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r than expected future zero rates Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull 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, 5th edition 169。 20xx by John C. Hull 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 Tyear rate R T RT? ??Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Calculation of Forward Rates Table , page 98 Zero Rate for Forward Rate an n year Investment for n th Year Year ( n ) (% per annum) (% per annum) 1 2 3 4 5 Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Zero Curve Calculated from the Data (Figure , page 98) 91011120 0 .5 1 1 .5 2 2 .5Zero Rate (%) Maturity (yrs) Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull The Bootstrapping the Zero Curve ? An amount can be earned on during 3 months. ? The 3month rate is 4 times % with quarterly pounding ? This is % with continuous pounding ? Similarly the 6 month and 1 year rates are % and % with continuous pounding Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull 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 and A is the present value of an annuity of $1 on each coupon date Amdc )1 0 01 0 0( ??Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull 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 b