(1) The force of interest changes from moment to moment according to the formula t = 0.10e 0.012t . You invest $ 1,000 at time 0. Find the accumulated value at time 6.

(2) The force of interest changes from moment to moment according to the formula t = 0.008t 2 , where t is measured in years. You expect to receive a payment of 500,000 at time 5. Find the present value of this payment at time 0.

(3) Find the present value of a 30-year annuity-immediate which pays $2,000 each year. Assume an effective annual interest rate of 8%.

(4) Find the present value of a 20-year annuity-due which pays $4,000 each year. Assume that the effective annual interest rate is 6%.

(5) Suppose that Abby offers to pay you $1,000 at the end of each year for the next 10 years. Ben offers to pay you $10,000 at time t. The effective annual interest rate is 7%. Find the value of t (in years, to two decimal places) such that you are indifferent between Abby's and Ben's offers.

(6) Suppose that Melissa offers to pay you $2,000 at the end of each year for the next 5 years. Matthew offers to pay you $7,000 at time t. The effective annual interest rate is 9%. Find the value of t (in years, to two decimal places) such that you are indifferent between Melissa's and Matthew's offers. Interpret your answer. 2

(7) An annuity pays 10,000 every year for 30 years. Find the accumulated value of this annuity 3 years after the last payment. Assume that the effective annual interest rate is 10%.

(8) An annuity pays 5,000 every year for 25 years. Find the present value of this annuity 2 years before the first payment. Assume that the effective annual interest rate is 12%.

(9) A perpetuity will pay X every year. The effective annual interest rate is 7.5%. The present value of this perpetuity 4 years before the first payment is 125,000. Find X.

(10) Suppose you deposit $20,000 at the end of each of the next 30 years into a retirement account. Immediately after your last deposit, you take the entire accumulated value in your account and purchase a 20-year annuity, which will pay you X at the beginning of each year for 20 years. The price of this 20-year annuity is equal to the present value (at the time you purchase the annuity) of the 20 annual cash flows. The effective annual interest rate throughout the entire 50-year period is 10%. Find X.