How to calculate expected return and standard deviation
About this lesson
In this video, we will discuss a stock’s mean or expected return and the standard deviation of returns, which is the measure of asset-specific risk. Finally, I will use two numeric examples to show how expected returns and standard deviations are calculated.
Expected return
The expected return is the probability-weighted average of the possible returns:
E(r) = Σ p(s) × r(s)
Standard deviation
The variance measures how far returns spread around the expected return:
σ² = Σ p(s) × [r(s) − E(r)]²
The standard deviation (σ) is the square root of the variance. It’s in the same units as the return, so it’s easier to interpret.
Example
| Scenario | Probability | Return |
|---|---|---|
| Boom | 0.25 | 30% |
| Normal | 0.50 | 10% |
| Recession | 0.25 | −10% |
Expected return:
0.25 × 30% + 0.50 × 10% + 0.25 × (−10%) = 7.5% + 5% − 2.5% = 10%
Variance:
0.25 × (30 − 10)² + 0.50 × (10 − 10)² + 0.25 × (−10 − 10)² = 100 + 0 + 100 = 200
Standard deviation: √200 = 14.14%
With historical returns
When you have past returns instead of scenarios, each year counts equally:
- Average return = sum of returns ÷ number of years. In Excel:
=AVERAGE(B2:B11) - Sample standard deviation divides by n − 1 instead of n. In Excel:
=STDEV.S(B2:B11)
To see this applied to a real stock, watch how to measure the risk of Apple stock in Excel.
Interpreting the numbers
If returns are roughly normally distributed, about two-thirds of outcomes fall within one standard deviation of the average (here, 10% ± 14.14%, or −4.1% to 24.1%), and about 95% within two.