Joe must pay liabilities of 1,000 due 6 months from now and another 1,000 due one year from now. There are two available investments which may be purchased in any amount: 6-month bond with face amount of 1,000, 8% nominal annual coupon rate convertible semiannually, and 6 nominal annual yield rate convertible semiannually.
1-year bond with face amount of 1,000, 5% nominal annual coupon rate convertible semiannually, and 7% nominal annual yield rate convertible semiannually.

What is the annual effective yield rate for the investment in the bonds required to exactly (absolutely) match the liabilities?

a. 6.5%
b. 6.6%
c. 6.7%
d. 6.8%
e. 6.9%

d. 6.8%

Explanation:

The one year bond has a coupon rate of 5% per year.

- For 6 months in order to have a payment of 1000 we need a face value of 1000/1.025 = 975.61.

- Coupon payments of 975.61 x .025 = 24.39 at time 0.5 and time 1,

- Payment of the par amount of 975.61 at time 1.

Total cash flow = 1000.

We compute the price of these bonds by discounting their cash flows at their yields:

1-year bond: Price = 24.39 x 1.035^-1 + 1000 x 1.035^-2 = 957.08

6-month bond: Price = 975.61 x 1.03^-1 = 947.19

Total Price = 957.08 + 947.19

Total Price = 1904.27

So, the required yield for the combined cash flows can be found by solving:

1904.27 = 1000v + 1000v^2

EffectIve rate per 6 months = 3.3332%

Annual effective yield basis = 1.03332^2 - 1

Annual effective yield basis = 1.0677502224 - 1

Annual effective yield basis = 0.0677502224

Annual effective yield basis = 6.8%

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