Question

Suppose a 6-year zero-coupon bond with a face value of $100 trades at $76.235. If the yield increases by 125 basis points, what is the magnitude of the error between the exact new bond price and the first-order approximation of the new bond price using the Modified Duration?

Answer 1

Given about a 6-year zero-coupon bond,

Face value = $100

price = $76.235

So, Yield on the bond = (FV/price)^(1/t) - 1 = (100/76.235)^(1/6) - 1 = 4.63%

When Yield increases by 125 basis points,

new yield = old yield + 1.25% = 4.63 + 1.25 = 5.88%

So, new exact price of the bond = FV/(1+Yield)^t = 100/1.0588^6 = $70.991

Bond's Macaulay duration = years to maturity = 6 years

So, modified duration D = Macaulay duration/(1+Old yield) = 6/1.0463 = 5.735 year

Using first-order approximation of the new bond price using the Modified Duration

Change in price dP = -D*P*dy = -5.735*76.235*0.0125 = -$5.465

So, new approx price = 76.235 - 5.465 = $70.770

So, the magnitude of the error between the exact new bond price and the first-order approximation of the new bond price using the Modified Duration = 70.991 - 70.770 = $0.221