Question

Joe must pay liabilities of 1,000 due six 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%

Answer 1

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Answer:

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 = 1904.27

So the required yield for the combined cash flows can be found by solving:
1904.27 = 1000v + 1000v^2
effect rate per 6 months = 3.3332%

t=>nominal semiannual yield basis then we say 2 x 0.03332 = 6.664%
=>annual effective yield basis then 1.03332^2 - 1 = 6.775% ~ 6.8%