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23). 22) can be simpliﬁed by substituting Df for the part of the expression representing the discount factor. 24). 24) states that the market value of a risk-free bond on any date is determined by the discount function on that date. 57 Bond Pricing and Spot and Forward Rates Forward Rates An investor can combine positions in bonds of differing maturities to guarantee a rate of return that begins at a point in the future. The trade ticket is written at time t to cover the period T to T + 1 where t < T.
22). 22) where Pc = the price of a coupon bond C = the bond coupon tn = the coupon payment dates, with n ≤ N , and t = 0 at the time of valuation w = the coupon frequency (annual or semiannual for plain vanilla bonds; monthly for certain ﬂoating-rate notes and asset-backed securities), expressed in number of times per year T = the maturity date Note that “100” on the right-hand side captures the fact that prices are quoted per 100 of the bond’s principal, or nominal, value. 23). 22) can be simpliﬁed by substituting Df for the part of the expression representing the discount factor.
13). 13) where r = the yield to maturity of an annual-coupon-paying bond As stated above, Macaulay duration equals modiﬁed duration multiplied by (1+r). 13) expresses the approximate percentage change in price as modiﬁed duration multiplied by the change in yield. Modiﬁed duration is a measure of the approximate change in bond price for a 1 percent change in yield. 14). A negative is used in this equation because the price movement is inverse to the interest rate movement, so a rise in yields produces a fall in price, and vice versa.