Expected variance

What Is Expected Variance?

Expected variance is a fundamental concept in quantitative analysis within the field of portfolio theory. It represents the anticipated dispersion or spread of a set of possible outcomes around their mean, or expected value. In finance, expected variance specifically quantifies the anticipated level of volatility or risk associated with an investment or a portfolio of assets⁵². A higher expected variance indicates a greater degree of uncertainty about future returns, suggesting that actual returns could deviate significantly from the predicted average⁵¹.

History and Origin

The concept of expected variance as a measure of investment risk gained prominence with the development of Modern Portfolio Theory (MPT). Pioneered by economist Harry Markowitz, MPT was introduced in his seminal 1952 paper, "Portfolio Selection," published in The Journal of Finance.⁴⁹, ⁵⁰ Markowitz's work provided a mathematical framework for constructing investment portfolios to maximize expected return for a given level of risk, or equivalently, minimize risk for a given expected return.⁴⁸ Central to his theory was the use of variance (and its square root, standard deviation) to quantify the risk of both individual assets and entire portfolios. This groundbreaking approach offered a quantitative explanation for the benefits of diversification, illustrating how combining assets whose returns do not move perfectly in sync can reduce overall portfolio volatility.⁴⁷ For his contributions, Markowitz was awarded the Nobel Memorial Prize in Economic Sciences in 1990.⁴⁶

Key Takeaways

• Expected variance measures the anticipated spread of returns around an investment's expected value.⁴⁵
• It is a core component of Modern Portfolio Theory, used to quantify investment risk.⁴³, ⁴⁴
• A higher expected variance implies greater potential fluctuations in an asset's or portfolio's returns.⁴²
• Expected variance is expressed in squared units, often making its square root, standard deviation, more intuitive for direct interpretation of risk.
• It helps investors and analysts make informed decisions about asset allocation and portfolio construction.⁴¹

Formula and Calculation

Expected variance is calculated as the probability-weighted average of the squared deviations of possible outcomes from the expected value.³⁹, ⁴⁰

For a single discrete random variable (X) (e.g., an asset's return) with possible outcomes (x_i) and their respective probabilities (P(x_i)), and an expected value (E(X)) (or mean (\mu)):

Where:

• (\text{Var}(X)) = Expected variance of the random variable (X).
• (E(X)) or (\mu) = The expected return (mean) of the asset.³⁸
• (x_i) = Each possible outcome or return of the asset.
• (P(x_i)) = The probability of each outcome (x_i) occurring.
• (n) = The number of possible outcomes.

For a portfolio of two assets (Asset A and Asset B), with weights (w_A) and (w_B), individual variances (\sigma_A^{2) and (\sigma_B}2), and covariance (\text{Cov}(R_A, R_B)) between their returns:³⁷

Where:

• (\text{Var}(R_P)) = Expected variance of the portfolio's returns.
• (w_A, w_B) = The respective weights of Asset A and Asset B in the portfolio.
• (\sigma_A^{2, \sigma_B}2) = The expected variances of Asset A and Asset B, respectively.
• (\text{Cov}(R_A, R_B)) = The covariance between the returns of Asset A and Asset B.³⁶

This portfolio formula highlights that portfolio risk is not merely the sum of individual asset risks but also depends on how the assets move together, as captured by their covariance.

Interpreting Expected Variance

Interpreting expected variance involves understanding its implications for investment risk and potential outcomes. A higher expected variance suggests that an asset's or portfolio's actual returns are likely to be widely dispersed around its average, indicating greater volatility. Conversely, a lower expected variance implies that returns are expected to cluster more closely around the mean, suggesting more predictable outcomes and lower volatility.³⁵

For investors, this means that investments with high expected variance carry a greater chance of both higher-than-expected gains and lower-than-expected losses, relative to their average return. Investments with low expected variance, on the other hand, are generally considered more stable, with returns less likely to deviate drastically from their anticipated value.³⁴ Understanding this measure is crucial for aligning investment choices with an individual's risk tolerance and financial objectives.³³

Hypothetical Example

Consider an investor, Sarah, who is evaluating two potential stocks for her portfolio: TechGrowth and StableUtility. She wants to understand their expected variance based on potential future economic states.

Assumptions:

Step 1: Calculate Expected Return ((\mu)) for each stock.

• TechGrowth:
  (\mu_{\text{TechGrowth}} = (0.30 \times 0.25) + (0.50 \times 0.10) + (0.20 \times -0.15))
  (\mu_{\text{TechGrowth}} = 0.075 + 0.050 - 0.030 = 0.095 = 9.5%)
• StableUtility:
  (\mu_{\text{StableUtility}} = (0.30 \times 0.08) + (0.50 \times 0.05) + (0.20 \times 0.02))
  (\mu_{\text{StableUtility}} = 0.024 + 0.025 + 0.004 = 0.053 = 5.3%)

Step 2: Calculate Squared Deviations from the Mean for each stock.

• TechGrowth:

  • Boom: ((0.25 - 0.095)^{2 = (0.155)}2 = 0.024025)
  • Normal: ((0.10 - 0.095)^{2 = (0.005)}2 = 0.000025)
  • Recession: ((-0.15 - 0.095)^{2 = (-0.245)}2 = 0.060025)

• StableUtility:

  • Boom: ((0.08 - 0.053)^{2 = (0.027)}2 = 0.000729)
  • Normal: ((0.05 - 0.053)^{2 = (-0.003)}2 = 0.000009)
  • Recession: ((0.02 - 0.053)^{2 = (-0.033)}2 = 0.001089)

Step 3: Calculate Expected Variance by weighting squared deviations by probability.

• TechGrowth ((\text{Var}_{\text{TechGrowth}})):
  (\text{Var}{\text{TechGrowth}} = (0.30 \times 0.024025) + (0.50 \times 0.000025) + (0.20 \times 0.060025))
  (\text{Var}{\text{TechGrowth}} = 0.0072075 + 0.0000125 + 0.012005 = 0.019225)

• StableUtility ((\text{Var}_{\text{StableUtility}})):
  (\text{Var}{\text{StableUtility}} = (0.30 \times 0.000729) + (0.50 \times 0.000009) + (0.20 \times 0.001089))
  (\text{Var}{\text{StableUtility}} = 0.0002187 + 0.0000045 + 0.0002178 = 0.000441)

Conclusion:

The expected variance for TechGrowth is 0.019225, while for StableUtility it is 0.000441. This quantitative financial modeling demonstrates that TechGrowth has a significantly higher expected variance, implying greater potential for its returns to deviate from its 9.5% expected average, reflecting a higher level of risk compared to StableUtility.

Practical Applications

Expected variance serves as a cornerstone in various real-world financial applications, particularly within investment management and risk assessment.

• Portfolio Optimization: Expected variance is central to constructing optimal portfolios. Investors use it, alongside expected returns and covariance between assets, to build portfolios that achieve the highest possible return for a given level of risk, or the lowest risk for a desired return. This process is a core component of Modern Portfolio Theory and helps define the efficient frontier.³²
• Risk Management: Financial institutions and individual investors employ expected variance to quantify and manage the risk exposure of their holdings. By calculating the expected variance of a portfolio, managers can assess the potential for unexpected fluctuations and implement strategies to mitigate adverse movements. Regulatory bodies, such as the Federal Reserve, provide guidance on robust risk management frameworks that implicitly rely on such quantitative measures.²⁷, ²⁸, ²⁹, ³⁰, ³¹
• Performance Evaluation: When evaluating the performance of investment managers or strategies, expected variance (or standard deviation) is often used to calculate risk-adjusted returns, providing a more comprehensive view than raw returns alone.
• Derivatives Pricing: In the pricing of options and other derivatives, expected variance (or, more commonly, implied volatility, which is derived from it) is a critical input that reflects the market's expectation of future price movements.
• Budgeting and Forecasting: Beyond investment, expected variance can be applied in corporate finance to analyze deviations between budgeted and actual financial outcomes, such as revenue or costs. This "variance analysis" helps businesses identify operational inefficiencies and adjust their strategies.²⁵, ²⁶

Limitations and Criticisms

While expected variance is a widely used measure of risk, it comes with several limitations and has faced criticisms:

• Symmetry Assumption: Expected variance treats upside and downside deviations from the mean identically.²⁴ However, investors are typically more concerned about negative deviations (losses) than positive ones (gains).²³ This symmetrical treatment can lead to an incomplete picture of an investor's true risk exposure.²²
• Reliance on Historical Data: Calculating expected variance often relies on historical data as a proxy for future probabilities and outcomes.²¹ Past performance is not indicative of future results, and rapidly changing market conditions or unprecedented events ("black swans") may not be adequately captured by historical data, limiting the predictive power of the measure.¹⁹, ²⁰
• Normal Distribution Assumption: For many practical applications, particularly within Modern Portfolio Theory, expected variance assumes that asset returns follow a normal distribution.¹⁷, ¹⁸ In reality, financial returns often exhibit "fat tails," meaning extreme positive or negative outcomes occur more frequently than predicted by a normal distribution, rendering variance a less accurate measure of tail risk.¹⁶
• Unit of Measurement: Expected variance is expressed in squared units (e.g., if returns are in percent, variance is in percent-squared), which can make it less intuitive to interpret directly compared to standard deviation, which is in the same units as the original data.¹⁵
• Incomplete Picture of Risk: Expected variance does not account for all types of investment risk, such as liquidity risk, credit risk, or operational risk.¹⁴ Critics argue that risk should be defined more broadly by probabilities and consequences rather than solely by variance.¹³
• Sensitivity to Outliers: Extreme values or outliers in the data set can disproportionately influence the calculated variance, potentially skewing the risk assessment.¹²

These limitations necessitate that expected variance be used as part of a broader statistical analysis toolkit, complemented by other risk measures and qualitative assessments, especially in sophisticated financial modeling. The Bogleheads investment forum, for instance, discusses these drawbacks, noting that variance and standard deviation may not capture the full complexity of investment risk, such as liquidity or counterparty risk.¹⁰, ¹¹

Expected Variance vs. Standard Deviation

Expected variance and standard deviation are both measures of dispersion or spread within a data set, and they are closely related. However, they differ in their calculation and interpretation.

• Expected Variance: As discussed, expected variance is the probability-weighted average of the squared deviations of outcomes from the expected value.⁹ It quantifies the overall spread of data points from the mean.
• Standard Deviation: Standard deviation is simply the square root of the expected variance.⁸

The key distinction lies in their units of measurement and ease of interpretation. Because expected variance involves squaring the deviations, its units are squared (e.g., if returns are in percent, variance is in percent-squared). This can make direct interpretation less intuitive.

In contrast, taking the square root to derive standard deviation brings the measure back to the original units of the data (e.g., percent for returns).⁶, ⁷ This makes standard deviation more directly comparable to the expected return and easier for investors to grasp the typical magnitude of deviation from the mean. For example, stating that a stock has an expected return of 10% with a standard deviation of 15% is often more understandable than saying it has an expected variance of 0.0225 (15%(^2)). Therefore, while expected variance is crucial for its mathematical properties in portfolio calculations, standard deviation is more commonly reported and used for communicating volatility to investors.⁵

FAQs

What does a high expected variance imply?

A high expected variance implies that the actual returns of an investment or portfolio are anticipated to be widely dispersed around its expected (average) return. This indicates a higher level of volatility and, consequently, greater investment risk, meaning that outcomes could deviate significantly from the forecast.⁴

Is expected variance the same as historical variance?

No, expected variance is not the same as historical variance. Expected variance is a forward-looking measure, an anticipation of future dispersion based on probabilities of various outcomes. Historical variance, on the other hand, is a backward-looking measure, calculated from past observed data. While historical variance is often used to estimate expected variance, it relies on the assumption that past patterns will continue into the future, which may not always hold true.³

Why is expected variance squared?

Expected variance is squared to ensure that both positive and negative deviations from the mean contribute positively to the measure of dispersion. If deviations were not squared, positive and negative deviations would cancel each other out, potentially resulting in a misleading zero value even for highly volatile assets. Squaring also gives larger deviations a disproportionately greater weight, emphasizing the impact of extreme outcomes.²

How does expected variance relate to Modern Portfolio Theory?

Expected variance is a cornerstone of Modern Portfolio Theory (MPT). MPT uses expected variance (or standard deviation) as its primary measure of risk. By combining assets with varying expected returns, expected variances, and crucially, different covariances, MPT demonstrates how investors can construct diversified portfolios to optimize the trade-off between risk and return, aiming to achieve the highest possible expected return for a given level of expected variance.¹