# Present Value Factor

Present value factor is the equivalent value today of $1 in future or a series of $1 in future. A table of present value factors can be used to work out the present value of a single sum or annuity.

There are multiple ways to find present value of a single value or an annuity: using the present value formula, using Microsoft Excel PV function, using some financial calculator or using present value tables. Present value tables list present value factor for multiple interest rates and time periods. The interest rates are normally listed in the top row and time periods are tabulated in the first column and we need to find the value that is at the intersection of our given interest rate and time period.

## Formula

Present value factors can be manually calculated using the following formulas:

$$ Present\ Value\ Factor\ of\ Single\ Sum=\frac{1}{{(1+\frac{r}{m})}^{n\times m}} $$

$$ Present\ Value\ Factor\ of\ Annuity=\frac{1-{(1+\frac{r}{m})}^{-n\times m}}{\frac{r}{m}} $$

$$ Present\ Value\ Factor\ of\ Annuity\ Due=\frac{1-{(1+\frac{r}{m})}^{-n\times m}}{\frac{r}{m}}\times(1+\frac{r}{m}) $$

Where **r** is the annual percentage interest rate, **n** is the number of years and **m** is the number of compounding periods per year.

## Example

If the interest rate is 10%, the present value factor for an amount received 10 years with semi-annual compounding is 0.3769:

$$ Present\ Value\ Factor\ of\ Single\ Sum\\=\frac{1}{{(1+\frac{10\%}{2})}^{10\times2}}\\=0.3769 $$

We just need to multiply the factor with the amount received to get the relevant present value:

$$ Present\ Value\ of\ $5,000\\=$5,000\times0.3769\\=$1,884 $$

The present value factor for an (ordinary) annuity with monthly compounding for 5 years at 10% APR is 47.0654:

$$ Present\ Value\ Factor\ of\ Annuity\\=\frac{1-{(1+\frac{10\%}{12})}^{-5\times12}}{\frac{10\%}{12}}\\=\ 47.0654\ $$

The present value factor for an annuity due with monthly compounding for 5 years at 10% APR is 47.4576:

$$ Present\ Value\ Factor\ of\ Annuity\ Due\\=\frac{1-{(1+\frac{10\%}{12})}^{-5\times12}}{\frac{10\%}{12}}\times\left(1+\frac{10\%}{12}\right)\\=\ 47.4576\ $$