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, = growth rate and n = number of periods.

### Examples of present value of growing annuity calculation

• 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?

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