The key risk facing the holder of a portfolio of bonds is the risk associated with a change in spot rates. As sport rates change, the value of each of the bonds in his portfolio will change, perhaps to his disadvantage. This exercise introduces a couple of concepts that will allow the holder of a bond portfolio to assess and measure the risk associated with interest rate changes. Throughout we make the simplifying assumption that sport rate are identical for all maturities, and denote the sport rate with r. The fat that the sport rate is the same for all maturities implies that all bonds have yields at rate r as well. Duration and elasticity Consider a bond with T periods to maturity. Its cash flow to the holder at date t is denoted C_1. As we have shown, its price can be written: P = sigma^T _t-1 C_1/(1 + r)^r In economics, the concept of an elasticity, usually denoted with an sigma, measures the percentage by which a variable y changes when another variable, x, is changed by a single percentage point. The mathematical formula for the elasticity of y with respect to x is: sigma = dy x/dx y Using simple differential calculus, show that the elasticity of our bond's price with respect to the gross spot rate, (1 + r), can be written as follows: Describe the implications of the preceding derivation for the linkage between the interest rate sensitivity of the bond and it's time to maturity. Derive the value of the elasticity derived above for a zero-coupon bond with T periods to maturity. Interpret this expression. Macaulay Duration and Modified Duration The Macaulay Duration, D, of a bond is simply the negative of the elasticity derived in question (1). Thus, for a bond with coupon rate c, face value X and T years to maturity it can be written as