You probably don't enjoy the idea of parting with a chunk of your money every month for a mortgage or car payment. But if you refer to the task as "fulfilling an installment on an ordinary annuity," it makes it sound more high-finance, and that might pep you up a little. An ordinary annuity is just a series of payments of equal value made at regular intervals, like a car payment. A perpetuity is the same thing, except that it goes on forever, like car payments seem to.

## Ordinary Annuities

Though the word "annuity" means, literally, "yearly amount," the interval between payments can be anything -- a week, a month, a year, whatever. Traditional pension payments are ordinary annuities, as are the interest payments on corporate and government bonds. Your car loan payment is an ordinary annuity. Big lottery winners can choose to receive their jackpot as an annuity, spread out over 20 years or so. And you can purchase an annuity for your retirement plan, putting up money now to collect more later.

## What's So Ordinary About Them

What makes an ordinary annuity "ordinary" is the timing of the payments. In an ordinary annuity, each payment comes at the end of the specified interval. When you buy a home, for example, your first mortgage payment usually isn't due until the end of the first full month you've been in the house -- the payment covers the past month. Same with a car payment; you pay for the time you've already used the car. The alternative to an ordinary annuity is an "annuity due," in which the payment comes at the beginning of the interval. Rent is typically an annuity-due situation. You pay your first month's rent when you move in, and on the first of each month, you pay the rent for the upcoming month.

## Perpetuities

Annuities, be they ordinary annuities or annuities due, don't continue forever. They have an end date. That date can be a concrete deadline, such as the 48th monthly payment on a four-year car loan, or determined by events, such as the death of a pension recipient or you moving out of your apartment. A perpetuity, by contrast, is a series of equal payments at regular intervals that continues forever. Charitable foundations and university endowments are commonly set up as perpetuities. A class of securities known as perpetual bonds are also designed to act as perpetuities. If you buy such a bond for, say, $1,000, you never get the $1,000 back, but you -- and later your heirs, or whoever winds up with the bond when you pass on -- get to collect interest payments forever.

## Annuity Present Value

If you were looking at an ordinary annuity that paid you $1,000 a month for 20 years, you might do a little math on the back of an envelope and conclude that it's worth $240,000 -- $1,000 times 12 months times 20 years. But actually, it's worth considerably less. When an annuity is set up, whoever is managing that annuity invests a certain amount of money upfront, with the aim of generating a return sufficient to make the annuity payments. For example, say you have an annuity based on a 3-percent annual return. That equates to 0.25 percent a month. If you put $997.51 into that investment today, you'd have $1,000 a month from now. So, assuming a 3-percent return, the "present value" of that first $1,000 payment is $997.51. The farther away each individual payment is, the lower its present value, because your gains are compounded. The present value of the annuity is the sum total of the present values of all the payments. That $1,000-a-month annuity mentioned earlier? Assuming a 3-percent annual return, the present value is $183,311. Assume a 6-percent interest rate, and the present value drops to $139,581. If you've ever wondered why lottery winners who choose to take their money in one lump sum get so much less than those who take it as an annuity, it's because they're getting the present value of the annuity -- the amount that would have to be invested right now to produce the full jackpot over 20 years.

## Perpetuity Present Value

In theory, a perpetuity will deliver an infinite amount of money, but its present value is far from infinite. In fact, it's easier to calculate the present value of a perpetuity than an annuity. Assuming the money was invested for a 3-percent annual return -- 0.25 percent a month -- and you invested $400,000 today, in one month you'd have $401,000. Take away your first $1,000 perpetuity payment, and the principal drops to $400,000. Wait another month, and the return makes the total $401,000 again. Take the next payment, lather, rinse, repeat until the sun burns out. The present value of that "infinite" amount of money is $400,000. Assume a 6-percent annual return, and it drops to $200,000.

### References

- David R. Frick & Co., CPA: Time Value of Money Concepts
- New York University Stern School of Business: Annuities and Perpetuities - Present Value
- "Financial Accounting for MBAs," Fourth Edition; Peter Easton, et al; 2010

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