# How we calculate average rate of return

If assumptions are Arithmetic

#### 'Arithmetic mean' values - in the absence of any volatility - will overstate the return received by the client(s), year-on-year. The degree of overstatement is a function of the assumed volatility, i.e. the SD value of the asset class/portfolio in question.

The average rate of return (ARR), also known as the arithmetic mean return, is calculated by finding the mean of a set of periodic returns over a specific period. The formula for calculating the average rate of return is relatively straightforward:

Average Rate of Return (ARR)=Sum of ReturnsNumber of Periods

Here's a step-by-step breakdown:

1. Calculate Returns:

• Determine the return for each period. The return is typically expressed as a percentage and is calculated using the formula: Return=(Ending Value−Beginning ValueBeginning Value)×100
2. The sum of Returns:

• Add up all the individual returns calculated in step 1.
3. Number of Periods:

• Determine the total number of periods for which returns were calculated.
4. Average Rate of Return:

• Divide the sum of returns by the number of periods to get the average rate of return.

Here's the formula again with more detail: ARR=Return1+Return2+…+Return

Where:

• Return1,Return2,…,Return are the returns for each period.
• is the total number of periods.

It's important to note that the average rate of return is a simple average and assumes that the investment grows or declines at a constant rate over each period. This method doesn't account for compounding, and it might not accurately reflect the actual geometric mean return over the investment period. For a more accurate representation of the overall return, especially for investments with volatile returns, the geometric mean or compound annual growth rate (CAGR) is often used.

#### If assumptions are Geometric

The average rate of return (ARR) can be calculated using geometric assumptions by using the geometric mean. The geometric mean takes into account the compounding effect of returns over multiple periods and is considered a more accurate measure when investment returns vary significantly. The formula for calculating the geometric mean is as follows:

Here's a step-by-step breakdown:

1. Calculate Returns:

• Determine the return for each period. The return is typically expressed as a decimal, where 1+Return represents the growth factor.
2. Multiply Growth Factors:

• For each period, add 1 to the return and multiply these growth factors together. This product represents the cumulative effect of compounding over all periods.

Product of Growth Factors=(1+Return1)×(1+Return2)×…×(1+Return)

3. Calculate Geometric Mean:

• Take the -th root of the product obtained in step 2. Subtract 1 from the result to obtain the geometric mean.

Geometric Mean=(Product of Growth Factors)1−1

The geometric mean is a more accurate representation of the average rate of return when there is significant variability in returns over time. It reflects the compounded growth rate that, if applied uniformly over the investment period, would result in the same cumulative return.

It's important to note that, unlike the arithmetic mean, the geometric mean accounts for the order of returns and is generally a better measure for evaluating the performance of an investment over time.