# Ordinary Annuity: The Present Value

An ordinary annuity, also called an immediate annuity, is an investment tool in which an individual (the annuitant) makes a single payment, at a certain time, in exchange for a series of smaller payments in the future. Ordinary annuities are generally sold by insurance companies and can be designed to pay for a certain period (e.g., 10 years) or for the remainder of the annuitant’s life. An annuity is effectively a loan by the annuitant to the issuer. The annuitant will collect payments that are made up of principal repayment and interest. The principal is not taxable and the interest portion is taxed as earned income. Since annuities are commonly used by retirees to guarantee income for life, the taxes are relatively low because the interest collected is small compared to the total payment.

#### Present Value of an Investment

An investment’s present value is the sum of each of its discounted future payments. The discount rate is sometimes very difficult to determine, however, in the context of fixed income securities, the typical discount rate is equal to the interest rate. Thus, if a loan is issued at 5 percent interest with yearly payments, each of the future payments will be discounted by 5 percent compounded per year after the loan was issued. As time goes on, the same dollar payment becomes less valuable in the present because the money can be invested elsewhere. In general, an individual would rather have \$100 today instead of \$100 one year from now. That is the basis of present value determination.

#### Present Value of an Annuity

Annuities that are issued for some certain amount of time are much easier to value because they are exactly like standard loans. The present value represents the amount of money that the investor should pay to match the discounted future payments. If the interest rate on the annuity is fixed, this is very easy to do.

Suppose an annuity is issued for 5 years and makes payments of \$1000 per year with an interest rate of 10 percent. Each payment is divided by the formula below, where t is the number of years that have passed.

• (1 + 10 percent)t

#### Barriers to Accurate Valuation

Determining the present value of an annuity that does not have a fixed interest rate is more challenging since it requires estimating the interest rate for each payment. This is less of an issue for short-term annuities, but it can have a huge impact on long-term annuities. For example, loan interest rates 30 years ago were around 15 percent, but today they are closer to 5 percent. In the same way, a life annuity is challenging for an individual to price because they do not know how long they will live. Insurance companies who issue many life annuities are protected because they price the annuity based on life expectancies.