5. (27pts) An insurance company collects 100 million USD from insurance premiums and needs to decide how to invest the money. Suppose there is a bond fund (B), a stock fund (S) and a risk-free asset to consider. Their statistics are given below.

Table 3: Mean returns, volatilies, correlation and the risk-free rate (rf )

Bond Stock

mean volatility 4% 10% 12% 25%

correlation 0.15 rf 1%

1. (1) (10pts) Find the optimal risky portfolio for the insurance company.

2. (2) (7pts) Suppose the insurance company cannot invest more than 20% in the stock fund and it hopes to achieve a mean return as high as possible. What is the complete portfolio?

3. (3) (5pts) Suppose the bond fund invests in two bonds: a 1-year zero coupon bond and a 4-year coupon bond (annual coupon with 5% coupon rate). The current YTM for both equals 6% and the pro- portions of the two bonds are 30% and 70%, respectively. What is the duration of the bond fund?

4. (4) (5pts) Suppose the yield suddenly increases 1% and there is no change in the stock fund value and in the value of the risk-free asset. What is the approximate change in the value of the complete portfolio held by the insurance company?