Question

Immunization problem. A liability of $20,000 is to be paid at t=6. years. Assume the current interest rate is 5%. Fully immunize by buying two zero coupon bonds with maturities at times t=2 and t=12.

Answer 1

The principle of immunisation requires that the present value of liabilities should be invested in a portfolio of bonds in such a manner that duration of assets is equal to duration of liabilities.

Present value of liability of $20000 to be paid at Year 6 = 20000*Present value interest factor(5%,6) = 20000*0.7462= 14924

It means that we need to invest $14924 in two zero coupon bonds with maturities at times t=2 and t=12 . Let us calculate the proportion of funds to be invested and amount to be invested in these two zero coupon bonds available.

We have:

Weight of Bond 1 = W1

Duration of Bond 1 = D1 (t=2)

Weight of Bond 2 = W2

Duration of Bond 2 = D2 (t=12)

Duration of assets = Duration of liabilities

W1*D1 + W2*D2 = DL

(1 - W2)D1 + W2*D2 = DL

(1 - W2)2 + W2*12 = 6

2 - 2W2 + 12W2 = 6

2 + 10W2 = 6

10W2 = 6 - 2

10W2 = 4

W2 = 4/10 = 0.4

W1 = 1 - 0.4 = 0.6

Amount to be invested in Bond 1 = 14924*0.4 = 5969.6

Amount to be invested in Bond 2 = 14924*0.6 = 8954.4