Question

A newly issued bond has a maturity of 10 years and pays a 7% coupon rate (with coupon payments coming once annually). The bond sells at par value. a. What are the convexity and the duration of the bond? Use the formula for convexity in footnote 7. (Round your answers to 3 decimal places.) Convexity Duration years b. Find the actual price of the bond assuming that its yield to maturity immediately increases from 7% to 8% (with maturity still 10 years) Assume a par value of 100. (Round your answer to 2 decimal places.) Actual price of the bond

Answer 1

a). Duration = 7.515 years; Convexity = 64.933

Formula Time until payment (n) Cash flow (CF) CF/(1+YTM)^n PV/BP wen PV of CF Weight (w) Weighted Time 6.54 0.07 0.07 6.11 0.06 0.12 5.71 0.06 0.17 5.34 0.05 0.21 4.99 0.05 0.25 4.66 0.05 0.28 4.36 0.04 0.31 4.07 0.04 0.33 3.81 0.04 0.34 54.39 0.54 5.44 100.00 Duration in years) 7.52 Bond price (BP)

Convexity calculation:

Formula Time until payment (n) Cash flow (CF) CF/(1+YTM)^n PV of CFL 6.54 6.11 5.71 5.34 4.99 4.66 4.36 4.07 3.81 54.39 PV(n^2 + n) 13.08 36.68 68.57 106.81 149.73| 195.90 244.12 293.33 342.68 5983.27 7434.17 Bond price (BP) 100.00

Convexity = 7,434.17/(Price*(1+y)^2) = 7,434.17/(100*(1+7%)^2) = 64.933

b). Actual price with YTM 8%: FV = 100; PMT = 70; N = 10; rate = 8%, solve for PV.

Price = 93.29

c). Modified duration (D*) = duration/(1+y) = 7.515/(1+0.07) = 7.02

Change in Price = - Price*(d*)*change in yield = -100*7.02*0.01 = -7.02

New price = 100 -7.02 = 92.98

%age price change = (new price/old price) -1 = (92.98/100) -1 = -7.02%

%age error = (predicted price-actual price)/actual price = (92.98/93.29)-1 = -0.34%

d). Change in price = Price*[(-D**change in yield) + 0.5*(Convexity*change in yield^2)]

= 100*[(-7.02*0.01) + 0.5*(64.933*0.01^2)] = -6.70

New price = 100 - 6.70 = 93.30

%age price change = (93.30/100) -1 = -6.70%

%age error = (93.30/93.29) -1 = 0.01%