The future value of annuity formula is used to calculate the value of a series of periodic payments at a future date.
This can be useful in determining how much you would have in future if you know how much you’re able to invest per period. It can also be helpful to compute the total cost of a loan knowing the loan payments since the loan payment formula is based on an ordinary annuity formula.
How to Calculate the Future Value of an Ordinary Annuity?
FV\; of\; Annuity = P\bigg[\frac{(1 + r)^{n} - 1}{r}\bigg]Where P = periodic payment, r = interest rate per period, and n = number of periods.
This formula assumes that the rate and periodic payments do not change. If the interest rate or periodic payments are different at different time periods, the Future Value of Annuity must be calculated as a sum of future values of each payment.
The Future Value of Annuity is always greater than the Present Value of Annuity, this happens due to Time Value of Money, according to which, the same amount of money available at the present time is worth more than the same amount in the future, because money available at the present time can be used to create more money.
How to Calculate the Future Value of Annuity Due?
The Future Value of Annuity Due is calculated as:
FV\; of\; Annuity\; Due = (1 + r ) x P\bigg[\frac{(1 + r)^{n} - 1}{r}\bigg]The difference between FV of Annuity Due and FV of an Ordinary Annuity is the timing of payments. In the case of the Annuity Due, we need to discount the formula one period back since the payments are made at the beginning of each period.
Examples of Future Value of Annuity Calculation
Example 1. Let’s consider you plan to pay $1,000 each year to your savings account for 5 years. The interest rate is 4%. The payments are made at the end of each year.
- FV of P1 =$1,000 × (1+4%)5-1=$1,169.86
- FV of P2 =$1,000 × (1+4%)5-2=$1,124.86
- FV of P3 =$1,000 × (1+4%)5-3=$1,081.60
- FV of P4 =$1,000 × (1+4%)5-4=$1,040.00
- FV of P5 =$1,000 × (1+4%)5-5=$1,000.00
Total: $5,416.32
With this example, we can see how the value of money changes over time. The first periodic payment will grow by 16.9% over 4 years. However, the fourth payment grows only by 4%, and the fifth payment doesn’t change its value, because the payment is made at the end of the fifth year.
This happens due to Compound Interest, which is the amount of interest earned on the investment where the amount earned is reinvested rather than being paid out. By using a compound interest an investor gets a faster increase in the value of capital than by using simple interest, where interest is not added to the principal, so there is no compounding.
Example 2. Let’s consider you plan to pay $1,000 each year to your savings account for 5 years. The interest rate is 4%. This time the payments are made at the beginning of each year.
- FV of P1 =$1,000 × (1+4%)5=$1,216.65
- FV of P2 =$1,000 × (1+4%)4=$1,169.86
- FV of P3 =$1,000 × (1+4%)3=$1,124.86
- FV of P4 =$1,000 × (1+4%)2=$1,081.60
- FV of P5 =$1,000 × (1+4%)1=$1,040.00
Total: $5,632.97
We can conclude, that Annuity Due is more favourable in terms of savings since the total Future Value of Annuity Due is greater than the Future Value of an Ordinary Annuity.