Variances

What Are Variances?

Variances, in the realm of statistical analysis, serve as a fundamental measure of the dispersion or spread of a set of data points around their mean. In finance, variances are critically important for quantifying risk, particularly the volatility of an investment's return. A high variance indicates that data points are widely spread from the average, suggesting greater volatility and, consequently, higher risk. Conversely, a low variance indicates that data points are clustered closely around the average, implying lower volatility and less risk. Understanding variances is essential for investors and analysts seeking to make informed decisions about asset allocation and portfolio construction.

History and Origin

While the concept of measuring dispersion has roots in earlier statistical thought, the formal development and widespread application of variances in finance largely trace back to the mid-20th century. A pivotal moment was the work of economist Harry Markowitz. In his seminal 1952 paper, "Portfolio Selection," published in The Journal of Finance, Markowitz introduced Modern Portfolio Theory (MPT), which mathematically formalized the relationship between risk and return for a portfolio of assets¹¹. Within MPT, variance became the cornerstone for quantifying portfolio risk, enabling investors to optimize their holdings based on a desired balance of expected return and risk. His work demonstrated how combining different financial assets could reduce overall portfolio risk through diversification, a concept central to the theory.¹⁰

Key Takeaways

• Variances measure how far a set of numbers are spread out from their average value.
• In finance, high variances in asset returns typically indicate higher volatility and risk.
• Low variances suggest more stable and predictable returns, implying lower risk.
• Variances are a foundational component of Modern Portfolio Theory, used in portfolio optimization and risk management.
• While widely used, variances have limitations, particularly their assumption of normally distributed returns and equal treatment of upside and downside movements.

Formula and Calculation

The calculation of variance depends on whether one is analyzing an entire population of data or a sample from that population.

For a population, the variance ($\sigma^2$) is calculated as the average of the squared differences from the Mean.

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

Where:

• $x_i$ = each individual data point
• $\mu$ = the population mean of the data points
• $N$ = the total number of data points in the population

For a sample, the sample variance ($s^2$) is calculated slightly differently to provide an unbiased estimate of the population variance, using $N-1$ in the denominator.

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

Where:

• $x_i$ = each individual data point in the sample
• $\bar{x}$ = the sample mean of the data points
• $n$ = the total number of data points in the sample

These formulas quantify the spread by squaring the differences, which ensures that both positive and negative deviations contribute positively to the measure of dispersion.

Interpreting Variances

Interpreting variances in a financial context revolves around understanding its implications for volatility and risk. A larger variance signifies that an investment's returns have historically deviated significantly from its average, indicating higher volatility. This means the actual returns are more likely to be far from the expected return. For investors, higher volatility implies a greater degree of uncertainty and a wider range of potential outcomes, both positive and negative.

Conversely, a smaller variance suggests that an investment's returns have historically stayed closer to its average, indicating lower volatility. Such investments are generally considered less risky because their future returns are anticipated to be more predictable and less prone to sharp fluctuations. When evaluating variances, it's crucial to compare them within the same asset class or against a relevant benchmark, as what constitutes "high" or "low" can be relative.

Hypothetical Example

Consider two hypothetical investments, Fund A and Fund B, over five years, with their annual returns:

• Fund A Returns: 8%, 10%, 9%, 12%, 11%
• Fund B Returns: 5%, 15%, 2%, 18%, 10%

Step 1: Calculate the Mean Return for Each Fund

• Fund A Mean: $(8+10+9+12+11) / 5 = 50 / 5 = 10%$
• Fund B Mean: $(5+15+2+18+10) / 5 = 50 / 5 = 10%$

Both funds have the same average annual return.

Step 2: Calculate the Squared Deviations from the Mean for Each Fund

• Fund A:

  • $(8 - 10)^{2 = (-2)}2 = 4$
  • $(10 - 10)^{2 = (0)}2 = 0$
  • $(9 - 10)^{2 = (-1)}2 = 1$
  • $(12 - 10)^{2 = (2)}2 = 4$
  • $(11 - 10)^{2 = (1)}2 = 1$
  • Sum of Squared Deviations = $4+0+1+4+1 = 10$

• Fund B:

  • $(5 - 10)^{2 = (-5)}2 = 25$
  • $(15 - 10)^{2 = (5)}2 = 25$
  • $(2 - 10)^{2 = (-8)}2 = 64$
  • $(18 - 10)^{2 = (8)}2 = 64$
  • $(10 - 10)^{2 = (0)}2 = 0$
  • Sum of Squared Deviations = $25+25+64+64+0 = 178$

Step 3: Calculate the Sample Variance for Each Fund

(Assuming these are samples, divide by $n-1 = 5-1 = 4$)

• Fund A Variance: $10 / 4 = 2.5%$
• Fund B Variance: $178 / 4 = 44.5%$

Conclusion: Although both funds yielded the same average return, Fund B has a significantly higher variance (44.5% vs. 2.5%). This indicates that Fund B's returns were much more spread out from its average, making it a much riskier investment than Fund A, which exhibited more consistent returns.

Practical Applications

Variances are a cornerstone in numerous financial applications, guiding decision-making in various sectors. In portfolio optimization, variances (and covariance between assets) are central to constructing diversified portfolios that balance risk and return. Investors use variances to determine the optimal asset allocation to meet their financial objectives while adhering to their risk tolerance. For instance, high-variance assets might be included in a portfolio alongside low-variance assets to achieve a desired overall risk profile, leveraging the benefits of diversification.

Beyond portfolio construction, variances are vital in risk management, helping analysts and institutions assess and monitor the stability of various financial instruments and the broader market. Financial institutions and regulators, such as the Federal Reserve, routinely assess market volatility in their analyses of financial stability.⁹ For example, risk metrics derived from variances inform regulatory capital requirements for banks and other financial entities. Variances are also applied in financial modeling, derivative pricing, and performance attribution, where understanding the dispersion of returns is crucial for accurate valuations and strategic planning. Many investment research firms, like Morningstar, use variations in returns (often via standard deviation, the square root of variance) to assign risk ratings to mutual funds and exchange-traded funds, helping investors gauge a fund's historical price fluctuations relative to its peers.⁸

Limitations and Criticisms

Despite their widespread use, variances as a measure of risk face several notable limitations and criticisms. One primary critique is that variances treat both positive (upside) and negative (downside) deviations from the mean equally. Investors, however, are typically more concerned with downside risk – the potential for losses – than with upside volatility, which represents unexpected gains. This symmetric treatment can lead to a less intuitive understanding of true investment risk for many.

An⁶, ⁷other significant drawback is the assumption of normally distributed returns. In reality, financial asset returns often exhibit "fat tails" (kurtosis) and skewness, meaning extreme events (both large gains and large losses) occur more frequently than a normal distribution would predict. When returns are not normally distributed, variance may not fully capture the true extent of potential extreme losses, leading to an underestimation of risk.

Fu⁴, ⁵rthermore, the reliance of variances on historical data is a key limitation. Historical volatility, while indicative, is not a guarantee of future performance or risk. Rapidly changing market conditions, unforeseen economic events, or shifts in company fundamentals can render past variances an unreliable predictor of future price movements. This "backward-looking" nature can be particularly problematic during periods of market stress or significant economic change. Las³tly, variances alone do not capture all dimensions of risk, such as liquidity risk, credit risk, or geopolitical risk, offering only a partial picture of an investment's overall exposure.

##¹, ² Variances vs. Standard Deviation

While often discussed interchangeably or in close relation, Variances and Standard deviation are distinct but mathematically linked measures of data dispersion. The key difference lies in their units and interpretability.

Variances represent the average of the squared differences from the mean. Because the differences are squared, the unit of variance is the square of the original data's unit (e.g., if returns are in percentage points, variance is in "percentage points squared"). This can make direct interpretation less intuitive for non-statisticians.

Standard Deviation, on the other hand, is simply the square root of the variance. This transformation brings the measure back to the same units as the original data. For example, if returns are in percentage points, standard deviation is also in percentage points. This makes standard deviation much more intuitive and easier to interpret in practical terms; it represents the typical distance of data points from the mean.

In finance, both are used to quantify volatility, but standard deviation is generally preferred for communicating risk to investors because its value directly relates to the average expected deviation in returns. For calculations within Modern Portfolio Theory, particularly involving portfolio diversification and the relationships between assets, variances are often used internally, especially when calculating covariance and correlation.

FAQs

Why are variances important in finance?

Variances are crucial in finance because they quantify how much an asset's returns deviate from its average over time, providing a statistical measure of its volatility or risk. This information is essential for investors and analysts to assess the potential range of returns and make informed decisions about managing their portfolio risk.

Can variances be negative?

No, variances can never be negative. Since the calculation involves squaring the differences from the mean, all results are non-negative. A variance of zero would indicate that all data points are identical and there is no dispersion.

How do outliers affect variances?

Outliers (extreme values) can significantly impact variances. Because the differences from the mean are squared, a single outlier far from the average will contribute disproportionately to the overall variance, potentially skewing the perception of the data's true dispersion and underlying risk.

Is a high variance always bad?

Not necessarily. While a high variance indicates greater volatility and, therefore, higher risk, it also implies the potential for higher positive returns. Some investors with a higher risk tolerance might seek assets with higher variances in pursuit of greater potential rewards. However, it also means a higher chance of significant losses.

How is variance used in portfolio diversification?

In portfolio diversification, variances of individual assets, along with their covariance, are used to calculate the overall risk of a portfolio. By combining assets whose returns do not move perfectly in sync (i.e., have low or negative covariance), investors can reduce the total portfolio variance, achieving a more efficient balance between risk and return.