# Number of Time Periods in TVM

If we have information about present value, future value, periodic cash flows, and interest rate, we can calculate the number of time periods involved algebra, financial calculator, or Excel NPER function.

The exact formula that we can use depends on whether we are dealing with simple interest or compound interest. Simple interest is when interest is calculated only on the principal balance and compound interest is when interest rate is applied to the sum of principal amount and any interest already accrued.

## Simple interest

If case of simple interest, number of time periods (t) can be calculated as follows:

$$\text{t}=\frac{\text{I}}{\text{PV}\times \text{i}}=\frac{\text{FV}\ -\ \text{PV}}{\text{PV}\times \text{i}}$$

Where FV is the future value, PV is the present value, I is the total interest amount, i is the periodic simple interest rate and t is the number of time periods.

### Understanding the math

We can use the expression for future value in case of simple interest to arrive at simple interest rate equation:

$$\text{FV}\ =\ \text{PV}\times(\text{1}+\text{i}\times \text{t})$$

Step 1: divide both sides by PV

$$\frac{\text{FV}}{\text{PV}}\ =\frac{\ \text{PV}}{\text{PV}}\times(\text{1}+\text{i}\times \text{t})=\text{1}+\text{i}\times \text{t}$$

Step 2: subtract 1 from both sides:

$$\frac{\text{FV}}{\text{PV}}-\text{1}\ =\text{i}\times \text{t}$$

$$\frac{\text{FV}\ -\ \text{PV}}{\text{PV}}=\text{i}\times \text{t}$$

Step 3: divide both sides by i

$$\text{t}\ =\ \frac{\text{FV}\ -\ \text{PV}}{\text{PV}\times \text{i}}$$

Since FV – PV equals I, the total interest amount, we can write:

$$\text{t}\ =\ \frac{\text{I}}{\text{PV}\times \text{i}}$$

## Compound interest

In case of compound interest, number of periods can be derived using the basic time value of money equation:

$$\text{FV}=\text{PV}\times{(\text{1}+\text{r})}^\text{n}$$

Where FV is the future value, PV is the present value, r is the periodic compound interest rate and n is the total number of periods.

### Understanding the math

Step 1: divide both sides with PV

$$\frac{\text{FV}}{\text{PV}}=\frac{\text{PV}}{\text{PV}}\times{(\text{1}+\text{r})}^\text{n}=\text{1}\times{(\text{1}+\text{r})}^\text{n}$$

Step 2: take log of both sides

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

Step 3: rearranging the above equation using quotient and power rules of logarithm:

$$\text{log}\ (\text{FV})\ -\ \text{log}\ (\text{PV})=\text{n}\times \text{log}\ (\text{1}+\text{r})$$

Step 4: dividing both sides with log (1 + r) and rearranging:

$$\text{n}\ =\frac{\text{log}\ (\text{FV})\ -\ \text{log}\ (\text{PV})}{\text{log}\ (\text{1}\ +\ \text{r})}$$

## Example

You write a promissory note which is based on annual simple interest rate of 6.67% with maturity value of $1,200 for$1,000 and invest the proceeds in a certificate of deposit at interest rate of 8.94% compounded semi-annually with maturity value of \$1,300. Find out the number of periods in both financial instruments.

### Solution

Since the promissory note is based on simple interest, the number of time periods (t) can be worked out as follows:

$$\text{t}=\frac{\text{\1,200}\ -\ \text{\1,000}}{\text{1,000}\times\text{6.67%}}=\text{3}$$

Since the interest rate is in years, we get t in years too.

The number of time periods involved in the certificate of deposit can be computed as follows:

$$\text{n}\ =\frac{\text{log}\ (\text{\1,300})\ -\ \text{log}\ (\text{\1,000})}{\text{log}\ (\text{1}\ +\ \text{4.47%})}$$

$$\text{n}\ =\frac{\text{3.114}\ -\ \text{3}}{\text{0.019}}=\text{6}$$

Since the interest is compounded semi-annually, we have used periodic interest rate of 4.47% (=8.94% divided by 2). In case of semi-annual compounding, the number of periods (NPER) that we get is also in number of six-monthly periods.

Total number of years can be calculated by dividing n with number of compounding periods per year i.e. 6 divided 2. This tells us that the certificate of deposit will mature in 3 years.