Question

2. Alpha Insurance Company is obligated to make payments of $2 million, $3 million, and $4 million at the end of the next three years, respectively. The market interest rate is 8% per annum. Suppose the company’s payment obligations are fully funded and immunized using both 6-month zero coupon bonds and perpetuities.

Determine how much of each of these bonds the company will hold in the portfolio.

Answer 1

For immunization, average duration of assets has to equal the average duration of liabilities.

Average duration of liabilities = 2.17 years

For assets: Duration of a zero coupon bond equals its time to maturity so duration here, is 0.5 years.

Duration of a perpetuity =(1+ interest rate)/interest rate = (1+8%)/8% = 13.5 years

Let the weight of the zero coupon bond be x. Then, weight of perpetuity is (1-x).

2.17 = 0.5x + 13.5(1-x)

(13.5-0.5)x = 13.5-2.17

x = 87.12% (weight of ZCB)

1-x = 12.88% (weight of perpetuity)

Amount to be invested in ZCB = 87.12% of PV of liabilities = 87.12%*7.60 = 6.62 million

Amount to be invested in perpetuity = 12.88%*7.60 = 0.98 million