1. Interest Rates Finance (Derivative Securities) 312 Tuesday, 8 August 2006 Readings: Chapter 4

2. Types of Rates  Treasury Rates Short-term government securities  LIBOR London Interbank Offer Rate Rate applicable to wholesale deposits between banks  Repo Rates Repurchase agreements

3. Measuring Rates  Compounding frequency is unit of measurement  Increased frequency leads to continuous compounding $100 grows to $ 100e RT when invested at a continuously compounded rate R for time T $100 received at time T discounts to $ 100e –RT at time zero when continuously compounded discount rate is R

4. Conversion Formula R c : continuously compounded rate R m : same rate with compounding m times per year

5. Bond Pricing  Relies on interest rates on zero-coupon bonds (zero rates) Interest is realised only at maturity date To calculate the price of a two year coupon bond paying 6% semi-annually:

6. Bond Yield  Single interest rate that discounts remaining CFs to equal the price  Using the previous example, solve the following equation for y : y = 0.0676 or 6.76%

7. Par Yield  Coupon rate that equates a bond’s price to its face value  Using previous example:

8. Par Yield  If: m = no. of coupon payments per year P = present value of $1 received at maturity A = present value of an annuity of $1 on each coupon date then:

9. Calculating Zero Rates BondTime toAnnualBond PrincipalMaturityCouponPrice (dollars)(years)(dollars) 1000.25097.5 1000.50094.9 1001.00090.0 1001.50896.0 1002.0012101.6

10. Bootstrap Method  2.5 can be earned on 97.5 after three months  3-month rate is 4 times 2.5/97.5 or 10.256% with quarterly compounding, and 10.127% with continuous compounding  Similarly the 6-month and 1-year rates are 10.469% and 10.536% with continuous compounding

11. Bootstrap Method  To calculate 1.5-year rate, solve: to get R = 0.10681 or 10.681%  Similarly the two-year rate is 10.808%

12. Zero Curve Zero Rate (%) Maturity (yrs) 10.127 10.46910.536 10.681 10.808

13. Forward Rates  Future zero rates implied by the current term structure Zero Rate forForward Rate an n -year Investmentfor n th Year Year ( n )(% per annum) 110.0 210.511.0 310.811.4 411.011.6 511.111.5

14. Calculating Forward Rates  Suppose that the zero rates for time periods T 1 and T 2 are R 1 and R 2 with both rates continuously compounded  The forward rate for the period between times T 1 and T 2 is:

15. Slope of Yield Curve  For an upward sloping yield curve: Fwd Rate > Zero Rate > Par Yield  For a downward sloping yield curve: Par Yield > Zero Rate > Fwd Rate

16. Forward Rate Agreements  Agreement that a fixed rate will apply to a certain principal during a specified future time period  Equivalent to agreement where interest at a predetermined rate, R K, is exchanged for interest at the market rate  Can be valued by assuming that the forward interest rate will be realised

17. Forward Rate Agreements  Let: R K = interest rate agreed to in FRA R F = forward LIBOR rate for period T 1 to T 2 calculated today R M = actual LIBOR rate for period T 1 to T 2 observed at T 1 L = principal underlying the contract

18. Forward Rate Agreements  If X lends to Y under the FRA, then: Cashflow to X at T 2 = L( R K – R M )( T 2 – T 1 ) Cashflow to Y at T 2 = L( R M – R K )( T 2 – T 1 )  Since FRAs are settled at T 1, payoffs must be discounted at [ 1 + R M ( T 2 – T 1 )]  Value of FRA is the payoff, based on forward rates, discounted at R 2 T 2 Value X = L ( R K – R F )( T 2 – T 1 )e –R 2 T 2 Value Y = L ( R F – R K )( T 2 – T 1 )e –R 2 T 2

19. Theories of Term Structure  Expectations Theory: forward rates equal expected future zero rates  Market Segmentation: short, medium and long rates determined independently of each other  Liquidity Preference Theory: forward rates higher than expected future zero rates