The Distribution of Rate of Return

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Suppose an investment fund (or a development company) promises you an average return of 14% per year. To convince you, they show that over the last 5 years their returns were: 15%, 20%, 30%, –20%, and 25%. The average of these is indeed 14%. Does the promise sound credible? Let’s check mathematically.

The promise:
A 5-year investment of 100 GEL at 14% annually:
100 GEL × (1 + 14%)⁵ = 192.54 GEL

Reality:
If you had actually invested 100 GEL for those 5 years:
100 GEL × 1.15 × 1.20 × 1.30 × 0.80 × 1.25 = 179.40 GEL → 12.4% lower

The point is that when returns compound continuously (or in general, when you reinvest), the effective return is not the arithmetic average but the geometric average. A similar concept applies to stocks.

Let’s dive deeper:

Start with the simple return formula:

Sₜ = S₀ × (1 + r)ᵀ,
and its continuously compounded version:
Sₜ = S₀ × eˣᵀ
(where r is replaced with X because X is unknown and this is what we want to solve for).

If we solve this formula for X and take the logarithm, we get:
X = ln(Sₜ / S₀) / T

We also know that ln(Sₜ / S₀) follows a normal distribution:
(See: Lognormal Property – Example)

Accordingly, the distribution of returns looks like this:

For example, if we take a stock with an average annual return of 17% and a volatility of 20%, then after three years the expected average return becomes 15%, and the standard deviation becomes 11.55%.

This gives the following 95% confidence interval for the 3-year return:

(Excel File: Return Distribution)

P.S.

The difference between expected return and average return often causes confusion. This is essentially the difference between arithmetic and geometric returns—just like in the fund example at the start.

• μ is the expected return, which is conceptually the arithmetic average, but expressed in continuously-compounded terms.
• μ − σ²/2 is the geometric average return—the return actually produced when returns compound continuously.

P.P.S.

There is also an interesting point about the variance of return distribution: σ² / T.
This tells us that as the time interval increases, the range of possible outcomes narrows. The longer the horizon, the smaller the standard deviation becomes.

In other words, over a 10-year period, we can be more confident about estimating average returns than over a 1-year period. For example, we can say with confidence that the S&P 500 generates about 12% annually over a 20-year horizon—but we cannot predict a 12% return for next year.

Adapted from:
Options, Futures & Other Derivatives, John C. Hull

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