Question

A company must pay a liability of $1,000 in 2 years. Zero coupon bonds with terms of 1 year and 4 years are available for investment. The effective rate of interest is 7.5%. How much of each bond should the company buy in order to achieve full immunization?

Answer 1

To immunize against interest rate fluctuation risk, company has to invest in bond or bond portfolio, so that duration of bond or bond portfolio is equal to the investment horizon i.e. 2 years in this case.

A zero coupon bond always has a duration equal to its maturity.

Hence the duration of one year term zero coupon bond would be 1 year and the duration of four year term zero coupon bond would be 4 years.

But the company investment horizon is 2 years. So, we have to invest in 1 year and 4 year term zero coupon bonds in such a proportion that the total duration of the bond portfolio would be 2 years.

Let us assume that to maintain portfolio duration of 2 years, the proportion of total fund to be invested in one year term bond be p and in four year term bond be 1-p.

So, we can write the following:

(p*1)+[(1-p)*4]=2

Therefore p=0.67

It means, to maintain the portfolio duration of 2 years, 67% sould be invested in 1 year term bond and 33% sould be invested in 4 year term bond.

Total amount to be invested= $1,000/1.075*1/1.075= $865

Investment to be made in 1 year term bond = $865*67%= $579.55

Investment to be made in 4 year term bond = $865*33%= $285.45

It means to immunize the company should buy $579.55 of 1 year term bond and $285.45 of 4 year yerm bond.