**Initial payment of growing annuity = FV [(r – g) ÷ { (1 + r)**^{n}** – (1 + g)**^{n}** } ]**

FV = future value

r = rate

g = growth rate

n = # of periods

Growing annuity payment with future value formula is utilized to determine the first cash flow of a growing annuity. This formula assumes the investment is a series of cash flow which grows at a constant rate. Another name for a growing annuity is ‘increasing annuity’ among groups of people.

An essential concept to keep in mind when using Growingannuity payment formula with future value is that it only calculates the firstpayment or cash flow for an annuity. All other subsequent cash flows must becalculated using a different approach and each cash flow separately. Detailsfor this concept are available below.

While on the other side growing annuity payment with future value formula is only operational where the future value of the growing annuity is known. Finalizing either to use growing annuity payment formula with future value or to use growing annuity payment formula with present value varies with case to case. If the balance of annuity is increasing then growing annuity payment with FV formula is used while on the other side if balance is decreasing with time then growing annuity payment with PV will be the used.

## How growing annuity payment with future value is useful for analysts

To understand the increasing and decreasing balance conceptlet’s consider an example. An individual is looking to calculate the cashoutflow per annum from an interesting account having $25,000 balance for the next7 years. The balance will earn up to 6% per year.

Moreover, the individual wants to increase cash outflows with3% per annum to consider inflation. As the balance is decreasing with time andwe know the present value (i.e. $25,000) therefore, we will use the growingannuity payment with present value formula.

Similarly, if an individual is looking to determine theamount to save per year in a saving account in order to get an ending balanceof $25,000 after 7 years. The deposit amount will increase by 3% per annum and therate of return will be 6% per annum. In this scenario, the balance isincreasing with time and future value (i.e. $25,000 is known) therefore thegrowing annuity payment formula will be used.

### How growing annuity payment with FV formula is induced?

If you look at the future value of a growing annuityformula you will see the portion of growing annuity payment with FV formulain it;

**Future value of growing annuity = P [(1 + r)**^{n}** – (1 + g)**^{n}**] ÷ [r – g]**

As we want to calculate the initial payment ** P**or say the present value of the growing annuity in the above equationhence we can rearrange it as;

**P = FV ÷ [ {(1 + r)**^{n}** – (1 + g)**^{n}**} ÷ {r – g} ]**

Now the above equation can bemultiplied with reciprocal of the denominator to get the equation available atthe top of the page.

### Calculating subsequent growing annuity payments

As we discussed earlier the growing annuity payment with FVformula only calculates the first cash flow. All the subsequent payments mustbe calculated separately as the payments are growing with time with a constantrate so each cash flow will be different. Any subsequent future payment can becalculated by using the formula below;

**P**_{t}** = P**_{1}** (1 + g)**^{t –1}

**P**_{t}** = payment at time t**

**P**_{1}** = first cash flow**

Combining growing annuity payment with future value formulawith above equation gives us an alternative for calculating subsequent paymentsat any specific time ** t,** see below;

**P**_{t}** = FV [ (r – g) ÷ { (1 + r)**^{n}** – (1 + g)**^{n}** } ] × (1 + g)**^{t –1}