Question

As of 12/31/03, an insurance company has a known obligation to pay $1,000,000 on

12/31/2007. To fund this liability, the company immediately purchases 4-year 5% annual

coupon bonds totaling $822,703 of par value. The company anticipates reinvestment interest

rates to remain constant at 5% through 12/31/07. The maturity value of the bond equals

the par value. Under the following reinvestment interest rate movement scenarios effective

1/1/2004, what best describes the insurance companys profit or (loss) as of 12/31/2007 after

the liability is paid?

(a) Interest rates drop by 1/2%

(b) Interest rates increase by 1/2%

Answer 1

The future value of the bond, including coupons reinvested at 5%, is

$822,703?1.054 ? $1,000,000.64,

more than enough to pay the liability. The redemption value of the bond is $822,703, and the rest is paid by the accumulated value of the coupons. If interest rates drop by 0.50% the value of the bond will increase initially, but its coupons will be reinvested at 4.5%, and its value will overall grow at 4.5%. The accumulated value on 12/31/2007 with the new interest rate of 4.5% will be (note that 5% of $822,703 is the coupon amount, and $822,703 is the redemption value

0.05 ?$822,703?s4 4.5% + $822,703 ? $998,687.03.

The amount of the liability payment on 12/31/2007 is $1,000,000, so there is a shortfall of $1,312.97. If interest rates increase by 0.50%, the coupons will be reinvested at 5.5%, leading to an accumulation of

0.05 ?$822,703?s4 5.5% + $822,703 ? $1,001,322.78.

As the liability is $1,000,000, this gives a surplus of $1,322.78