Standard deviations

What Are Standard Deviations?

Standard deviations are a fundamental statistical measure that quantifies the amount of dispersion or variability in a set of data points. In plain English, it indicates how much individual data points typically deviate from the average (mean) of the set. Within portfolio theory and risk management, standard deviation is widely used as a primary indicator of an investment's volatility or risk. A low standard deviation suggests that data points tend to be close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range.

History and Origin

While the concept of measuring the spread of data has roots in earlier statistical work, the term "standard deviation" was formally introduced and extensively utilized by the English mathematician and statistician Karl Pearson in 1893. His work helped standardize statistical methods for analyzing dispersion, building upon earlier notions such as Gauss's "mean error" in the context of normal distribution and errors in astronomical observations. Pearson's contributions were pivotal in establishing the mathematical rigor for this crucial measure of variability, cementing its place in modern statistics and, subsequently, in finance.¹

Key Takeaways

Standard deviation measures the dispersion of data points around the mean, indicating the data's volatility.
In finance, it is a common measure of an investment's risk, with higher values implying greater price fluctuations.
It is a core component in calculating risk-adjusted return metrics like the Sharpe ratio.
A key assumption when interpreting standard deviation in finance is that asset returns follow a normal distribution, though real-world returns often exhibit "fat tails" or skewness.
Lower standard deviation for a given return is generally preferred by risk-averse investors.

Formula and Calculation

The standard deviation is calculated as the square root of the variance. For a population data set, the formula is:

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

Where:

(\sigma) (sigma) represents the population standard deviation.
(x_i) represents each individual data point.
(\mu) (mu) represents the population mean.
(N) represents the total number of data points in the population.

For a sample data set, a slightly modified formula is used (Bessel's correction) to provide an unbiased estimate of the population standard deviation:

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

Where:

(s) represents the sample standard deviation.
(x_i) represents each individual data point.
(\bar{x}) (x-bar) represents the sample mean.
(n) represents the number of data points in the sample.

The (n-1) in the denominator accounts for the degrees of freedom when estimating the population standard deviation from a sample.

Interpreting Standard Deviations

Standard deviations provide a quantifiable measure of how much an asset's price or return has historically deviated from its average. A higher standard deviation indicates greater volatility and, therefore, higher perceived risk. For instance, an investment with an average annual return of 8% and a standard deviation of 2% implies that its returns typically fall between 6% and 10% (8% ± 2%). Conversely, an investment with an average return of 8% and a standard deviation of 10% suggests returns could range more widely, from -2% to 18%. Investors use this insight to gauge the potential fluctuations in an investment and determine if the level of risk aligns with their tolerance. It is particularly relevant for understanding the potential spread of expected return.

Hypothetical Example

Consider two hypothetical stocks, Stock A and Stock B, over five years.

Stock A Annual Returns: 10%, 12%, 9%, 11%, 8%
Stock B Annual Returns: 25%, -5%, 30%, -10%, 20%

First, calculate the mean return for each:

Mean for Stock A: ((10+12+9+11+8) / 5 = 50 / 5 = 10%)
Mean for Stock B: ((25-5+30-10+20) / 5 = 60 / 5 = 12%)

Next, calculate the deviations from the mean, square them, sum them, and apply the standard deviation formula (using the sample formula, n-1, as this is a sample of returns).

For Stock A:

Deviations: (0, 2, -1, 1, -2)
Squared Deviations: (0, 4, 1, 1, 4)
Sum of Squared Deviations: (0+4+1+1+4 = 10)
Standard Deviation A: (\sqrt{10 / (5-1)} = \sqrt{10 / 4} = \sqrt{2.5} \approx 1.58%)

For Stock B:

Deviations: (13, -17, 18, -22, 8)
Squared Deviations: (169, 289, 324, 484, 64)
Sum of Squared Deviations: (169+289+324+484+64 = 1330)
Standard Deviation B: (\sqrt{1330 / (5-1)} = \sqrt{1330 / 4} = \sqrt{332.5} \approx 18.23%)

Despite Stock B having a higher average return, its significantly higher standard deviation ((18.23%) vs. (1.58%)) indicates much greater volatility and potential for widely fluctuating returns. An investor seeking stable growth might prefer Stock A, while one willing to take on substantial risk for higher potential returns might consider Stock B.

Practical Applications

Standard deviations are extensively used across various facets of finance and investing:

Investment Risk Assessment: It is a primary metric to gauge the historical price fluctuations of stocks, bonds, mutual funds, and other investments. A higher standard deviation indicates a more volatile investment.
Modern Portfolio Theory (MPT): Developed by Harry Markowitz, MPT uses standard deviation as its measure of risk to construct optimal portfolios. The goal is to find the Efficient Frontier, which represents portfolios offering the highest expected return for a given level of risk, or the lowest risk for a given expected return.
Sharpe Ratio: This widely used risk-adjusted return metric divides a portfolio's excess return (return minus the risk-free rate) by its standard deviation. A higher Sharpe ratio suggests better risk-adjusted performance.
Behavioral Finance: While standard deviation quantifies historical volatility, it also implicitly influences investor behavior. Understanding potential deviations from the mean helps investors set realistic expectations and manage their emotional responses to market swings.
Option Pricing Models: Standard deviation, often referred to as implied volatility in this context, is a critical input in models like the Black-Scholes formula, used to price options contracts.
Performance Benchmarking: Fund managers often compare their portfolio's standard deviation to that of a relevant benchmark index to assess their risk-taking relative to the market.

Limitations and Criticisms

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

Assumes Normal Distribution: Standard deviation works best when data points are symmetrically distributed around the mean, like in a bell curve. However, financial market returns often exhibit "fat tails" (more extreme positive or negative events than a normal distribution predicts) and skewness (asymmetric distribution), which standard deviation may not adequately capture.
Treats Upside and Downside Volatility Equally: Standard deviation measures deviation from the average regardless of direction. For investors, large positive deviations (upside volatility) are generally desirable, whereas large negative deviations (downside volatility) are undesirable. Standard deviation doesn't distinguish between these, potentially overstating the "risk" associated with positive price movements. This is a common critique, particularly highlighted in discussions around the limitations of the Sharpe ratio, where it can penalize desirable upside volatility.
Reliance on Historical Data: Standard deviation is calculated using past performance data. While historical volatility can be indicative, it does not guarantee future performance or risk. Market conditions can change rapidly, rendering historical measures less relevant.
Doesn't Account for Tail Events: Extreme market events, often called "Black Swan" events, are rare and difficult to predict using historical standard deviation, as they fall far outside the typical distribution. Other risk measures, like Value at Risk (VaR) or Conditional Value at Risk (CVaR), attempt to address these tail risks more directly.
Context Dependency: A given standard deviation figure must be interpreted within the context of the asset class and market conditions. A high standard deviation might be acceptable for a small-cap stock but concerning for a large-cap blue-chip.

Standard Deviations vs. Variance

While closely related and often used interchangeably in discussions of volatility, standard deviations and variance represent distinct statistical concepts.

Feature	Standard Deviations	Variance
Definition	Average distance of data points from the mean.	Average of the squared differences from the mean.
Unit	Same units as the original data (e.g., percentage).	Units are squared (e.g., percentage squared).
Interpretability	Easier to interpret as it is in the original units.	Less intuitive to interpret due to squared units.
Calculation	Square root of the variance.	Sum of squared deviations from the mean, divided by N or n-1.
Use Case	Widely used as a direct measure of risk and volatility.	Intermediate step in calculating standard deviation; used in advanced statistical models.

The main difference lies in their units and interpretability. Standard deviations revert the squared units of variance back to the original units of the data, making it a more intuitive and directly comparable measure of dispersion. Variance, while mathematically crucial for calculations (especially in portfolio optimization given its additive property for independent variables), is less commonly cited on its own as a direct measure of risk due to its squared units.

FAQs

What does a high standard deviation mean for an investment?

A high standard deviation for an investment indicates that its historical returns have fluctuated significantly around its average return. This implies higher volatility and, therefore, higher perceived risk. Investors considering such an asset should be prepared for potentially wider swings in its value.

How is standard deviation used in portfolio management?

In portfolio management, standard deviation is used to quantify the overall risk of a portfolio, as well as the individual assets within it. Portfolio managers use it as a key input in Modern Portfolio Theory to optimize the mix of assets, aiming to achieve the best possible expected return for a given level of risk, or to minimize risk for a target return through effective diversification.

Can standard deviation predict future risk?

Standard deviation is a historical measure and calculates past volatility. While past volatility can offer insights into an asset's typical behavior, it does not guarantee future risk or performance. Market conditions, economic environments, and asset-specific factors can change, affecting future deviations. It is best used as a tool for understanding historical tendencies rather than a precise predictor of the future.

Is standard deviation the only measure of investment risk?

No, standard deviation is a widely used but not the sole measure of investment risk. Other measures include Beta (which measures systematic risk relative to the market), Value at Risk (VaR), Conditional Value at Risk (CVaR), and drawdown analysis. Each measure provides a different perspective on risk, and a comprehensive risk assessment typically involves looking at multiple metrics.