Question

An insurance company expects to make a one-time payout of $10 million dollars in exactly 6 years to fulfil contractual agreements for an investment product. To create a portfolio that immunizes this liability, using only a 10-year zero-coupon bond and a 3 year 8% annual coupon bond with a yield to maturity of 10% and a Duration of 2.78 years, one would have to invest ________ of the portfolio value in the zero-coupon bond.

Answer 1

Sol:

Expected pay out value = $10 million

Period (n) = 6 years

Duration of 3 year 8% annual coupon bond = 2.78 years

Duration of zero coupon bond = Life of bond.

Assume weight (W) invested in zero coupon bond, then weight in 3 year 8% annual coupon bond will be = 1-W

Portfolio duration = weighted average duration of each asset class.

6 = 10W + (1 - W) x 2.78

6 = 10W + 2.78 - 2.78W

6 - 2.78 = 10W - 2.78W

3.22 = 7.22W

W = 3.22/7.22 = 0.4460 or 44.60%

Now amount to be invested in zero coupon bond will be = Expected pay out value x Weight of zero coupon bond

Now amount to be invested in zero coupon bond will be = $10 million x 0.4460 = $4.46 million.