5.2 Measuring Risk: Variance and Standard Deviation

Risk and return are two sides of the investment coin. In this section, we dive into measuring risk using variance and standard deviation. These tools help investors quantify the volatility 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
• Portfolio variance is calculated as the weighted average of individual asset variances plus the weighted covariances between each pair of assets
  • Portfolio standard deviation 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 "fat tails" or higher frequencies of extreme outcomes than predicted by a normal distribution (stock market crashes, sudden spikes in volatility)
• Variance and standard deviation are based on historical data 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 downside risk (potential losses) than upside potential (gains)
• These measures do not capture the full complexity of investment risk, such as liquidity risk, counterparty risk, or the potential for catastrophic events or "black swans" (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 coefficient of variation (CV), 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 Sharpe ratio or other risk-adjusted performance metrics to evaluate the risk-return tradeoff 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