Question

Suppose that a pension fund has to make a payment of $1 million in 2 years time. It can hold a one year pure discount bond that will pay $1000 in 1 year; or a 3 year bond that will pay a coupon of $100 each year, will be redeemed for $1,000 at the end of the third year. The yield on each bond is 15%. (a) What is the price of each bond? (b) What is the duration of each bond. (c) How much of each bond should the fund hold to immunize itself against interest rate risk? Explain.

Answer 1

(a) Yield, y = 15%

Price of one year pure discount bond that will pay $1000 in 1 year, P₁ = $ 1,000 / (1 + y) = 1000 / (1 + 15%) = $  869.57

Price of a n = 3 year bond that will pay a coupon of C = $100 each year, will be redeemed for FV, P₃ = $1,000 at the end of the third year = C / y x [1 - (1 + y)^-n] + FV x (1 + y)^-n = 100 / 15% x [1 - (1 + 15%)^-3] + 1,000 x (1 + 15%)^-3 = $  885.84

(b)

Duration of 1 year discount bond, D₁ = 1 year

Duration of three year coupon bond has been calculated below:

Part (c)

Let's assume it contains V₁ and V₃ amount of the two bonds respectively.

PV of liability = $1 million in 2 years time = 1,000,000 / (1 + y)² = 1,000,000 / (1 + 15%)² =  756,144

Duration of liability = 2 years

Hence, V₁ + V₃ = PV of liability = 756,144

Or, V₁ + V₃ =  756,144 -------------------Equation (1)

Duration of the portfolio of bonds = Duration of the liability

Hence, D₁ x V₁ / (V₁ + V₃) + D₃ x V₃ / (V₁ + V₃) = 2

Or, D₁ x V₁ + D₃ x V₃ = 2 x (V₁ + V₃)

Or, 1 x V₁ +  2.7183 x V₃ = 2 x 756,144 = 1,512,287

Or, V₁ +  2.7183V₃ = 1,512,287 ------------ Equation (2)

Hence, Equation (2) - Equation (1) gives:

(2.7183 - 1)V₃ = 1,512,287 - 756,144 = 756,144

Hence, V₃ = 756,144 / 1.7183 =  $ 440,049

and V₁ = 756,144 - V₃ =  $ 316,094

Hence, the immunized portfolio should comprise of $ 316,094 amount of 1 year discount bond and $ 440,049 amount of 3 year coupon bond.