Question

An insurance company must make payments to a customer of $10 million in 5 years and $25 million in 30 years. The yield curve is flat at 8%. a) What is the present value and duration of its obligation? b) If it wants to fully fund and immunize its obligation to this customer with a single issue of a zero- coupon bond, what maturity bond must it purchase? Suppose you buy a zero-coupon bond with value and duration equal to your obligation, and that rates immediately increase to 9%. What happens to your net position, that is, to the difference between the value of the bond and that of your tuition obligation?

Answer 1

Please see the table below. Please be guided by the second row to understand the mathematics.Figures in parenthesis, if any, mean negative values. All financials are in $ million.

Liabilities	Term	PV	n x PV
L	n	L x 1.08^(-n)
10	5	6.81	34.03
25	30	2.48	74.53
Total		9.29	108.56

Part (a)

The present value of the obligations = $ 9.29 million

Duration = 108.56 / 9.29 = 11.69 years

Part (b)

Maturity of the zero coupon bond (ZCB) = duration of the liabilities calculated above = 11.69 years

Part (c)

Face value of the ZCB that you will buy = 9.29 x (1 + 8%)^11.69 =  22.84

When YTM changes to 9%;

PV of the bonds = 22.84 / (1 + 9%)^11.69 =   8.34

and the PV of the obligations = 10 / (1 + 9%)⁵ + 25 / (1 + 9%)³⁰ = 6.50 + 1.88 = 8.38

Hence, the net position = 8.34 - 8.38 = - $ 0.04 million

Thus the net position is a loss (reduction in value of) of $ 0.04 million