How to calculate the present and future value of annuities
In the world of finance, an annuity is a contract between you and a life insurance company in which you give the company a lump sum or series of payments, and in return, the insurer promises to provide you with a stream of income, either now or in the future.
But annuities can also be more of a general concept that describes anything that’s broken up into a series of payments. For example, a lottery winner may opt to receive a series of payments over time instead of a single lump sum distribution. This can also be called an annuity.
Two terms related to annuities are present value and future value.
Here’s what you need to know.
Present value of an annuity vs. future value of an annuity: What’s the difference?
While future value tells you how much a series of investments will be worth in the future, present value takes the opposite approach. It calculates the current amount of money you’d need to invest today to generate a stream of future payments, considering a specific interest rate.
The future value should be worth more than the present value since it’s earning interest and growing over time.
Ordinary annuity vs. annuity due: What’s the difference?
When using the general term “annuity,” there are two types of annuities: ordinary and period due.
- Ordinary annuity: Payments are due at the end of the period.
- Annuity due: Payments are due at the beginning of the period.
This seemingly minor difference in timing can impact the future value of an annuity because of the time value of money. Money received earlier allows it more time to earn interest, potentially leading to a higher future value compared to an ordinary annuity with the same payment amount.
You can use an online calculator to figure both the present and future value of an annuity, so long as you know the interest rate, payment amount and duration.
How to calculate future value of an ordinary annuity
The future value tells you how much a series of regular investments will be worth at a specific point in the future, considering the interest earned over time.
In simpler terms, it tells you how much money the annuity will be worth after all the payments are received and compounded with interest.
It’s a tool for planning how much you’ll accumulate by consistently contributing to a retirement plan or understanding the total repayment amount for a loan with regular installments.
Imagine you plan to invest a fixed amount, say $1,000, every year for the next five years at a 5 percent interest rate. The time value of money comes into play here. The first $1,000 you invest earns interest for a longer period compared to subsequent contributions. So, the earlier contributions have a greater impact on the final value.
To calculate the future value of these regular investments, we can use the following formula for ordinary annuities:
FV = C x [((1 + i)^n – 1) / i]
where:
FV = Future Value
C = Cash flow per period (your regular investment amount – $1,000 in this example)
i = Interest rate (expressed as a decimal)
n = Number of compounding periods (number of years you invest)
This formula considers the impact of both regular contributions and interest earned over time. By using this formula, you can determine the total value your series of regular investments will reach in the future, considering the power of compound interest.
Using the example above:
FV = $1,000 x [((1 + 0.05)^5 – 1) / 0.05]
Solving for FV:
Calculate (1 + 0.05)^5:
(1 + 0.05)^5 = 1.2762815625
Subtract 1 and divide by the interest rate (0.05):
(1.2762815625 – 1) / 0.05 = 5.52563125
Multiply the result by the cash flow per period (C):
$1,000 x 5.52563125 ≈ $5,525.63
Therefore, the future value of your regular $1,000 investments over five years at a 5 percent interest rate would be about $5,525.63.
Note: This calculation assumes equal annual contributions and compound interest applied at the end of each year. In reality, accumulated interest might be different depending on how often interest is compounded.
How to calculate the present value of an ordinary annuity
Present value of an annuity refers to how much money must be invested today in order to guarantee the payout you want in the future.
Essentially, it asks: How much money do you need to invest now to generate a specific amount of money down the road?
Using the same example of five $1,000 annual payments, the present value calculation would determine the single upfront investment required to generate this future income stream, assuming a certain interest rate, in this case, 5 percent.
The formula for calculating the present value of an ordinary annuity is:
PV = C x [(1 – (1 + i)^-n) / i]
where:
PV = Present Value
C = Cash flow per period (amount of each annual payment – $1,000 in this example)
i = Interest rate (expressed as a decimal)
n = Number of compounding periods (number of years)
By plugging in the values and solving the formula, you can determine the amount you’d need to invest today to receive the future stream of payments. In this example, with a 5 percent interest rate, the present value might be around $4,329.48.
This concept helps you compare future income streams with current investment opportunities, allowing you to make informed financial decisions.
If you own an annuity, the present value represents the cash you’d get if you cashed out early, before any fees, penalties or taxes are taken out. You can usually find the current present value of your annuity on your policy statements or your online account.
How to calculate the future value of an annuity due
Now let’s explore annuity due, where payments happen at the beginning of each period.
This slight difference in timing impacts the future value because earlier payments have more time to earn interest. Imagine investing $1,000 on Oct. 1 instead of Oct. 31 — it gains an extra month of interest growth.
To account for this time advantage, the formula for the future value of an annuity due is:
FVAnnuity Due = C x [((1 + i)^n – 1) / i] x (1 + i)
where:
FVAnnuity Due = Future Value of the annuity due
C = Cash flow per period (your regular investment amount)
i = Interest rate (expressed as a decimal)
n = Number of compounding periods (number of years)
This formula incorporates both the time value of money within the period and the additional interest earned due to earlier payments.
Using the same example:
C = $1,000 (regular investment)
i = 0.05 (5 percent interest rate)
n = 5 (number of years)[((1 + 0.05)^5 – 1) / 0.05] ≈ 5.53
Multiply this result by (1 + i):
5.53 x (1 + 0.05) ≈ 5.8019
Therefore, the future value of your annuity due with $1,000 annual payments at a 5 percent interest rate for five years would be about $5,801.91.
Remember: Annuity due results in a higher future value compared to an ordinary annuity because of earlier payments.
How to calculate the present value of an annuity due
Similar to the future value, the present value calculation for an annuity due also considers the earlier receipt of payments compared to ordinary annuities. This reduces the present value needed to generate the same future income stream.
Think about it this way: Having $1,000 today is more valuable than having $1,000 a year from now because you can invest it and earn interest. So for an annuity due where you receive payments at the beginning of each period, you’d need to invest less today to generate the same future cash flow compared to an ordinary annuity with payments at the end.
The formula for the present value of an annuity due is:
PVAnnuity Due = C x [(1 – (1 + i)^-n) / i] x (1 + i)
where:
PVAnnuity Due = Present Value of the annuity due
C = Cash flow per period (your regular payment amount)
i = Interest rate (expressed as a decimal)
n = Number of compounding periods (number of periods)(1 + 0.05)^-5 ≈ 0.783526
Subtract the result from 1 and divide by the interest rate (0.05):
[1 – (1 + 0.05)^-5] / 0.05 ≈ 4.33
Multiply this result by the cash flow per period (C) and then by (1 + i):
$1,000 x 4.33 x (1 + 0.05) ≈ $4,545.95
So the present value you’d need to invest today to cover five $1,000 payments, assuming a 5 percent interest rate, would be about $4,545.95.
As a reminder, this calculation assumes equal monthly payments and compound interest applied at the beginning of each month. In reality, interest accumulation might differ slightly depending on how often interest is compounded.
By calculating the present value, you can understand the effective cost in today’s dollars, potentially helping you with budgeting or financial planning.
Bottom line
Understanding annuities, both in concept and through the calculations of present and future values, can help you make informed decisions about your money. There are tools available to simplify the calculations for both the present and future value of annuities, ordinary or due. These online calculators typically require the interest rate, payment amount and investment duration as inputs.
