1. © K.Cuthbertson, D. Nitzsche FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001) K. Cuthbertson and D. Nitzsche LECTURE INTEREST RATE DERIVATIVES LECTURE Dynamic Hedging and Portfolio Insurance

2. © K.Cuthbertson, D. Nitzsche Caplet, Cap, Floorlet, Floor,Collar Swaption Forward Swap Mortgage Backed Securities Topics

3. © K.Cuthbertson, D. Nitzsche LECTURE Dynamic Hedging and Portfolio Insurance Caplet, Caps Floorlets, Floors and Collars

4. © K.Cuthbertson, D. Nitzsche Interest rate option gives holder the right but not the obligation to receive one interest rate (eg. floating\LIBOR) and pay another (eg. the fixed strike rate K%). Caplet (payoff at maturity) (Excel T15.1): [15.1]Q { max (0,LIBOR T - K c ) days/360 } Floorlet (payoff at maturity)(Excel T15.2) : [15.2]Q max (0,K FL - LIBOR T ) days/360 } LECTURE Dynamic Hedging and Portfolio Insurance Caplet and Floorlet

5. © K.Cuthbertson, D. Nitzsche Fig 15.1: Payoff Caplet on 90 - day LIBOR 0T=30t=120 days Expiry \ Valuation of option, (LIBOR T - K c ) Cash Payout Strike rate K c fixed in the contract 90 days Caplet fixes effective max cost of loan at K c

6. © K.Cuthbertson, D. Nitzsche Fig15.2:Planned Borrowing+ Caplet (Call) Loan only Loan plus long call INPUT IS FROM EXCEL FILE T15.1

7. © K.Cuthbertson, D. Nitzsche Figure 15.3 : Loan+ Interest Rate Floorlet (Put) Loan plus interest rate put Return on loan only Note : Payoff profile is like a protective put or long call. INPUT IS FROM EXCEL FILE T15.2

8. © K.Cuthbertson, D. Nitzsche See Excel files T15.3 and 15.4 Payoffs to (Loan+Cap) and (Deposit+Floors)

9. © K.Cuthbertson, D. Nitzsche Comprises a long cap and short floor. It establishes both a floor and a ceiling on a corporate or bank’s (floating rate) borrowing costs. Effective Borrowing Cost with Collar (at T = t – 90) = [LIBOR t-90 + max[{0,LIBOR t-90 - K cap } -max {0, K FL –LIBOR (t-90) }]Q (90/360) = K cap Q(90/360)if LIBOR t-90 > K cap = K FL Q(90/360)if LIBOR t-90 < K FL = LIBOR t-90 (90/360)if K FL < LIBOR t-90 < K cap Collar involves borrowing cost at each payment date of either K cap = 10% or K FL = 8% or LIBOR if the latter is between K FL = 8% and K c = 10%. LECTURE Dynamic Hedging and Portfolio Insurance Collar (Excel T15.5)

10. © K.Cuthbertson, D. Nitzsche Swaption Forward Swap and MBS

11. © K.Cuthbertson, D. Nitzsche OTC option to enter into a swap either as a fixed rate payer and floating rate receiver (ie. payer swaption) or vice versa US corporate may need to borrow $10m over 3 years at a floating rate, beginning in 2 years time. Wishes to swap the floating rate payments for fixed rate Corporate therefore needs a $10m swap, to pay fixed and receive floating beginning in 2 years time and an agreement that swap will last for further 3 years Swaption

12. © K.Cuthbertson, D. Nitzsche Suppose the corporate thinks that interest rates will rise over the next 2 years and hence the cost of the fixed rate payments in the swap will be higher than at present. The corporate can hedge by purchasing a 2 year European payer swaption, on a 3 year “pay fixed-receive floating” swap, at say K = 10%. Payoff is the annuity value of Q max{cp T – K, 0} Value of Swaption at T [15.15] V swpo (T=2) = $10m [cp T – K] [(1+r 23 ) -1 + (1+r 24 ) -2 + (1+r 25 ) -3 ] Swaption

13. © K.Cuthbertson, D. Nitzsche Figure 15.4 : Forward Swap 012345 f 23 f 24 f 25 Swap’s Life Enter into forward swap Expiration of forward swap A long forward swap is “pay fixed-receive floating” swap that will start in the future but at a swap rate agreed today. It ‘locks in’ a swap rate, agreed today or, can be used to speculate on future swap rates(see below)

14. © K.Cuthbertson, D. Nitzsche Example Long a 2-year forward contract on a 3-year swap, on a notional principal of Q=$10m. How do we price this swap at time t=0 (see figure 15.4) ? Pricing a Forward Swap

15. © K.Cuthbertson, D. Nitzsche The forward swap rate at t=0 is the fixed coupon rate cp f that makes the swap have zero expected value at T=2. [15.16]Q = C e -f23(1) + C e -f24(2) + C e -f25(3) + Q e -f25(3) f ij =forward rates ( known at t=0) Fixed coupon rate cp f = C/Q, hence 15.17]At t=0,cp f (2-5) = [ 1 - ] / AN 2-5 AN 2-5 = ~ annuity value of $1 using the forward rates at t=0. ~ cp f is the forward swap rate agreed at t=0. Pricing a Forward Swap

16. © K.Cuthbertson, D. Nitzsche Value at expiration (T=2) to the fixed rate payer After 2 years, current swap rate is cp 2 (2-5) Value is 3-yr annuity value of (cp 2 -cp f ) per $1 principal. [15.18] V fs (at T=2) = (cp 2 – cp f ) [e -r23(1) + e -r24(2) + e -r25(3) ] Cash value at expiry = $(Q.V fs ), paid to “the long”. Note: the r 2j are the actual spot rates (ex-post) known at t=2 which is now ‘the present’, that is, two years after inception of the forward swap. Speculation: If at t=0 you believe cp 2 will exceed cp f then go long a forward swap Value of Forward Swap at T

17. © K.Cuthbertson, D. Nitzsche Mortgages bundled up into portfolio and sold to investors in the form of mortgaged backed securities (MBS). Interest only (IO) strip entitles the investor to receive only the interest payment from the portfolio of mortgages. Principal only (PO) strip, only receives payments of principal [15.22]PV PT = PV IO + PV PO [Table 15.6 here - Excel] LECTURE Dynamic Hedging and Portfolio Insurance MBS: Mortgage Pass-Throughs and Strips

18. © K.Cuthbertson, D. Nitzsche LECTURE Dynamic Hedging and Portfolio Insurance End of Slides