An annuity, just like a perpetuity, is a shortcut used while making present value calculations. Unlike a perpetuity, which is very difficult to find in real life, we find examples of annuities all around us. The monthly mortgage payments we make, and the car loan or student loan that we pay off, are all annuities. Annuities play a very important role in corporate finance. They form the basis for valuation of bonds and other financial instruments. This article explains what an annuity is, the different types of annuity calculations, and the important distinction between an ordinary annuity and an annuity due.
What is an Annuity?
Finite Stream: The primary difference between an annuity and a perpetuity is that an annuity has a finite life. Unlike perpetuities, annuities do not go on forever. It is for this reason that they are conceptually more intuitive and easier to understand.
Equal Amounts: A stream of payments can be called an annuity if, and only if, all the payments in that stream of future cash flows are of equal amounts. For instance, if the future cash flows for 4 consecutive years from now are $100 in each year, then this stream is called an annuity. On the other hand, if the future cash flows for the next 3 years are $100 and the 4th year is $110, then this stream of cash flows cannot be called an annuity (it is an annuity if you consider only years 1 to 3).
Equal Time Lag: Every payment in the stream of cash flows should be equally spaced. This means that if payments are being made on a monthly basis, all payments should be made on a monthly basis. If the time lag between payments changes, then the cash flow schedule cannot be classified as an annuity, because the annuity formula assumes that the cash flows are evenly spaced out.
Same Interest Rate: A stream of cash flows can be called an annuity if the interest rate being charged throughout the period is the same. For instance, if the rate of interest across the entire duration of a 10 year loan is 10%, then the stream of payments can be classified as an annuity. On the other hand, if the rate of interest keeps varying from year to year, it cannot be valued as an annuity, because the annuity calculation formula assumes a constant interest rate.
Amortization Concept: The payments in an annuity represent amortization of a lump sum amount. This means that although the amount paid in installments is constant, its internal components are changing.
Let’s understand this with an example. Say there is a $100 payment per month for the next 5 years. The $100 amount will remain constant for the next 5 years, but the internal components will change. The first payment may represent an $80 interest charge and $20 repayment of principal, while the last payment may represent only $10 interest and $90 repayment of principal. This is called amortization. The first few payments in an annuity have very high interest components; with the passage of time, the interest component becomes smaller and the repayment of principal becomes larger.
Types of Annuity Calculations
In the article on present and future value, we learn that the value of a dollar today is not the same as it will be 10 years from now. Annuities are a powerful mechanism that ensure the nominal value of the payments remains the same throughout the years, whereas the internal components (interest and principal) keep changing. Annuities therefore give us a very useful way to work with a schedule of payments. There are various types of payment schedules possible while working with an annuity. Here are the important types:
Lump Sum to Annuity Payments: Annuities can convert a lump sum payment today into a series of future cash flows that have the exact same value as of today. This is useful in business because the outlays required usually have to be made immediately in a lump sum, whereas the benefits arrive at a later date and in installments. Annuities enable us to draw a comparison between these values and evaluate if they are beneficial to us.
Example: Assuming a 12% rate of return for the next 5 years, an annual payment of $27.74 has the same present value as a $100 payment today. So we can choose between making a $100 payment upfront or choosing a 5-year annuity of $27.74.
Annuity Payments to Lump Sum: The reverse calculation is also useful. Annuities help us take a series of future equal payments, made at equal periodic intervals, and arrive at a lump sum present value that is equal to those payments. Let’s say you are scheduled to make mortgage payments for the next 5 years, but instead you choose to pay upfront and close the loan. What is the amount you should pay the lender? Annuity calculations help us come up with that amount.
Example: Assuming a 14% interest rate for the next 5 years and an annual payment of $100, the present value of this stream of payments is $343.31.
Partial Lump Sum: Annuity calculations can also be used for a combination of the two cases above. The payment may be made partially in equal installments and partially as a lump sum. For instance, if you owed the bank $500, you could pay $200 upfront and convert the balance into an annuity.
Annuity calculations allow you to convert any lump sum or stream of cash flows into any other lump sum, stream of cash flows, or combination of both. These calculations form the backbone of finance, and it is difficult to imagine the financial world without them.
| Calculation Type | What It Does | Typical Use Case |
|---|---|---|
| Lump Sum to Annuity | Converts a present lump sum into a series of equal future payments of the same value | Deciding between paying $100 today or a 5-year annuity of $27.74 |
| Annuity to Lump Sum | Converts a series of equal future payments into an equivalent present-day lump sum | Calculating the payoff amount to close a mortgage early |
| Partial Lump Sum | Splits a payment obligation between an upfront lump sum and an annuity for the remainder | Paying $200 upfront on a $500 debt and financing the rest |
Ordinary Annuity vs. Annuity Due
Annuities can be divided into two types based on the exact time when the payments occur in a given period. The payments could either occur at the beginning of every period or at the end of every period. For instance, when you rent a house, the rent is usually paid in advance, whereas mortgage payments are usually made at the end of every period. So the payments made at the end of every period are called an ordinary annuity. This is because an ordinary annuity is the usual state of affairs — most annuities are paid at the end of the period.
Alternatively, when annuity payments are made in advance, we call them an annuity due. The difference in the formula used to calculate the two types of annuities is very small, and the difference in the final calculated value can be substantial depending on the interest rate and duration. However, to be precise, a student of finance must know the difference between an ordinary annuity and an annuity due, and know when to use each formula.
One Extra Period
As we’ve seen, ordinary annuity payments are made at the end of each period, whereas payments for an annuity due are made at the beginning of each period. Hence, the difference between an ordinary annuity and an annuity due is one extra period. An adjustment therefore needs to be made for this one extra period when calculating both the present value and the future value of an annuity due.
Future Value of an Annuity Due: Let’s say we want to calculate the future value of an annuity that pays $100 for 5 years, with payments beginning at the start of the first period. The rate of interest is 10%.
If we used the regular annuity formula or table, we would get the future value of the above case as $610.51. However, this is the value if the payments were made at the end of each period. To convert this into an annuity due, we need to account for the one extra period, so we further multiply the answer by (1+i). Since the interest rate is 10% per annum, we multiply by 1.1. So the future value of the same example, as an annuity due, is $610.51 x 1.1 = $671.56.
Calculating the future value of an annuity due is a simple 2-step procedure:
- First, calculate the future value as a regular (ordinary) annuity.
- Second, compound that future value for one additional period.
Present Value of an Annuity Due: Let’s say you were to receive 5 annual payments of $100 each for the next 5 years, beginning at the start of each period, and your required rate of return is 10% per annum.
If we used the regular annuity formula or table, we would get the present value of the above case as $379.08. However, this is the value if the payments were made at the end of each period. Because payments under an annuity due arrive one period earlier, each one is discounted for one less period, which makes them worth more, not less. To convert this into an annuity due, we need to account for the one extra period by further multiplying the answer by (1+i). Since the interest rate is 10% per annum, we multiply by 1.1. So the present value of the same example, as an annuity due, is $379.08 x 1.1 = $416.99.
Calculating the present value of an annuity due is a simple 2-step procedure:
- First, calculate the present value as a regular (ordinary) annuity.
- Second, compound that present value for one additional period.
Please note the pattern: because payments arrive earlier under an annuity due, both its future value and its present value are always compounded forward for one extra period relative to the ordinary annuity — in both cases we multiply by (1+i), not divide.
The concept of an annuity due will often be hidden in a question — it will not be explicitly stated. One must pay attention to when the payments are being made to determine whether a problem describes an ordinary annuity or an annuity due.
| Feature | Ordinary Annuity | Annuity Due |
|---|---|---|
| When payments are made | End of each period | Beginning of each period |
| Typical real-world example | Mortgage payments | Rent payments |
| Future value | Base annuity formula/table value | Base future value x (1+i) |
| Present value | Base annuity formula/table value | Base present value x (1+i) |
| Which is worth more, all else equal | Lower | Higher (payments received/made sooner) |
Frequently Asked Questions
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What is the main difference between an annuity and a perpetuity?
An annuity has a finite life — it pays out for a fixed number of periods — while a perpetuity is a stream of payments that never terminates.
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Is a monthly loan repayment an ordinary annuity or an annuity due?
Most loan repayments, including mortgages, are paid at the end of each period, which makes them ordinary annuities.
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Why is an annuity due always worth more than an equivalent ordinary annuity?
Because each payment under an annuity due is made or received one period earlier, it has one less period of discounting (for present value) or one more period of compounding (for future value) applied to it, so both its present value and future value are higher.
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Can an annuity have a changing interest rate?
No. The annuity calculation formulas assume a constant interest rate throughout the term; if the rate varies from period to period, the cash flow stream cannot be valued as an annuity.q
