Question

1. You manage a pension fund that promises to pay out $10 million to its contributors in five years. You buy $7472582 worth of par-value bonds that make annual coupon payments of 6% and mature in five years. Right after you make the purchase, the interest rate on same-risk bonds decreases to 4.1%. If the rate does not change again and you reinvest the coupon payments that you receive in same-risk bonds, how much will you fall short of the money that you promised? Write your answer as a positive number and round it to the nearest dollar.

2. Your pension fund is invested in $40 million worth of bonds with a duration of 5.5 years and $60 million worth of bonds with a duration of 8 years. The "target date" (the date that the fund needs to pay its contributors) is 6.978 years from now. To become duration-matched, the fund needs to shift how much of its money from 8-year duration bonds into 5.5-year duration bonds? Round your answer to the nearest dollar.

Answer 1

1. At the end of 5 years, the par value of the bonds will be recovered, i.e. $7,472,582.

To determine the future value of annual coupon payments received, we will use the FV of ordinary annuity's formula:

Here, P = 6% of $7,472,582 = $448,354.92

r = Interest rate = 4.10%

n = 5 years

FV of annuity = $448,354.92 * [ {(1+4.10%)^5-1}/4.10% ] = $2,433,292.74

Shortfall at end of 5 years = $10,000,000 - $7,472,582 - $2,433,293 = $94,125

2. Total sum invested (Dp) = $100 million

Let w1 be the sum invested in duration 5.5 years bond

& (1-w1) be the sum invested in duration 8 years bond

Duration of a portfolio (Dp) = w1*D1 + (1-w1)*D2 = 6.978

w1*5.5 + (1-w1)*8 = 6.978

w1 = 0.4088 = 40.88%

1-w1 = 0.5912 = 59.12%

Therefore, 40.88% of the total sum invested should be in duration 5.5 years bond. i.e. an additional $0.88 million needs to be taken out of duration 8-year bonds and put into duration 5.5 years bond.