Variance",

Variance is a statistical measure that quantifies the dispersion of a set of data points around their mean, or average value. In the context of finance, it is a foundational concept within Portfolio Theory and Risk Management, serving as a common metric for assessing the Volatility and Risk associated with investments or portfolios. A high variance indicates that data points are widely spread out from the mean, suggesting greater volatility, while a low variance means data points are closely clustered around the mean, indicating lower volatility.

History and Origin

While the broader concepts of error measurement and least squares methods trace back centuries to mathematicians like Laplace and Gauss, the specific term "variance" was formally introduced by the eminent British statistician Ronald Fisher in 1918. Fisher defined variance as the square of the Standard Deviation in his groundbreaking paper, "The Correlation Between Relatives on the Supposition of Mendelian Inheritance," published in the Transactions of the Royal Society of Edinburgh.²⁹ He noted the utility of using the square of the standard deviation as a measure of variability because the variances of independent random variables can be added directly. This innovation provided a robust framework for analyzing the causes of variability in various fields, including genetics and, later, finance.²⁷, ²⁸

Key Takeaways

• Variance quantifies the spread of data points from their Mean, serving as a key measure of risk in finance.
• A higher variance implies greater price fluctuations and potential for larger losses or gains.
• It is a core component in modern Portfolio theory, particularly in evaluating the risk-return trade-off.
• Variance does not differentiate between upward and downward movements, treating both as deviations from the mean.
• The concept has limitations, especially when asset returns do not follow a normal distribution.

Formula and Calculation

Variance is calculated as the average of the squared differences from the mean. For a population, the formula is:

For a sample, the formula is slightly adjusted to account for bias:

Where:

• (\sigma^2) (sigma squared) represents the population variance.
• (s^2) represents the sample variance.
• (X_i) is the individual data point (e.g., an individual Return for an investment).
• (\mu) (mu) is the population mean.
• (\bar{X}) (X-bar) is the sample mean.
• (N) is the total number of data points in the population.
• (n) is the total number of data points in the sample.

The sum of squared deviations is divided by (N) for a population and by (n-1) for a sample to ensure an unbiased estimate of the population variance.

Interpreting the Variance

In finance, variance is typically used to gauge the potential dispersion of an investment's returns around its Expected Value. A higher variance suggests that an asset's price movements are more spread out, indicating higher volatility and, consequently, higher risk. Conversely, a lower variance implies that an asset's returns are more clustered around the mean, suggesting lower volatility and risk.

For example, an investment with a historical average return of 8% and a high variance indicates that its actual returns have historically deviated significantly from 8%, ranging widely from losses to much higher gains. An investment with the same 8% average return but a low variance suggests its historical returns have consistently been close to 8%. Investors often seek a balance between potential return and the level of variance they are willing to accept, a concept central to Asset Allocation.

Hypothetical Example

Consider two hypothetical stocks, Stock A and Stock B, and their annual returns over five years:

• Stock A Returns: 10%, 12%, 8%, 9%, 11%
• Stock B Returns: 25%, -5%, 30%, 2%, 18%

Step 1: Calculate the Mean Return for each stock.

• Stock A Mean: ((10 + 12 + 8 + 9 + 11) / 5 = 10%)
• Stock B Mean: ((25 + (-5) + 30 + 2 + 18) / 5 = 14%)

Step 2: Calculate the squared deviation from the mean for each return.

• Stock A:
  • ((10 - 10)^2 = 0)
  • ((12 - 10)^2 = 4)
  • ((8 - 10)^2 = 4)
  • ((9 - 10)^2 = 1)
  • ((11 - 10)^2 = 1)
  • Sum of squared deviations = (0 + 4 + 4 + 1 + 1 = 10)
• Stock B:
  • ((25 - 14)^2 = 121)
  • ((-5 - 14)^2 = 361)
  • ((30 - 14)^2 = 256)
  • ((2 - 14)^2 = 144)
  • ((18 - 14)^2 = 16)
  • Sum of squared deviations = (121 + 361 + 256 + 144 + 16 = 898)

Step 3: Calculate the Variance (using sample formula, n-1).

• Stock A Variance: (10 / (5 - 1) = 10 / 4 = 2.5)
• Stock B Variance: (898 / (5 - 1) = 898 / 4 = 224.5)

In this example, despite Stock B having a higher average return, its much higher variance (224.5 vs. 2.5) clearly indicates significantly greater Volatility and risk compared to Stock A. An investor focused on consistent Return might prefer Stock A, while one seeking higher potential returns and willing to accept more risk might consider Stock B.

Practical Applications

Variance is a cornerstone of quantitative finance and finds numerous practical applications across investing, markets, and financial analysis:

• Portfolio Optimization: Harry Markowitz's Modern Portfolio Theory (MPT) fundamentally relies on variance to construct efficient portfolios that maximize expected return for a given level of risk, or minimize risk for a given expected return. It considers the variance of individual assets and their Covariance to determine overall portfolio variance.
• Risk Assessment: Investors and analysts use variance to measure the historical volatility of individual securities or entire portfolios, helping them understand the potential range of outcomes. Regulatory bodies, such as the U.S. Securities and Exchange Commission (SEC), require investment companies to provide clear Risk disclosures to investors, often implicitly or explicitly referencing volatility measures.²³, ²⁴, ²⁵, ²⁶
• Option Pricing Models: Variance (or its square root, standard deviation, also known as volatility) is a critical input in models like the Black-Scholes formula, which calculates the theoretical price of options.
• Performance Measurement: Variance is used in various risk-adjusted performance metrics, such as the Sharpe Ratio, to evaluate how much return an investment generates per unit of risk taken.
• Financial Modeling: In simulations and stress testing, variance helps model the potential range of financial outcomes under different market conditions. For instance, historical data on market volatility, such as the CBOE Volatility Index (VIX), can provide insights into expected market dispersion.²⁰, ²¹, ²²

Limitations and Criticisms

While widely used, variance as a measure of Risk has several notable limitations:

• Symmetry Assumption: Variance treats upside and downside deviations from the Mean identically.¹⁹ This means it penalizes positive price movements (gains) just as much as negative ones (losses). However, most investors are typically more concerned with downside risk (potential losses) than upside potential (gains).¹⁶, ¹⁷, ¹⁸
• Normal Distribution Assumption: Variance and Standard Deviation are most effective when returns follow a normal (bell-shaped) distribution.¹⁴, ¹⁵ However, real-world financial returns often exhibit "fat tails" (more extreme events than a normal distribution would predict) and Skewness (asymmetrical distribution), meaning that large positive or negative events are more probable than assumed, and the distribution is not perfectly symmetrical.¹², ¹³ This can lead to an underestimation of true risk, particularly tail risk.⁹, ¹⁰, ¹¹ The Federal Reserve Bank of San Francisco has published research highlighting how asset returns often display characteristics like skewness and kurtosis that go beyond what variance alone can capture.⁷, ⁸
• Reliance on Historical Data: Variance is calculated using historical data, which may not always be indicative of future volatility. Rapidly changing market conditions or unprecedented events can limit the predictive power of historical variance.⁶
• Incomplete Picture of Risk: Variance does not capture all aspects of investment risk. It doesn't account for liquidity risk, counterparty risk, or the impact of "black swan" events (rare, unpredictable, and high-impact occurrences).⁵

For these reasons, more sophisticated Risk Management techniques, such as Value at Risk (VaR) and Expected Shortfall (ES), are often used in conjunction with or as alternatives to variance, especially for complex Investment Strategy applications.³, ⁴

Variance vs. Standard Deviation

Variance and Standard Deviation are both measures of data dispersion, but they differ in their units and interpretability.

While variance provides the mathematical foundation for measuring dispersion and is crucial in many statistical models (e.g., in calculating Covariance and Correlation for portfolio analysis), standard deviation is generally preferred when discussing risk in practical terms with investors. This is because standard deviation is expressed in the same units as the average return, making it easier to directly compare with the mean and understand the typical spread of returns.

FAQs

Why is variance considered a measure of risk?

Variance measures how much an asset's actual returns tend to deviate from its average return. In finance, this deviation represents uncertainty. The greater the deviation (higher variance), the more uncertain the future returns, implying a higher level of Risk that the actual return will differ significantly from the expected return.²

Can variance be negative?

No, variance cannot be negative. It is calculated by summing squared differences from the mean, and the square of any real number (positive or negative) is always non-negative. A variance of zero would indicate that all data points are identical to the Mean, meaning no Volatility or dispersion.

Is a high variance good or bad?

A high variance is neither inherently good nor bad; rather, it indicates higher Volatility and potential for larger fluctuations in Return. For some investors, a high variance might be "bad" if they seek stability and predictable returns, as it suggests greater downside risk. For others, particularly those with a higher Risk tolerance, high variance might be "good" because it also implies the potential for higher positive returns.¹ It depends entirely on an individual's financial goals and risk appetite.

How does diversification affect portfolio variance?

Diversification aims to reduce portfolio variance. By combining assets whose returns are not perfectly positively correlated, the overall Portfolio fluctuations can be smoothed out. When one asset performs poorly, another might perform well, offsetting the negative impact and leading to a lower overall portfolio variance than the sum of individual asset variances. This is a fundamental principle of modern portfolio theory.

What is the difference between variance and volatility?

In finance, "volatility" is often used interchangeably with Standard Deviation, which is the square root of variance. While variance measures the average of squared deviations, volatility (standard deviation) expresses this measure in the same units as the original data, making it more intuitive for practical interpretation of price movements. Both are quantitative measures of dispersion and Risk.