To calculate the purchase price of the annuity, we need to determine the present value of the future cash flows.
For the first eleven years, Stephen receives $3,000 every six months, which is equivalent to 2 payments per year. The interest rate is 3.1% compounded quarterly, so the effective interest rate per six-month period is (1 + 0.031/4)^2 - 1.
Using the formula for the present value of an annuity:
PV = Payment * [(1 - (1 + r)^(-n)) / r],
where PV is the present value, Payment is the periodic payment, r is the interest rate per period, and n is the number of periods.
Calculating the present value for the first eleven years:
PV1 = $3,000 * [(1 - (1 + 0.031/4)^(-2*11)) / (0.031/4)].
For the next four years, Stephen receives $400 per month, which is equivalent to 12 payments per year. The interest rate is 3.1% compounded quarterly, so the effective interest rate per month is (1 + 0.031/4)^3 - 1.
Calculating the present value for the next four years:
PV2 = $400 * [(1 - (1 + 0.031/4)^(-12*4)) / (0.031/4)].
To find the purchase price of the annuity, we sum the present values of both periods:
Purchase Price = PV1 + PV2.
Calculating the purchase price using the given information will provide the answer.