Average Return

The average return is the arithmetic mean of an investment's returns over a series of periods, such as months, quarters or years. It is calculated by adding the periodic returns and dividing by the number of periods. Investors, analysts and researchers use it to describe typical performance, to estimate expected returns and to compare assets or strategies.

Average return is the simplest of several ways to summarise performance. It differs from the geometric or compound return, which reflects how money actually grows over time, and from the internal rate of return, which accounts for the timing of cash added or withdrawn. When returns are quoted per year, the measure is often called the average annual return. Average returns are also central to finance theory, where they are paired with measures of risk such as standard deviation.

If a fruit stall makes $100 profit one week, $50 the next and $150 the week after, its average weekly profit is $100. Average return applies the same idea to percentage gains and losses: add them up and divide by the number of periods.

For investments, though, a simple average can be misleading. If your savings rise 50% and then fall 50%, the average return is zero, yet you have lost a quarter of your money. That is because the fall applied to a bigger amount than the rise did. The average describes a typical period; it does not always describe what happened to your wealth.

The arithmetic average return over $n$ periods is:

• $\bar{r}$ is the average return.
• $r_t$ is the return in period $t$.
• $n$ is the number of periods.

Suppose a portfolio returns 8%, −4%, 12% and 2% over four years. The average return is (8 − 4 + 12 + 2) ÷ 4 = 4.5%. The compound return is $(1.08 \times 0.96 \times 1.12 \times 1.02)^{1/4} - 1$, which works out to about 4.32% a year. The arithmetic average is higher, as it always is when returns vary.

The gap between the two averages grows with volatility. A useful approximation is:

where $\sigma$ is the standard deviation of returns. This difference is sometimes called volatility drag. Each measure has its place:

• Arithmetic average. The best estimate of a single future period's expected value when returns are independent, so it is used in forecasting and in models such as the capital asset pricing model.
• Geometric average. The correct measure of historical growth of wealth over many periods, used in performance reporting, as in the compound annual growth rate.

• Risk and return analysis. Average returns are compared with risk in the risk-return trade-off and in ratios such as the Sharpe ratio.
• Asset class history. Long-run averages for shares, bonds and cash are used to set expectations, with care over which average is quoted.
• Portfolio construction. Average returns of assets, with their correlations, feed mean-variance optimisation in building a portfolio.
• Performance comparison. Funds and strategies are compared on average returns over the same periods.

Crypto assets show very high volatility, so the gap between arithmetic and geometric averages can be large. A token with returns of +300% one year and −75% the next has an arithmetic average of +112.5%, yet an investor who held both years would be exactly back where they started. Return distributions in crypto also have fat tails, with extreme moves more common than a normal distribution would suggest, which makes simple averages a weak guide to future results.

Average return ignores the order of returns and the effect of compounding, does not show how severe losses were, and can be distorted by a single extreme period. It also ignores the timing of deposits and withdrawals, which matters for investors who add money regularly, for example through dollar cost averaging. Past average returns do not guarantee future results, and averages calculated over short or carefully chosen periods can mislead.