Question

The yield on all bonds is currently 2% per annum (continuously compounded).Find the duration and convexity of a portfolio consisting of the following bonds:

i. One zero-coupon bond with a face value of £5,000 and a maturity of one year
ii. One zero-coupon bond with a face value of £10,000 and a maturity of 10 years

Answer 1

Given that,

Yield on bonds y = 2% compounded continuously,

Portfolio consist of

one zero-coupon bond with a face value of £5,000 and a maturity of one year

& one zero-coupon bond with a face value of £10,000 and a maturity of 10 years

So, For one zero-coupon bond with a face value of £5,000 and a maturity of one year

Price = FV*e^(-y*t) = 5000*e^(-0.02) = $4900.99

Duration of a zero coupon bond is t, so its duration = 1

Convexity = t^2, is convexity = 1

So, For one zero-coupon bond with a face value of £10,000 and a maturity of 10 year

Price = FV*e^(-y*t) = 10000*e^(-0.02*10) = $8187.31

Duration of a zero coupon bond is t, so its duration = 10

Convexity = t^2, is convexity = 10^2 = 100

Weight of bond 1 W1 = price/(sum of price of both bonds) = 4900.99/(4900.99+8187.31) = 0.3745

Weight of bond 2, W2 = 1-W1 = 1-0.3745 = 0.6255

So, Portfolio duration = (W1*D1 + W2*D2) = 0.3745*1 + 0.6255*10 = 6.63 years

Convexity of portfolio (W1*C1 + W2*C2) = 0.3745*1 + 0.6255*100 = 62.93