Question

Company ABC has liabilities of 20,000, 50,000 and 70,000 due at the end of years 1, 2 and 3 respectively. The company would like to exactly (absolutely) match these liabilities using the following assets:a one-year zero coupon bond with a yield of 4%a two-year zero coupon bond with a yield of 5%a three-year coupon bond with annual coupons of 6% and a yield of 5.5%What is the total cost of the asset portfolio that will exactly match the liabilities?

Answer 1

Answer:

Ans. Bond A =$20,800; Bond B =$55,125; Bond C =$69,068.29

Explanation:

Hello, First, we have to find the price of the bond, or in other words what is the percentage of its nominal rate that will match its face value, that is , for bond A (zero coupon, yield=4%, 1 year) $20,000, for bond B (zero coupon, yield=5%, 2 years) $50,000, and bond C (coupon=6%, yield=5.5%, 3 year) $70,000. The equation is as follows.

$Price=\frac{PresentValueCashFlows}{Liability}$

For the first 2 bonds, the math is as follows

$PriceBondA=\frac{\frac{20000}{(1+0.04)^{1} } }{20000} =0.961538462$

$PriceBondB=\frac{\frac{50000}{(1+0.05)^{2} } }{50000} =0.907029478$

For Bond C, remember that this is a coupon bond so we have to find the present value of this instrument by using the following equation.

$PriceBondC=\frac{\frac{Coupon((1+yield)^{n-1}-1) }{yield(1+yield)^{n-1} } +\frac{(4200+70000)}{(1+yield)^{n} } }{Liability}$

$PriceBondA=\frac{\frac{4200((1+0.055)^{2}-1) }{0.055(1+0.055)^{2} } +\frac{(4200+70000)}{(1+0.055)^{3} } }{70000} =1.013489667$

So, the amount in Bond value for each one that will match each debt is:

Bond A = 20000/0.961538462=$20,800

Bond B = 50000/0.907029478=$55,125

Bond C = 70000/1.013489667=$69,068.29

Best of luck.