Finance

What Is Standard Deviation?

Standard deviation is a widely used quantitative measure in finance that quantifies the amount of dispersion or variability of a set of data points around their mean. Within the broader context of portfolio theory, it serves as a common proxy for investment risk, reflecting the historical volatility of an asset's or a portfolio's return over time. A higher standard deviation indicates that the data points are spread out over a wider range, implying greater price fluctuations and, consequently, higher risk. Conversely, a lower standard deviation suggests that data points are clustered closely around the mean, indicating less volatility and lower risk. Standard deviation is a foundational concept in assessing the consistency of returns and is crucial for investors aiming to understand potential price movements.

History and Origin

The application of standard deviation to finance gained prominence with the pioneering work of Harry Markowitz. In 1952, Markowitz published his seminal paper, "Portfolio Selection," which laid the groundwork for Modern Portfolio Theory (MPT).¹⁴ Before Markowitz, investment decisions often focused on selecting individual securities with the highest anticipated returns. Markowitz revolutionized this approach by demonstrating that investors should not only consider the expected return of individual assets but also how those assets behave in relation to one another within a portfolio.,¹³ His theory highlighted the importance of diversification and introduced a mathematical framework for quantifying portfolio risk using standard deviation, allowing investors to construct portfolios that optimize the balance between risk and return.¹²,¹¹

Key Takeaways

• Standard deviation measures the dispersion of an asset's or portfolio's returns around its average (mean) return.
• It serves as a widely accepted quantitative indicator of investment risk, with higher values signifying greater volatility.
• The concept was formalized for financial applications by Harry Markowitz in his 1952 work on Modern Portfolio Theory.
• While useful, standard deviation assumes returns follow a normal distribution, which may not always hold true for financial markets, especially during extreme events.
• It is a key input in various financial metrics and models, including the Sharpe Ratio and Value at Risk (VaR).

Formula and Calculation

The standard deviation for a sample of historical returns is calculated using the following formula:

Where:

• (\sigma) (sigma) = Standard Deviation
• (R_i) = Individual return in the data set
• (\bar{R}) = The mean (average) return of the data set
• (n) = The number of data points in the sample
• (\sum) = Summation (sum of all values)

This formula calculates the square root of the variance, which is the average of the squared differences from the mean. For instance, to calculate the standard deviation of historical stock returns, one would first find the average of those returns, then subtract this average from each individual return, square the result, sum all squared differences, divide by the number of observations minus one (for sample standard deviation), and finally take the square root. The mean represents the expected return of the asset over the period.

Interpreting the Standard Deviation

Interpreting standard deviation in finance involves understanding what its value implies about the investment's behavior. A low standard deviation suggests that an investment's returns have historically stayed close to its average return, indicating predictable and stable performance. This is generally preferred by risk-averse investors. Conversely, a high standard deviation means that returns have historically been widely dispersed from the average, implying greater volatility and less predictability. This higher volatility can translate to larger potential gains but also larger potential losses, appealing more to investors with a higher risk tolerance.

For example, a stock with an average annual return of 10% and a standard deviation of 5% is less volatile than a stock with the same 10% average return but a 20% standard deviation. The latter suggests that its actual returns in any given year could deviate significantly from the 10% average, potentially ranging from a 30% loss to a 50% gain, assuming a normal distribution of returns.

Hypothetical Example

Consider two hypothetical investment funds, Fund A and Fund B, over a five-year period.

Fund A Annual Returns: 8%, 10%, 9%, 11%, 12%
Fund B Annual Returns: -5%, 25%, 5%, 30%, 15%

Step 1: Calculate the Mean Return ((\bar{R})) for each fund.

• Fund A: ((8 + 10 + 9 + 11 + 12) / 5 = 50 / 5 = 10%)
• Fund B: ((-5 + 25 + 5 + 30 + 15) / 5 = 70 / 5 = 14%)

Step 2: Calculate the difference of each return from the mean, square it, and sum the results.

• Fund A:

  • ((8 - 10)^{2 = (-2)}2 = 4)
  • ((10 - 10)^{2 = 0}2 = 0)
  • ((9 - 10)^{2 = (-1)}2 = 1)
  • ((11 - 10)^{2 = 1}2 = 1)
  • ((12 - 10)^{2 = 2}2 = 4)
  • Sum of squared differences = (4 + 0 + 1 + 1 + 4 = 10)

• Fund B:

  • ((-5 - 14)^{2 = (-19)}2 = 361)
  • ((25 - 14)^{2 = 11}2 = 121)
  • ((5 - 14)^{2 = (-9)}2 = 81)
  • ((30 - 14)^{2 = 16}2 = 256)
  • ((15 - 14)^{2 = 1}2 = 1)
  • Sum of squared differences = (361 + 121 + 81 + 256 + 1 = 820)

Step 3: Divide by (n-1) and take the square root.

• Fund A Standard Deviation: (\sqrt{10 / (5-1)} = \sqrt{10 / 4} = \sqrt{2.5} \approx 1.58%)
• Fund B Standard Deviation: (\sqrt{820 / (5-1)} = \sqrt{820 / 4} = \sqrt{205} \approx 14.32%)

Conclusion: Fund A has a much lower standard deviation (approximately 1.58%) compared to Fund B (approximately 14.32%). Despite Fund B having a higher average return (14% vs. 10%), Fund A demonstrates significantly less volatility. This example highlights how standard deviation helps an investor assess the consistency of returns for different investment strategies.

Practical Applications

Standard deviation is a cornerstone in many areas of finance and investing. Its practical applications span portfolio management, risk assessment, and regulatory compliance:

• Portfolio Management: Portfolio managers use standard deviation to measure the overall risk of a portfolio. By analyzing the standard deviation of individual assets and their correlation, they can apply principles of asset allocation to construct portfolios that offer the desired balance between risk and expected return. This is a central tenet of Modern Portfolio Theory and helps in achieving portfolio optimization.
• Risk Assessment and Measurement: Beyond portfolios, standard deviation helps gauge the market risk of individual financial instruments like stocks, bonds, or commodities. It is a key component in calculating other risk-adjusted performance measures, such as the Sharpe Ratio, which evaluates the return earned per unit of risk.
• Regulatory Disclosures: Regulatory bodies, such as the U.S. Securities and Exchange Commission (SEC), require companies to provide disclosures about their market risk exposures.¹⁰ While not always explicitly demanding standard deviation, these regulations often necessitate quantitative and qualitative information about sensitivity to market changes, for which standard deviation can be a contributing measure in risk models and disclosures. The SEC's rules require disclosures about policies for derivatives and quantitative/qualitative information regarding market risk inherent in derivatives and other financial instruments.⁹
• Fund Evaluation: Investors often compare the standard deviation of mutual funds or exchange-traded funds (ETFs) to evaluate their historical volatility. A fund with a lower standard deviation for a given return might be considered more efficient or stable.

The Federal Reserve Board's Financial Stability Report also assesses vulnerabilities within the financial system, underscoring the importance of understanding and monitoring various risk indicators, implicitly including those where standard deviation is a component.⁸

Limitations and Criticisms

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

• Assumption of Normal Distribution: Standard deviation assumes that asset returns follow a normal distribution (a bell-shaped curve), where returns are symmetrically distributed around the mean.⁷ However, financial market returns often exhibit "fat tails," meaning extreme positive or negative events (known as outliers) occur more frequently than a normal distribution would predict.⁶ This can lead to an underestimation of the probability of severe losses.⁵
• Treats Upside and Downside Volatility Equally: Standard deviation measures all deviations from the mean in the same way, regardless of whether they are positive or negative.⁴ Investors, however, typically perceive downside volatility (losses) as risk, while upside volatility (gains) is generally welcome. This can be a significant drawback, as a high standard deviation might be inflated by beneficial large positive returns, leading to a misperception of risk.
• Historical Basis: Standard deviation is calculated using historical data, and past performance is not indicative of future results. Market conditions can change, rendering historical standard deviation less relevant for predicting future volatility.³
• Sensitivity to Outliers: As a measure that squares deviations, standard deviation is highly sensitive to extreme data points. A few large outliers can disproportionately increase the standard deviation, potentially distorting the perceived risk of an asset or portfolio.²

Critics argue that for certain financial applications, especially those involving non-normally distributed returns or a focus on downside risk, other measures like Value at Risk (VaR) or conditional Value at Risk (CVaR) may offer a more nuanced perspective than standard deviation.¹

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 volatility of an asset or portfolio's returns, indicating how much those returns deviate from their average. It is an absolute measure of risk, reflecting both systematic and unsystematic risk components.

In contrast, Beta measures an asset's or portfolio's sensitivity to the overall market risk. It is a relative measure, indicating how much an asset's price tends to move in relation to the broader market index. A Beta of 1 suggests the asset moves in line with the market, while a Beta greater than 1 implies higher sensitivity (more volatile than the market), and a Beta less than 1 suggests lower sensitivity (less volatile than the market). The key distinction lies in what they aim to capture: standard deviation quantifies total price fluctuation, whereas Beta quantifies market-related price fluctuation.

FAQs

What does a high standard deviation mean for an investment?

A high standard deviation means an investment's historical returns have been highly volatile, fluctuating significantly around its average return. This indicates a higher level of risk but also potentially higher rewards.

Can standard deviation predict future returns?

No, standard deviation is a backward-looking measure based on historical data. While it can provide insights into an asset's past volatility and consistency, it does not guarantee future performance or predict specific returns.

How is standard deviation used in portfolio construction?

In portfolio construction, standard deviation is used to measure the overall risk of a diversified group of assets. By combining assets with different standard deviations and correlation to each other, investors can use principles of Modern Portfolio Theory to potentially achieve a desired level of return for a given level of risk.

Is a low standard deviation always better?

Not necessarily. A low standard deviation indicates lower volatility and generally less risk. However, investments with very low standard deviations may also have lower expected return potential. The "better" standard deviation depends on an investor's individual risk tolerance and financial goals.

What are "fat tails" in relation to standard deviation?

"Fat tails" refer to a characteristic of some probability distributions where extreme events occur more frequently than predicted by a normal distribution. When financial returns exhibit fat tails, standard deviation, which assumes a normal distribution, may underestimate the actual likelihood of large, infrequent price swings, particularly severe losses.