Annuities
buttons.Calculating Annuities
Annuities Defined
To understand how to calculate an annuity, it is useful to understand the variables that impact the calculation. An annuity is essentially a loan, a multi-period investment that is paid back over a fixed (or perpetual, in the case of a perpetuity) period of time. The amount paid back over time is relative to the amount of time it takes to pay it back, the interest rate being applied, and the principal (when creating the annuity, this is the present value).
Generally
speaking, annuities and perpetuities will have consistent payments over
time. However, it is also an option to scale payments up or down, for
various reasons.
Variables
This gives us six simple variables to use in our calculations:
- Present Value (PV) – This is the value of the annuity at time 0 (when the annuity is first created)
- Future Value (FV) – This is the value of the annuity at time n (i.e., after the life of the annuity).
- Payments (A) – Each period will require individual payments that will be represented by this amount.
- Number of Payments (n) - The number of payments (A) will equate to
the number of expected periods of payment over the life of the annuity.
- Interest (i) – Annuities occur over time, and thus, a given rate of return (interest) is applied to capture the time value of money.
- Growth (g) – For annuities that have changes in payments, a growth rate is applied to these payments over time.
Calculating Annuities
With all of the inputs above at hand, it is fairly simple to value various types of annuities. Generally, investors, lenders, and borrowers are interested in the present and future value of annuities.
Present Value
The present value of an annuity can be calculated as follows:
\(PV(A)=\dfrac{A}{i} ⋅ [1−\dfrac{1}{(1+i)^n}]\)
For a growth annuity (where the payment amount changes at a
predetermined rate over the life of the annuity), the present value can
be calculated as follows:
\( PV=\dfrac{A}{(i−g)}[1−(\dfrac{1+g}{1+i})^n] \)
Future Value
The future value of an annuity can be determined using this equation:
\(FV(A)=A⋅\dfrac{(1+i)^n−1}{i}\)
In a situation where payments grow over time, the future value can be determined using this equation:
\(FV(A)=A⋅\dfrac{(1+i)^n−(1+g)^n}{i−g}\)
Various Formula Arrangements
It is also possible to use existing information to solve for missing information.
Which is to say, if you know interest and time, you can solve for the following (given the following):
| Find | Given | Formula |
|---|---|---|
| Future value (F) | Present value (P) | \( F = P \cdot (1 + i)^n \) |
| Present value (P) | Future value (F) | \( P = F \cdot (1 + i)^{-n} \) |
| Repeating payment (A) | Future value (F) | \( A = F \cdot \frac{i}{(1 + i)^n - 1} \) |
| Repeating payment (A) | Present value (P) | \( A = P \cdot \frac{i(1 + i)^n}{(1 + i)^n - 1} \) |
| Future value (F) | Repeating payment (A) | \( F = A \cdot \frac{(1 + i)^n - 1}{i} \) |
| Present value (P) | Repeating payment (A) | \( P = A \cdot \frac{(1 + i)^n - 1}{i(1 + i)^n} \) |
| Future value (F) | Gradient payment (G) | \( F = G \cdot \frac{(1 + i)^n - in - 1}{i^2} \) |
| Present value (P) | Gradient payment (G) | \( P = G \cdot \frac{(1 + i)^n - in - 1}{i^2(1 + i)^n} \) |
| Fixed payment (A) | Gradient payment (G) | \( A = G \cdot \left[ \frac{1}{i} - \frac{n}{(1 + i)^n - 1} \right] \) |
| Future value (F) | Exponentially increasing payment (D) | \( F = D \cdot \frac{(1 + g)^n - (1 + i)^n}{g - i} \quad (\text{for } i \neq g) \) |
| Increasing percentage (g) | \( F = D \cdot \frac{n(1 + i)^n}{1 + g} \quad (\text{for } i = g) \) | |
| Present value (P) | Exponentially increasing payment (D) | \( P = D \cdot \frac{\binom{1+g}{1+i}^n - 1}{g - i} \quad (\text{for } i \neq g) \) |
| Increasing percentage (g) | \( P = D \cdot \frac{n}{1 + g} \quad (\text{for } i = g) \) |
Annuities Equations This table is a useful way to view the calculation of annuities variables from a number of directions. Understanding how to manipulate the formula will underline the relationship between the variables, and provide some conceptual clarity as to what annuities are.
Key Points
- Annuities are basically loans that are paid back over a set period of time at a set interest rate with consistent payments each period.
- A mortgage or car loan are simple examples of an annuity. Borrowers agree to pay a given amount each month when borrowing capital to compensate for the risk and the time value of money.
- The six potential variables included in an annuity calculation are the present value, the future value, interest, time (number of periods), payment amount, and payment growth (if applicable).
- Through integrating each of these (excluding payment growth, if payments are consistent over time), it is simple to solve for the present of future value of a given annuity.
Term
- Annuity – A right to receive amounts of money regularly over a certain fixed period in repayment of a loan or investment (or perpetually, in the case of a perpetuity).