# Calculator of the Present Value of a Growing Perpetuity

Instructions: Use this Growing Perpetuity calculator to compute the present value ($$PV$$) of a growing perpetuity by indicating the yearly payment ($$D$$), the interest rate ($$r$$), the growth rate ($$g$$) and the payment received right now ($$D_0$$), if any (leave empty otherwise): Yearly Payment $$(D)$$ = Interest Rate $$(r)$$ = Growing Rate $$(g)$$ = Amount paid now (if any, leave blank otherwise) $$(D_0)$$ =

## Calculator of the Present Value of a Growing Perpetuity

More about the this growing perpetuity calculator so you can better understand how to use this solver: The present value ($$PV$$) of a growing perpetuity payment $$D$$ depends on the interest rate $$r$$, the growth rate $$g$$ and whether or not the first payment is right now or at the end of the year. 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 perpetuity, and its present value ($$PV$$) can be computed using the following growing perpetuity formula:

$PV = \displaystyle \sum_{n = 1}^{\infty} \frac{D \times (1+g)^{n-1}}{(1+r)^n} = \frac{D}{r-g}$

The derivation of the perpetuity formula is related with the calculation of a geometric series with a ratio that has an absolute value that is less than 1, which holds in this case.

On the other hand, if the first payment $$D_0$$ is made now, then we have a growing perpetuity due, and its present value ($$PV$$) can be computed using the following formula.

$PV = D_0 + \displaystyle \sum_{n = 1}^{\infty} \frac{D \times (1+g)^{n-1}}{(1+r)^n} = D_0 + \frac{D}{r-g}$

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

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