# Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.1 Interest Rates Chapter 4.

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Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.1 Interest Rates Chapter 4

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.2 Types of Rates For any given currency, many different types of interest rates are regularly quoted (mortgage rates, deposit rates, prime rates...) Treasury rates LIBOR rates Repo rates

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.3 Types of Rates Treasury rates are treated as risk-free rates  Derivatives traders do not usually use Treasury rates as risk- free rates. Instead they use LIBOR rates. LIBOR rates: 1 mo, 3 mo, 6 mo, 12 mo. in all major currencies.  Bank must have AA credit rating in order to operate in LIBOR market  LIBOR is not totally risk-free  In this book LIBOR will be interpreted as risk-free  LIBOR and LIBID trade in what is known Eurocurrency market Repo rates, also with very little credit risk.  Most common is the overnight REPO  Term (long term) REPOs also exist.

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.4 Measuring Interest Rates The compounding frequency used for an interest rate is the unit of measurement The difference between quarterly and annual compounding is analogous to the difference between miles and kilometers

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.5 Effect of the compounding frequency on the value of \$100 at the end of 1 year when the interest rate is 10% p.a. Compounding frequency Annually (m = 1) Semiannually (m = 2) Quarterly (m = 4) Monthly (m = 12) Weekly (m = 52) Daily (m = 365) Value of \$100 at the end of year 110.00 110.25 110.38 110.47 110.51 110.52

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.6 Generalization: Terminal value of the investment is: A – amount, T – no. of years, R – interest rate, m – compounded frequency per year

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.7 Continuous Compounding (Page 79) In the limit as we compound more and more frequently we obtain continuously compounded interest rates \$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 the continuously compounded discount rate is R

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.8 Conversion Formulas (Page 79) Define R c : continuously compounded rate R m : same rate with compounding m times per year

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.9 Zero Rates or Spot Rates A zero rate (or spot rate), for maturity T is the rate of interest earned on an investment that provides a payoff only at time T

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.10 Example (Table 4.2, page 81)

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.11 Bond Pricing To calculate the cash price of a bond we discount each cash flow at the appropriate zero rate In our example, the theoretical price of a two- year bond providing a 6% coupon semiannually is

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.12 Bond Yield The bond yield is the discount rate that makes the present value of the cash flows on the bond equal to the market price of the bond Suppose that the market price of the bond in our example equals its theoretical price of 98.39 The bond yield (continuously compounded) is given by solving to get y =0.0676 or 6.76%.

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.13 Par Yield The par yield for a certain maturity is the coupon rate that causes the bond price to equal its face value. In our example we solve

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.14 Par Yield continued In general if m is the number of coupon payments per year, d is the present value of \$1 received at maturity of the bond and A is the present value of an annuity of \$1 on each coupon date

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.15 Sample Data (Table 4.3, page 82) BondTime toAnnualBond Cash PrincipalMaturityCouponPrice (dollars)(years) (dollars) 1000.25097.5 1000.50094.9 1001.00090.0 1001.50896.0 1002.0012101.6

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.16 The Bootstrap Method An amount 2.5 can be earned on 97.5 during 3 months. The 3-month rate is 4 times 2.5/97.5 or 10.256% with quarterly compounding This is 10.127% with continuous compounding Similarly the 6 month and 1 year rates are 10.469% and 10.536% with continuous compounding

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.17 The Bootstrap Method continued To calculate the 1.5 year rate we solve to get R = 0.10681 or 10.681% Similarly the two-year rate is 10.808%

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.18 Zero Curve Calculated from the Data (Figure 4.1, page 84) Zero Rate (%) Maturity (yrs) 10.127 10.46910.536 10.681 10.808

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.19 Forward Rates The forward rate is the future zero rate implied by today’s term structure of interest rates

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.20 Calculation of Forward Rates Table 4.5, page 85 Zero Rate forForward Rate an n -year Investmentfor n th Year Year ( n )(% per annum) 13.0 24.05.0 34.65.8 45.06.2 55.36.5

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.21 Formula for 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

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.22 Instantaneous Forward Rate The instantaneous forward rate for a maturity T is the forward rate that applies for a very short time period starting at T. It is where R is the T -year rate

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.23 Upward vs Downward Sloping 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

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.24 Forward Rate Agreement A forward rate agreement (FRA) is an agreement that a certain rate will apply to a certain principal during a certain future time period

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.25 Forward Rate Agreement continued An FRA is equivalent to an agreement where interest at a predetermined rate, R K is exchanged for interest at the market rate An FRA can be valued by assuming that the forward interest rate is certain to be realized

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.26 Valuation Formulas (equations 4.9 and 4.10 page 88) Value of FRA where a fixed rate R K will be received on a principal L between times T 1 and T 2 is Value of FRA where a fixed rate is paid is R F is the forward rate for the period and R 2 is the zero rate for maturity T 2 What compounding frequencies are used in these formulas for R K, R M, and R 2?

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.27 Duration of a bond that provides cash flow c i at time t i is where B is its price and y is its yield (continuously compounded) This leads to Duration (page 89)

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.28 Duration Continued When the yield y is expressed with compounding m times per year The expression is referred to as the “modified duration”

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.29 Convexity The convexity of a bond is defined as

Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 4.30 Theories of the Term Structure Page 93 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