Measurement tools: Meaning, Criticisms & Real-World Uses

What Is Standard Deviation?

Standard deviation is a fundamental statistical measure that quantifies the amount of dispersion or variability in a set of data points around their mean, or average. In the context of financial markets, particularly within the realm of portfolio theory and risk management, standard deviation is widely used as a proxy for volatility and, consequently, risk. A low standard deviation indicates that data points tend to be close to the mean, suggesting less volatility. Conversely, a high standard deviation implies that data points are spread out over a wider range, indicating greater volatility and higher risk.

History and Origin

The concept of standard deviation has its roots in the broader development of statistics and the study of variation. While related ideas were explored earlier, the formal mathematical formula and widespread adoption of standard deviation are largely attributed to English mathematician and biostatistician Karl Pearson. Pearson developed the concept from related ideas introduced by Francis Galton in the 1880s, applying it to statistical analysis in the nascent field of biometry.¹⁷ Its application in finance gained significant traction with the advent of Modern Portfolio Theory (MPT), pioneered by economist Harry Markowitz in his 1952 paper "Portfolio Selection." Markowitz's work revolutionized portfolio management by explicitly incorporating standard deviation as a key measure of risk when optimizing portfolios for expected returns.¹⁶

Key Takeaways

Standard deviation quantifies the dispersion of data points around their average, serving as a primary measure of volatility in finance.
A higher standard deviation indicates greater price fluctuations and higher risk, while a lower standard deviation suggests more stable prices and lower risk.
It is a core component of Modern Portfolio Theory, helping investors evaluate and manage portfolio risk.
While widely used, standard deviation assumes a normal distribution of returns, which may not always hold true for financial data.
It does not differentiate between upward and downward price movements, treating both as equally risky.

Formula and Calculation

The standard deviation is calculated as the square root of the variance. For a population, the formula for standard deviation ((\sigma)) is:

\sigma = \sqrt{\frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}}

Where:

(x_i) = each individual data point
(\mu) = the population mean (average of all data points)
(N) = the total number of data points in the population

For a sample, the formula for sample standard deviation ((s)) is:

s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}}

Where:

(x_i) = each individual data point in the sample
(\bar{x}) = the sample mean (average of the data points in the sample)
(n) = the total number of data points in the sample

The use of (n-1) in the denominator for sample standard deviation provides an unbiased estimate of the population standard deviation, a correction known as Bessel's correction.

Interpreting the Standard Deviation

In finance, standard deviation is typically interpreted as a measure of an asset's or portfolio's historical volatility. A higher standard deviation suggests that an investment's returns have historically deviated more significantly from its average return, implying greater price swings and higher perceived risk. Conversely, a lower standard deviation indicates that returns have been more consistent, implying less risk. For instance, if an investment has an average annual return of 8% with a standard deviation of 15%, its returns have historically fluctuated more widely than an investment with an 8% average return and a 5% standard deviation. Investors can use this measure to gauge potential fluctuations and align investments with their individual risk tolerance.¹⁵

Hypothetical Example

Consider two hypothetical mutual funds, Fund A and Fund B, over the past five years. Their annual returns are:

Fund A: 10%, 12%, 8%, 11%, 9%
Fund B: 25%, -5%, 18%, 2%, 10%

To calculate the standard deviation for Fund A:

Calculate the Mean: ((10 + 12 + 8 + 11 + 9) / 5 = 50 / 5 = 10%)
Calculate Deviations from the Mean: (0, 2, -2, 1, -1)
Square the Deviations: (0, 4, 4, 1, 1)
Sum the Squared Deviations: (0 + 4 + 4 + 1 + 1 = 10)
Calculate Variance (sample): (10 / (5-1) = 10 / 4 = 2.5)
Calculate Standard Deviation: (\sqrt{2.5} \approx 1.58%)

Now for Fund B:

Calculate the Mean: ((25 - 5 + 18 + 2 + 10) / 5 = 50 / 5 = 10%)
Calculate Deviations from the Mean: (15, -15, 8, -8, 0)
Square the Deviations: (225, 225, 64, 64, 0)
Sum the Squared Deviations: (225 + 225 + 64 + 64 + 0 = 578)
Calculate Variance (sample): (578 / (5-1) = 578 / 4 = 144.5)
Calculate Standard Deviation: (\sqrt{144.5} \approx 12.02%)

Even though both funds had the same average return of 10%, Fund A's standard deviation of approximately 1.58% is significantly lower than Fund B's 12.02%. This indicates that Fund A's returns were much more consistent and less volatile, while Fund B experienced substantial swings, making it a riskier investment from a volatility perspective.

Practical Applications

Standard deviation is a widely used tool across various financial disciplines:

Investment Risk Assessment: Investors and analysts use standard deviation to gauge the historical volatility of individual stocks, bonds, or mutual funds. A higher standard deviation suggests higher risk, helping investors make informed decisions based on their risk appetite.¹⁴
Portfolio Diversification: In Modern Portfolio Theory, standard deviation is crucial for constructing diversified portfolios that optimize returns for a given level of risk. By combining assets with different correlations, the overall portfolio standard deviation can be reduced, leading to a more efficient risk-return trade-off.¹³
Performance Evaluation: Standard deviation is a key component in calculating risk-adjusted returns metrics like the Sharpe Ratio. The Sharpe Ratio measures the excess return per unit of risk (standard deviation), allowing for a more comprehensive comparison of investment performance.¹²
Regulatory Filings: The U.S. Securities and Exchange Commission (SEC) has considered requiring mutual funds and other investment companies to provide specific measures of total risk, including standard deviation, to help investors better understand fund volatility.¹¹ This highlights its importance in transparent financial reporting.
Options Pricing: Standard deviation, particularly in the form of implied volatility, is a critical input in options pricing models like the Black-Scholes model, as it represents the expected future volatility of the underlying asset.

Limitations and Criticisms

Despite its widespread use, standard deviation has several limitations as a sole measure of risk in finance:

Assumption of Normal Distribution: Standard deviation assumes that investment returns follow a normal distribution (bell-shaped curve). However, financial market returns often exhibit "fat tails" (more frequent extreme events) and skewness (asymmetric distribution), meaning large positive or negative deviations occur more often than a normal distribution would predict.⁹, ¹⁰ This can lead to an underestimation of the likelihood of extreme losses.
Treats Upside and Downside Volatility Equally: Standard deviation measures all deviations from the mean equally, regardless of whether they represent positive gains or negative losses. Investors, however, are typically more concerned about "downside risk" (losses) than "upside volatility" (gains). Other metrics, like semi-variance, attempt to address this by focusing only on deviations below the mean.⁷, ⁸
Sensitivity to Outliers: Standard deviation is highly sensitive to extreme values or outliers. A single exceptionally large positive or negative return can significantly inflate the calculated standard deviation, potentially misrepresenting the typical volatility of an asset.⁶
Historical Nature: Standard deviation is a backward-looking measure, calculated based on historical data. It does not inherently predict future volatility, which can change rapidly due to market conditions or unforeseen events.⁵
Doesn't Capture Correlation Fully: While essential for portfolio management when combined with correlation, standard deviation for individual assets does not fully capture how assets move together within a portfolio.⁴

Academic research, such as "The Limitations of Standard Deviation as a Measure of Bond Portfolio Risk," has highlighted these challenges, particularly when applying standard deviation to certain asset classes like fixed-income portfolios.², ³

Standard Deviation vs. Variance

Standard deviation and variance are closely related statistical measures of dispersion. Variance is the average of the squared differences from the mean, while standard deviation is simply the square root of the variance.

Feature	Standard Deviation	Variance
Unit of Measurement	Same unit as the original data (e.g., dollars, percent)	Squared unit of the original data (e.g., dollars squared)
Interpretability	More intuitive and easier to interpret in real-world terms	Less intuitive due to squared units, harder to relate to original data
Calculation Basis	Square root of the variance	Average of the squared deviations from the mean
Use in Finance	Widely used as a direct measure of volatility/risk	Often an intermediate step in calculating standard deviation or other statistical models

The key distinction for financial professionals is that standard deviation is expressed in the same units as the returns or prices being measured, making it much more interpretable than variance, which is in squared units. For example, a stock return volatility stated as a 10% standard deviation is more meaningful than a variance of 0.01.

FAQs

What does a high standard deviation mean for an investment?

A high standard deviation for an investment indicates that its returns have historically fluctuated significantly from its average return. This suggests that the investment is more volatile and generally carries a higher level of risk because its future returns are less predictable.

Is standard deviation a good measure of risk?

Standard deviation is a widely accepted and useful measure of risk, particularly for assessing volatility. However, it has limitations, such as assuming a normal distribution of returns and not distinguishing between positive and negative fluctuations. Therefore, it is often used in conjunction with other risk metrics like Beta or the Sharpe ratio for a more comprehensive view of risk.¹

How does standard deviation relate to Modern Portfolio Theory?

Standard deviation is a cornerstone of Modern Portfolio Theory (MPT). MPT uses standard deviation to quantify the risk of individual assets and, critically, of entire portfolios. By analyzing how the returns of different assets correlate, MPT aims to build diversified portfolios that minimize the overall standard deviation (risk) for a given expected return, or maximize expected return for a given risk, leading to the efficient frontier.

Can standard deviation predict future returns?

No, standard deviation is a historical measure and does not predict future returns or guarantee specific outcomes. It quantifies the past variability of returns. While historical volatility can offer insights into potential future behavior, actual market conditions and investment performance can deviate significantly from historical trends.

What is the "empirical rule" in relation to standard deviation?

For data that follows a normal distribution (bell curve), the "empirical rule" (also known as the 68-95-99.7 rule) states that:

Approximately 68% of data falls within one standard deviation of the mean.
Approximately 95% of data falls within two standard deviations of the mean.
Approximately 99.7% of data falls within three standard deviations of the mean.
This rule helps in interpreting the spread of returns for normally distributed assets.