An insurance company must pay liabilities of 99 at the end of one year, 102 at the end of two years and 100 at the end of three years. The only investments available to the company are the following three bonds. Bond A and Bond C are annual coupon bonds. Bond B is a zero-coupon bond.

Bond Maturity (in years) Yeild to Maturity (Annualized) Coupon Rate

A 1 6% 7%

B 2 7% 0%

C 3 9% 5%

All three bonds have a par value of 100 and will be redeemed at par. Calculate the number of units of Bond A that must be purchased to match the liabilities exactly

a. 0.8807

b. 0.8901

c. 0.8975

d. 0.9524

e. 0.9724

The correct answer is option (a) 0.8807

Explanation:

Solution

Given that:

We start from the liability of bond in 3 years.

Thus, the $100 liability can be an offset by Bond C.

The cash flow of Bond C and the payment of final coupon in year 3 is given as:

100 + (5%*100) = 105

Now,

the number of Bond C which will offset a liability of $100 which is = 100/105 = 0.9524 (All cash flows of Bond C is multiplied by this)

So, the remaining liability becomes

Time Liabilities cash flow Cash flow from Bond C Remaining liabilities

1 99 4.76 94.24

2 102 4.76 97.24

3 100 100.00

Thus,

The year 2 liability offset is $97.24

For Bond B, this can be the offset which contains a cash flow of $100 (which is a zero coupon bond)

The Bond number which are required for this offset is = 97.24/100 = 0.974

The remaining cash flow is computed as follows:

Time = 1,2, 3

Liabilities cash flow = 99, 102, 100

Cash flow from Bond C = 4.76, 4.76. 100.00

Remaining liabilities = 94.24, 97.24

Cash flow from Bond B = 0, 97.24

Remaining liabilities = 97.24

What this suggest is that The Bond A has to offset at approximately $94.24 in year 1.

The Cash flow from Bond A = 100 + (7%*100) = 107

Hence,

The number of Bond A's needed = 94.24/107 = 0.8807