5.2 Measuring Risk: Variance and Standard Deviation
4 min read•august 14, 2024
Risk and return are two sides of the investment coin. In this section, we dive into measuring risk using and . These tools help investors quantify the of their investments and make informed decisions.
Understanding variance and standard deviation is crucial for comparing different investments. By calculating these metrics, investors can assess how much an investment's returns might deviate from the expected average, helping them gauge potential risks and rewards.
Variance and Standard Deviation of Returns
Calculating Variance and Standard Deviation
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- Variance measures the average squared deviation of each number from the mean of a data set
- Calculated as the sum of squared deviations from the mean divided by the number of observations minus 1
- Standard deviation is the square root of variance and measures the dispersion of a data set relative to its mean
- Represents a measure of volatility and a commonly used measure of investment risk
- is calculated as the weighted average of individual asset variances plus the weighted covariances between each pair of assets
- is the square root of portfolio variance
- The variance and standard deviation of a portfolio depend on the variances and standard deviations of the individual assets in the portfolio, as well as the correlations between the returns of those assets
Interpreting Variance and Standard Deviation
- In investment analysis, variance and standard deviation measure the volatility or dispersion of returns around the average return
- Higher values indicate greater variability and risk
- Standard deviation quantifies and compares the level of risk associated with different investments or portfolios
- Provides a standardized measure of risk in the same units as the original data
- Assuming returns are normally distributed, approximately 68% of observations fall within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations
- Investors can use standard deviation to assess whether the potential returns offered by an investment are commensurate with the level of risk involved
Risk Interpretation of Variance and Standard Deviation
Normal Distribution Assumptions
- Standard deviation assumes that returns are normally distributed, which may not always be the case in reality
- Asset returns often exhibit "" or higher frequencies of extreme outcomes than predicted by a (stock market crashes, sudden spikes in volatility)
- Variance and standard deviation are based on and may not accurately predict future risk, especially in rapidly changing market conditions or for assets with limited return histories
Investor Preferences and Risk Perception
- Variance and standard deviation are symmetrical measures that treat positive and negative deviations from the mean identically
- However, investors may have asymmetric preferences, being more concerned about (potential losses) than (gains)
- These measures do not capture the full complexity of investment risk, such as , , or the potential for catastrophic events or "" (sudden market collapses, geopolitical events)
Limitations of Variance and Standard Deviation
Symmetrical Treatment of Deviations
- Variance and standard deviation treat positive and negative deviations from the mean identically
- Investors may be more concerned about downside risk (potential losses) than upside potential (gains)
- These measures do not distinguish between upside and downside volatility, which may have different implications for investors
Assumption of Normal Distribution
- Standard deviation assumes that returns are normally distributed, which may not always be the case in reality
- Asset returns often exhibit "fat tails" or higher frequencies of extreme outcomes than predicted by a normal distribution (stock market crashes, sudden spikes in volatility)
- Non-normal return distributions can lead to underestimation of risk using variance and standard deviation
Reliance on Historical Data
- Variance and standard deviation are based on historical data and may not accurately predict future risk
- Rapidly changing market conditions or assets with limited return histories can limit the predictive power of these measures
- Historical data may not capture the full range of potential future outcomes, especially in the case of rare or unprecedented events
Incomplete Picture of Risk
- Variance and standard deviation do not capture the full complexity of investment risk
- Other types of risk, such as liquidity risk (difficulty buying or selling an asset), counterparty risk (risk of default by another party), or the potential for catastrophic events or "black swans" (sudden market collapses, geopolitical events) are not accounted for
- Investors should consider a more comprehensive set of risk measures and qualitative factors when assessing investment risk
Risk Comparison of Investments
Comparing Standard Deviations
- Investors can calculate and compare the standard deviations of different assets or portfolios to assess their relative levels of risk
- Higher standard deviation generally indicates higher risk
- When comparing investments, it's important to consider both risk and return
- The , calculated as standard deviation divided by mean return, provides a standardized measure of risk per unit of return
- Investors can use risk measures like standard deviation in conjunction with the or other risk-adjusted performance metrics to evaluate the of different investments
Considerations for Risk Comparison
- It's important to compare the risk of investments over similar time horizons
- Risk measures can vary depending on the length of the investment period (short-term vs. long-term)
- Investors should consider the potential impact of any differences in the distributions of returns that may not be captured by standard deviation alone
- Skewness (asymmetry of returns) and kurtosis (likelihood of extreme outcomes) can provide additional insights into the risk profile of an investment
- When comparing portfolios, investors should also consider the diversification benefits and potential correlations between assets
- Lower correlation between assets can help reduce overall portfolio risk
Key Terms to Review (16)
Black Swans: Black swans are rare and unpredictable events that have a massive impact on the world. They are often beyond the realm of normal expectations, making them difficult to predict using standard forecasting methods like variance and standard deviation, which typically measure typical risks. Understanding black swans is crucial because they can significantly affect financial markets and investment strategies, illustrating the limitations of traditional risk assessment approaches.
Coefficient of variation (cv): The coefficient of variation (cv) is a statistical measure that represents the ratio of the standard deviation to the mean, expressed as a percentage. It provides a standardized way to assess the relative variability of different datasets, allowing for comparison across different contexts and units. This measure is particularly useful in finance and risk assessment, where it helps gauge the risk associated with an investment relative to its expected return.
Counterparty risk: Counterparty risk is the possibility that a party involved in a financial transaction may default on their obligations, leading to potential losses for the other party. This risk is particularly significant in trading agreements like forward and futures contracts, where each party relies on the other to fulfill their end of the deal. It also plays a crucial role in hedging strategies and risk management, as parties must assess the reliability of their counterparts to effectively manage their own financial exposure.
Downside risk: Downside risk refers to the potential loss or negative return on an investment, specifically the likelihood of an asset's value falling below a certain benchmark or expected return. It is a critical concept in finance, as it helps investors understand the potential for losses and the need for effective risk management strategies. Measuring downside risk often involves statistical methods like semi-variance and value at risk (VaR), which focus on negative deviations from the mean return.
Fat Tails: Fat tails refer to the phenomenon in probability distributions where extreme outcomes (or outliers) have a higher probability of occurring than what is predicted by a normal distribution. This characteristic is crucial for risk assessment because traditional metrics like variance and standard deviation can underestimate the likelihood of extreme events, leading to potentially significant miscalculations in risk management strategies.
Historical Data: Historical data refers to the collection of past financial information or performance metrics that have been recorded over time. This data is crucial for analyzing trends, making informed predictions, and measuring risk in financial contexts. By examining historical data, investors and analysts can assess the volatility of assets and evaluate potential future performance based on past behavior.
Liquidity Risk: Liquidity risk refers to the potential difficulty of converting an asset into cash without significantly affecting its price. This risk is important for financial institutions and investors because it can impact their ability to meet obligations or capitalize on opportunities in a timely manner, ultimately influencing the overall efficiency and stability of financial markets.
Normal Distribution: Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. This concept is crucial when measuring risk using variance and standard deviation, as it helps in understanding how values are dispersed and the likelihood of different outcomes.
Portfolio standard deviation: Portfolio standard deviation is a measure of the total risk associated with a portfolio of investments, reflecting how much the returns on the portfolio are expected to fluctuate over time. It combines the variances of individual assets and their covariances, indicating the overall volatility of the portfolio. A higher standard deviation implies greater risk, which is crucial for investors when assessing their risk tolerance and making investment decisions.
Portfolio variance: Portfolio variance is a statistical measure that quantifies the dispersion of returns of a portfolio of assets, reflecting the degree to which individual asset returns deviate from the overall portfolio return. It helps investors understand the risk associated with holding multiple investments, as it considers both the volatility of individual assets and how they interact with each other through their correlations.
Risk-return tradeoff: The risk-return tradeoff is the principle that potential return rises with an increase in risk. It indicates that investors must weigh the level of risk they are willing to accept against the potential gains they expect to receive from an investment. Understanding this balance helps investors make informed decisions about where to allocate their resources, taking into account both the variability of returns and the expected profit.
Sharpe Ratio: The Sharpe Ratio is a measure that helps investors understand the return of an investment compared to its risk, calculated by subtracting the risk-free rate from the investment's return and dividing by the investment's standard deviation. It’s essential for evaluating the performance of an investment relative to its risk, making it particularly useful in assessing diversified portfolios, understanding market equilibrium in asset pricing models, and implementing effective risk management strategies.
Standard Deviation: Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation means that the values are spread out over a wider range. This concept is essential for understanding risk, as it helps investors assess the volatility of individual assets and portfolios, making it easier to achieve optimal diversification and manage potential returns effectively.
Upside potential: Upside potential refers to the possibility of an investment or asset increasing in value, providing a positive return to the investor. This concept is important in assessing the attractiveness of an investment, especially when considering its risk-reward profile. In the context of measuring risk, upside potential is often evaluated alongside metrics like variance and standard deviation to understand how much an investment can grow compared to its expected volatility.
Variance: Variance is a statistical measurement that indicates the degree of spread or dispersion in a set of data points. It reflects how much individual data points deviate from the mean of the dataset, providing insight into the level of risk associated with an investment or financial decision.
Volatility: Volatility refers to the degree of variation of a trading price series over time, often measured by the standard deviation of returns. It indicates how much the price of an asset, such as stocks or bonds, is expected to fluctuate, which can influence investor decisions and market behavior. In financial markets, higher volatility signifies greater risk and uncertainty, especially in a globalized environment where interconnected markets can amplify these price swings and lead to more significant impacts on investments.