# Growing Annuity

A growing annuity is a finite stream of equal cash flows that occur after equal interval of time and grow at a constant rate. It is also called an increasing annuity. It differs from ordinary annuity and annuity due in that the periodic cash flows in a growing annuity grow at a constant rate but stays constant in an annuity.

Many cash flows stream constitute a growing annuity. For example, rental contracts may stipulate an increase in annual rental at a constant rate. The multi-stage dividend growth model might include a stage in which a company’s dividend may be expected to grow at a constant rate over a certain period.

## Present Value of a Growing Annuity

The present value of a growing annuity can be calculated by (a) finding each cash flow by growing the first cash flow at the given constant rate, (b) individually discounting each cash flow to time 0 and (c) summing up the component present values.

It can also be worked out directly by using the following formula:

$${\rm \text{PV}} _ {\text{GA}}=\frac{\text{C}}{\text{r}-\text{g}}\times\left(\text{1}-\left(\frac{\text{1}+\text{g}}{\text{1}+\text{r}}\right)^\text{n}\right)$$

The present value of a growing annuity due can be worked out by multiplying the above equation with (1 + r).

$${\rm \text{PV}} _ {\text{GAD}}=\frac{\text{C}}{\text{r}-\text{g}}\times\left(\text{1}-\left(\frac{\text{1}+\text{g}}{\text{1}+\text{r}}\right)^\text{n}\right)\times(\text{1}+\text{r})$$

Where PVGA is the present value of growing annuity, PVGAD is the present value of annuity due, C is the periodic cash flow, r is the periodic discount rate, g is the periodic growth rate and n is the total number of cash flows.

## Future Value of a Growing Annuity

The future value of a growing annuity can be calculated by working out each individual cash flow by (a) growing the initial cash flow at g; (b) finding future value of each cash flow at the interest rate r and (c) then summing up all the component future values.

The future value of a growing annuity can also be calculated by growing the present value of the growing annuity at the interest rate r for n periods. This can be expressed mathematically as follows:

$${\rm \text{FV}} _ {\text{GA}}={\rm \text{PV}} _ {\text{GA}}\times{(\text{1}+\text{r})}^\text{n}$$

Where FVGA is the future value of growing annuity, PVGA is the present value of growing annuity, r is the periodic discount rate and n is the number of cash flows.

We have effectively moved a single value at time 0 i.e. PVGA n number of years in future at the interest rate r.

Substituting the PVGA formula in the above equation, we get the following direct formula:

$${\rm \text{FV}} _ {\text{GA}}=\frac{\text{C}}{\text{r}-\text{g}}\times\left(\text{1}-\left(\frac{\text{1}+\text{g}}{\text{1}+\text{r}}\right)^\text{n}\right)\times{(\text{1}+\text{r})}^\text{n}$$

This can be simplified as follows:

$${\rm \text{FV}} _ {\text{GAD}}=\frac{\text{C}}{\text{r}-\text{g}}\times\left({(\text{1}+\text{r})}^\text{n}-{(\text{1}+\text{g})}^\text{n}\right)$$

Where FVGAD is the future value of growing annuity due.

The future value of a growing annuity due can be worked out by multiplying the above expression with (1 + r).

$${\rm \text{FV}} _ {\text{GAD}}=\frac{\text{C}}{\text{r}-\text{g}}\times\left({(\text{1}+\text{r})}^\text{n}-{(\text{1}+\text{g})}^\text{n}\right)\times(\text{1}+\text{r})$$

Where FVGAD is the future value of growing annuity due.

## Example

Your parents want to set-up a college fund for you to fund your 4-year bachelors program. The tuition fee is \$40,000 per semester payable in advance. The tuition fee is expected to grow at 4% and the college fund will earn 8% interest per annum. Calculate the amount the college fund must have when you start college.

You need to work out the present value of the growing annuity due in this case. There are two semesters in a year so periodic growth rate and rate of return are 2% and 4% respectively and there are 8 total semesters.

The following function works out the balance needed:

$${\rm \text{PV}} _ {\text{GAD}}=\frac{\text{\40,000}}{\text{4%}-\text{2%}}\times\left(\text{1}-\left(\frac{\text{1}+\text{2%}}{\text{1}+\text{4%}}\right)^\text{8}\right)\times(\text{1}+\text{4%})=\text{\299,270}$$