# Consider a 3-year bond from which you receive three coupon

Consider a 3-year bond from which you receive three coupon payments (C), one at the end of every year. The face value of the bond F is received at the end of year 3.

1. Write the formula for the price P of this bond, where C is the coupon payment, F is the face value and YTM is the Yield to Maturity.
2. Obtain the first derivative of the price of this bond with respect to the YTM (dP/dYTM). Write this derivative using the Macauley Duration (MD). The formula for the MD when there are n periods is:

MD = MD = 1.(C / P) 1+ YTM + 2.(C / P) 1+ YTM + …. + n.(C + F) / P 1+ YTM 2 n

(c)      Find the second derivative of P w.r.t. the YTM (d2P / dYTM2). That is, using the Chain Rule (showing each step) differentiate the first derivative to obtain the second derivative. That is, show how the following is obtained:

= d P d(YTM) 2 2 = (1.2).C (1+ YTM) + (2.3).C 1+ YTM +….+ (n)(n+1).(C + F) 1+ YTM 3 4 n+2

Then use your answer to fully define: Convexity =  = 1 2 d P d(YTM) 2 2 1 P

(d) Next use all your answers to write out the complete formula for the quadratic approximation for the change in the    price of a bond when YTM changes. That is, write out the extended version of the following equation:

dP = (MMD).P.d(YTM) + Convexity.P.[d(YTM)]2

Question 5 – using excel

(a) Consider the 3-year bond in QUESTION 4, if you were told that F = 1,250, C = 125 and the YTM = 0.04; what will be the price (P), Macauley Duration (MD) and Convexity of this bond?

(b) Using the information in part (a) find the actual change in P when the YTM changes from 0.04 to 0.03 AND the quadratic approximation to this change using the formula below. Briefly comment on the quality of this approximation: P = MMD.P.[YTM] + Convexity.P.[YTM]2 [MMD is the modified Macauley Duration]

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