Present Value Calculations for a Deferred Annuity

The "time value of money" means a dollar in your pocket today is worth more than a dollar you'll receive next month, because you can put today's dollar into a savings account and earn interest on it for the month. After a month, the account is worth more than a dollar, which why getting the dollar today is a better deal than waiting for it. Another way to look at it is today's value of a future dollar is equal to some amount less than a dollar, say 99 cents. In other words, put 99 cents into a savings account, earn a penny of interest (say at monthly rate of 0.01 percent) and after a month you'll have one dollar. Calculate present value by reducing, or discounting, the value of the future dollar using a discount factor equal to the interest rate you can earn on the savings account. In this case, the future value of $1 was discounted by a factor of 0.01 percent for one month to calculate the present value of 99 cents. In a completely rational world, you would be equally happy to earn 99 cents today or $1 in one month assuming you would save the 99 cents rather than spend it.

Present Value Calculations for a Deferred Annuity
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Annuities 101

An annuity is a series of cash payments, also called cash flows, that occur at regular intervals. An annuity contract is an agreement you make with an insurance company in which you give the insurance company an amount of money, and it sends you regular cash payments. The payments continue until the annuity expires either in a preset number of years or when you die. The present value of the annuity is the amount of money you'd need today that, if invested at the annuity's interest rate, would equal the sum of all the cash flows you'd receive from the annuity over its lifetime.

Annuity Types

In an immediate annuity, you deposit a lump sum and begin receiving payments right away. In a deferred annuity, you can contribute one or more cash payments up to a future date, called the annuity date, when you stop contributing and begin receiving your payments. An example of a deferred annuity would be to contribute $10,000 into an annuity account with a fixed interest rate of 9.6 percent annual, (0.8 percent monthly) and then, in three years, start receiving monthly payments of $93.87 for the following 20 years. Each succeeding payment is worth less in today's dollars than the one before it because of the time value of money.

Immediate Annuity Calculation

The calculation of an immediate annuity is straightforward, because it is simply the present value of the future cash flows, discounted at the annuity's interest rate. The formula is PV = P {[1 - (1 / ((1+i)^n)] / i}. In this formula, P represents the amount of each payment, i is the annuity's monthly interest rate, and n is the number of payments. Assuming P equals $93.87, i equals 0.8 percent and n equals 240 (a 20-year annuity), then the present value works out to be $10,000. However, that's the present value three years from now, when the deferred annuity starts paying.

Deferred Annuity Calculation

The present value three years from now of $10,000 must be discounted again to find the present value as of today. You can use this formula: PV today = (PV in future) * [(1/(1+i))^t], where PV in future is the present value in three years ($10,000), i is the monthly interest rate (0.8 percent), and t is the number of periods that payment is deferred (36 months). The result is $7,506, which is the amount you would need to deposit on the date you open the annuity contract. This sum will grow to $10,000 on the annuity date in three years, and then generate monthly payments of $93.87 for 20 years, a total of $22,529 in cash flows from an initial investment of $7,506.