Standard Deviation of Returns Calculator
Separate values with commas, spaces, or new lines (e.g., 5, 12, -3.5, 8)
Sample Standard Deviation (Standard for historical returns)
Population Standard Deviation (Entire universe of data)
Mean Return (Average):
Variance:
Standard Deviation:
Data Count (n):
Understanding the Standard Deviation of Rates of Return
In finance and investment analysis, the standard deviation of returns is the primary measure used to quantify market volatility and investment risk. It tells an investor how much the actual return on an investment deviates from the expected average return over a specific period.
Why Calculate Standard Deviation?
A high standard deviation indicates high volatility, meaning the investment's price can fluctuate wildly in either direction. Conversely, a low standard deviation suggests a more stable investment with returns that stay close to the historical average. This metric is the cornerstone of the Modern Portfolio Theory (MPT) and is used to calculate the Sharpe Ratio.
Sample SD Formula: σ = √[ Σ(xi – x̄)² / (n – 1) ]
Where:
σ = Standard Deviation
xi = Each individual return
x̄ = Mean (average) of all returns
n = Number of data points
Where:
σ = Standard Deviation
xi = Each individual return
x̄ = Mean (average) of all returns
n = Number of data points
How to Calculate Step-by-Step
- Find the Mean: Add all your percentage returns together and divide by the total number of periods.
- Calculate Deviations: Subtract the mean from each individual return.
- Square the Deviations: Square each result from the previous step to eliminate negative signs.
- Sum of Squares: Add all the squared values together.
- Divide: Divide that sum by (n – 1) for a sample, or (n) for a population. This gives you the Variance.
- Square Root: Take the square root of the variance to find the Standard Deviation.
Realistic Example:
Suppose a stock had the following annual returns over 4 years: 10%, 20%, -5%, and 7%.
1. Mean = (10+20-5+7) / 4 = 8%
2. Squared Deviations:
(10-8)² = 4
(20-8)² = 144
(-5-8)² = 169
(7-8)² = 1
3. Sum = 4 + 144 + 169 + 1 = 318
4. Sample Variance = 318 / (4-1) = 106
5. Standard Deviation = √106 = 10.29%
Suppose a stock had the following annual returns over 4 years: 10%, 20%, -5%, and 7%.
1. Mean = (10+20-5+7) / 4 = 8%
2. Squared Deviations:
(10-8)² = 4
(20-8)² = 144
(-5-8)² = 169
(7-8)² = 1
3. Sum = 4 + 144 + 169 + 1 = 318
4. Sample Variance = 318 / (4-1) = 106
5. Standard Deviation = √106 = 10.29%
Interpretation of Results
If you are comparing two mutual funds with the same 8% average return, but Fund A has a standard deviation of 5% and Fund B has a standard deviation of 15%, Fund A is considered significantly safer. Fund B provides the same "reward" for much higher "risk."