Problem $3$ Your client is a regional savings and loan institution. You have been hired to

assess the client's exposure to interest rate risk and advise on possible ways to reduce it$.$ You

have the following information on the S&L$\text{'}$s balance sheet: The total assets are morigages

and are worth $$3,220\mathrm{.}7M.$ The assets have a duration of $9\mathrm{.}23$ years. The liabilities are $5-$year

Certificates of Deposit $($CDs$)$ with a coupon rate of $8\%,$ which we assume is paid annually

$($CDs are coupon bonds, so all coupon bond formulas apply to them$).$ The total face value

of the CDs that have been issued by your client is $$2\mathrm{.}5$ billion. The yield curve is flat at

$8\%.$

a$)$ Compute the duration of the liabilities.

b$)$ Compute the duration of the S&L$\text{'}$s equity $($assets minus liabilities$).$

c$)$ Your recommendation to the client is to sell some of the morigage portfolio and to

use the proceeds to buy $1-$year CDs $($which pay an annual coupon rate of $8\%)$ with

the proceeds. How many dollars worth of the assets does your client have to sell

to reduce the duration of the $($net$)$ market value of the bank to $0($i$.$e$.,$ to achieve

immunization$)?$ Assume that the assets of S&L are homogeneous.