How Do the Moments of Return Distributions Shape Investment Analysis?

In financial analysis, understanding the moments of a return distribution is critical for assessing the behavior of investment returns. These moments provide insight into the shape, risk, and potential outcomes of a probability distribution, particularly in alternative investments where the behavior of returns can differ significantly from traditional assets.

The first four moments—mean, variance, skewness, and kurtosis—are used to describe the characteristics of return distributions, helping investors and analysts better understand the risks and rewards of a particular investment.

This article will break down these four moments, their significance in return distribution analysis, and how they are applied to evaluate investment risks and potential outcomes.

1. The Mean: First Moment of the Distribution

The mean, or expected value, is the first moment of a return distribution and represents the central location or average of the returns. In a simple sense, the mean indicates the average return that an investor can expect over a certain period. It is calculated by averaging the observed returns over a specific time frame:

Mean (expected value)=1n∑i=1nRitext{Mean (expected value)} = frac{1}{n} sum_{i=1}^{n} R_iMean (expected value)=n1​i=1∑n​Ri​

Where RiR_iRi​ represents the individual returns, and nnn is the number of observations.

In investment analysis, the mean is often used to estimate the long-term average return of a portfolio or asset. For alternative investments, the mean helps set expectations regarding the performance of investments that may not have daily observable prices, such as private equity or real estate.

2. Variance and Standard Deviation: Second Moment of the Distribution

The variance is the second moment and measures the dispersion of returns around the mean. It quantifies the degree to which individual returns deviate from the average return. Variance is calculated as the probability-weighted average of the squared deviations from the mean:

Variance (V(R))=1n−1∑i=1n(Ri−Rˉ)2text{Variance (V(R))} = frac{1}{n - 1} sum_{i=1}^{n} (R_i - bar{R})^2Variance (V(R))=n−11​i=1∑n​(Ri​−Rˉ)2

Where RiR_iRi​ is the individual return, and Rˉbar{R}Rˉ is the mean return.

The square root of variance is called the standard deviation, which is widely used in finance as a measure of volatility. Standard deviation reflects the level of uncertainty or risk associated with an investment's returns. In the context of alternative investments, volatility is often used to describe the degree of risk and price fluctuation, with higher standard deviations indicating greater uncertainty.

Standard Deviation=Variancetext{Standard Deviation} = sqrt{text{Variance}}Standard Deviation=Variance​

3. Skewness: Third Moment of the Distribution

Skewness is the third moment and measures the asymmetry of a return distribution. It tells investors whether the distribution of returns is biased toward one side of the mean, indicating the likelihood of larger positive or negative outcomes. Skewness is calculated as:

Skewness=1n∑i=1n(Ri−Rˉ)3Standard Deviation3text{Skewness} = frac{frac{1}{n} sum_{i=1}^{n} (R_i - bar{R})^3}{text{Standard Deviation}^3}Skewness=Standard Deviation3n1​∑i=1n​(Ri​−Rˉ)3​

A skewness value can be:

• Positive: A positively skewed distribution has a longer right tail, indicating that extreme positive returns are more likely. This suggests that most returns are concentrated on the left side, with the potential for occasional large gains.

• Negative: A negatively skewed distribution has a longer left tail, meaning that extreme negative returns are more likely. This implies that the mass of the distribution is concentrated on the right side, but there is a greater risk of significant losses.

• Zero: A skewness of zero typically indicates a symmetrical distribution, where the left and right tails are balanced, as seen in a normal distribution.

In alternative investments, understanding skewness helps investors identify potential asymmetric risk. Investments with negative skewness may have higher risks of large losses, which is particularly important in illiquid assets like real estate or private equity.

4. Kurtosis and Excess Kurtosis: Fourth Moment of the Distribution

Kurtosis is the fourth moment of a return distribution and measures the fatness of the tails. It captures the frequency of extreme deviations (either gains or losses) from the mean. A higher kurtosis indicates a greater likelihood of extreme outcomes, while lower kurtosis suggests more moderate variations.

Kurtosis=1n∑i=1n(Ri−Rˉ)4Standard Deviation4text{Kurtosis} = frac{frac{1}{n} sum_{i=1}^{n} (R_i - bar{R})^4}{text{Standard Deviation}^4}Kurtosis=Standard Deviation4n1​∑i=1n​(Ri​−Rˉ)4​

Excess kurtosis is kurtosis relative to a normal distribution, which has a kurtosis value of 3. Excess kurtosis is calculated by subtracting 3 from the kurtosis value:

Excess Kurtosis=Kurtosis−3text{Excess Kurtosis} = text{Kurtosis} - 3Excess Kurtosis=Kurtosis−3

• Positive excess kurtosis (leptokurtic) indicates fat tails, suggesting a higher probability of extreme events—either very large gains or significant losses.

• Negative excess kurtosis (platykurtic) indicates thin tails, suggesting fewer extreme outcomes and a more rounded distribution.

• Zero excess kurtosis (mesokurtic) implies a distribution similar to the normal distribution, with moderate tail behavior.

Understanding kurtosis is essential for evaluating the risk of extreme outcomes, especially in alternative investments, where liquidity and price fluctuations can lead to higher probabilities of rare but significant losses or gains.

Skewness and Kurtosis in Risk Management

In the analysis of alternative investments, both skewness and kurtosis are critical in identifying tail risks—the likelihood of extreme losses or gains. Investors may prefer distributions with positive skewness and low excess kurtosis, which indicate a higher probability of moderate positive outcomes and a lower risk of significant losses. Conversely, distributions with negative skewness and high kurtosis can be dangerous, as they suggest a higher risk of catastrophic outcomes.

Two investments may have the same mean and variance, but if one has high negative skewness and leptokurtosis, it may expose the investor to greater risks of extreme losses, which is particularly relevant in alternative assets with illiquid markets.

Platykurtic, Mesokurtic, and Leptokurtic Distributions

In practice, the shape of the tails of a distribution is an important consideration in portfolio construction and risk management. There are three types of distributions based on kurtosis:

• Platykurtic: A distribution with negative excess kurtosis (kurtosis < 3) is platykurtic, meaning it has thin tails. This suggests a lower likelihood of extreme outcomes and more moderate variability. Platykurtic distributions are often preferred for conservative investments.

• Mesokurtic: A distribution with zero excess kurtosis (kurtosis ≈ 3) is mesokurtic, which is typical of a normal distribution. In this case, the probability of extreme outcomes is neither too high nor too low.

• Leptokurtic: A distribution with positive excess kurtosis (kurtosis > 3) is leptokurtic, meaning it has fat tails. This indicates a higher probability of extreme returns (both gains and losses), making these distributions riskier, particularly for portfolios that are sensitive to large, unexpected moves.

Conclusion: Moments of Distribution in Investment Analysis

The first four moments of a return distribution—mean, variance, skewness, and kurtosis—provide vital information about the behavior of asset returns and are especially critical when analyzing alternative investments. While the mean and variance help define the central tendency and dispersion, skewness and kurtosis offer deeper insights into asymmetry and the likelihood of extreme outcomes. Understanding these moments allows investors to better manage risk, construct diversified portfolios, and make informed decisions based on the specific characteristics of the investments they hold.

At Orgon Bank, we provide expert insights into risk management and portfolio construction using advanced statistical measures like skewness and kurtosis. Whether you're managing private equity, hedge funds, or real estate assets, our tailored solutions ensure that you fully understand the risks and rewards of your investments.