: transaction carried out at prespecified terms Potential Problems? 7 Examples Forward contract on Treasury forward contract on zero: analytic interest when-issued market Forward Rate Agreement: FRA contract on a future loan/deposit standardized, especially in Euromarkets components: 3 dates, 2 quantities, 2 prices, 1 rate Loan commitment, forward deposit 8

5 Definition: Forward Rates A forward rate is a yield quoted now on a zerocoupon bond to be delivered in the future For example, a 2-year rate 1-year forward is the yield quoted today on a 2-year zero starting one year from now and maturing 3 years from now Mechanics: Forward Rates Forward rates are embedded in the term structure. Suppose you own a zero that matures in 1 year and yields 6%: two alternative strategies Interest could accumulate at the same rate over the entire year or It could accumulate at one rate for the first half year and at another rate for the second half year such that the average is 6%. 10

6 Two Strategies, One Result: Finding 6M Forward Rates 6M forward rates are implied by the term structure: why? 0 1/ Strategy 1 Investing 0 P 1 at 6% for 1 year interest accumulates at the same rate throughout at the end of a half year you will have: 0P at the end of a year you will have 0 P

7 Strategy 2 Alternatively, invest 0 P 1 at say 4% for 6M, and then reinvest the proceeds at another rate, say 1 r 1, at the end of a half year you will have: 0P at the end of a year you will have 0P r The Arbitrage Argument These two strategies both have to result in a 6% yield for the year if that is, if and generally r 1 = sr 12 = 2 = r ( ) 2 ( ) 0P s P s

8 Forward Rates from Term Structure The spot-rate curve implies forward rates under the assumption of no arbitrage having $100 to invest for three years do we want a three-year spot return? how about a one-year spot followed by two more years of spot returns.. The Yield Curve yr 2 yr 5yr 10 yr 20yr S(1) S(2) S(5) S(10) S(20) Recall that the yield curve is a collection of current spot rates 16

9 Forward Rates Forward rates are interest rates for contracts to be written in the future. (F) F(1,1) = Interest rate on 1 year loans contracted 1 year from now F(1,2) = Interest rate on 2 yr loans contracted 2 years from now F(2,1) = interest rate on 1 year loans contracted 2 years from now S(1) = F(0,1) 17 Spot and Forward Rates S(3) S(2) S(1) Now 1yr 2yrs 3yrs 4yrs F(0,1) F(1,1) F(2,1) Spot Rates 5yrs F(0,2) F(1,2) F(2,2) Forward Rates F(1,3) 18

10 Forward Rates: Data Forward rates are not observed, but are implied in the yield curve Consequence? Suppose the current annual yield on a 2YTreasury is 2.61% while a 1Y Treasury pays an annual rate of 2.12% S(2) 2.61%/yr S(1) 2.12%/yr Now 1yr 2yrs 3yrs 4yrs F(1,1) 5yrs 19 Calculating Forward Rates S(2) 2.61%/yr S(1) 2.12%/yr Now 1yr 2yrs 3yrs 4yrs F(1,1) 5yrs Strategy #1: Invest $1 in a two year Treasury $1(1.0261)(1.0261) = (5.3%) Strategy #2: Invest $1 in a 1 year Treasury and then reinvest in 1 year $1(1.0212)(1 + F(1,1)) For these strategies, to pay the same return, the one year forward rate would need to be 3.1% $1(1.0261)(1.0261) = $1(1.0261)(1+F(1,1) 1+F(1,1) = $1(1.0261)(1.0261) $1(1.0212) =

11 Calculating Spot Rates We can also carry out the reverse calculation. If we knew the path for forward rates, we can calculate the spot rates: S(3)??? S(2)??? S(1) 2% Now 1yr 2yrs 3yrs 4yrs 2% 3.3% 2.9% 5yrs 21 Calculating Spot Rates S(2)??? Now 1yr 2yrs 3yrs 4yrs 2% 3.3% 5yrs Strategy #1: Invest $1 in a two year Treasury $1(1+(S(2))(1+S(2)) For these strategies, to pay the same return, the two year spot rate would need to be 2.6% Strategy #1: Invest $1 in a 1 year Treasury and then reinvest in 1 year $1(1.02)(1.033) = (5.4%) $1(1.02)(1.033) = $1(1+S(2)) 2 1+S(2) = ((1.02)(1.033)) 1/2 =

12 Arithmetic vs. Geometric Averages S(2) 2.6% Now 1yr 2yrs 3yrs 4yrs 2% 3.3% 5yrs In the previous example, we calculated the Geometric Average of expected forward rates to get the current spot rate 1/2 1+S(2) = ((1.02)(1.033)) =1.026 (2.6%) The Arithmetic Average is generally a good approximation S(2) = 2% + 3.3% 2 = 2.65% 23 Expectation Hypothesis Spot rates are equal to the averages of the corresponding forward rates S(3) 2.73% S(2) 2.65% S(1) 2% Now 1yr 2yrs 3yrs 4yrs 2% 3.3% 2.9% 5yrs S(2) = 2% + 3.3% 2 = 2.65% S(3) = 2% + 3.3% + 2.9% 3 = 2.73% 24

13 However, the expectations hypothesis assumes that investing in long term bonds is an equivalent strategy to investing in short term bonds S(2) 2.65 Now 1yr 2yrs 3yrs 4yrs 2% Preferred Habitat Hypothesis 3.3% % This rate is flexible at time 0 Long term bondholders should be compensated for inflexibility of their portfolios by adding a liquidity premium to longer term rates This rate is locked in at time 0 5yrs 25 Statistical Models Now 1yr 2yrs 3yrs 4yrs F(0,1) F(1,1) F(2,1) F(3,1) 3.3% F(4,1) 5yrs First, write down a model to explain movements in the forward rates Then, calculate the yield curve implied by the forward rates. Does it look like the actual yield curve? S(3) S(2) S(1) Now 1yr 2yrs 3yrs 4yrs 5yrs 26

14 Spot Rates Graphically, we need to assess the time line of interest to us: z 2 z 1 1 f 1 The current 1Y spot rate is z 1 = % and the current 2Y spot rate z 2 = % % % 2/15/ Forward Rates Robert B.H. Hauswald Implied Forward Rates In the absence of arbitrage we need to have: (1+ z 1 )*(1+ 1 f 1 ) = (1+z 2 ) 2 ( )*(1+ 1 f 1 ) = ( ) 2 ( )*(1+ 1 f 1 ) = f 1 = f 1 = % % % % 28

15 The Meaning of Forward Rates If we knew the future short (1Y) rate: 0(Today) 8% 1 10% 2 11% 3 11% then the price of a 4Y zero would be $1,000 P = ( )( )( )( ) slide 29 Short vs. Spot Rate r 1 = 8% r 1 = 10% r 3 = 11% r 4 = 11% Spot rate: the yield to maturity on zerocoupon bonds Short rate: the yield to maturity on a 1Y zero-coupon bond above we have the current and future short rates slide 30

16 From Future to Spot Rates r 1 = 8% r 1 = 10% r 3 = 11% r 4 = 11% y 1 = 8% ( ) = y 2 = 8.995% 3 ( ) 1 1 =? y 3 = 9.66% y 4 = 9.993% 31 Manufacturing a Future Loan Suppose you will need a loan in two years from now for one year. How one can create such a loan today? Go short a three-year zero coupon bond. Go long a two-year zero coupon bond. Cash flows are then slide 32

17 Future Loan Rates (1 + y n ) n = (1 + y n-1 ) n-1 (1 + f n ) (1 + y n ) n (1 + y n-1 ) n f n 3 slide 33 Locking in Future Loan Rates In other words we can lock now interest rate for a loan which will be taken in future To specify a forward interest rate one should provide information about today s date beginning date of the loan end date of the loan slide 34

18 The Arbitrage Argument Again Buy a two years bond Buy a one year bond and then use the money to buy another bond the price can be fixed today (1+r 2 )=(1+r 1 )(1+f 12 ) Term structure of instantaneous forward rates (1+r 3 ) 3 =(1+r 1 )(1+f 13 ) 2 = (1+r 1 )(1+f 12 )(1+f 23 ) slide 35 slide 36

19 Pricing Forward Contracts General principle: forward price = spot price x cost-of-carry otherwise? Holds for all forward transactions: FI FX commodities Other costs of carry items? ( 1 ) t F = P + r t 37 Valuation of a Forward Marking-to-market: fundamental valuation principle of financial positions only valid accounting approach: alternatives? principle: value positions at their fair market price Computation principle: reverse the transaction: conceptual squaring of position, i.e., compute the cost/gain arising from the offsetting transaction 38

20 Forward T-Bill Transaction What price is right? What principle is appropriate? Forward purchase of a 1Y T-bill for in 6M: data: 6M spot rate 10%, current price 80 forward price: 88, 84, 80? Mispricing of forward: forward < spot: spot > forward: Cost-of-carry: r 39 Marking-to-Market t Example continued: F0 P0 ( 1 r t ) = + = +. = after 3 months: still 3M to go, price and rate changes spot price of 9M bill $90, interest rate 12% t current forward price: F90 P90 ( 1 r t ) = + = +. = Selling contract nets what? valuation? marked to market? Buy side of forward: marked-to-market V 40 t = F F r t 0 1+ T t

21 Repurchase Agreements Definition: purchase of an asset with promise to reverse (sell back) the asset later Different forward contract: why? combination of spot and forward transaction spot: forward: Function: collateralized (or: secured) lending inventory finance short selling 41 Repo Rate Calculations Repo: $10m of 7.25%, 5/15/2016 T-bond protection against adverse price changesand default margin: hair-cut of 0.5% of market price 5/10/1986: full price = hair cut: $47, amount borrowed: 9,467, /13/1986: delivery and sold for full repo rate of 6%: forward rate that equates the underlying hedging of position 42

22 Forward Rate Agreement (FRA) Forward contract on interest rates Not a commitment of borrowing or lending Like any forward, value for buyer increases as underlying increases Here the underlying is an interest rate Payoff based on difference between market interest rate at settlement and the contract rate protection against interest-rate uncertainty 43 A forward rate agreement (FRA) is a forward contract based on interest rates The buyer of an FRA agrees to pay a fixed-rate coupon payment (at the exercise/contract rate) and receive a floating-rate payment against a notional principal amount at a specified future date. The buyer of an FRA will receive (pay) cash when the actual interest rate at settlement is greater (less) than the exercise/contract rate (specified fixed-rate). The seller of an FRA agrees to make a floating-rate payment and receive a fixed-rate payment against a notional principal amount at a specified future date. The seller of an FRA will receive (pay) cash when the actual interest rate at settlement is less (greater) than the exercise rate. 44

23 Forward Rate Agreements OTC contract for the exchange of payments between two parties over a single future period one party pays fixed rate, the other variable one why would parties wish to exchange obligations? Example: $100m, 3/9 FRA, 5% fixed, 6M LIBOR /9: party A pays: party B pays: 45 FRA Time Structure deferment period contract period dealing date spot date fixing date settlement date maturity date contract rate agreed reference rate determined settlement sum paid 46

24 Application of FRAs Life insurer has fixed policy loan rate that resets every year Exposure is that in 2nd half of policy year, rates will increase and policyholders withdraw cash to earn higher interest elsewhere Purchase of FRA with 6 month settlement will mitigate risk Rates increase, insurer receives cash Rates decrease, FRA payment from bond gains 47 Implicit Forward Rates Pricing an plain-vanilla forward: 1 year loan or deposit in 1 year s time transaction details what are the alternatives? arbitrage reasoning: NFL The two legs: 2 1 two year investment: ( 1 + r2 ) 2 one year investments: ( 1+ r1 )( 1+ r 1, 2) > the arbitrage relation: ( 1+ r )( 1+ r ) ( 1 r ) 2 1 1, < +

25 Forward Rates and Prices Generalize preceding example current future rate: 1 ( 1+ 0( )) = + t yt f s, t, t > s 0 s ( 1+ ys) later future rate, i.e., future rate at a future date? work it out! Forward price on zeros: + = ( 1 ft( T1, T2 )) ( ) ( 1+ y2 ) ( 1+ y ) T t 2 T1 t 1 1 T T 2 1 (, 1) (, ) P t T = P t T 49 3 Summary First derivative: forwards Key concepts: forward price marked-to-market implicit forward rates FRAs and FRNs: related derivatives synthetic security YC application 50

26 Appendix More on forwards legal issues methodological problem Continuous forward rates simple generalization Forward rate curve More on FRAs the pricing of floaters 51 Problems with Forwards Legal problem: fundamental default risk reneging potential insurance: posting margins, legal arrangements alternatives: exchange traded instruments: Analytic problem: implicit YC assumption common YC assumption underlying the valuation? is it realistic? alternatives? 52

27 Continuous Forward Rates Let P(t,s) be the price at time t of a pure discount bond maturing at time s > t. the yield to maturity R(t,T) is the internal rate of return at time t on a bond maturing at t+t. Then P(t, t+t) = exp[-r(t,t)*t] R(t,T) = - log[p(t, t+t)]/t slide 53 From Forward Rates to YTM The integral of the forward rates gives the yield to maturity: t+ T 1 R ( t, T ) = F( t, s) ds T Alternatively, after substituting in and taking (partial) derivatives F( t, s) = log P( t, s) s t slide 54

28 Forward Rate Curve Use of the yield curve underlying assumption realistic? Construct forward rate curve from current YC familiarization with forward rates prediction tool for future (spot) YC: what hypothesis implicitly underlies such an approach to YC prediction important analytic tool, especially for more advanced yield curve modelling 55 FRA Contract Settlement Value On 1/1, an insurer enters an FRA contract for the period of 7/1-12/31, the contract rate is 5%, and the notional amount is $1 million If on 7/1 rates are 6%, a borrower without an FRA would pay $1,030,000 on 12/31 With an FRA, the amount of payment on 7/1 reflects PV of excess interest throughout contract period $5, 000 = $4, ( ) 56 2

29 FRAs and Forward Rates A contract entered at t=0, where the parties (a lender and a borrower) agree to let a certain interest rate R*, act on a prespecified principal, K, over some future time period [S,T]. Assuming continuous compounding we have at time S: -K at time T: Ke R*(T-S) Calculate the FRA rate R* which sets PV=0 hint: it is equal to forward rate slide 57 FRA Pricing Use YC to infer future LIBOR from implied forward rates necessary data: 6M rate for in 3M: fixed rate what else needs to be done? Settlement: at the end of the reference period: 3M netted: just the then prevailing net flow what is paid at start of contract? 58

30 Floaters Floating rate notes (FRNs): FIS (!) paying coupons linked to some other interest rate LIBOR, Treasury, prime + index linked security with premium Valuation: extend FRA idea calculate reset rates as implied forward rates from current YC add the spread discount back using, again, YC 59

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