Explore how skewness and kurtosis affect return distributions, focusing on the risk management implications for portfolios.
Skewness and kurtosis might sound a little intimidating at first—like terms only a mathematician would love. But, trust me, they are actually super-important in everyday portfolio management. You might be thinking, “Aren’t mean and standard deviation enough?” Well, not always. Standard deviation focuses on the average spread of returns, but it doesn’t do a great job of telling you about big, unexpected moves. That’s where skewness and kurtosis come in. They help you figure out whether the distribution of returns is lopsided (skewed) or prone to extreme ups or downs (kurtosis). As you’ll soon see, ignoring these can lead to serious misjudgments in risk management.
Picture a distribution of daily returns on your portfolio. If you only consider standard deviation, you might assume the returns are all nicely “clustered” around the mean, with symmetrical tails. But in reality, market returns can be lumpy, with more large negative (or positive) events than expected. Understanding skewness (asymmetry) and kurtosis (“tail thickness”) helps you see how your portfolio might handle stress during unexpected market events.
In other words:
• Skewness answers: “Are rare events more likely to be big gains or big losses?”
• Kurtosis answers: “How extreme can those rare events get?”
I once worked with a small investment fund that was consistently enjoying moderate gains—until a single week saw enormous negative returns that nearly wiped out the entire year’s progress. When we examined the distribution of returns, it turned out that the returns were negatively skewed and leptokurtic; so large losses, though infrequent, were significantly more probable than a normal distribution would suggest. If we had taken skewness and kurtosis more seriously from day one, we might have hedged some of that tail risk a bit sooner.
Skewness tells us about the asymmetry of the distribution of returns around the mean. A distribution can be:
• Perfectly symmetrical (no skew)
• Positively skewed (right-skewed): The right-hand tail is longer
• Negatively skewed (left-skewed): The left-hand tail is longer
In a perfectly symmetrical distribution (like an idealized normal distribution), the mean, median, and mode all coincide. That’s nice in theory, but financial returns often do not conform to that. Instead, you might see heavier tails on one side or the other.
If returns are positively skewed, you have a longer tail on the right side of the distribution. This typically implies the possibility—though sometimes a small likelihood—of outsize positive gains. The mean is usually larger than the median. This scenario might sound ideal, but it’s not always free of complications, because if most returns cluster around a smaller positive range and the big outliers are extremely big winners, your main challenge is that your normal “average risk measures” might underestimate just how valuable these rare events could be. (And ironically, risk managers often don’t fear large positive returns as much as large negative ones.)
Negative skew is the scarier one—at least for most investors—because it indicates that the left tail is longer. The mean is often less than the median. Practically, you might see frequent small gains or stable outcomes, with an occasional huge drop. That’s tricky because standard deviation alone doesn’t fully capture the potential meltdown in the left tail. Investors sometimes refer to assets with negative skew as “picking up pennies in front of a steamroller.”
It’s like a friend telling you, “Most of the time, this strategy churns out nice, modest returns.” But every once in a while? You might see a catastrophic loss. If you only look at average returns and standard deviation, that tail risk can be hidden, leading you to underestimate the chance of extreme negative returns.
You might recall the general formula for sample skewness (though there are multiple definitions often used in practice):
• \( X_i \) are the observed returns.
• \( \overline{X} \) is the sample mean.
• \( s \) is the sample standard deviation.
• \( n \) is the number of observations.
A negative skewness implies negative skew (longer left tail). A positive skewness implies positive skew (longer right tail). A skewness near zero suggests a more or less symmetrical distribution—although that doesn’t guarantee normality, of course.
Kurtosis measures the “peakedness” or “tailedness” of a distribution. If a distribution has very tall peaks and very fat tails, it has high kurtosis. In finance, high kurtosis is often interpreted as having ‘fat tails,’ meaning that extreme returns (positive or negative) are more likely than the standard normal distribution suggests.
Leptokurtic (Kurtosis > 3):
Mesokurtic (Kurtosis ≈ 3):
Platykurtic (Kurtosis < 3):
Naturally, from a risk-management viewpoint, leptokurtic distributions are the ones that raise eyebrows. They remind us that catastrophic losses might show up more frequently than we’d expect under a normal model.
One common version of the sample excess kurtosis (commonly used in statistics software) is:
When the excess kurtosis is positive, it suggests leptokurtic. When it’s negative, it suggests platykurtic. When it’s around zero, you’re near the “mesokurtic” (normal-like) baseline.
Below is a simple Mermaid diagram that conceptualizes a flow of how to classify skewness and kurtosis and interpret their implications:
flowchart TB
A["Distribution of <br/> Asset Returns"] --> B["Calculate Skewness"]
A --> C["Calculate Kurtosis"]
B --> D{Skewness <br/> 0?}
D --> E["Positive Skew <br/>(Mean > Median)"]
D --> F["Zero Skew <br/>(Symmetrical)"]
D --> G["Negative Skew <br/>(Mean < Median)"]
C --> H{Kurtosis <br/> vs 3?}
H --> I["> 3: <br/>Leptokurtic"]
H --> J["= 3: <br/>Mesokurtic"]
H --> K["< 3: <br/>Platykurtic"]
This is obviously simplified, but it illustrates the key steps:
• A negatively skewed portfolio might generate consistent small profits but is vulnerable to large drawdowns. Think of some popular option-writing strategies (like writing out-of-the-money puts) that produce small gains most of the time but occasionally get hammered when markets plunge.
• A positively skewed distribution might see more frequent small losses but occasionally big windfalls, e.g., holding long options or assets that benefit from volatility spikes.
• Leptokurtic distributions have higher tail risk. You should consider robust tail hedging or other forms of risk mitigation (such as systematic put-buying strategies or dynamic hedging).
• Options-based hedging: Buying puts (or put spreads) can help protect against the left-tail risk in negatively skewed assets. Though it comes at a premium cost, it’s often a crucial insurance.
• Diversification: Allocating across asset classes or strategies with different skewness and kurtosis profiles can help offset extremes, though correlation dynamics also matter heavily.
• Stress testing and scenario analysis: If your distribution is leptokurtic, standard risk metrics (like VaR based on a normal assumption) can be misleading. Scenario analysis or historical stress tests might reveal deeper vulnerabilities.
Imagine you hold a high-yield bond fund that produces relatively stable monthly returns. It might have a negative skew, because in times of credit market stress, defaults spike, and the fund can suddenly drop in value sharply. If you’re worried about a credit event, you could:
• Allocate a bit toward Treasury bonds or other safe havens (lower correlation).
• Buy put options on a high-yield bond ETF.
• Purchase credit default swaps (CDS) to hedge corporate default risk.
All of these strategies aim to flatten that left tail—i.e., reduce negative skewness—so that catastrophic losses become less likely or at least less severe. This may in turn reduce the overall portfolio’s leptokurtic tendencies as well.
Below is a small Python snippet that illustrates how you might compute skewness and kurtosis from a simple return series:
1import numpy as np
2from scipy.stats import skew, kurtosis
3
4returns = [0.01, 0.02, -0.03, 0.01, 0.05, -0.10, 0.03, 0.02, 0.04, -0.02]
5
6sample_skew = skew(returns) # default is sample skewness
7sample_kurtosis = kurtosis(returns, fisher=False)
8
9print("Sample Skewness:", sample_skew)
10print("Sample Kurtosis:", sample_kurtosis)
You might see, for instance, a negative skew if those occasional large losses (like -10%) overshadow smaller gains. And if big negative outliers are frequent, you might find a kurtosis well above 3.
• Chapter 3.1–3.2 introduced basic measures of central tendency and dispersion. Skewness and kurtosis are more advanced descriptors that refine our view of distributions.
• Chapters 4 and 5 dive into scenario analysis and portfolio mathematics. When constructing an optimal portfolio or performing a stress test, factoring in asymmetry and heavy tails can be critical to avoid nasty surprises.
• In advanced risk modeling (see Chapter 6 on simulation methods), you can incorporate non-normal distributions or run Monte Carlo simulations that account for higher skewness and kurtosis.
When performing investment analysis, the CFA Institute’s Code and Standards emphasizes diligence and thoroughness. Overlooking tail risk or misrepresenting the risk profile of your portfolio could violate the ethical requirement to provide a fair, objective analysis. Especially in the context of preparing portfolios for clients, practitioners have a responsibility to represent the full range of potential outcomes, including those lurking in the tails.
Understanding skewness and kurtosis can save you from the common trap of relying too heavily on a “normal” viewpoint. Negative skew or leptokurtic distributions signal that extreme events are more likely or more severe than normal assumptions would suggest, and ignoring that is a recipe for being blindsided by deep drawdowns. Here are a few final pointers to keep in mind:
• For exam purposes, memorize the differences between types of skew (positive vs. negative) and the meaning of leptokurtic, platykurtic, and mesokurtic.
• Be ready to interpret the sign of sample skewness, especially in short-answer or item-set questions that might present data on sample moments.
• Know that “excess kurtosis” refers to kurtosis minus 3, so pay attention to whether the exam question uses that or the raw kurtosis measure.
• For scenario-based questions, be prepared to discuss risk management strategies to mitigate negative skew or leptokurtic risk.
Remember: Markets often exhibit fatter tails than normal. If something looks “too good to be true” based on standard deviation alone, it might well be. When you see high kurtosis, be mindful of tail-risk hedging strategies, especially if the distribution is also negatively skewed and prone to large losses.
• CFA Institute Level I Curriculum, Quantitative Methods: “Skewness, Kurtosis, and Descriptive Statistics.”
• Mandelbrot, Benoit (2004). The (Mis)Behavior of Markets. Basic Books.
• The Journal of Portfolio Management and The Journal of Risk (various papers discussing advanced measures of tail risk and hedging techniques).
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