# Present Value of a Perpetuity Calculator

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Instructions:
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Use this Perpetuity Calculator to compute the present value (\(PV\)) of a perpetuity by indicating the yearly payment (\(D\)), the interest rate (\(r\)), and the payment received right now (\(D_0\)), if any (leave empty otherwise):

## Present Value of a Perpetuity Calculator

More about the
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this perpetuity calculator
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so you can better understand how to use this solver

### How do you compute the present value of a perpetuity?

The present value (\(PV\)) of a perpetuity payment \(D\) depends on the interest rate \(r\) 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 perpetuity, and its present value (\(PV\)) can be computed using the following formula:

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

\[ PV = D_0 + \displaystyle \sum_{n = 1}^{\infty} \frac{D}{(1+r)^n} = \frac{D}{r} \]Notice that the above formula does not apply if there a growth in the payment. In that case, you to use a specific perpetuity formula with growth.

If you are trying to compute the present value of a perpetuity in which the yearly payment increases, use the following calculator of growing perpetuities . For payments with infinite number of payments, you can use this present value of perpetuity calculator .

### Example

**Question**: Find the present value of a perpetuity, which pays out $12,000 every year, at the end of the year, if the
discount rate is 3.5%.

Solution:

This is the information we have been provided with:

• The yearly payment is \(D = 12000\), the yearly interest rate is \(r = 0.035\).

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

\[ \begin{array}{ccl} PV & = & \displaystyle \sum_{n = 1}^{\infty} {\frac{D}{(1+r)^n}} \\\\ \\\\ & = & \displaystyle \frac{D}{1+r} + \frac{D}{(1+r)^2 } + ... \\\\ \\\\ & = & \displaystyle \frac{D}{r} \\\\ \\\\ & = & \displaystyle \frac{12000}{0.035} \\\\ \\\\ & = & 342857.14 \end{array} \]Therefore, we get that the present value for this annuity type is \( \text{\textdollar}342857.14 \).