13 A 13.15-year maturity zero-coupon bond selling at a yield to maturity of 8% (

Question

13 A 13.15-year maturity zero-coupon bond selling at a yield to maturity of 8% (effective annual yield) has convexity of 160.1 and modified duration of 11.91 years. A 40-year maturity 6% coupon bond making annual coupon payments also selling at a yield to maturity of 8% has nearly identical modified duration -11.75 years-but considerably higher convexity of 280.0. a. Suppose the yield to maturity on both bonds increases to 9%. What will be the actual percentage capital loss on each bond? What percentage capital loss would be predicted by the duration-with- convexity rule? (Do not round intermediate calculations. Round your answers to 2 decimal places.) b. Suppose the yield to maturity on both bonds decreases to 7%. What will be the actual percentage capital gain on each bond? What percentage capital gain would be predicted by the duration-with convexity rule? (Do not round intermediate calculations. Round your answers to 2 decimal places.)

Explanation / Answer

To determine the actual and predicted loss, we need to calculate the bond price for both types of bonds at different YTMs. The price can be determined with the use of PV function/formula of EXCEL/Financial Calculator. The formula/function of PV is PV(Rate,Nper,PMT,FV) where Rate = YTM, Nper = Period, PMT = Interest Payment and FV = Face Value of Bonds

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Price of Zero-Coupon Bond at 8% YTM

Here, Rate = 8%, Nper = 13.15 Years, PMT = 0 and FV = $1,000

Using these values in the above function/formula for PV, we get,

Bond Price = PV(8%,13.15,0,1000) = $363.48

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Price of Zero-Coupon Bond at 9% YTM

Here, Rate = 9%, Nper = 13.15 Years, PMT = 0 and FV = $1,000

Using these values in the above function/formula for PV, we get,

Bond Price = PV(9%,13.15,0,1000) = $321.99

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Price of 6% Coupon Bond at 8% YTM

Here, Rate = 8%, Nper = 40 Years, PMT = 6%*1000 = $60 and FV = $1,000

Using these values in the above function/formula for PV, we get,

Bond Price = PV(8%,40,60,1000) = $761.51

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Price of 6% Coupon Bond at 9% YTM

Here, Rate = 9%, Nper = 40 Years, PMT = 6%*1000 = $60 and FV = $1,000

Using these values in the above function/formula for PV, we get,

Bond Price = PV(9%,40,60,1000) = $677.28

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Part A)

The formulas for calculating actual loss/gain and predicted loss/gain are given below:

Actual Loss/Gain = (Yield to Maturity at 9% - Yield to Maturity at 8%)/Yield to Maturity at 8%*100

Predicted Loss/Gain (Duration-with-Convexity Rule) = [(-Modified Duration)*(Change in Yield)] + [.5*Convexity*(Change in Yield)^2]

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Using the values calculated above and information provided in the question, we get,

Actual Loss (Zero Coupon Bond) = (321.99 - 363.48)/363.48*100 = -11.41%

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Predicted Loss (Zero Coupon Bond) = [-11.91*(1%)] + [.50*160.1*(1%)^2] = -11.11%

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Actual Loss (6% Coupon Bond) = (677.28 - 761.51)/761.51*100 = -11.06%

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Predicted Loss (6% Coupon Bond) = [(-11.75)*(1%)] + [.5*280*(1%)^2] = -10.35%

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Part B)

Price of Zero-Coupon Bond at 7% YTM

Here, Rate = 7%, Nper = 13.15 Years, PMT = 0 and FV = $1,000

Using these values in the above function/formula for PV, we get,

Bond Price = PV(7%,13.15,0,1000) = $410.77

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Price of 6% Coupon Bond at 7% YTM

Here, Rate = 7%, Nper = 40 Years, PMT = 6%*1000 = $60 and FV = $1,000

Using these values in the above function/formula for PV, we get,

Bond Price = PV(7%,40,60,1000) = $866.68

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Using the values calculated above and information provided in the question, we get,

Actual Gain (Zero Coupon Bond) = (410.77 - 363.48)/363.48*100 = 13.01%

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Predicted Gain (Zero Coupon Bond) = [-11.91*(-1%)] + [.50*160.1*(-1%)^2] = 12.71%

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Actual Gain (6% Coupon Bond) = (866.68 - 761.51)/761.51*100 = 13.81%

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Predicted Gain (6% Coupon Bond) = [(-11.75)*(-1%)] + [.5*280*(-1%)^2] = 13.15%

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