A fixed annuity is a financial investment in which one gives an amount A to an insurance/investment company in return for fixed, guaranteed periodic payments P for a period of time (either a set period, or the recipients lifetime, which is converted to an estimated time period by the insurance company).

When an annuity contract is signed, the insurance company specifies the periodic (usually monthly) payment, which is based on the investment amount, the known or estimated duration of the payments, and the annual interest rate a that the insurance company is giving to the investor. Obviously, the insurance company will invest the amount A and expects that it will earn a higher rate of return on it than a; a is the portion of the expected rate of return that the company is giving back, and the difference is the companys compensation for risk, administrative costs, and profit.

a)      Assuming that the annual interest rate, more commonly referred to as the annual percentage yield (APY) , that the insurance company returns to the annuity holder is a and that payments will be made for N years, determine a formula for the amount of each annual payment P. Assume that payments are made at the end of each year.

b)      Because of inflation, the value of a fixed payment P declines over time. An annuity with an adjusted payment schedule increases P by a fixed percentage b each year, so that the payments made are P1 during the first year, P2 = (1+b)P1 during the second year, etc. Obviously, the earlier payments will be smaller (and the later ones larger) than the constant payment P calculated in Question (1) for the same values of A, a, and N. Determine a closed formula for the sequence Pn under the same assumptions as Question (a).

c)      Investors typically prefer receiving a payment more often than a yearly lump sum, so the usual annuity contract provides monthly payments. Because part of the payment is given to the investor earlier than the end of the year, that part earns interest for less than a full year. Construct a formula for the monthly payment M corresponding to the constant yearly payment P, and for the adjusted monthly payments Mn corresponding to the payments Pn. (M1 is the monthly payment during the first twelve months, M2 the monthly payment during the next twelve months, and so on.) In both cases, assume that the payments are made at the end of each month.