Variance

What Is Variance?

Variance is a statistical measurement that quantifies the degree of spread or dispersion of a data set around its mean. In finance, it is a key metric used in statistics, portfolio theory, and risk management to gauge the volatility and, by extension, the risk associated with an investment's return. A high variance indicates that individual data points tend to deviate significantly from the average, suggesting greater unpredictability or wider fluctuations. Conversely, a low variance implies that data points are clustered closely around the mean, indicating more consistency and less volatility.

History and Origin

The concept of variance as a formal statistical measure was introduced by Sir Ronald Aylmer Fisher, a prominent British statistician and geneticist. Fisher first formally defined variance in his 1918 paper, "The Correlation Between Relatives on the Supposition of Mendelian Inheritance," and later elaborated on it with his development of the Analysis of Variance (ANOVA). His work at the Rothamsted Experimental Station in the early 20th century, where he sought to make sense of vast agricultural data, led to groundbreaking contributions in statistical theory, including the systematic use of variance to analyze experimental results.⁴

In the financial realm, variance gained significant prominence with the advent of Modern Portfolio Theory (MPT), pioneered by Harry Markowitz. Markowitz's seminal 1952 paper, "Portfolio Selection," laid the groundwork for quantifying investment risk and demonstrating the benefits of diversification. He proposed using variance as the primary measure of a portfolio's risk, showing that investors could optimize their portfolios by considering both expected return and variance. For his foundational work, Markowitz was awarded the Nobel Memorial Prize in Economic Sciences in 1990.³

Key Takeaways

Variance measures the dispersion of a set of data points around their average value, serving as a quantitative indicator of volatility or risk.
In finance, higher variance typically signifies greater investment risk, as it suggests wider swings in returns.
The concept of variance is fundamental to Modern Portfolio Theory, where it helps in constructing portfolios that balance risk and return.
It is calculated by taking the average of the squared differences from the mean, giving more weight to larger deviations.
While widely used, variance has limitations, particularly its symmetrical treatment of both positive and negative deviations from the mean.

Formula and Calculation

Variance is calculated by averaging the squared differences between each data point and the mean of the data set. Squaring the differences ensures that negative deviations do not cancel out positive ones, and it also penalizes larger deviations more heavily.

For a population data set, the formula for variance ((\sigma^2)) is:

\sigma^2 = \frac{\sum_{i=1}^{N} (X_i - \mu)^2}{N}

For a sample data set, the formula for variance ((s^2)) is:

s^2 = \frac{\sum_{i=1}^{n} (X_i - \bar{x})^2}{n-1}

Where:

( \sigma^2 ) = Population variance
( s^2 ) = Sample variance
( X_i ) = Each individual data point (e.g., individual stock returns)
( \mu ) = Population mean
( \bar{x} ) = Sample mean
( N ) = Total number of data points in the population
( n ) = Total number of data points in the sample
( \sum ) = Summation (adding up all values)

The (n-1) in the sample variance formula is known as Bessel's correction, which helps provide an unbiased estimate of the population variance from a sample, especially when the sample size is small. In finance, variance is typically calculated from historical return data to estimate future volatility. The relationship between the returns of different assets is captured by covariance, which is also used in portfolio variance calculations.

Interpreting the Variance

Interpreting variance in finance centers on its role as a measure of risk. A higher variance indicates that an investment's historical returns have been widely scattered around its average, implying greater unpredictability in its future performance. For example, an investment with a high variance suggests that its actual return could be significantly different from its expected return, either positively or negatively. This wide range of potential outcomes is directly linked to higher volatility.

Conversely, an investment with low variance has historically delivered returns that consistently stick close to its average. This suggests a more stable and predictable asset, often associated with lower risk. While a stable return might be appealing, it also typically means a more limited potential for outsized gains. Investors use variance to compare the riskiness of different assets or portfolios, aiming to select investments that align with their risk tolerance.

Hypothetical Example

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

Year	Stock A Return (%)	Stock B Return (%)
1	10	2
2	12	18
3	8	5
4	15	20
5	5	0

Step 1: Calculate the Mean Return for each stock.

Stock A Mean: ((10 + 12 + 8 + 15 + 5) / 5 = 50 / 5 = 10%)
Stock B Mean: ((2 + 18 + 5 + 20 + 0) / 5 = 45 / 5 = 9%)

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

Stock A:
- ((10 - 10)^2 = 0)
- ((12 - 10)^2 = 4)
- ((8 - 10)^2 = 4)
- ((15 - 10)^2 = 25)
- ((5 - 10)^2 = 25)
Stock B:
- ((2 - 9)^2 = 49)
- ((18 - 9)^2 = 81)
- ((5 - 9)^2 = 16)
- ((20 - 9)^2 = 121)
- ((0 - 9)^2 = 81)

Step 3: Sum the Squared Deviations.

Stock A Sum: (0 + 4 + 4 + 25 + 25 = 58)
Stock B Sum: (49 + 81 + 16 + 121 + 81 = 348)

Step 4: Calculate the Variance (using sample variance, dividing by (n-1)).

Stock A Variance: (58 / (5 - 1) = 58 / 4 = 14.5)
Stock B Variance: (348 / (5 - 1) = 348 / 4 = 87)

In this data set, Stock B has a significantly higher variance (87) than Stock A (14.5). Despite Stock B having a slightly lower average expected return (9% vs. 10%), its returns fluctuated much more widely, indicating greater risk.

Practical Applications

Variance is a cornerstone of quantitative finance, utilized across various applications in investing and market analysis:

Portfolio Management: In Modern Portfolio Theory (MPT), variance is the primary measure of portfolio risk. Portfolio managers use it to construct efficient portfolios that offer the highest return for a given level of risk, or the lowest risk for a desired return. The goal is often to reach the efficient frontier by combining assets whose returns are not perfectly correlated, thereby reducing overall portfolio variance through diversification.
Risk Assessment: Investors and analysts use historical variance to assess the inherent risk of individual securities or asset classes. A higher variance suggests a more volatile investment, which is crucial for determining suitable asset allocation strategies. Regulatory bodies, such as the U.S. Securities and Exchange Commission (SEC), emphasize transparent risk disclosure for investors, underscoring the importance of understanding measures like variance in evaluating potential investment outcomes.²
Performance Evaluation: Variance is used to adjust investment performance metrics. For instance, the Sharpe Ratio divides a portfolio's excess return by its standard deviation (the square root of variance) to measure risk-adjusted return, providing a more comprehensive view than return alone.
Derivatives Pricing: Models for pricing options and other derivatives, such as the Black-Scholes model, heavily rely on the concept of future price volatility, which is often estimated using historical variance.
Capital Asset Pricing Model (CAPM): While CAPM primarily uses Beta as its risk measure, Beta itself is derived from the covariance of an asset's return with the market's return, and covariance is directly related to variance.

Limitations and Criticisms

Despite its widespread use, variance as a measure of risk has several limitations and has drawn criticism from financial professionals and academics:

Symmetrical Treatment of Deviations: Variance treats both upside (positive) and downside (negative) deviations from the mean equally. In finance, investors are generally more concerned with negative deviations (losses) than positive ones (gains). This symmetrical approach means variance doesn't distinguish between desirable and undesirable volatility.¹
Sensitivity to Outliers: Because variance involves squaring the deviations, extreme data points (outliers) can disproportionately inflate the calculated variance, potentially misrepresenting the typical spread of the data. This can lead to an overestimation of risk in situations where extreme events are rare but significant.
Assumption of Normal Distribution: Many financial models that use variance implicitly assume that returns are normally distributed. However, real-world financial returns often exhibit "fat tails" (more extreme positive or negative events than a normal distribution would predict) and skewness, meaning the distribution of returns is not perfectly symmetrical. This can lead to an underestimation of extreme risks.
Historical Data Dependence: Variance is typically calculated using historical return data. While historical performance can offer insights, it is not always a reliable predictor of future volatility. Market conditions, economic environments, and asset characteristics can change, affecting future risk profiles differently from the past.
Lack of Intuitive Interpretation: The squared nature of variance means its units are not the same as the original data. For instance, if returns are in percentages, variance is in "percentage squared," which can be difficult to interpret intuitively. This is why standard deviation, the square root of variance, is often preferred for interpretation as it is in the same units as the mean.

Variance vs. Standard Deviation

Variance and standard deviation are both measures of dispersion, but they differ in their calculation and interpretation. Variance represents the average of the squared differences from the mean, providing a measure in squared units. This makes it mathematically convenient for certain calculations, such as in portfolio theory when combining assets, but less intuitive for direct understanding of data spread.

Standard deviation, on the other hand, is the square root of the variance. By taking the square root, standard deviation reverts the measure back to the original units of the data. This makes it directly comparable to the mean and much easier to interpret as it represents the typical distance of data points from the average. For instance, if the average daily return of a stock is 0.1% and its standard deviation is 2%, it means daily returns typically deviate by about 2% from the average. While variance penalizes extreme deviations more heavily due to squaring, standard deviation provides a more direct and understandable measure of volatility for investors.

FAQs

Why is variance important in finance?

Variance is crucial in finance because it quantifies the risk or volatility of an investment. It helps investors understand how much an asset's actual returns might deviate from its expected return, which is essential for making informed investment decisions and building diversified portfolios.

What does a high variance mean for an investment?

A high variance implies that an investment's historical returns have fluctuated widely around its mean. This indicates a higher degree of unpredictability and greater potential for large swings in value, both up and down, making it a riskier investment.

How does variance relate to Modern Portfolio Theory?

In Modern Portfolio Theory (MPT), developed by Harry Markowitz, variance is used as the quantitative measure of portfolio risk. MPT demonstrates how combining different assets in a portfolio can reduce the overall variance (and thus risk) through diversification, leading to a more efficient allocation of capital and helping to identify the efficient frontier.

Is variance always a good measure of risk?

While variance is widely used, it has limitations as a sole measure of risk. It treats positive and negative deviations equally, even though investors are typically more concerned about losses. It can also be heavily influenced by outliers and assumes a normal distribution of returns, which isn't always accurate in financial markets. Therefore, it is often used in conjunction with other risk metrics.

How can investors use variance to make decisions?

Investors can use variance to compare the relative risk of different investment options. By analyzing the historical variance of various assets, they can choose those that align with their risk tolerance. For example, a conservative investor might prefer assets with lower variance, while an aggressive investor might accept higher variance for the potential of greater returns. It's also vital in asset allocation to balance risk across a portfolio.