Question

Jolene must pay liabilities of 1000 due 6 months from now and another 3000 due one year from now. There are two available investment:

Bond I is a 6-month bond with face amount and redemption value of 1000, a 7% semiannual coupons, and a 5% nominal annual yield rate convertible semiannually.

Bond II is a one-year bond with face amount and redemption value of 1000, a 4% semiannual coupon, and a 6% nominal annual yield rate convertible semiannually.

Suppose that Jolene buys x of Bond I and y of Bond II in order to exactly (absolutely) match the liabilities.

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

(A) 5.91% (B) 5.93% (C) 5.95% (D) 5.97% (E) 5.99%

Answer 1

$1000 is due 6 months from now

$3000 is due 12 months from now

Bond II

Since FV = 1000

Coupon Payment = 4% paid semiannually

Therefore if 1 unit of bond II is bought it will give at the end of 1 year

= 1000 + 2% of 1000

= $ 1020

for return = 1000, he must buy

   = 1000/1020 = 0.98039 units

For payment of 3000

Y    = 3*0.98039 = 2.94117

If he buys 2.94117 unit for 1 year he will get 1 coupon payment at 6 months, which is

   = 2.94117 * (20)

   = $ 58.8234

To fulfil total liability of $ 1000 at time = 6 months he needs

   = 1000 - 58.8234 = $ 941.1766

This amount will be sourced from bond I

Bond I has FV = 1000

Coupon Yield = 7% paid semiannually

For each unit of Bond I, he will get

= 1000 + 3.5% of 1000

= 1035

But he needs only $ 941.1766, therefore he will buy

x   = 941.1766/1035

x   = 0.90935 units

For 6-months bond

n = 1, PMT = 35 , FV = 1000, and yield i = 2.5% per semiannual period.

By formula PV = FV/(1+r)ⁿ

Therefore PV = 1009.76

Simillarly, For 12-months bond

n = 2, PMT = 20 , FV = 1000, and yield i = 3% per semiannual period.

Therefore PV = 980.87

Therefore total investment required

= (1009.76 * 0.90935 ) + ( 980.87 * 2.94117)

= 918.23 + 2884.91

= $ 3803.14

The annual Effective yield rate

3803.14 = 1000/(1+r/2)¹ + 3000 / (1+ r/2)²

Solving this we get,

r = 5.86 %

Please check options as none of it matches.