Standard deviation of returns

What Is Standard Deviation of Returns?

Standard deviation of returns is a statistical measure that quantifies the amount of dispersion or variability of an investment's historical returns around its average, or mean, return. Within the broader field of portfolio theory, it is a widely used metric to gauge an investment's risk. A higher standard deviation of returns indicates that the investment's actual returns have historically deviated more significantly from its average return, suggesting greater volatility and, by extension, higher risk. Conversely, a lower standard deviation suggests that the returns have been more consistent and closer to the average, implying lower risk. This measure is crucial for investors seeking to understand the potential fluctuations in their investment portfolios.

History and Origin

The concept of using standard deviation as a measure of investment risk gained prominence with the advent of Modern Portfolio Theory (MPT). MPT was pioneered by economist Harry Markowitz, whose seminal 1952 paper, "Portfolio Selection," laid the mathematical groundwork for understanding how to construct an optimal portfolio by considering both expected return and the statistical variance of returns.¹⁷,¹⁶,¹⁵ Before Markowitz's work, risk in investing was often discussed in general terms without a precise quantitative measure.¹⁴ His breakthrough was in quantifying this "undesirable thing" an investor seeks to avoid, by using the historical variability of returns, specifically standard deviation, as a proxy for risk.¹³ This allowed for a more rigorous and scientific approach to diversification and asset allocation, transforming how investors and financial professionals approached portfolio construction.

Key Takeaways

• Standard deviation of returns is a statistical measure of an investment's historical volatility or risk.
• It quantifies how much an investment's returns have deviated from its average return.
• A higher standard deviation implies greater price swings and higher risk.
• It is a foundational component of Modern Portfolio Theory, used in assessing portfolio risk and optimizing returns.
• While widely used, it has limitations, particularly for returns that do not follow a normal distribution.

Formula and Calculation

The standard deviation of returns is derived from the variance of returns. To calculate it, follow these steps:

1. Calculate the arithmetic mean (average) of the historical returns over a specific period.
2. Subtract the mean from each individual return to find the deviation of each return.
3. Square each deviation to make all values positive and emphasize larger deviations.
4. Sum the squared deviations.
5. Divide the sum of the squared deviations by the number of observations (for population standard deviation) or by the number of observations minus one (for sample standard deviation). This result is the variance.
6. Take the square root of the variance to get the standard deviation.

The formula for the sample standard deviation of returns is:

Where:

• (\sigma) = Standard deviation of returns
• (R_i) = Individual return in the dataset
• (\bar{R}) = Arithmetic mean (average) of the returns
• (n) = Number of observations (returns) in the dataset
• (\sum) = Summation

Financial services often annualize monthly standard deviation by multiplying it by the square root of 12.¹²

Interpreting the Standard Deviation of Returns

Interpreting the standard deviation of returns involves understanding its relationship to the expected consistency of an investment's performance. A higher standard deviation suggests that an investment's returns are more spread out from the average, meaning there's a wider range of potential outcomes, both positive and negative. For instance, an investment with an average annual return of 10% and a standard deviation of 15% would typically see its annual returns fall between -5% and 25% about two-thirds of the time.¹¹

This measure helps investors gauge how volatile a security or portfolio might be. For example, a bond fund typically has a much lower standard deviation than a stock fund, reflecting its generally more stable returns.¹⁰ When evaluating different investment options, investors often compare their standard deviations to assess their inherent risk profiles and decide if the potential for higher returns justifies the increased variability.

Hypothetical Example

Consider two hypothetical investments, Fund A and Fund B, over the past five years:

Step 1: Calculate the Mean Return for each fund.

• Fund A Mean: (12 + 10 + 11 + 9 + 8) / 5 = 50 / 5 = 10%
• Fund B Mean: (25 + (-5) + 18 + 30 + (-8)) / 5 = 60 / 5 = 12%

Step 2: Calculate Deviations from the Mean for each fund.

• Fund A Deviations: (2, 0, 1, -1, -2)
• Fund B Deviations: (13, -17, 6, 18, -20)

Step 3: Square the Deviations.

• Fund A Squared Deviations: (4, 0, 1, 1, 4) = Sum: 10
• Fund B Squared Deviations: (169, 289, 36, 324, 400) = Sum: 1218

Step 4: Calculate Variance (Sum of Squared Deviations / (n-1)).

• Fund A Variance: 10 / (5-1) = 10 / 4 = 2.5
• Fund B Variance: 1218 / (5-1) = 1218 / 4 = 304.5

Step 5: Calculate Standard Deviation (Square Root of Variance).

• Fund A Standard Deviation: (\sqrt{2.5} \approx 1.58%)
• Fund B Standard Deviation: (\sqrt{304.5} \approx 17.45%)

Even though Fund B had a higher average expected return, its much higher standard deviation of returns indicates significantly greater risk or volatility compared to Fund A.

Practical Applications

Standard deviation of returns is a cornerstone metric with numerous practical applications across finance and investment analysis.

• Portfolio Management: Fund managers and financial advisors use standard deviation to assess the overall risk of a portfolio. It helps them construct portfolios that align with a client's risk tolerance and investment objectives. For example, a conservative investor might prefer a portfolio with lower standard deviation, while an aggressive investor might accept a higher standard deviation for the potential of greater returns.
• Fund Analysis: When comparing mutual funds or exchange-traded funds (ETFs), investors often look at the standard deviation of returns to understand their historical price swings. It is a commonly reported metric by services like Morningstar.⁹,⁸
• Risk-Adjusted Performance Measures: Standard deviation is a key input for calculating important risk-adjusted performance metrics such as the Sharpe Ratio, which evaluates the return earned per unit of risk taken.⁷,⁶
• Regulatory Disclosures: Financial institutions and companies often use standard deviation as part of their risk disclosures to regulatory bodies like the Securities and Exchange Commission (SEC), providing quantitative insights into market risk exposures.⁵
• Market Analysis: Analysts use the standard deviation of market indexes (e.g., S&P 500) to understand overall market volatility. Periods of higher standard deviation often coincide with increased market uncertainty.

Limitations and Criticisms

Despite its widespread use, the standard deviation of returns has several limitations as a sole measure of risk.

• Assumes Normal Distribution: Standard deviation is most effective when returns are normally distributed (i.e., follow a bell curve). However, financial returns, especially during extreme market events, often exhibit "fat tails" or skewness, meaning large positive or negative deviations occur more frequently than a normal distribution would predict. This can lead standard deviation to underestimate true risk during periods of crisis or rapid change.⁴,³
• Treats Upside and Downside Equally: Standard deviation measures dispersion in both positive and negative directions. From an investor's perspective, large positive deviations (higher returns) are generally desirable, while large negative deviations (losses) are undesirable. Critics argue that it doesn't distinguish between these two, thus not fully capturing the specific concern about downside risk. Metrics like downside deviation or Sortino Ratio address this limitation.²
• Historical Data Reliance: Standard deviation is calculated using historical return data. While past performance can offer insights, it is not necessarily indicative of future results, particularly if market conditions or an investment's underlying characteristics change significantly.
• May Not Capture Tail Risk: It may not adequately account for "tail risk" – the risk of rare, extreme events (often called "Black Swan" events) that fall many standard deviations away from the mean but can have catastrophic impacts.

These limitations highlight the importance of using standard deviation in conjunction with other risk measures and qualitative analysis to form a comprehensive view of an investment's risk profile.

Standard Deviation of Returns vs. Volatility

The terms "standard deviation of returns" and "volatility" are often used interchangeably in finance, and for most practical purposes, they refer to the same concept. Both measure the degree of variation of a series of returns around its average. A higher standard deviation indicates greater volatility, implying that returns are more spread out and thus less predictable.

However, while standard deviation is a specific statistical calculation, volatility can sometimes be used in a broader, more qualitative sense to describe the general tendency of an asset's price to fluctuate. For instance, a stock might be described as "highly volatile" based on general market observations, even without a precise standard deviation calculation. Yet, when financial professionals quantify this fluctuation, they almost always refer to the standard deviation of returns. Other related concepts, like Beta in the Capital Asset Pricing Model (CAPM), also measure a form of volatility but specifically relate an asset's price movements to the overall market.

¹## FAQs

Is a higher standard deviation of returns always bad?

Not necessarily. A higher standard deviation indicates greater price swings, meaning there's potential for both larger gains and larger losses. While it signifies higher risk, it can also be associated with higher potential return. Investors with a higher risk tolerance and longer investment horizons might accept higher standard deviation in pursuit of greater long-term growth.

How is standard deviation used in portfolio construction?

In portfolio construction, standard deviation helps quantify the risk of individual assets and the portfolio as a whole. Modern Portfolio Theory, which utilizes standard deviation, suggests that combining assets with low correlation can reduce the overall portfolio's standard deviation without necessarily sacrificing expected return, leading to more efficient diversification.

What is a typical standard deviation for a stock or mutual fund?

There isn't a single "typical" standard deviation, as it varies widely depending on the asset class and specific investment. For instance, a short-term bond fund might have an annualized standard deviation below 1%, while a diversified equity fund could range from 10% to 20%, and individual stocks or more volatile sectors might exceed 30% or more. Comparing an investment's standard deviation to its peers or a relevant benchmark is often more insightful than looking at the number in isolation.

Does standard deviation predict future returns?

No, standard deviation of returns is a measure of historical volatility and does not predict future returns or risk with certainty. It provides an indication of how much an investment's returns have fluctuated in the past, which can be a useful guide for understanding potential future behavior, but market conditions can change, affecting future outcomes.