1. © K. Cuthbertson, D. Nitzsche1 Lecture Forward Rates and the Yield Curve The material in these slides is taken from Chapter 8 Investments: Spot and Derivatives Markets (J. Wiley, 2001) K. Cuthbertson and D. Nitzsche These slides provide the introductory material required for Chapters 5 and 6 of Financial Engineering Version 1/9/2001

2. © K. Cuthbertson, D. Nitzsche2 Uses of Forward Rates Calculation of Forward Rates Calculation of ‘no arbitrage’ (equilibrium) forward rate Forward Rate Agreement, FRA Yield Curve - expectations hypothesis - slope of the yield curve - forecasting inflation Topics

3. © K. Cuthbertson, D. Nitzsche3 Uses of Forward Rates 1)Today, you can “lock in” an interest rate which will apply between two periods in the future (e.g. between end of year-1 and end of year-2, denoted f 12 ) ‘Analytically’ this would simply involve ringing up a bank TODAY and it would quote you a forward rate f 12 = 11%, say for a deposit of $100. You would then be obliged to give the bank $100 IN ONE YEARS TIME and the bank would pay you $110 one year later (ie. at the end of year- 2) In practice, things are a little more complicated than this but the same result is achieved by using a Forward Rate Agreement, FRA.

4. © K. Cuthbertson, D. Nitzsche4 Uses of Forward Rates 2)Pricing : Forward Agreements -strictly ‘forward-forward aggmts(FFA) Forward Rate Agreements, FRA’s (hedging- practice) 3) Forecasting future inflation 4) Also used in Pricing -Floating Rate Notes, FRN’s -Interest Rate Futures Contracts -floating rate receipts, in an interest rate swap

5. © K. Cuthbertson, D. Nitzsche5 Calculation of Forward Rates

6. © K. Cuthbertson, D. Nitzsche6 The bank calculates the forward rate by using two spot rates which it observes on its dealers screen., r 02 = interest rate on money lent today for 2-years (ie. 2-year spot rate) - often just denoted as ‘ r 2 ‘. Consider a fixed 2-year investment horizon. Choices 1) Invest $1 for 2-years at r 02 1) Receipts at t=2 (with certainty) are $1 ( 1 + r 02 ) 2 2) Invest $1 for 1-year at r 01 and today also enter into a forward agreement to invest between t=1 and t=2 at a quoted rate f 12 2) Receipts at t=2 (with certainty) are $1( 1 + r 01 ) (1 + f 12 ) These transactions are riskless hence the amounts received at t=2, must be equal. HOW DOES THE BANK KNOW WHAT FORWARD RATE TO QUOTE?

7. © K. Cuthbertson, D. Nitzsche7 What is the relationship between the (no arbitrage) forward rate and current spot rates ? Equating 1 and 2 $1( 1 + r 2 ) 2 = $1( 1 + r 1 ) (1 + f 12 ) Therefore ( 1 + f 12 ) = ( 1 + r 2 ) 2 / ( 1 + r 1 ) Or, approximately (Let r 1 = 9% p.a. and r 2 = 10% pa ) f 12 = 2. r 2 - r 1 = 2 (10) - 9 = 11% 1) The correct forward rate is derived from current spot rates (yield curve) 2) f 12 is the rate a bank should quote if it does not want to be ‘ripped off’ - see later 3) Also it can be shown (later) that f 12 is the market’s best forecast of what the “the one-year rate in one-years time” (denoted Er 1t+1 ) will be

8. © K. Cuthbertson, D. Nitzsche8 Algebra of General Calculation of Forward Rates Calculate other forward rates from today’s spot rates is pretty ‘intuitive’ since the superscripts and subscripts ‘add up’ to the same amount on each side of the equals sign ( 1 + r 03 ) 3 = ( 1 + r 02 ) 2. (1 + f 23 ) 1 ( 1 + r 03 ) 3 = ( 1 + r 01 ) 1. (1 + f 13 ) 2 In general (there is no need to memorise this!) f m,n = [ n / (n -m) ] r n - [ m / (n -m) ] r m e.g. f 1,3 = [ 3 / 2 ] r 3 - [ 1 / 2 ] r 1

9. © K. Cuthbertson, D. Nitzsche9 Spot Rates and Forward Rates 0123 r1r1 r2r2 r3r3 1 year forward rates 0123 f 12 f 23 Next years spot curve (forecast at t = 0) 0123 f 12 f 13

10. © K. Cuthbertson, D. Nitzsche10 Calculation of the No-arbitrage or (Equilibrium) Forward Rate

11. © K. Cuthbertson, D. Nitzsche11 Market behaviour behind the determination of the ‘no-arbitrage’ forward rate Problem: You will RECEIVE $100 in one years time ( t=1) from payment on defence contract and wish to place the money on deposit for a further year. You are worried that in 1-years time 1-year interest rates will have fallen Can you TODAY, “lock in” an interest rate on your future deposit ? YES : use a “forward” (FFA) agreement with the bank (- loosely speaking this is equivalent to an FRA)

12. © K. Cuthbertson, D. Nitzsche12 Fig 1: Actual Cash Flows -‘Real’ Forward Agreement Will RECEIVE $100 at t=1 from payment on defence contract. So: Agree today, to lend (pay out) $100, in 1-years time and receive the principle plus interest at end year-2. Bank’s Quoted forward rate is, f 12 = 10.5% p.a. (Note: No “own funds” are used at t=0) f 12 = 10.5 % 110.5 2 1 0 100 (pay to bank)

13. © K. Cuthbertson, D. Nitzsche13 Should you take the bank’s offer or, can you manufacture/engineer cash flows which exactly match the bank’s FRA, but at a lower interest cost ? The ‘home made’ FRA we call the “synthetic forward” The “synthetic” is a simple example of “financial engineering” It provides a way of obtaining the “fair” or “true” “(forward) rate for the actual forward contract offered by the bank. If you “engineer” your synthetic to have exactly the same pattern of cash flows as the actual forward contract then actual forward rate f 12 quoted by the bank must equal the “interest cost” of creating your synthetic contract - if not then you can ‘rip off’ the bank. Great! Calculation of the ‘no-arbitrage’ forward rate

14. © K. Cuthbertson, D. Nitzsche14 Creating a “synthetic” forward contract What can we use ‘to make’ our synthetic forward ? We borrow and lend using the 1-year and 2-year money market spot rates”. We will borrow and lend equal amounts at t=0 and hence use no ‘own funds’ - just as in the real FRA. Let r 1 = 9% and r 2 = 10%

15. © K. Cuthbertson, D. Nitzsche15 Figure 8.2 :Synthetic Forward Rate: sf 12. r 2 =10% r 1 = 9% sf 12 = ?? = 11% 1) Pay out $100 2) Borrow $91.74 at r 1 = 9% 3) Lend $91.74 at r 2 =10% 4) Receive $111 To reproduce the pattern of cash flows in the actual FRA, follow the steps 1-4 Create cash flows with “timing” equivalent to actual FRA

16. © K. Cuthbertson, D. Nitzsche16 Synthetic Forward 1)Borrow $91.743 at t=0, for 1-year at r 1 : Owe $100 at t=1 (paid for from proceeds of the defence contract) 2)Use this $91.743 at t=0 to 3) Invest for 2-years at r 2 : receive $111 at t=2 You have not used any of your “own funds” at t=0 The above cash flows are equivalent to Lending $100 at t=1 with payment $111 at t=2 Hence Synthetic forward rate sf 12 = 11 % (also note sf 12 = 2 r 2 - r 1 ) Don’t take the bank’s offer of f 12 = 10.5 % !!

17. © K. Cuthbertson, D. Nitzsche17 Synthetic Forward,Algebra:(for afficionados only) 1)At t=0 borrow 100/(1 + r 1 ) = $91.74 for 1-year (where r 1 = 0.09 ) This means you will have a cash outflow of £100 at t=1 (to match that in real FRA) 2) Then lend the $91.74 you have just borrowed, for 2 years This means you will receive in 2 years time: $91.74 (1+ r 2 ) 2 = £111 (at t=2)(where r 2 = 10%) 3)The synthetic futures rate between t=1 and t=2 must therefore be given by (1 + sf 12 ) = £111/100 = $91.74 (1+ r 2 ) 2 / $91.74 (1 + r 1 ) = (1+ r 2 ) 2 / (1 + r 1 ) Simplifying: sf 12 = 2 r 2 - r 1 (we have set (r x f) or (r x r) to zero)

18. © K. Cuthbertson, D. Nitzsche18 Can we “Rip-Off” the Bank ? When sf = 11% > f = 10.5%, can we ‘rip-off’ the bank? Can we also ‘dispense with’ the defence contract cash ? YES WE CAN !! 1)Today, go to Bank-A and agree a real FRA at f 12 = 10.5% which involves a) BORROW(ie. receive) $100 at t=1 and hence b) PAY OUT $110.5 at t=2 2) Create your own synthetic forward contract as in Fig. 8.2 above, which implies no net cash flows at t=0 but you have to a) PAY OUT $100 at t=1 (use “1a” to pay this) b)YOU WILL RECEIVE $111 at t=2 3). Net result is a RISKLESS PROFIT of 111 - 110.5 = $0.5 at t=2 and you have used non of your own money(capital) at t=0, 1 or 2 !

19. © K. Cuthbertson, D. Nitzsche19 ARBITRAGE: Equalises f and sf sf = 11% and f = 10.5% Everyone is buying FRA (ie borrowing) from Bank-A, so as the demand for its (forward) loans has increased, it raises its quoted rate f Everyone is also using the synthetic forward route that is Borrowing at r 1 and lending at r 2 (see fig 2 above) Hence: r 1 will rise, and r 2 will fall Fall in r 2 means less $’s at t=2 from the investment in the synthetic forward and hence a fall in sf ( = 2r 2 - r 1 ). This completes the argument of why it must be the case that at all times: ( 1 + r 2 ) 2 = ( 1 + r 1 ) (1 + f 12 )

20. © K. Cuthbertson, D. Nitzsche20 Forward Rate Agreement, FRA

21. © K. Cuthbertson, D. Nitzsche21 Forward Rate Agreement, FRA Analytically this is the same as a forward(-forward) agreement at a rate f = 11% agreed today ( t = 0) However in practice only “differences” are exchanged (and there are some other nuances, which need not concern us)

22. © K. Cuthbertson, D. Nitzsche22 Forward Rate Agreement, FRA Example: You will receive $100 in one years time and want to then place it on deposit for a further year. You fear a fall in interest rates over the coming year. You therefore take out an FRA at f = 11%. At the end of the year suppose interest rates have fallen to say 8%. The FRA pays out: f - 8% = 3% on $100 = $3 You only get 8% from the bank deposit but the 3% from the FRA means you have “locked in” 11%. (Actually you get the PV of $3, but this need not concern you)

23. © K. Cuthbertson, D. Nitzsche Figure 8.4 : Buy 3m x 6m FRA f 3,6 Recieve $1m plus interest at f 3,6 Notional Cash Flows 3m 6m 0 FRA negotiated Lend/Deposit of $1m 90 days

24. © K. Cuthbertson, D. Nitzsche24 Summary : Forward Rates Forward rates apply between two periods in the future but the rate is agreed today ‘Locks in’ an interest rate on a loan or deposit between two future dates It can be shown that f 12 is the market’s best forecast of what the “the one- year interest rate in one-years time” will be. This can then be used to give an inflation forecast. For example if f 12 = 11% and the real interest rate is 3% (= growth rate of the economy) then the forecast for annual inflation in one years time would be 8% (=11 - 3). Forward rates are used to “price” FRAs, ( also FRNs, interest rate futures and swaps)

25. © K. Cuthbertson, D. Nitzsche25 The Yield Curve and the Expectations Hypothesis

26. © K. Cuthbertson, D. Nitzsche26 THE YIELD CURVE Why are long rates of interest often higher than short rates of interest ? - can long rates be lower than short rates ? Yes ! = Expectations Hypothesis If we know the shape of the yield curve (ie. All the spot rates) then we can calculate forward rates for all maturities

27. © K. Cuthbertson, D. Nitzsche27 Figure 5 :YIELD CURVE We can show that if the yield curve is upward sloping then (one-period) short rates in the future are expected to rise. Yield 6 4 1 2 Time to maturity A A B B 3 7

28. © K. Cuthbertson, D. Nitzsche28 Expectations Hypothesis (EH): Term Structure ‘Risky’ Arbitrage but assuming risk neutrality $1.( 1+ r 2 ) 2 = $1. (1+r 1 ). [ 1 + Er 12 ] Approx. r 2 = ( 1 / 2 ). [ r 1 + Er 12 ] EH implies 1.Long-rate r 2 is weighted average of current (r 1 ) and expected future (one-period) short rates Er 12

29. © K. Cuthbertson, D. Nitzsche29 Upward Sloping Yield Curve Three Period Rate : r 3t r 03 = ( 1 /3 ). [ r 01 + Er 12 + Er 23 ] Suppose r 1 = 4 and future (one-period) rates are expected to rise, Er 12 = 8, Er 23 = 9 then r 01 = 4 r 02 = ( 1 /2 ). [ r 01 + Er 12 ] = 6 r 03 = ( 1 /3 ). [ r 01 + Er 12 + Er 23 ] = 7 Hence, yield curve will be upward sloping (4<6<7)

30. © K. Cuthbertson, D. Nitzsche30 Inflation Prediction from the Yield Curve Rising yield curve implies that short rates are expected to be higher in the future and this is probably because inflation is expected to rise in future years Observe the current yield curve r 2 = 10%, r 1 = 5%, then f 12 = 15.2% If real rate = 3%, then ( from Fisher effect) Expected inflation in 1-years time = 15.2 - 3 = 12.2% = Bank of England inflation forecast ?

31. © K. Cuthbertson, D. Nitzsche31 Liquidity Preference Lenders like to lend at short horizon, borrowers like to borrow for long horizon, so long rates contain positive “liquidity premia” Expectations Hypothesis liquidity premium L.Preference Maturity Yield

32. © K. Cuthbertson, D. Nitzsche32 Expectations Hypothesis and Forward Rates EH mathematically is: 1) r 02 = ( 1 /2 ). [ r 01 + Er 12 ] Definition/ Calculation of ‘no arbitrage’ forward rate: 2) r 02 = ( 1 /2 ). [ r 01 + f 12 ] Hence the EH implies that the current forward rate is the market’s best predictor of the future spot rate (not 100% accurate though!) - hence from 1 and 2, above f 12 = Er 12 So why not disband the MPC on whom we spend a lot of money so they can try and forecast future inflation, better than an average of all market participants, (who at least ‘put their money where there mouth is’ when ‘playing the yield curve’ - ie. the EH).

33. © K. Cuthbertson, D. Nitzsche33 SLIDES END HERE