$2.$ As a passive manager, you are managing a pension fund which has an obligation to pay a

retiring employee $$50,000$ in year $3$ and $$40,000$ in year $5.$ The yield curve is flat at $8\%.$

a$)$ What is the present value of this pension fund?

b$)$ What is the duration of this pension fund?

c$)$ To immunize the interest rate risk, you plan to construct a bond portfolio so that the

duration of your assets is equal to the duration of the pension fund. Suppose only the

following zero$-$coupon bonds are available to invest, what are the right weights of each

bond to be invested in this portfolio?

Bond Maturity YTM Face Value

C $3$ years $8\%$ $$1,000$

D $7$ years $8\%$ $$1,000$

d$)$ What is the dollar amount to be invested in bond C and bond D$,$ respectively?