PastPaperHero | Statistical concepts and market returns - Skewness kurtosis and distributions

Learning Outcomes

This article explains the statistical concepts of skewness, kurtosis, and excess kurtosis in the context of financial return distributions and shows how they are examined in typical CFA Level 1 exam questions. It defines each measure, demonstrates how they are calculated from sample data, and emphasizes the interpretation of signs and magnitudes for real-world return series. The article distinguishes normal from non-normal distributions and highlights the visual and numerical signatures of positively and negatively skewed, leptokurtic, mesokurtic, and platykurtic distributions. It explains how distribution shape affects risk assessment, why reliance only on mean and standard deviation can understate tail risk, and how negative skewness and fat tails translate into greater downside risk for portfolios. Worked examples illustrate exam-style tasks such as interpreting summary statistics, comparing alternative investments with different skewness and kurtosis, and identifying hidden tail risk in apparently low-volatility strategies. The article also connects skewness and kurtosis to key risk tools—such as Value at Risk (VaR), Expected Shortfall, stress testing, and the use of non-normal distributions—so that candidates can relate distribution properties directly to risk management, asset pricing, and portfolio construction decisions.

CFA Level 1 Syllabus

For the CFA Level 1 exam, you are required to understand how statistical concepts are applied to analyze market returns, with a focus on the following syllabus points:

Calculate, interpret, and compare measures of central tendency, dispersion, skewness, and kurtosis for return distributions
Evaluate the shape and characteristics of normal and non-normal return distributions
Interpret the meaning and investment implications of positive/negative skewness and leptokurtic/platykurtic distributions
Recognize how these distributional properties affect risk assessment and investment decision-making

Test Your Knowledge

Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision.

What does a negative skew in a distribution of monthly equity returns most directly indicate?
1. Large positive returns are more likely than large negative returns.
2. Extreme returns are less likely than under a normal distribution.
3. The probability of very large negative returns exceeds that of very large positive returns.
4. The distribution of returns is symmetric around the mean.
A return series has a sample kurtosis of 6.5. Which statement about its excess kurtosis and risk characteristics is most accurate?
1. Excess kurtosis is 0, and extreme returns are as likely as under a normal distribution.
2. Excess kurtosis is 3.5, indicating more frequent extreme returns than under a normal distribution.
3. Excess kurtosis is −3.5, indicating fewer outliers than under a normal distribution.
4. Excess kurtosis cannot be determined from kurtosis.
An analyst models portfolio returns using only the mean and standard deviation, assuming normality, but the true distribution is leptokurtic with negative skew. Which risk is most likely to be understated?
1. The probability of achieving returns above the mean.
2. The probability of small losses close to the mean.
3. The probability of very large negative returns.
4. The probability of zero return.
Empirical evidence suggests that many equity indices have excess kurtosis significantly greater than zero. What does this most likely imply?
1. The standard normal distribution is an excellent description of equity returns.
2. Extreme equity market moves occur less often than the normal distribution predicts.
3. Extreme equity market moves occur more often than the normal distribution predicts.
4. Equity returns are symmetric and thin‑tailed.

Introduction

Statistical measures like skewness and kurtosis describe how the returns of assets or portfolios are distributed. While the mean and standard deviation provide a baseline for understanding returns and volatility, real-world return patterns usually display asymmetry and extreme values not captured by standard deviation alone. CFA candidates must appreciate how these features complicate risk assessment—and how interpreting them enables more robust investment decisions.

Key Term: skewness
Skewness measures the degree of asymmetry of a distribution around its mean, based on the standardized third power of deviations from the mean. Positive skew indicates a longer or fatter right tail; negative skew a longer or fatter left tail.

Key Term: kurtosis
Kurtosis measures how much probability mass lies in the tails of a distribution relative to its center, based on the standardized fourth power of deviations from the mean. High kurtosis signals more frequent extreme events and a more peaked center. The kurtosis of a normal distribution is 3.

Key Term: excess kurtosis
Excess kurtosis is kurtosis minus 3. Positive excess kurtosis (leptokurtic) means fatter tails than the normal distribution, increasing the probability of extreme outcomes; negative excess kurtosis (platykurtic) means thinner tails.

Key Term: tail risk
Tail risk is the risk of outcomes far from the mean, typically very large losses or gains that lie in the extreme tails of the return distribution.

In many finance models (for example, CAPM or basic VaR under normality), we implicitly assume that returns are normally distributed and can be fully summarized by mean and standard deviation. In practice, however, daily and monthly returns on equities, commodities, and many portfolios often show:

Non‑zero skewness (asymmetry)
Positive excess kurtosis (fat tails)

These empirical facts mean that relying solely on mean and standard deviation will often understate the probability and size of extreme return realizations—especially large losses.

Key Term: normal distribution
A bell-shaped, symmetric, continuous distribution fully described by its mean and standard deviation, with zero skewness and kurtosis of 3 (excess kurtosis of 0).

Skewness and Distribution Asymmetry

Return distributions may not be symmetric. Positive skewness (right-skewed) distributions have frequent small losses or small gains and a few extreme gains. Negative skewness (left-skewed) indicates frequent small gains but rare, severe losses—a particularly important risk for investors. A distribution with zero skewness is symmetric, like the normal distribution.

For a large sample of size $n$ , an approximate formula for sample skewness is:

\text{Skewness} \approx \frac{1}{n} \sum\_{i=1}^{n} \frac{(X_i - \bar{X})^3}{s^3}

where:

$\bar{X}$ is the sample mean
$s$ is the sample standard deviation
$X_i$ are the observed returns

Because each deviation is standardized by $s$ , skewness is scale-free: it does not change if returns are measured in percentages or basis points, or if all returns are multiplied by a constant.

Interpretation guidelines commonly used in practice:

Skewness close to 0 (e.g., between −0.5 and +0.5): distribution is approximately symmetric.
Skewness significantly positive: relatively more probability in the right tail; potential for occasional very large positive returns.
Skewness significantly negative: relatively more probability in the left tail; potential for occasional very large negative returns.

In skewed distributions, the mean, median, and mode are not equal:

For a positively skewed distribution: mode < median < mean
For a negatively skewed distribution: mean < median < mode

Investment risk analysis often focuses on negative skewness, because it represents a pattern of “many small gains, few very large losses,” as seen in some option-writing strategies or carry trades.

Key Term: outlier
An outlier is an observation that lies an unusually large distance from the bulk of the data and can have a strong influence on estimated skewness and kurtosis.

Because skewness depends on the cube of deviations, a single extreme outlier can strongly affect the skewness measure. In practice, analysts should check whether extreme values are genuine market moves or data errors before interpreting skewness.

Key Term: normal distribution
A bell-shaped, symmetric, continuous distribution fully described by its mean and standard deviation, with zero skewness and kurtosis of 3.

Worked Example 1.1

Suppose the monthly returns of Asset A show frequent values near the mean, but a few months have exceptionally large negative returns. An analyst calculates Asset A’s skewness and finds it is –1.2. What does this signal?

Answer:
A skewness of –1.2 indicates Asset A’s returns are negatively skewed. Large negative return outliers (“tail risk”) are more probable than large positive ones—implying higher downside risk. A mean–variance model assuming normality would understate the chance of rare, severe losses.

Kurtosis and the Likelihood of Extreme Outcomes

Whereas skewness addresses asymmetry, kurtosis measures the tails’ thickness and the concentration of observations around the mean.

The kurtosis of the normal distribution is 3.
If kurtosis exceeds 3 (leptokurtic), extreme returns—both positive and negative—are more common than under normality.
If kurtosis is less than 3 (platykurtic), extreme returns are less common; tails are thinner.

For a large sample, an approximate measure of excess kurtosis is:

\text{Excess kurtosis} \approx \left(\frac{1}{n} \sum\_{i=1}^{n} \frac{(X_i - \bar{X})^4}{s^4}\right) - 3

A normal distribution has excess kurtosis of 0. Many statistical packages report excess kurtosis but label it simply as “kurtosis,” so exam questions may require you to infer that a value of 0 represents normal tails.

Key Term: leptokurtic distribution
A distribution with excess kurtosis greater than 0. It has fatter tails and a more pronounced peak than the normal distribution, leading to more frequent extreme outcomes and more observations near the mean.

Key Term: platykurtic distribution
A distribution with excess kurtosis less than 0. It has thinner tails than the normal distribution, with fewer extreme outcomes and a more flat-topped center.

Key Term: mesokurtic distribution
A distribution with kurtosis equal to 3 (excess kurtosis 0), having tail thickness similar to the normal distribution.

Key Term: fat-tailed distribution
A fat-tailed distribution has more probability mass in the tails than the normal distribution (positive excess kurtosis), implying more frequent extremely large deviations from the mean.

Key Term: thin-tailed distribution
A thin-tailed distribution has less probability mass in the tails than the normal distribution (negative excess kurtosis), implying fewer extremely large deviations from the mean.

Conceptually, a leptokurtic distribution:

Places more probability very close to the mean (extremely common “typical” returns)
Places more probability far in the tails (extreme events more common)
Places less probability in the intermediate range between the center and tails

This “peaked center plus fat tails” pattern is typical of many financial return series.

Worked Example 1.2

The annual returns of Index B exhibit a kurtosis of 7.1. What does this indicate for risk?

Answer:
Kurtosis well above 3 implies excess kurtosis of 4.1, so the return distribution is leptokurtic. Index B’s returns more frequently produce outliers/extreme returns (both large gains and losses) than predicted by a normal distribution. Risk will be underestimated if an analyst relies solely on standard deviation and assumes normality.

Empirical Example: Equity Index with Skewness and Kurtosis

Consider a daily equity index whose summary statistics (similar to the EAA Equity Index in the curriculum) are:

Mean daily return: 0.0347%
Standard deviation: 0.8341%
Skewness: –0.4260
Excess kurtosis: 3.7962

Interpreting these numbers:

The distribution is negatively skewed, indicated by the negative skewness. There are more or larger negative deviations from the mean than positive ones.
Excess kurtosis around 3.8 indicates strongly fat tails. Both very negative and very positive returns occur more often than a normal distribution with the same mean and standard deviation would imply.
Graphically, we would observe:
- Very high frequency of returns close to the mean (peaked center)
- Thick tails (more extreme returns)
- Fewer observations in the intermediate ranges

For risk management, this combination means downside tail risk is substantially greater than a normal model would suggest.

Worked Example 1.3

An analyst compares two portfolios:

Portfolio X has daily return skewness of 0.05 and excess kurtosis of –0.2.
Portfolio Y has daily return skewness of –0.6 and excess kurtosis of 2.4.

Which portfolio is more prone to unexpected, extreme losses?

Answer:
Portfolio Y, with negative skewness and high positive excess kurtosis, is more likely to suffer rare, large negative returns (tail risk). Portfolio X’s distribution is close to symmetric (skew ≈ 0) and has slightly thinner-than-normal tails, so its outlier risk is comparatively low. Given similar means and standard deviations, Y has more hidden downside risk.

Calculating and Classifying Distribution Shape

When you are given summary statistics on the exam, a typical task is to classify the distribution and interpret the implications.

Skewness near 0 and excess kurtosis near 0: distribution approximately normal (mesokurtic and symmetric).
Skewness positive, excess kurtosis positive: positively skewed, fat-tailed distribution (occasional very large positive returns).
Skewness negative, excess kurtosis positive: negatively skewed, fat-tailed distribution (occasional very large negative returns—especially concerning for risk management).
Excess kurtosis negative: platykurtic, thin tails; extreme observations are less likely than under normality.

Worked Example 1.4

Three hedge funds have the following estimated monthly return characteristics:

Fund A: skewness = –0.3, excess kurtosis = 0.1
Fund B: skewness = –0.9, excess kurtosis = 2.8
Fund C: skewness = 0.2, excess kurtosis = –0.5

Assuming similar means and standard deviations, which fund most likely has the greatest downside tail risk?

Answer:
Fund B has the greatest downside tail risk. Its skewness is the most negative (–0.9) and its excess kurtosis is strongly positive (2.8). This combination implies a higher probability of very large negative returns. Fund A is only mildly non-normal, and Fund C has thin tails (negative excess kurtosis), making extreme deviations less frequent.

Interpreting Return Distributions in Practice

Most theoretical models (like CAPM) assume asset returns are normally distributed. However, empirical evidence shows equity returns often have negative skewness and positive excess kurtosis—meaning greater likelihood of tail events (market crashes or rallies) than the normal distribution predicts.

Common empirical patterns:

Broad equity indices: often slightly negatively skewed and leptokurtic
Individual stocks: may display stronger skewness and kurtosis than indices
Option‑writing strategies: often strongly negatively skewed and leptokurtic (steady small gains punctuated by large losses)
Trend‑following or option‑buying strategies: sometimes positively skewed (many small losses, occasional large gains)

These characteristics influence:

Risk premia: Investors may demand higher expected returns for holding assets with negative skewness and fat tails.
Portfolio construction: Combining assets with different skewness/kurtosis may improve the overall distribution of portfolio returns (for example, adding positively skewed assets to offset negative skew).

Key Term: Student’s t-distribution
A continuous probability distribution similar to the normal distribution but with heavier tails when degrees of freedom are low. It is often used to model return distributions with more extreme outcomes than the normal distribution.

In some risk models, a Student’s t-distribution with a small number of degrees of freedom is fitted to return data. Compared with a normal distribution, a t-distribution with, say, 5 degrees of freedom has longer tails: to attain the same tail probability (for example, 1% in each tail), the quantiles lie farther from the mean. This reflects higher tail risk.

Worked Example 1.5

An analyst fits two models to a stock index’s daily returns:

Model N: normal distribution with mean 0 and standard deviation 1%
Model T: Student’s t-distribution with 5 degrees of freedom, mean 0 and the same standard deviation 1%

Which model will assign a higher probability to a daily loss worse than –3%?

Answer:
Model T (the t-distribution) assigns a higher probability to a daily loss worse than –3%. With the same standard deviation, a t-distribution with 5 degrees of freedom has fatter tails than the normal distribution. Therefore, extreme losses (and gains) are more likely under Model T than under Model N.

Exam Warning

Skewness and kurtosis are often hidden in summary tables that also report mean and standard deviation. Ignoring them can lead to incorrect conclusions about risk.

A distribution with negative skewness and positive excess kurtosis may have the same mean and standard deviation as another, more “normal” distribution but much higher downside tail risk.
The CFA exam may present you with two investments that have identical means and standard deviations but different skewness and kurtosis, and ask you to identify which is riskier from a downside view.

Failure to adjust for skewness or kurtosis leads to underestimating tail risk and the probability of severe losses. The exam may require explicit identification of non-normal distribution risks when interpreting standard deviation or applying models like VaR.

Why Skewness and Kurtosis Matter for Investment Decisions

Ignoring the shape of the return distribution (non-zero skewness or non-zero excess kurtosis) results in underappreciated risk. Negative skewness and fat tails mean a higher chance of sharp losses than suggested by a simple analysis using mean and standard deviation.

Key implications:

Risk measurement choices:
- Portfolios with negative skew and high excess kurtosis require additional attention—risk metrics that incorporate these properties (for example, Value at Risk and Expected Shortfall) may be more appropriate.
- If VaR is calculated assuming normality when returns are leptokurtic, the reported VaR will often be too low for a given confidence level.
Asset pricing and risk premia:
- Investors facing negative skewness may demand a premium for holding such assets due to the psychological and financial cost of rare, devastating losses.
- Assets with positive skewness (lottery-like payoffs) may, in some cases, be priced to offer lower average returns because investors value the upside potential.
Diversification and portfolio construction:
- Combining assets solely to minimize variance may not be sufficient. A portfolio could have low standard deviation but still be exposed to severe negative tail risk if many components share similar skewness and tail behavior.
- Adding assets with different skewness and kurtosis properties can improve not just volatility but the overall shape of the portfolio return distribution.
Stress-testing and scenario analysis:
- Stress tests that simulate extreme negative scenarios are particularly important when skewness is negative and kurtosis is high.
- Scenario analysis helps highlight losses that would not be anticipated under a normal distribution.

Skewness and kurtosis are also linked to outliers. Very high kurtosis is often accompanied by extreme observations that may be genuine (for example, crisis periods) or may reflect data issues. When outliers are genuine, trimming or winsorizing them to reduce kurtosis may make the series look safer than it really is. For risk analysis, it is usually better to keep legitimate extreme observations in the dataset.

Revision Tip

Always review summary statistics for return series. If excess kurtosis is positive or skewness is markedly negative, the use of standard deviation as the only risk measure is not sufficient—tail risk is present. Look for:

Sign and magnitude of skewness
Sign and magnitude of excess kurtosis
Whether the investment’s payoff pattern (for example, selling options) is consistent with negative skew and fat tails

Key Point Checklist

This article has covered the following key knowledge points:

Define and interpret the meaning of skewness, kurtosis, and excess kurtosis in financial return distributions
Identify normal, leptokurtic (fat-tailed), mesokurtic, and platykurtic (thin-tailed) distributions
Assess the practical investment implications of negative skewness and positive excess kurtosis
Recognize that many asset return distributions deviate from normality, especially equities
Explain why reliance on mean and standard deviation alone can understate true investment risk for non-normal returns
Interpret summary statistics (mean, standard deviation, skewness, kurtosis) to compare the risk profiles of different assets or portfolios
Relate skewness and kurtosis to tail risk, VaR, and stress-testing in a risk management context

Key Terms and Concepts

skewness
kurtosis
excess kurtosis
tail risk
normal distribution
outlier
leptokurtic distribution
platykurtic distribution
mesokurtic distribution
fat-tailed distribution
thin-tailed distribution
Student’s t-distribution

Statistical concepts and market returns - Skewness kurtosis ...