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Extra info for Bloomberg - Fixed Income Securities And Derivatives Handbook Analysis And Valuation (Choudry, Cfa Level 2)
The strips corresponding to the coupon payments have face values that equal percentages of the nominal value of the bond itself, with successively longer maturity dates; the strip corresponding to the ﬁnal redemption payment has the face value and maturity date of the bond. A bond issued at time i and maturing at time T makes w payments (C1 … Cw) on w payment dates (t1, … tw-1, T ). K. 74 Bond Pricing and Spot and Forward Rates 51 function of time: C (t ), i < t ≤ T . Investors purchasing a bond at time t that matures at time T pay P(t, T ) and receive the coupon payments as long as they hold the bond.
How sensitive a bond’s price is to changes in its yield. This measure is obtained by applying a mathematical property known as a Taylor expansion to the basic equation. The relationship between price volatility and duration can be made clearer if the bond price equation, viewed as a function of r, is expanded as a Taylor series (see Butler, pp. 112–114 for an accessible explanation of Taylor expansions). 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).
A second-order measure of interest rate risk uses second-order derivatives. It measures the curvature of the price-yield graph and the degree to which this diverges from the straight-line estimation. Convex- 42 Introduction to Bonds ity can thus be regarded as an indication of the error made when using Macaulay and modiﬁed duration. A bond’s convexity is positively correlated to the dispersion of its cash ﬂows: all else being equal, a bond whose cash ﬂows are more spread out in time—that is, more dispersed—than another’s will have a higher convexity.