What is an annuity?
In the previous discussion of time value of money (present value and future value), we derived the formula for a single payment that would be discounted or compounded respectively over a period of time (n) at an interest rate r. What if we have a series of payments of a fixed amount for a specified number of periods? Then we need to use a present or future value of an annuity formula.
I am sure you may be asking yourself what an annuity is. Simply put, an annuity is a stream of equal annual or periodic cash flows. It is believed that the stream of payments occurs at the end of the period, but occasionally, they may occur at the beginning of the period. If the stream of payments occurs at the end of the period, we call it a regular or deferred annuity. If on the other hand they occur at the beginning of the period, we call it an annuity due. The difference between a regular annuity and annuity due is that in the computation of an annuity due, each receipt or payment is shifted back by one period. Don’t worry, we would have a look at their formulae and also take examples. Let us start with the future value of a deferred or regular annuity.
Future Value of a deferred or regular annuity
Let us assume that the receipts of equal amounts of cash from a pension fund are deposited in a savings account at each time period. You receive a payment C each year from your pension, and the first year’s payment and each subsequent payment is transferred into a savings account at an interest r for a period of 6 years. To illustrate how the future value would be calculated, we draw up a table below.
| year | cashflow | years to end in | future value |
| 0 | – | 6 | – |
| 1 | C | 5 | C(1+r)5 |
| 2 | C | 4 | C(1+r)4 |
| 3 | C | 3 | C(1+r)3 |
| 4 | C | 2 | C(1+r)2 |
| 5 | C | 1 | C(1+r)1 |
| 6 | C | 0 | C |
From the above, we see that the annuity pays C dollars 6 times. Time 0 is a specific point n time, nothing happens there. We start our counting from 0 to 1, and year 1 above would be the end of period 1. The cash flow received in year 1 is at the end of year 1, so it can be placed in the savings account for only 5 years within the period. Those occurring at the end of the second, third, fourth and fifth years will be in the account for 4,3,2 and 1 year(s) respectively. The cash flow for year 6 will not be ripe for interest since it is placed at the end of year 6.
The idea is that the first payment is placed in the savings account for five years and it is compounded for n period which in this case would be 5. We take it separately from the others and we compound the single payment for 5 years at the interest rate r. For the second year, the cash flow is deposited at the end of year 2 which means it cannot receive interest for the previous year 1 and the current year 2. So this particular payment has 4 years to maturity. That means this single payment can be compounded and earn interest for the next four years. We thereby take the period (n) for which it would be compounded to as 4 years.
The mathematical formula for finding the future value of a regular annuity is:
Where FVA = Future Value of an annuity
A = Amount of regular stream of payments
n = number of years to be deposited
r = interest rate
Let us take an example: Tayo just got laid off from work and he plans to start a business in five years’ time. The money for the startup would be gotten from the regular payment of his pension of $2,000 annually in a savings account paying 8% annual compound interest rate. How long would Tayo have to start his business at the end of the 5 year waiting period?
Using the formula we have:
Where
FVA = ?
A = $2,000
r = 8%
n = 5 years
2000 (5.866)
= $11,732
This means that Tayo would have $11732 to start his business at the end of year 5.
Using the table, we can derive it thus:
| year | Amt | No of yrs | FVIFA @ 8% | FV | |
| compounded | |||||
| 0 | – | 5 | 1.469 | – | |
| 1 | 2000 | 4 | 1.360 | 2720 | |
| 2 | 2000 | 3 | 1.260 | 2520 | |
| 3 | 2000 | 2 | 1.166 | 2332 | |
| 4 | 2000 | 1 | 1.08 | 2160 | |
| 5 | 2000 | 0 | 1 | 2000 | |
Future Value of an Annuity due
Remember we said that an annuity which starts the stream of equal payments at the beginning of the year is an annuity due. Supposing we take the previous example of the receipts of equal amounts of cash from a pension fund deposited in a savings account at each time period for 6 years, and you receive a payment C which starts at the beginning of each of the 6 year period at an interest rate of r.
To illustrate how the future value will be, we can draw up a table:
| year | cash flow | years to end : n | Future Value |
| 0 | C | 6 | C (1+r)6 |
| 1 | C | 5 | C (1+r)5 |
| 2 | C | 4 | C (1+r)4 |
| 3 | C | 3 | C (1+r)3 |
| 4 | C | 2 | C (1+r)2 |
| 5 | C | 1 | C (1+r)1 |
| 6 | – | 0 | – |
Unlike the previous type of annuity (regular annuity) where we start counting interests from year 1, an annuity due starts counting interests from year zero (0). This is because in annuity due, payment starts from the beginning of the year; we believe that our payment starts from a specific point in time (0), and we take 0 as the payment of the first year which has 6 years left to earn interest. This is why we say (1+r)6 for year 0 which represents the first payment. It follows in this order, and we take year 1 on the table as the payment of the second year, and this has 5 years to earn interest. That is why we say C(1+r)5. We do this for the remaining 4 years i.e payments for the third year is taking as 2 on the table, payment for the fourth year is taken as 3 on the table, payments for the fifth year is taken as 4 on the table, and payment of for the sixth year is taken as 5 on the table. That is why year 6 on the table has no payment C.
C starts from 0 on this table because from the moment the first of the series of payment is made, it can start earning interest. Remember, 0 is a point in time, so we take it at the point the payment is made (at the beginning of the year), then we can start earning interest on the payment from then.
The mathematical formula for an annuity due is
Where
A = Stream of regular payments
r = interest rate
n = number of years
let us take an example Mr P is to start receiving a regular payment into his savings account at the beginning of the year for four years. The amount is a $1000 each year into savings account paying 7% annual compound interest. Mr P would like to find out the future value of the payments at the end of the four year period.
Possible solution:
Recall the formula for the future value of an annuity due
Where
FVA(d) = ?
A = $1,000
R = 7 % or 0.07
n= 4 year
FVd = 1000 (4.7505)
FVd = $4,750.7
Using the table, we have:
| year | Amt | No of yrs | FVIFA @ | FV |
| compounded | 7% | |||
| 0 | 1000 | 4 | 1.31079 | 1310.79 |
| 1 | 1000 | 3 | 1.2250 | 1225 |
| 2 | 1000 | 2 | 1.1449 | 1144.9 |
| 3 | 1000 | 1 | 1.07 | 1070.0 |
| 4 | – | 0 | 1 | – |
| 4750.7 | ||||
Note: FVIFA means future value of interest factor of an annuity. In the above table since we are calculating the FVIFA of each payment, we take it ndividually and multiply the single payment with (1+r)n
Where n is the number of years compounded and r is the interest rate.
Present Value of a deferred or regular annuity
Present value of a deferred annuity or regular annuity is the opposite of future value because in calculating the present value, we wish to know how much we need to pay now in order to have a guaranteed stream of equal amount of cash receipts for n number of years at an opportunity cost r. Remember that our aim in the calculation of the future value was to know how much an equal stream of payments would amount to after n number of years with an annual compound interest rate r.
The present value of an annuity can be defined as the lump sum money that one would need to invest today in order to be able to withdraw constant or equal amounts of each period and end up with a balance exactly equal to zero at the end of a specified period. Let us illustrate with an example to make it clearer.
Suppose we wished to make sufficient payments into a mutual fund today to ensure that for the next 4 years an annual receipt of $1000 will be guaranteed. If the cash receipts were to occur at the end of each year and the interest rate is 9% per annum, what should the lumpsum be?
Using the table, we have:
| year | Cashflow | PVIFA @ | PV |
| 9% | |||
| 1 | 1000 | 0.9174 | 917.4 |
| 2 | 1000 | 0.8417 | 841.7 |
| 3 | 1000 | 0.7722 | 772.2 |
| 4 | 1000 | 0.7084 | 708.4 |
| 3239.7 |
The present value for a stream of payments into the mutual fund for 4 years at an interest rate of 9@ would be $3,239.7
Note: To get the PVIFA (Present value interest factor), we use the formula: (1+r)-n
So, for the first year, we are paying into the mutual fund in a year’s time from now (i.e end of year 1). To determine the present value now of that payment, we take our n to be 1. This is because we are actually going to make that payment into the mutual fund. We then go ahead to discount the $1000 to be paid at the underlying interest rate of 9% to determine the present value.
For the payment in the second year, we repeat the same process. We are paying that particular $1000 n year 2 on the table 2 years from now. That means our n in this case would be 2. The next step after identifying n in year 2 is to discount that single $1000 to be paid in year 2 at the interest rate of 9% with n as 2. We simply say: 1000(1.09)-2. We repeat the same process for year 3 and 4.
If you do not want to use the table, then you can make use of the formula which is
Where:
A = Amount
r = interest rate
n = number of years
From the previous question, our
A = 1,000
r = 9% or 0.09
n = 4
So we have :
PVA = 1000 (3.2397)
PVA = $3,239.7
Present Value of an annuity due
The present value of an annuity due can be determined mathematically with the following formula:
Where A = Cash Flow
r = interest rate
n = number of years
Using the previous example in the present value of an ordinary annuity, lets assume that the payment to the mutual fund begins immediately. To solve this, we say:
Where
A= $1000
r = 9% or 0.09
n = 4 years
PVD = 1000 (3.53111)
PVD = $3,5311
Alternatively, we can use the formula:
Where
A 0 = $1000
An = Constant stream of cash flow in subsequent tears.
r = interest rate
n = number of periods less by 1.
In this formula, we see that the first cash flow is added back without discounting. This is because the payment is made immediately; therefore that in its own sense is present value. To offset this, we subtract the first cash flow from the discounting formula by saying (n-1) because n includes the first cash flow.
Recommended Readings
Jae sim., Joel siegel., Schaum’s outline of financial management
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