**Interest Rate Calculation Formula **

Here, we’ll use a **future value equation** to solve for how much a deposit today can grow to in the future. There are times, however, when we already know both the future and **present values** and want to know either the **interest rate** or the number of time periods it took to earn the future balance.

We can use the following formula to solve for the interest rate (when it is the unknown):

\[\begin{align}& F{{V}_{n}}=P{{V}_{0}}\times {{\left( 1+i \right)}^{n}} \\& {}^{F{{V}_{n}}}/{}_{P{{V}_{0}}}={{\left( 1+i \right)}^{n}} \\& {{\left[ {}^{F{{V}_{n}}}/{}_{P{{V}_{0}}} \right]}^{{}^{1}/{}_{n}}}={{\left[ {{\left( 1+i \right)}^{n}} \right]}^{{}^{1}/{}_{n}}}=\left( 1+i \right) \\& i={{\left[ {}^{F{{V}_{n}}}/{}_{P{{V}_{0}}} \right]}^{{}^{1}/{}_{n}}}-1 \\\end{align}\]

**Example 1** Future Value, Solving for the Interest Rate

Just as you were about to enter college, you discovered that a great uncle had established an education trust fund for you when you were born, 20 years ago. He made a deposit of $5,000 in the beginning. If the total balance is now $19,348.42, what average compounded rate of return has been earned?

We can use the following equation to solve for the unknown interest rate:

\[i={{\left[ {}^{F{{V}_{n}}}/{}_{P{{V}_{0}}} \right]}^{{}^{1}/{}_{n}}}-1={{\left[ {}^{19,348.42}/{}_{5,000} \right]}^{{}^{1}/{}_{20}}}-1=7%\]

**Number of Time Periods Calculation Formula **

If you know the PV, FV, and the interest rate you can compute the number of time periods.

**Example 2** Future Value, Solving for n

How long will it take to double the money at 3% interest rate? For instance, if inflation averages 3% per annum, then total how many years will it take for the price of a pint of beer to double from $1.50 to $3.00?

**Solution**

\[\begin{align} & P{{V}_{0}}=\$1.50\\&F{{V}_{0}}=\$3\\&n=?\\&i=0.03\\\end{align}\]

First, we set this up as a future value problem where the unknown is the number of compounding periods, n:

\[\begin{align}& F{{V}_{n}}=P{{V}_{0}}\times {{\left( 1+i \right)}^{n}} \\& {}^{F{{V}_{n}}}/{}_{P{{V}_{0}}}={{\left( 1+i \right)}^{n}} \\& n=\frac{\ln \left( {}^{F{{V}_{n}}}/{}_{P{{V}_{0}}} \right)}{\ln \left( 1+i \right)} \\\end{align}\]

The excel sheet above shows how to solve for n when other variables are known.