Question

A company has 3 future liabilities of 1000 at time 2, 1200 at time 4 and 1000 at time 6.

The only investment vehicles available are one-year zero-coupon bonds and five-year Zero-coupon bonds. If the interest rate is 10%, how much money should be invested in each bond to achieve local immunization?

Answer 1

Step 1: Calculation of Duration of liabilities :

Years	Liability Amount	Present value @ 10%	Present Value x Years
2	1000	1000 x 0.8264 = 826.4	826.4 x 2 = 1652.8
4	1200	1200 x 0.683 = 819.6	819.6 x 4 = 3278.4
6	1000	1000 x 0.5644 = 564.4	564.4 x 6 = 3386.4
	TOTAL	2210.4	8317.6

Duration of Liabilities = Total of Present Value x Years / Total of Present value

= 8317.6 / 2210.4 = 3.76 years

Step 2 : Immunisation

Let one-year zero-coupon bonds is x and five-year Zero-coupon bonds is y.

Let weight of Bond x is Wx and weight of Bond Y is Wy.

Similiarly, duration of Bond X is Dx and duration of Bond Y is Dy.

We know that weight of both bonds should be 1. So, if weight of Bond X is Wx then weight of Bond Y will be 1 - Wx.

Duration of Bond X i.e. Dx = 1

Duration of Bond Y i.e. Dy = 5

As per equation,

Duration of assets = Duration of liabilities

Wx Dx + WyDy = Duration of liabilities

Wx1 + Wy5 = 3.76

Wx1 + (1 - Wx)5 = 3.76

Wx + 5 - 5Wx = 3.76

-4Wx = 3.76 - 5

-4Wx = -1.24

4Wx = 1.24

Wx = 1.24/4 = 0.31

Wy = 1 - Wx = 1 - 0.31 = 0.69

Step 3 : Calculation of amount to be invested in each bond :

Total investment amount = Present value of liabilities as calculated in table = 2210.4

So, amount to be invested in one-year zero-coupon bonds = 2210.4 x 0.31 = 685.224

Amount to be invested in five-year Zero-coupon bond = 2210.4 x 0.69 = 1525.176