3. Consider an insurance company that has been selling insurance contracts for many years. A year from today, you expect the aggregate payment for the maturing contracts to be $20 million. Thereafter, you project payments for the maturing contracts to grow forever at an annual rate of 1%. The term structure is flat at 9%, and the current assets of the company consist of $300 million in cash. Rates are with annual compounding. (a) What is the present value of the company's liabilities? What is the company's net worth? (b) Suppose the company keeps its assets in cash. Explain why this is a dangerous choice. To provide an example, consider a parallel shift in the term structure to 5% (the term structure remains flat). (c) Quantify the company's interest-rate risk by calculating Macaulay duration and modified duration of the liabilities. Show that the Macaulay duration is 13.625 years. Hint: Derive an analytical formula for the duration of a growing perpetuity by linking duration to the derivative of the present value. You can check your result using an approximate numerical calculation in Excel by assuming a large number of years. (d) Explain why the company cannot exactly offset interest-rate risk (say, by purchasing finite-maturity bonds that exactly replicate the present value of the liabilities). How can the company reduce interest-rate risk? (e) You 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 face value of 100, and coupon payments are annual. Calculate the price, Macaulay duration, and modified duration, for each of these bonds. (f) Set up a portfolio of the two bonds that has the same price and approximate price changes as the liabilities. Use modified duration to obtain the approximate price changes. Does the company currently have enough assets to purchase this portfolio?