A 12.75-year maturity zero-coupon bond selling at a yield to maturity of 8% (eff

Question

A 12.75-year maturity zero-coupon bond selling at a yield to maturity of 8% (effective annual yield) has convexity of 150.3 and modified duration of 11.81 years. A 30-year maturity 6% coupon bond making annual coupon payments also selling at a yield to maturity of 8% has nearly identical modified duration, 1 1.79 years, but considerably higher convexity of 23.2. , a. Suppose the yield to maturity on both bonds increases to 8.5%. i. ii. What will be the actual percentage capital loss on each bond? What percentage capital loss would be predicted by the duration-with-convexity rule?

Explanation / Answer

i) Current Price of zero coupon bond, P0 = FV / (1 + r)^n = 1,000 / (1 + 8%)^12.75 = $374.84

When yield = 8.5%, then Price, P1 = 1,000 / (1 + 8.5%)^12.75 = $353.40

Actual % Change = P1 / P0 - 1 = - 5.72%

Current Price of coupon bond can be calculated using PV function

N = 30, I/Y = 8%, PMT = 60, FV = 1000 => Compute PV = $774.84 = P0

When I/Y = 8.5% => PV = $731.33 = P1

Actual % Change = P1 / P0 - 1 = - 5.62%

ii) Using duration-convexity rule, % Change = - Duration x Chg + 0.5 x Convexity x (Chg)^2

% Change in zero coupon = - 11.81 x 0.5% + 0.5 x 150.3 x 0.5%^2 = -5.72%

% Change in coupon bond = -11.79 x 0.5% + 0.5 x 231.2 x 0.5%^2 = -5.61%

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