# Annualized Total Return Formula and Calculation

## What Is Annualized Total Return?

An annualized total return is the geometric average amount of money earned by an investment each year over a given time period. The annualized return formula is calculated as a geometric average to show what an investor would earn over a period of time if the annual return was compounded.

An annualized total return provides only a snapshot of an investment's performance and does not give investors any indication of its volatility or price fluctuations.

### Key Takeaways

• An annualized total return is the geometric average amount of money earned by an investment each year over a given time period.
• The annualized return formula shows what an investor would earn over a period of time if the annual return was compounded.
• Calculating the annualized rate of return needs only two variables: the returns for a given period and the time the investment was held.
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## Understanding Annualized Total Return

To understand annualized total return, we'll compare the hypothetical performances of two mutual funds. Below is the annualized rate of return over a five-year period for the two funds:

• Mutual Fund A Returns: 3%, 7%, 5%, 12%, and 1%
• Mutual Fund B Returns: 4%, 6%, 5%, 6%, and 6.7%

Both mutual funds have an annualized rate of return of 5.5%, but Mutual Fund A is much more volatile. Its standard deviation is 4.2%, while Mutual Fund B's standard deviation is only 1%. Even when analyzing an investment's annualized return, it is important to review risk statistics.

## Annualized Return Formula and Calculation

The formula to calculate annualized rate of return needs only two variables: the returns for a given period of time and the time the investment was held. The formula is:

﻿ \begin{aligned} \text{Annualized Return} = &\big ( (1 + r_1 ) \times (1 + r_2) \times (1 + r_3) \times \\ &\dots \times (1 + r_n) \big ) ^ \frac{1}{n} - 1 \\ \end{aligned}﻿

For example, take the annual rates of returns of Mutual Fund A above. An analyst substitutes each of the "r" variables with the appropriate return, and "n" with the number of years the investment was held. In this case, five years. The annualized return of Mutual Fund A is calculated as:

﻿ \begin{aligned} \text{Annualized Return} &= \big ( (1 + .03) \times (1 + .07) \times (1 + .05) \times \\ &\quad \quad (1 + .12) \times (1 + .01) \big ) ^ \frac{1}{5} -1 \\ &= 1.309 ^ {0.20} - 1 \\ &= 1.0553 - 1 \\ &= .0553, \text{or } 5.53\% \\ \end{aligned}﻿

An annualized return does not have to be limited to yearly returns. If an investor has a cumulative return for a given period, even if it is a specific number of days, an annualized performance figure can be calculated; however, the annual return formula must be slightly adjusted to:

﻿ \begin{aligned} &\text{Annualized Return} = ( 1 + \text{Cumulative Return} ) ^ \frac {365}{ \text{Days Held} } - 1 \\ \end{aligned}﻿

For example, assume a mutual fund was held by an investor for 575 days and earned a cumulative return of 23.74%. The annualized rate of return would be:

﻿ \begin{aligned} \text{Annualized Return} &= ( 1 + .2374) ^ \frac{365}{575} - 1 \\ &= 1.145 - 1 \\ &= .145, \text{or } 14.5\% \\ \end{aligned}﻿

## Difference Between Annualized Return and Average Return

Calculations of simple averages only work when numbers are independent of each other. The annualized return is used because the amount of investment lost or gained in a given year is interdependent with the amount from the other years under consideration because of compounding.

For example, if a mutual fund manager loses half of her client's money, she has to make a 100% return to break even. Using the more accurate annualized return also gives a clearer picture when comparing various mutual funds or the return of stocks that have traded over different time periods.

## Reporting Annualized Return

According to the Global Investment Performance Standards (GIPS)—a set of standardized, industry-wide principles that guide the ethics of performance reporting—any investment that does not have a track record of at least 365 days cannot "ratchet up" its performance to be annualized.

Thus, if a fund has been operating for only six months and earned 5%, it is not allowed to say its annualized performance is approximately 10% since that is predicting future performance instead of stating facts from the past. In other words, calculating an annualized rate of return must be based on historical numbers.

## How Is Annualized Total Return Calculated?

The annualized total return is a metric that captures the average annual performance of an investment or portfolio of investments. It is calculated as a geometric average, meaning that it captures the effects of compounding over time. The annualized total return is sometimes referred to as the compound annual growth rate (CAGR).

## What Is the Difference Between an Annualized Total Return and an Average Return?

The key difference between the annualized total return and the average return is that the annualized total return captures the effects of compounding, whereas the average return does not.

For example, consider the case of an investment that loses 50% of its value in year 1 but has a 100% return in year 2. Simply averaging these two percentages would give you an average return of 25% per year. However, common sense would tell you that the investor in this scenario has actually broken even on their money (losing half its value in year one, then regaining that loss in year 2). This fact would be better captured by the annualized total return, which would be 0.00% in this instance.

## What Is the Difference Between the Annualized Total Return and the Compound Annual Growth Rate (CAGR)

The annualized total return is conceptually the same as the CAGR, in that both formulas seek to capture the geometric return of an investment over time. The main difference between them is that the CAGR is often presented using only the beginning and ending values, whereas the annualized total return is typically calculated using the returns from several years. This, however, is more a matter of convention. In substance, the two measures are the same.

## The Bottom Line

Annualized total return represents the geometric average amount that an investment has earned each year over a specific period. By calculating a geometric average, the annualized total return formula accounts for compounding when depicting the yearly earnings that the investment would generate over the holding period. While the metric provides a useful snapshot of an investment's performance, it does not reveal volatility and price fluctuations.

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