Question

1. A company must pay a liability of $1,000 in 2 years. Zero coupon bonds with term of 1 year and 4 years are available for investment. The effective rate of interest is 7.5%.

   1. (a) How much of each bond should the company buy in order to achieve full immunization?

   2. (b) Show empirically that immunization has been achieved even for large changes in the interest rate. Take as an example a decrease in the interest rate to 0% and an increase to 100%.

Answer 1

Solution:

A)

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.

I am giving answer to only fist part .

Please post b) as new question ..