8. Answer the following questions:

a) What is the duration of a 16 year 5% bond trading at par?

b) What is the modified duration of a 28 year 5% bond yielding 6.8%? (to nearest 0.01 years)

c) Suppose 7 years from today you are to receive $4,000, and you will continue to receive $4,000 annually for each year thereafter, in perpetuity. Given a continously compounded interest rate of 0.2%, what is the convexity of these cash flows? (to the nearest 0.01 years^2) (Hint: Work out a closed form solution, rather than try to calculate numerically.)

d) Given a bond that trades at price 108 with modified duration 17 years and convexity 492 years^2, if the bond's yield decreases by 21 basis points, what is your estimate of the new bond price (to nearest 0.01)?

e) Suppose you have liabilities with present value $8 million and duration 11 years. You can invest in two zero coupon bonds of maturities 2 and 25 years respectively. To immunizes your liabilities, how much should you invest in the zero coupon bond of maturity 2 years? (to nearest $0.01)

The answer to these problems are a) 11.20 b) 13.21 c) 506,542.34 d) 111.97 e) 4,869,565.22.

I want the equations to get these answers.

Thank you!