Suppose a bank just sold a 5-year, $100,000 annuity to a client. The client transferred $100,000 to the bank immediately while the bank promised to make semi-annual flat payments to the client, equivalent to a 3% yield (APR, semi-annually compounded) over the next five years. The first annuity payment is due in 6 months.

Assume that the yield curve is flat, with all yields equal to 3.25% (APR, semi-annually compounded) at the time of the bank-client transaction.

(a) In present value terms, how much profit is the bank expected to make from this transaction?

(b) After setting aside the upfront profit from the clients $100,000 initial transfer, the bank keeps the remaining amount in a separate account and plans to implement an immunization strategy so that it can make the promised annuity payments over the next five years using funds from the account. The funds are immediately invested in two Canadian Treasury securities a 15-year bond with 5% coupon and $1,000 par and a six-month T-bill with $1,000 par such that the duration of the Treasury portfolio is matched exactly with the duration of the annuity initially and subsequently over the next five years. The bank plans to reevaluate the duration matching operation at six-month intervals (immediately after each annuity payment) and rebalance the portfolio in order to keep durations matched between the portfolio and the annuity. While the same Treasury bond (albeit with a shorter maturity as time goes by) will be kept in the portfolio, the matured T-bill will be replaced with a new 6-month T-bill.

Assume that the yield curve remains flat in the future but will, however, experience a parallel shift every 6 months subsequently, immediately before the coupon/annuity payment date. The interest rate shock will be a 25-basis point increase every 6 months in the first half of the five- year period (from month 6 to month 30) and a 25-basis point decrease every 6 months in the second half (from month 36 to month 60).

How does the duration matching strategy work out over the five-year period? Make sure that you show all calculations, including the calculation of the total balance in the account, both before and after the annuity payment/interest rate shock, duration matching, and portfolio rebalancing, allocation and holdings in each 6-month period.