!!!provide the answers rounded to the nearest ten-thousandth

(all explanations are needed)

3. You work for a bank and your client wants to purchase an annuity plan that pays a fixed amount of money every year for the next ten years. He plans to pay $250,000 for such a plan.

(3-a) Assuming a flat term structure at a continuously compounded interest rate of 2%, how much does the annuity plan should pay to the client every year for ten years? Note that you need an annually compounded interest rate to use the PV formula for the annuity.

(3-b) Your bank needs to hedge this commitment to pay for the annuity plan. First, the bank considers cash flow matching. Explain the cash flow matching method.

(3-c) [15pts] Instead, the bank decides to implement a dynamic immunization strategy for the hedging. The bank will construct a portfolio of 10-year T-note with 1.5% coupon rate and cash in a deposit account. It will rebalance its portfolio every year. Assume the term structure is flat at 2% (continuously compounded) and it does not change for the next ten years. Also, assume that the annuity payments and the T-note coupon payments occur on the same date. The below are the detailed steps to implement the strategy. Answer the relevant questions:

At t = 0, the bank receives $250,000 from the client, and the duration of 10-year T-note with 1.5% coupon rate is 9.3029.

• [i] Compute the duration of the annuity described in (3-a).

• [ii] How much of $250,000 should be invested in the T-note?

At t = 1, the bank collects the coupon payments from the T-note, the interest earned from the deposit account, and pays the annuity payment.

• [iii] Calculate the total coupon payments from the T-note for the first year. Note that there are two coupon payments for the year. Just provide the sum of these two coupon payments.

• [iv] Calculate the interest earned from the deposit account. (Assume the interest rate for the deposit account is the same as the term structure; the deposit account pays continuously compounded interests.)

• [v] What is the value of the portfolio after taking into account the coupon payments, interest income, capital gain on the T-note, and the annuity payment?