A life insurance company expects to pay out cash flows on policies that it has written according to the following schedule.

1-year $1 million

2-year $2 million

3-year $3 million

4-year $10 million

The insurance company wants to invest in some combination of 1-year and 5-year bonds such that the liability is immunized. Both bonds have a face value of $1,000 and a ytm of 4%. The 1-year bond pays a coupon of 2% and the 5-year bond pays a coupon of 6%. You may assume annual payments. 

a)  What should be the price of the 5-year bonds? What is the duration of the cash flows to the life insurance company?

b) The insurance company wants to invest in some combination of 1-year and 5-year bonds such that the liability is immunized. How many 1-year bonds should they buy?