The annualized rate of return is a measure used to express the yearly rate of return on an investment over a period of time, typically presented as a percentage. It’s a way to standardize returns on investments with different time frames. This measure is especially useful when comparing the performance of investments that have varying holding periods.

The formula for calculating the annualized rate of return is as follows:

\[
\text{Annualized Rate of Return} = \left( \frac{\text{Ending Value}}{\text{Beginning Value}} \right)^{\frac{1}{\text{Number of Years}}} – 1
\]

Or, in a more general form:

\[
\text{Annualized Rate of Return} = \left( \frac{\text{Ending Value}}{\text{Beginning Value}} \right)^{\frac{1}{\text{Number of Periods}}} – 1
\]

Here’s a breakdown of the components:

– \(\text{Ending Value}\) is the value of the investment at the end of the period.
– \(\text{Beginning Value}\) is the initial value of the investment at the beginning of the period.
– \(\text{Number of Years}\) is the total number of years the investment was held.

This formula assumes that the investment grows or declines at a steady rate over the entire period.

For example, if you invested $1,000 at the beginning of the year and it grew to $1,200 by the end of the year, the annualized rate of return would be calculated as follows:

\[
\text{Annualized Rate of Return} = \left( \frac{1,200}{1,000} \right)^{\frac{1}{1}} – 1 = 0.2 \text{ or } 20\%
\]

This would represent a 20% annualized rate of return for that investment.

It’s important to note that this calculation assumes a consistent rate of return over the entire period. In reality, investment returns can vary, and this method provides a simplified way to express a compound rate of return on an annual basis.