culate the price, Macaulay duration, and modified duration for each of these bonds. (e) Set up a portfolio of the two bonds that has the same price and approximate price changes as the GIC liabilities. 3. (20 points) You are an actuary at an insurance company that sells to customers contracts that pay a given amount at some point in the future. Customers purchase these contracts at a discount and hold them until they mature. Although these contracts are essentially zero-coupon bonds issued by an insurance company, they are usually called guaranteed investment contracts (GIC) by the insurance industry. These contracts are often purchased by investors for their retirement accounts. Your insurance company has been selling these contracts for several years. Six months from today, you expect the aggregate payment for the maturing GICs to be $11 million. Thereafter, you expect payments for maturing GICs to grow forever at the rate of 0.5% every six months. Your company has always made semiannual GIC payments, and you expect this to continue in the future. Lately, management has been concerned about adverse profitability effects due to the interest rate risk the company faces in its GIC liabilities. You have been asked to explore immunization methods to reduce the impact of interest rates on the present value of the GIC portfolio. In particular, management wants to offset such risk by making appropriate investments in government bonds. Your analysis requires answers to several questions. For all of these questions, you are to assume that the term structure is flat, and assume that the yield on all fixed- income cash flows is 8% semiannually compounded. (a) Explain how, in the absence of immunization, the interest rate risk of the GIC liabilities can adversely affect the insurance company's profitability. Explain why the company cannot exactly offset this risk (say, by purchasing Treasury bonds that exactly replicate the present value of the GIC liabilities). How will management's immunization plan reduce this risk? (b) What is the present value of the company's GIC liabilities? Note that the formula given in class must be adjusted for semiannual rates. (c) Quantify the interest-rate risk by calculating Macaulay duration, and modified duration of the GIC liabilities. Hint: one way to approach the problem is to com- pute and go from there. Remember that the modified duration D satisfies the relation -DMP. (d) You will consider two government bonds for your analysis. The first is a zero- coupon bond that matures in 18 years, and the second is a 10% coupon bond that also matures in 18 years. Both bonds have a par value of $1000, and any coupon payments are paid semiannually starting six months from today. Cal-