Question

1.Duration and Convexity

Your research department reports continuously compounded interest rates as

Maturity (Years) 0.5, 1.0, 1.5, 2.0

Interest Rate (%) 1.00, 1.50, 2.00, 2.00

(a) Use these rates to compute the prices P_z(0,1) and P_z(0,2) of one- and two-year zero coupon bonds, and the price P_c(0,2) of a two-year, 3% coupon bond. Coupons are paid semi-annually, and the face value of all bonds is 100.

(b) Obtain the coupon bond's duration and convexity.

(c) Suppose that the monthly changes in the interest rates have a mean of zero and a standard deviation of 0.5%. Obtain the monthly 95% Value at Risk and Expected Shortfall on the coupon bond.

(d) Construct a hedge portfolio of 1 coupon bond and k one-year zero coupon bonds that has zero duration. What is the value of k? What is the convexity of the hedge portfolio?

(e) Construct a hedge portfolio of 1 coupon bond, and k₁ one-year zero coupon bonds and k2 two-year zero coupon bonds, that has zero duration and convexity. What are the values of k₁ and k₂?

(f) Suppose that the yield curve shifts upward with dr = 1%. Recalculate P_c(0,2), P_z(0,1) and P_z(0,2) and use this to calculate the change in the values of the hedge portfolios constructed in (d) and (e). Comment on the result.

Answer 1

(a)

Zero Coupon Bond Price P_z(0,n) = Maturity Value / (1+r)ⁿ

r is the rate of interest required (or Market Rate)

Hence P_z(0,1) = 100/ (1.01) = $99.0099 or $99.01

and P_z(0,2) = 100/ (1.02)² = $96.1169 or $99.12

Price of Coupon Bond = Present Value of all Future Payments

Hence P_c(0,2) with 3% coupon Semi Annually at 1% discounting (r/2 as it is semi annual) =

===> = $101.95098 or $101.95

(b)

Duration of a bond is Weighted average times to each payment of the bond.

Periods	CashFlow	Present Value Factor	Weighted CashFlow	Present Value
1	1.5	0.9901	1.5	1.49
2	1.5	0.9803	3	2.94
3	1.5	0.9706	4.5	4.37
4	101.5	0.9610	406	390.16
Total				398.95

Macaulay Duration = 398.95 / 100 = 3.99 Periods or Half Years = 1.96 Years

Convexity =

P= Price

y = Yield to Maturity

t = period

T = time of Maturity

Periods	CashFlow	Present Value Factor	Weighted CashFlow	1+t	Present Value
1	1.5	0.9901	1.5	2	2.97
2	1.5	0.9803	3	3	8.82
3	1.5	0.9706	4.5	4	17.47
4	101.5	0.9610	406	5	1950.79
Total					1980.05

Convexity = 1980.05/ (Price * (1+y)²) = 1980.05 / (101.951 * 1.0201) = 1980.05 / 104 = 19.04