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What Is Standard Deviation?

Standard deviation is a statistical measure within the field of portfolio theory that quantifies the amount of dispersion or variation of a set of data values around their arithmetic mean. In finance, standard deviation is widely used to measure the volatility of an investment or an investment portfolio. A high standard deviation indicates that the data points are spread out over a wider range of values, implying greater volatility and, consequently, higher risk. Conversely, a low standard deviation suggests that the data points are clustered closely around the mean, indicating lower volatility and risk. It provides investors with a concrete number to understand the historical consistency of an investment's returns.

History and Origin

The concept of standard deviation was introduced by Karl Pearson in 1894. Its application in finance, however, became central with the advent of Modern Portfolio Theory (MPT). Pioneered by economist Harry Markowitz in his 1952 paper "Portfolio Selection," MPT established a mathematical framework for assembling a portfolio of assets to maximize expected return for a given level of risk. Markowitz's groundbreaking work recognized that an asset's risk should not be assessed in isolation but rather by how it contributes to a portfolio's overall risk and return. He used the variance of return, and by extension, its square root, standard deviation, as the measure of risk. This fundamental insight, which earned Markowitz a Nobel Memorial Prize in Economic Sciences in 1990, revolutionized portfolio management by demonstrating the benefits of diversification to reduce overall portfolio volatility.,17,16

Key Takeaways

  • Standard deviation quantifies the dispersion of data points around their average, serving as a key measure of volatility and risk in finance.
  • A higher standard deviation indicates greater price fluctuations and higher risk for an investment or portfolio.15
  • It is a foundational component of Modern Portfolio Theory (MPT), where it is used to assess portfolio risk and optimize asset allocation.
  • For investments whose returns follow a normal distribution, approximately 68% of returns will fall within one standard deviation of the mean, and 95% within two standard deviations.14,13
  • While useful, standard deviation has limitations, including its sensitivity to outliers and its assumption of normally distributed returns.12

Formula and Calculation

The formula for calculating the sample standard deviation (denoted as (s)) is:

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

Where:

  • (s) = Sample standard deviation
  • (\Sigma) = Summation symbol
  • (x_i) = Each individual data point in the data set
  • (\bar{x}) = The sample mean (average) of the data set
  • (n) = The number of data points in the sample

This formula essentially calculates the square root of the variance, providing a measure in the same units as the original data, which makes it easier to interpret.,11

Interpreting the Standard Deviation

Interpreting standard deviation involves understanding what its value signifies in the context of investment returns. A low standard deviation suggests that an investment's returns have historically been stable and predictable, staying close to its average return. Conversely, a high standard deviation indicates that returns have been widely dispersed, meaning the investment has experienced significant price swings, both up and down.

For investors, a higher standard deviation implies higher risk because the range of potential outcomes is wider, making future returns less predictable. For instance, a stock with an average annual return of 10% and a standard deviation of 2% is generally considered less risky than a stock with the same 10% average return but a 15% standard deviation. The first stock's returns are more likely to fall within a narrow band (8% to 12% approximately 68% of the time, assuming a normal distribution), while the second stock's returns could fluctuate much more dramatically.10 This understanding is crucial for investors to align their investment portfolio choices with their individual risk tolerance.

Hypothetical Example

Consider two hypothetical portfolios, Portfolio A and Portfolio B, over five years:

YearPortfolio A Return (%)Portfolio B Return (%)
11025
212-5
31130
49-10
51320

First, calculate the average return for each portfolio:

  • Average Return for Portfolio A: ((10 + 12 + 11 + 9 + 13) / 5 = 11%)
  • Average Return for Portfolio B: ((25 - 5 + 30 - 10 + 20) / 5 = 12%)

Next, calculate the standard deviation for each:

For Portfolio A:

sA=(10−11)2+(12−11)2+(11−11)2+(9−11)2+(13−11)25−1=1+1+0+4+44=104=2.5≈1.58%s_A = \sqrt{\frac{(10-11)^2 + (12-11)^2 + (11-11)^2 + (9-11)^2 + (13-11)^2}{5-1}} = \sqrt{\frac{1 + 1 + 0 + 4 + 4}{4}} = \sqrt{\frac{10}{4}} = \sqrt{2.5} \approx 1.58\%

For Portfolio B:

sB=(25−12)2+(−5−12)2+(30−12)2+(−10−12)2+(20−12)25−1=169+289+324+484+644=13304=332.5≈18.23%s_B = \sqrt{\frac{(25-12)^2 + (-5-12)^2 + (30-12)^2 + (-10-12)^2 + (20-12)^2}{5-1}} = \sqrt{\frac{169 + 289 + 324 + 484 + 64}{4}} = \sqrt{\frac{1330}{4}} = \sqrt{332.5} \approx 18.23\%

Even though Portfolio B has a slightly higher average return (12% vs. 11%), its standard deviation of approximately 18.23% is significantly higher than Portfolio A's 1.58%. This indicates that Portfolio A's returns are much more consistent and less volatile, making it a lower-risk option compared to Portfolio B, which exhibits substantial fluctuations. This helps an investor in risk-adjusted return analysis.

Practical Applications

Standard deviation is a cornerstone metric with numerous practical applications across finance and investing. In portfolio management, it helps investors assess and manage the risk level of their holdings. A portfolio manager might use standard deviation to construct an efficient frontier, identifying portfolios that offer the highest expected return for a given level of risk.

Beyond portfolio construction, standard deviation is used in:

  • Performance Evaluation: It helps evaluate the risk taken to achieve a certain return. For example, the Sharpe Ratio incorporates standard deviation to measure risk-adjusted returns.9
  • Asset Allocation: Investors consider the standard deviation of different asset classes (e.g., stocks, bonds, real estate) when determining their asset allocation, aiming to balance risk and return based on their personal objectives.
  • Option Pricing: Volatility, often measured by standard deviation, is a critical input in models like the Black-Scholes model for pricing options.
  • Market Analysis: Analysts use standard deviation to gauge overall market volatility or the volatility of specific sectors and industries. For instance, the Federal Reserve Bank of San Francisco has discussed how standard deviation is a commonly used measure of volatility for stock market indexes like the S&P 500.8

Limitations and Criticisms

While standard deviation is a widely accepted measure of risk, it is not without limitations and criticisms. One primary criticism is its assumption that data, particularly financial returns, follows a normal distribution. In reality, financial markets often exhibit "fat tails" and skewness, meaning extreme events (both positive and negative) occur more frequently than a normal distribution would predict. This can lead to an underestimation of risk, especially during periods of market stress.7,6

Another drawback is that standard deviation treats both positive and negative deviations from the mean equally. Investors, however, are typically more concerned with downside risk (losses) than upside volatility (gains). A large positive deviation might be seen as favorable, but standard deviation penalizes it just as much as a large negative deviation.5 This can obscure the true nature of risk for some investors.

Furthermore, standard deviation is highly sensitive to outliers. A single extreme return, whether a significant gain or loss, can disproportionately inflate the standard deviation, potentially giving a misleading impression of an investment's typical volatility.4 Despite these criticisms, it remains a common and valuable tool when understood within its context and used alongside other analytical methods.3

Standard Deviation vs. Beta

Standard deviation and beta are both measures of risk in finance, but they quantify different aspects. Standard deviation measures the total risk of an asset or portfolio, reflecting the overall volatility of its returns around its average. This total risk encompasses both systematic risk (market risk) and unsystematic risk (specific risk to an asset).

Beta, on the other hand, measures only the systematic risk of an asset. It indicates how sensitive an asset's returns are to movements in the overall market. A beta of 1 means the asset tends to move in line with the market. A beta greater than 1 suggests the asset is more volatile than the market, while a beta less than 1 indicates it is less volatile. The key difference is that standard deviation provides an absolute measure of volatility, while beta provides a relative measure of an asset's volatility compared to a benchmark market. For example, a highly diversified mutual fund might have a low beta because its unsystematic risk has been diversified away, but its standard deviation could still be high if the overall market it tracks is highly volatile.

FAQs

What is a good standard deviation for an investment?

There isn't a universally "good" standard deviation; it depends on the investor's risk tolerance and investment goals. Lower standard deviations are generally preferred for conservative investors seeking stable returns, while aggressive investors might accept higher standard deviations for the potential of greater returns. It's often evaluated relative to comparable investments or benchmarks.

How does standard deviation relate to risk?

Standard deviation is directly related to risk in finance. A higher standard deviation implies greater volatility in an investment's returns, meaning the actual returns are more likely to deviate significantly from the expected average. This increased variability is synonymous with higher risk.2

Can standard deviation predict future returns?

No, standard deviation is a historical measure and cannot predict future returns. It quantifies past volatility and helps estimate the range within which future returns might fall, assuming past patterns continue. It does not forecast the direction or magnitude of those returns.1,

Is standard deviation the only measure of investment risk?

No, standard deviation is one of several measures of investment risk. Other metrics include beta, Value at Risk (VaR), and Sortino Ratio. Each provides a different perspective on risk, and a comprehensive investment portfolio analysis often involves using multiple measures.