02) (14 points: 4+4+4+2) Par = $1,000 ; coupon-796, annually paid ; maturity-3 y

Question

02) (14 points: 4+4+4+2) Par = $1,000 ; coupon-796, annually paid ; maturity-3 years; yield to maturity-69. a) What is the above bond's duration? If the bond's YTM increases by 1% shortly after t=0, what is the (dollar) change in the bond price as predicted by modified duration? Why is the estimated change different from the actual change? b) How many dollars is the estimated price change in part (c) if the convexity of the bond is given to be 9? c) d) Is higher convexity a good attribute in a bond? Why?

Explanation / Answer

Par Value 1000 Coupon Rate 7% Maturity (years) 3 Years YTM 6% (a) Duration of bond - Year Cash flows PV Fator PV of cash flows W x X (x) (w) 1 70 0.94339623 66.03773585 66.03774 2 70 0.88999644 62.2997508 124.5995 3 1070 0.83961928 898.3926328 2695.178 1026.730119 2885.815 Duration = Sum WX/Sum W = 2885.815/1026.730 2.811 (b) Modified duration = D/1+r D = Duration r = Periodic YTM MD = 2.811/1.06 2.652 MD shows If there is 1% change in YTM then price is changed by MD Therefore if YTM is decreased by 1% Then price of bond will increase by 2.652% Estimated New price of bond = 1026.730 + 1026.73 x 2.652% 1053.954791 Calculation of actual price at YTM 5% - Year Cash flows PV Fator PV of cash flows (x) (w) 1 70 0.95238095 66.66666667 2 70 0.90702948 63.49206349 3 1070 0.8638376 924.3062304 1054.464961 Difference b/w Actual and Estiamted Estimated 1053.954791 Actual 1054.464961 0.510169999 This Difference is due to convexity. The formula for D and MD does not consider Convexity, it is formed assuming bond equation as a straight line. ( c) Change in price = Change in price with effect of MD + change in price due to convexity Change in price = (MD x (change in Y) x100) + (Convexity x (Change in Y)^2 x 100) Change in Y = + 1% or + 0.01 MD = 2.652 Change in price = (2.652 x 0.01 x 100) + (9 x (0.01)^2 x 100)            = 2.742% decrease in price estimated price change in $ = 0.02742 x 1026.73 = 28.15294 (d) If two bonds offer the same duration and yield but one exhibits greater convexity, changes in interest rates will affect each bond in a different manner. A bond with greater convexity is less affected by interest rates than a bond with less convexity. In addition, bonds with greater convexity will have a higher price than bonds with lower convexity, regardless of what’s happening with interest rates Therefore yes a bond with higher convexity is a good attribute of bond Please provide feedback.. Thanks in advance.. :-)

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