Question

An insurance company must make payments to a customer of 10$ million in one year and 5$ million in five years. The yield curve is flat at 10%.

a. If it wants to fully find and immunize its obligation to this customer with a single issue of a zero-coupon bond, what maturity bond must it purchase?

b. What amount should be invested in the zero-coupon bonds? What will be the maturity value of the zero-coupon bonds?

c. What will be the net position of the insurance, that is, the difference between the value of the zero-coupon bonds and that of the obligation if yields stay at 10%?

d. What will happen to the net position if the yield falls to 8%? What if the yield rises to 12%?

Answer 1

a) Present value of first payment =P1 = 10,000,000 * PV _(10,1) = 10,000,000 * 0.909=9,090,000

Present value of second payment = P2=4,000,000 * PV _{(10 , 4)} = 4,000,000 * 0.683 =2,732,000

Total Present Value of the bond = P1 +P2 = 11,822,000

W 1= 9,090,000/11,822,000 =0.7689

W 2 = 2,732,000/11,822,000=0.23109

Duration = T1 * W1 + T2 * W2 = 1*0.7689 + 5* 0.23109 = 0.7689 + 1.15545 = 1.92435 =1.92 years

b) Amount should be invested in this zero - coupon bond is the market value of bond and since it is a zero coupon bond its maturity value is equal to face value.

Face Value =Maturity Value = $ 10,000,000 + $ 4,000,000 = $ 14,000,000

Market Value = Amount invested = $ 14,000,000 * PV _{(1.92 , 10 )} = 14,000,000* (1 / 1.2008) = $ 11,658,895