The Return Distribution: Mean and Volatility

Suppose you buy one share. Next year's return is unknown. But its mean and its spread can be stated.

With a mean of 8%8\% and volatility of 15%15\%, returns wobble around 8%8\% by roughly 15%15\%. The formula below is the shape of that distribution. Move the cutoff and watch how the probability of falling below it changes.

For example, suppose you invest 100,000 won of allowance in a stock. With a mean return of 8%8\%, you expect about 108,000 won next year. But with volatility of 15%15\%, it could be 93,000 won or 123,000 won. The expectation and the wobble must be read separately.

Source: Fama, E. F. (1965). The Behavior of Stock-Market Prices. Journal of Business, 38(1), 34-105. That paper showed real stock returns have fatter tails than a normal distribution, so the normal curve in this scene is only an approximation.

f(x)=1σ2πe−(x−μ)22σ2f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}
P(X<c)=Φ(c−μσ)P(X < c) = \Phi\left(\frac{c - \mu}{\sigma}\right)

Part of these learning paths

Examples & Exercises

Compute

Returns follow a normal distribution with mean 8%8\% and volatility 15%15\%.

Find the probability that a return falls below 0%0\%.

Distribution with mean 8%8\%, volatility 15%15\%, cutoff 0%0\%
Show proof

  1. Standardize: z=(0−8)/15=−0.533z = (0 - 8) / 15 = -0.533.

  2. From the standard normal distribution, Φ(−0.533)≈0.297\Phi(-0.533) \approx 0.297.

Answer

About 0.2970.297, that is about 29.7%29.7\%

1. Statistics and Probability Basics