# Calculator of the Present Value of a Growing Annuity

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Instructions:
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Compute the present value (\(PV\)) of a growing annuity by indicating the yearly payment (\(D\)), the interest rate (\(r\)), the number of years (\(n\)), the growth rate (\(g\)) and the payment received right now (\(D_0\)), if any (leave empty otherwise):

## Calculator of the Present Value of a Growing Annuity

More about the
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this growing annuity calculator
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so you can better understand how to use this solver: The present value (\(PV\)) of
a growing annuity payment \(D\) depends on the interest rate \(r\), the growth rate \(g\),
the number of years the payment is received for \(n\), and whether or not the first payment is
right now or at the end of the year.

### How do you calculate the present value of a growing annuity?

If the first payment of a perpetual stream of payments of \(D\) is made at the end of the year, we then have a regular growing annuity, and its present value (\(PV\)) can be computed using the following formula:

### Present value of growing annuity formula

\[ PV = \displaystyle \sum_{k = 1}^{n} \frac{D \times (1+g)^{k-1}}{(1+r)^k} = \frac{D}{r-g}\left( 1 - \left( \frac{1+g}{1+r} \right)^n \right) \]On the other hand, if the first payment \(D_0\) is made now, then we have a growing annuity due, and its present value (\(PV\)) can be computed using the following formula.

\[ PV = D_0 + \displaystyle \sum_{k = 1}^{n} \frac{D \times (1+g)^{k-1}}{(1+r)^k} = \frac{D}{r-g}\left( 1 - \left( \frac{1+g}{1+r} \right)^n \right)\]### How do you calculate present value of a growing annuity in Excel?

Excel has an splendid set of financial functions to compute all you need to do with finance formulas. The only thing is that there is some kind of a learning curve to it, but once you learn how to use them, you find yourself in a good position.

Excel will provide you with "PV", "FV" and "RATE" which allows you to compute present values, future values and rates. For the calculation of the the present value of a growing annuity, you will need to use one flavor of the "PV" function.

### Present Value of a Growing Annuity Calculator Example

**Question**: You expect to retire within soon, and you expect to live 25 years after retirement. You want to receive, $20,000
a year at the end of those 25 years, but you expect some inflation in the future, and you would like your annual pension
to grow 2% per year. How much money you need to put in the bank today to fund that pension, if the bank gives
you an interest rate of \(r = 3%\) annually?

Solution:

Hence, the present value of the growth annuity is calculated using the following formula:

\[ \begin{array}{ccl} PV & = & \displaystyle \sum_{k = 1}^{n} \frac{D \times (1+g)^{k-1}}{(1+r)^k} \\\\ \\\\ & = & \displaystyle \frac{D}{1+r} + \frac{D \times (1+g)}{(1+r)^2} +...+ \frac{D \times (1+g)^{n-1}}{(1+r)^n}\\\\ \\\\ & = & \displaystyle \frac{D}{r-g} \left( 1- \left(\frac{1+g}{1+r}\right)^n \right) \\\\ \\\\ & = & \displaystyle \frac{20000}{0.04 - 0.02} \left( 1- \left(\frac{1+0.02}{1+0.04}\right)^{25} \right) \\\\ \\\\ & = & 384581.13 \end{array} \]Therefore, we get that the present value for this growth annuity type is \( \text{\textdollar}384,581.13 \).

### Other financial calculators you may be interesting in

If you are trying to compute the present value of an annuity in which the yearly payment remains constant, use the following calculator of a regular annuity , or simply use \(g = 0\).

Now, if what you need is a cash flow that comes at perpetuity, you may find it useful to use this present value of a perpetuity calculator.

On the other hand, if you need to deal with a growing perpetuity, you may find it useful to use this present value of an growing perpetuity calculator.