Question

You have a 25-year maturity, 10.4% coupon paid semi-annually, 10.4% YTM bond with a duration of 10 years and a convexity of 113.9976. If the interest rate were to fall 129 bps:

a) Show the total change in the bond price, Δ B, as a result of the decline in yields.

b) Show the bond price change due to duration, Bd.

c) Show the bond price change due to convexity, Bc.

d) Verify the accuracy of your responses by showing: BN = Bo + Δ Bd + Δ Bc

Answer 1

(a) Coupon Rate = 10.4 % and YTM = 10.4 %

As YTM = Coupon Rate, Current Bond Price = Par Value = $ 1000

Duration = 10 years and Convexity = 113.9976 years

% Change in Interest Rate = - 129 bps

Total % Change in Bond Price = - Duration x % Change in Interest Rate + 1/2 x (Convexity) x (% Change in Interest Rate)^(2) = - 10 x - 0.0129 + 1/2 x (113.9976) x (-0.0129)^(2) = 0.138485 or 13.8485 %

New Bond Price = (1.138485) *1000 = $ 1138.485

Change in Price = 1138.845 - 1000 = 138.845

(b) % Change in Price due to Duration = - 10 x - 0.0129 = 0.129 or 12.9 %

Change in Price = 1000 x (1.129) - 1000 = $ 129

(c) % Change in Price due to Convexity = 1/2 x 113.9976 x (-0.0129)^(2) = 0.009485 or 0.9485 %

Change in Price = 1000 x (1.009485) - 1000 = $ 9.485

(d) Total Price Change = $ 138.845 = Change Due to Duration + Change due to Convexity = 129 + 9.845 = $138.845