# INTEREST RATE(r) OF LUMP SUM CALCULATOR

Enter the present value, future value and the number of years. Don't forget to choose the compounding frequency.

Compounded annually Semi-annually Quarterly Monthly weekly daily

## Frequently Asked Questions on Interest rate in Lump sum

### Formula for calculating the rate of interest of a lump sum

Assuming present value, future value and the number of years is known, the annual rate of interest can be computed as follows:

$$r=m ×\left(\left(\frac{FV}{PV}\right)^{\frac{1}{nm}}-1\right)$$

Where,

m is the compounding frequently (yearly, semi-annually, quarterly, monthly, weekly or daily

FV is the future value of the lump sum

PV is the present value of the lump sum

## Proof of the rate of interest of lump sum formula

We can proof that the formula of annual rate of interest rate of using the future value of lump sum formula.

We know that FV is calculated as follow:

$$FV=PV\left(1+\frac{r}{m}\right)^{n×m}$$

To obtain the formula for rate of interest, we simply make r the subject of the formula

$FV=PV\left(1+\frac{r}{m}\right)^{n×m}$

First, we divide both side by PV

$\frac{FV}{PV}=\left(1+\frac{r}{m}\right)^{n×m}$

Next, we multiply the power of both sides by $\frac{1}{nm}$

$\left(\frac{FV}{PV}\right)^{\frac{1}{n×m}}=1+\frac{r}{m}$

Now, let's substract one from both sides

$\left(\frac{FV}{PV}\right)^{\frac{1}{nm}}-1=\frac{r}{m}$

Finally, we multiply both sides by m

$m\left(\left(\frac{FV}{PV}\right)^{\frac{1}{nm}}-1\right)=r$

$r=m\left(\left(\frac{FV}{PV}\right)^{\frac{1}{nm}}-1\right)$

If you like, you can open the bracket,

$r=m\left(\frac{FV}{PV}\right)^{\frac{1}{n×m}}-m$

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