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The present value of a growing annuity formula calculates the current, present day, value of a series of future periodic payments that are growing at a proportionate rate.

Put simply, a growing annuity is a series of payments that increase in amount with each payment. The payments are made periodically in equal amounts at regular intervals and can be made annually, semi-annually, quarterly, monthly or weekly.

For example, if the rate of growth is 10% that means that the first payment will be 10%, the second payment will be 10% more than the first one, the third payment will be 10% more than the second and so on.  If the first payment is $100, the first cash flow is $100, the second is $100 × (1+10%) = $110, the third $110 × (1+10%) = $121. This can be described with the following formula:

Cash\;flow_{n} = Cash\;flow_{1} \times (1 + Growth\;rate)^{n=1}

There are two types of annuities distinguished by the timing of payments:

  • Ordinary annuity – the payments are made at the end of a period.
  • Annuity due – the payments are made at the begging of a period.

How to calculate present value of growing annuity?

Present value of growing annuity is calculated as:

PV = \frac{P_{1}}{r-g} \times \bigg[1 - \Big(\frac{1+g}{1+r}\Big)^{n}\bigg]

Where P1 = first payment, r = interest rate per period, g = growth rate and n = number of periods.

Examples of present value of growing annuity calculation

Example 1

  • An individual pays $1,200 to their bank account every year for 5 years.
  • The interest rate is 3%.
  • The growth rate is 2%.

What is the present value of their payments?

PV = \frac{\$1,000}{0.03 - 0.02} \times \bigg[1 - \Big(\frac{1+0.02}{1+0.03}\Big)^{5}\bigg] = \$5,713.22

Example 2

  • An individual pays $100 to their bank account every month for 5 years.
  • The interest rate is 3% (0.0025 monthly).
  • The growth rate is 2% (0.0017 monthly).
  • In this example the rates are divided by 12 because the payments are made monthly.

What is the present value of their payments?

PV = \frac{\$100}{0.03 - 0.02} \times \bigg[1 - \Big(\frac{1+0.02}{1+0.03}\Big)^{60}\bigg] = \$5,998.43

Notice that this individual pays the same amount of money every year in these two examples, but because they pay only once a year in the first example and 12 times a year in the second example, the PV of the annuity is growing faster, because the payments are multiplied by the growth rate 12 times more.

Present value of growing annuity due

Annuity due is very similar to a regular annuity. The only difference is that the payments are made at the beginning of a period. Therefore, present value of a growing annuity due can be calculated as below.

If the interest rate and growth rate are not equal, the present value formula is:

PV = \frac{P_{1}}{r-g} \times \bigg[1 - \Big(\frac{1+g}{1+r}\Big)^{n}\bigg] \times (1+r)

If the interest rate and the growth rate are equal, then the formula is:

PV = P_{1} \times \frac{\bigg[1 - \Big(\frac{1+g}{1+r}\Big)^{n} \bigg]}{r-g} \times (1+r)

Example of present value of growing annuity due calculation

  • An individual pays $1,000 to their bank account every year for 5 years.
  • The payments are made at the beginning of each year.
  • The interest rate is 3%.
  • The growth rate is 2%.

What is the present value of their payments?

PV = \frac{$1,200}{0.03-0.02} \times \bigg[1 - \Big(\frac{1+0.02}{1+0.03}\Big)^{5}\bigg] \times (1+0.03) = \$5,884.62