Annuity payments are a very common financial arrangement. An annuity payout calculation is a series of equal periodic payments. The payments occur regularly, year.
Valuing Annuities
Annuities occur in many different financial transactions. Monthly payments on a car loan, a student loan, or a mortgage are annuities. Monthly rent is an annuity. A paycheck, with a fixed salary, is an annuity. Lease, interest and dividend payments are annuities. Any series of equal, periodic payments is an annuity.
The majority of annuities have end-of-period payments. For example, car loans usually require end-of-month payments. If it's a 48-month loan, the first payment is made at the end of the first month and the 48th (and last) is made at the end of month 48. This kind of annuity, where payments occur at the end of each period, is called an ordinary annuity.
Other annuities, such as for a rental, require beginning-of-period payments. For a 12 month apartment lease, the first rent payment is due at the beginning of the first month and the 12th (and last) is due at the beginning of the 12th month. This kind of annuity, where payments occur at the beginning of each period, is called an annuity due.
We know the timing of payments affects value. Therefore, it's critical to know whether you are dealing with an ordinary annuity or an annuity due. We'll start by analyzing the future and present values of an ordinary annuity. Later, we'll show you how to handle an annuity due.
The Future Value of an Annuity
We started our discussion of the time value of money in the previous posts with an example of depositing money in a savings account. Now consider a savings plan for depositing the same amount every period for n periods. How much will you have at the end of the n periods?
Let the periodic cash flow, PMT, be the amount deposited at the end of each time period (that is ,
-------------(1)
CF1=CF2 = ........=CFn=PMT). Figure 1 illustrates the future value of an n-period annuity.
Figure 1
The future value of an n-period annuity.
The future value of an annuity is the total value that will have accumulated at the end of the annuity if the annuity payments are all invested at r per period. The future value of an annuity can be computed using the future value formula to value each payment and then adding up the individual values to get the total. If we start with the last payment and then adding up the individual values to get the total. If we start with the last payment at time t = n and proceed backward to the first payment at time t = 1, the future value of the annuity at time n, FVAn, is
FVAn=PMT(1+r)0 + PMT(1+r)1
+ …………………..+PMT(1+r)n-1
Figure 1 illustrates this calculation. Note that the first payment (at t = 1) earns interest for (n-1) periods, not n periods. Each subsequent payment earns interest for one less period than the previous one. Not that the last payment occurs exactly at the end of the annuity, so it doesn't earn any interest; (1+r)0 = 1.
The equation for FVAn has a PMT in every term on the right-hand side. If the PMT is factored out, the equation can be rewritten as
where ∑ is a summation. This equation can be simplified to
-------------(1)
The quantity in large brackets in equation (1) is called the future-value-annuity factor. The future-value-annuity factor, FVAr,n, is the total future value of $1.00 per period for n periods invested at r per period. The particular values for PMT, n, and r along with equation (1) are all that's needed to determine the future value of the annuity, regardless of the number of payments.
The Present Value of an Annuity
The present value of an annuity is the amount that, if invested today at r per period, could exactly provide equal payments of PMT every period for n periods. The present value of an annuity. PVAn, is simply the sum of the present values of the n individual payments:
The present value of an n-period annuity is illustrated in figure 2. Because the cash flows or payments are all identical, we can rewrite this as
This equation for PVAn can also be simplified; it becomes
-------------(2)
The quantity in large brackets in equation (2) is called the present-value-annuity factor. The present-value-annuity factor, PVAFr,n, is the total present value of an annuity of $ 1.00 per period for n periods discounted at r per period. The particular values for PMT, n, and r are all that is needed to determine the present value of the annuity.
Now suppose you are getting ahead of the game and saving money regularly rather than paying off a loan. The accumulated amount is a future value. We can solve for the amount that must be saved regularly to accumulate a given future value, this time by rearranging equation (1):
To create an amortization schedule, start with the amount borrowed. To this amount add the first period's interest and then subtract the first period's payment. The result is the remaining balance, which is the starting amount for the second period. Repeat this procedure each period until the remainder becomes zero at the end of the last period.
Valuing Annuities Not Starting Today
Sometimes, annuities start at a time other than right away (where the first payment is at t=1). The present value of such an annuity can be computed from the difference between the present values of two other annuities. The first annuity goes from now until the end of the one in question. The second annuity goes from now until the start of the one in question. The difference between the two values is the value of the annuity in question.
As we showed in this figure 2 of the previous post, the present-value factor becomes smaller as n becomes larger. Therefore, later payments in a long annuity add little to the present value of the annuity.
For example, at a required return of 10% per year, the present value of getting $100 in 30 years is only $5.73. It is a mere 85 cents if payment is going to take 50 years. As it turns out, the present value of an annuity has a maximum value, no matter how many payments are expected. That maximum value is the value of a perpetuity.
To examine the present value of a perpetuity, we can start with the present value of an annuity and see what happens when the life of the annuity, n, becomes very large. Let's start by rewriting equation (2), the present-value-of-an-annuity formula:
Written this way, you can see what happens when n becomes large. The first term on the right-hand side of the bottom expression is not affected by n. But the second term gets smaller because (1+r)n gets larger when n increases. As n gets really big, the second term goes to zero. Therefore, the present value of a perpetuity is
The present value of an annuity is the amount that, if invested today at r per period, could exactly provide equal payments of PMT every period for n periods. The present value of an annuity. PVAn, is simply the sum of the present values of the n individual payments:
-------------(2)Figure 2
The present value of an n-period annuity.
The quantity in large brackets in equation (2) is called the present-value-annuity factor. The present-value-annuity factor, PVAFr,n, is the total present value of an annuity of $ 1.00 per period for n periods discounted at r per period. The particular values for PMT, n, and r are all that is needed to determine the present value of the annuity.
Calculating Annuity Payments
We have shown how to compute the present and future value of an annuity, given a set of payments and a discount rate. When you borrow money, the amount is the present value , and the annuity is the loan payments. We can solve for the payments by rearranging equation (2);
Amortizing a Loan
A loan amortization schedule shows how the loan is paid off over time. That is, it shows how the principal (the original amount borrowed) and interest are paid. Because an installment loan is an annuity, an amortization schedule for such a loan shows the relationships among the payments, principal and interest rate.To create an amortization schedule, start with the amount borrowed. To this amount add the first period's interest and then subtract the first period's payment. The result is the remaining balance, which is the starting amount for the second period. Repeat this procedure each period until the remainder becomes zero at the end of the last period.
Calculating the Discount Rate and Number of Annuity Payments
In addition to solving for the payments, future value, or present value of an annuity, we can solve for the discount rate or the number of annuity payments. However, unlike the payments, we cannot always rearrange our equation to solve for these variables. Instead, the equation must be solved using trial and error. So the calculator is especially convenient for calculating these variables because it performs the tedious trial-and-error calculations automatically.Tables 1
A Loan Amortization Schedule
Valuing Annuities Not Starting Today
Sometimes, annuities start at a time other than right away (where the first payment is at t=1). The present value of such an annuity can be computed from the difference between the present values of two other annuities. The first annuity goes from now until the end of the one in question. The second annuity goes from now until the start of the one in question. The difference between the two values is the value of the annuity in question.
Perpetuities
An annuity that goes on forever is called a perpetuity. Although perpetuitites actually exist in some situations, the most important reason for studying them is that they can be used as a simple and fairly accurate approximation of a long-term annuity.As we showed in this figure 2 of the previous post, the present-value factor becomes smaller as n becomes larger. Therefore, later payments in a long annuity add little to the present value of the annuity.
Figure 3
Duplicating the annuity cash flows for a "postponed" annuity.
For example, at a required return of 10% per year, the present value of getting $100 in 30 years is only $5.73. It is a mere 85 cents if payment is going to take 50 years. As it turns out, the present value of an annuity has a maximum value, no matter how many payments are expected. That maximum value is the value of a perpetuity.
To examine the present value of a perpetuity, we can start with the present value of an annuity and see what happens when the life of the annuity, n, becomes very large. Let's start by rewriting equation (2), the present-value-of-an-annuity formula:
Valuing an Annuity Due
The payments for an annuity due occur at the beginning of each period instead of at the end. Because each payment occurs one period earlier, an annuity due has a higher present value than a comparable ordinary annuity. Likewise, an annuity due has a higher future value than a comparable ordinary annuity because each payment has an additional period to compound. In fact, annuity payout calculation a simple way to value an annuity due is to multiply the value of a comparable ordinary annuity by (1 + r).
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