# Present Value of Growing Annuity

The annuity is a periodic payment that is made in the future. The interest rate of the annuity is usually defined based on various parameters and the periodic payment amount is usually fixed. However, in some cases, the annuity may also have a provision for a growth rate to account for the reduction in the value of the money due to inflation or other factors. The period of the annuity and corresponding interest rate, the growth rate is usually clearly defined when the annuity payment is finalized. It should be noted that the interest rate, the growth rate (annuity factor with growth) is defined only for the specific period only or derived from the annual rate which is specified.

Definition

The Present Value of Growing Annuity considers that the fact that the value of money increases over a period of time. It calculates the current value of an annuity which is increasing in value over a period of time at a rate that is defined. In a growing annuity, each annuity payment will be higher than the previous payment based on the growth rate which is defined for the annuity. The period varies depending on the annuity, it may be weekly, monthly, quarterly, six-monthly or annually, and the present value will vary accordingly.

The present value of growing annuity formula

The formula for the present value of an annuity which is growing is specified as follows

Present value = ( P /(r-g) (1 – ( (1+g)/(1+r)^n))

where P = first payment

r = rate per period of the annuity

g = growth rate specified for the annuity

n = number of periods in the annuity

The formula can be rewritten so that it denotes the discounted value of future cash flows from the annuity. It is the discounted value of the cash flows from each annuity payment at present. The formula can be written as a geometric series with (1+g) and (1+r) as a ratio which is common for the annuities. After canceling the common terms, and values, the above formula is obtained.

Explanation

Annuities can be classified into ordinary annuity and annuity due based on the timing of the annuity. In the ordinary annuity, the payment is made at the end of the period for which it is specified. In the annuity due, the payment is made at the beginning of the period for which it is calculated. There is a slight difference in the formula and value of the ordinary annuity when compared to annuity due since there is a time difference in the payment of these annuities.

It should be noted that the amount will vary greatly depending on the period of the annuity through the interest rate and the growth rate of the annuity may be the same. For example, if the annuity has a period of one month, due to compounding, the total amount paid and future value will be higher compared to the same amount if the period of the annuity is one year. It should be noted that for one month, the annual rate is divided by twelve, to derive the monthly rate which is used for calculating the future value.

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Igor has been a trader since 2007. Currently, Igor works for several prop trading companies. He is an expert in financial niche, long-term trading, and weekly technical levels. The primary field of Igor's research is the application of machine learning in algorithmic trading. Education: Computer Engineering and Ph.D. in machine learning. Igor regularly publishes trading-related videos on the Fxigor Youtube channel. To contact Igor write on: igor@forex.in.rs

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