CFA Level I Exam · Statistical Characteristics of Asset Returns

Skewness and Kurtosis of Returns for CFA Level I

Q: What is excess kurtosis and why is it measured against 3?

Excess kurtosis is kurtosis minus 3. The normal distribution has kurtosis of 3, so excess kurtosis shows how far the tails depart from normal. Above 0 is leptokurtic, below 0 is platykurtic.

Q: What does leptokurtic mean for investment risk?

Leptokurtic returns have fatter tails, so extreme gains and losses occur more often than a normal distribution predicts. A risk model that assumes normality will understate the chance of large losses.

Q: Is negative skewness bad for investors?

Negative skew means large losses are more likely than large gains of similar size, which most risk-averse investors dislike. Standard deviation alone would not show this asymmetry.

Updated 7 October 2026 · Fact-checked

Skewness measures how lopsided a return distribution is. Kurtosis measures how heavy its tails are. A normal distribution has skewness 0 and kurtosis 3 (excess kurtosis 0). Negative skew and excess kurtosis above 0 mean more frequent or larger extreme losses than a normal model predicts.

Understand Skewness and Kurtosis of Returns

A normal distribution is symmetric and bell-shaped. Mean, median and mode are equal. Real asset returns often do not behave this way, so you need two measures beyond mean and variance: skewness and kurtosis.

Skewness measures asymmetry. A positively skewed distribution has a long right tail: a few large gains pull the mean up. Order of the measures: mean > median > mode. A negatively skewed distribution has a long left tail: a few large losses pull the mean down. Order: mean < median < mode. The mean is dragged toward the tail, the median sits in the middle, and the mode sits at the peak.

A useful memory aid: the tail tells you the name, and the mean moves toward the tail. Positive skew means mean above median. Negative skew means mean below median.

Kurtosis measures how much probability sits in the tails (and peak) relative to a normal distribution. A normal distribution has kurtosis of 3. Excess kurtosis = kurtosis − 3. A leptokurtic distribution has excess kurtosis above 0: fatter tails and a higher, sharper peak. Extreme outcomes, both gains and losses, occur more often than normal predicts. A platykurtic distribution has excess kurtosis below 0: thinner tails. A mesokurtic distribution has excess kurtosis of 0, like the normal.

Why it matters for risk: standard deviation treats gains and losses alike and assumes tails are normal. If returns are negatively skewed and leptokurtic, a standard-deviation-based model understates the chance of large losses. Investors who dislike large losses should look beyond mean and variance. Note that skewness and kurtosis are both unaffected by the units of the data, because they are standardized measures. Also, kurtosis describes tail weight in both directions, so it does not say whether extreme outcomes are gains or losses. Skewness gives that direction.

Key formulas to remember

Normal distribution benchmarks: Skewness = 0; Kurtosis = 3; Mean = Median = Mode; The normal is symmetric, so every comparison is made against these values.
Excess kurtosis: Excess kurtosis = Kurtosis − 3; Many software packages and exam questions report excess kurtosis directly. Check which one a question gives you.
Positive skew ordering: Mean > Median > Mode; Long right tail. Skewness above 0.
Negative skew ordering: Mean < Median < Mode; Long left tail. Skewness below 0.
Sample skewness (idea): Skewness ≈ [1 ÷ n] × Σ (Xi − X̄)³ ÷ s³; Cubed deviations keep the sign, so a long tail on one side sets the sign. Level I rarely asks you to compute it; know its meaning.
Sample kurtosis (idea): Kurtosis ≈ [1 ÷ n] × Σ (Xi − X̄)⁴ ÷ s⁴; Fourth powers are always positive and magnify large deviations, so heavy tails raise the value.

How to solve Skewness and Kurtosis of Returns questions

Use this routine for any question on the shape of a return distribution.

1Identify what the question gives you: a skewness value, a kurtosis or excess kurtosis value, or the order of mean, median and mode.
2Decide whether kurtosis is stated as total or excess. Subtract 3 from total kurtosis to get excess kurtosis.
3For skewness, use the sign: positive means long right tail and mean > median; negative means long left tail and mean < median.
4For kurtosis, compare with 3 (or excess with 0): above means leptokurtic (fat tails), below means platykurtic (thin tails), equal means mesokurtic.
5Link the shape to risk: negative skew means large losses are more likely than gains of the same size; excess kurtosis means extreme outcomes occur more often than normal predicts.
6Eliminate options that mix up the tail direction, confuse total and excess kurtosis, or claim standard deviation fully captures risk.
7Pick the option that matches both the sign and the risk meaning.

Quickest way: Tail-pulls-the-mean shortcut

When to use it: Use this when a question gives you mean, median and mode, or asks you to name the skew from a description.

Find which of mean and median is larger.
If mean is larger, the skew is positive. If mean is smaller, the skew is negative.
For kurtosis, subtract 3 if the value is total. Positive result means leptokurtic, fat tails.
Remember that kurtosis says nothing about direction. Only skewness does.
Check the remaining option for a risk claim: fat tails or negative skew mean standard deviation understates downside risk.

Common mistakes in Skewness and Kurtosis of Returns

Calling a distribution with a long left tail positively skewed because the peak sits on the right.
Students look at where the bulk of the data is rather than where the tail is.
Fix: Name the skew by the tail. Long left tail is negative skew, with mean < median < mode.
Treating a kurtosis of 3 as fat-tailed.
Students forget that the normal distribution has kurtosis of 3, not 0.
Fix: Compare total kurtosis with 3, or compute excess kurtosis = kurtosis − 3 and compare with 0.
Saying leptokurtic means more frequent large losses only.
Students link kurtosis to downside risk and ignore that both tails are fatter.
Fix: Leptokurtic means more extreme outcomes in both directions. Use skewness to say which side is worse.
Reversing the order of mean, median and mode.
Students memorize the order without the logic.
Fix: The mean moves furthest toward the tail, the mode stays at the peak, and the median is between. Positive skew: mean > median > mode.
Assuming standard deviation captures all of an asset's risk when returns are non-normal.
Mean-variance analysis is taught first, so students treat it as complete.
Fix: Standard deviation does not reflect asymmetry or tail weight. With negative skew or excess kurtosis, downside risk can be larger than a normal model suggests.

Worked examples

Example 1

A fund's monthly returns have a sample mean of 0.80%, a median of 1.10% and a mode of 1.40%. Which describes the distribution? A. Positively skewed, B. Negatively skewed, C. Symmetric.

Show the solution

Compare the three measures: mean 0.80% < median 1.10% < mode 1.40%.
The order mean < median < mode matches negative skew.
The long tail is on the left, so occasional large losses pull the mean below the median.
Option A needs mean > median > mode. Option C needs all three equal.

Answer: B. Negatively skewed.

Example 2

A return series has a kurtosis of 5.2. What does this imply compared with a normal distribution? A. Thinner tails, B. The same tail weight, C. Fatter tails, so extreme returns are more likely.

Show the solution

Excess kurtosis = 5.2 − 3 = 2.2.
Excess kurtosis above 0 means leptokurtic.
A leptokurtic distribution has fatter tails, so returns far from the mean occur more often than a normal model predicts.
Thinner tails (A) would need kurtosis below 3. Same tail weight (B) would need kurtosis equal to 3.

Answer: C. Fatter tails, so extreme returns are more likely.

Exam tips

Always check whether the question gives total kurtosis or excess kurtosis before judging the tails.
Name skew by the tail, then place the mean toward the tail. This answers most ordering questions in seconds.
Kurtosis gives no direction. If an option says kurtosis shows losses are more likely than gains, eliminate it.
Risk-interpretation options usually say a normal-based measure understates risk. Fat tails and negative skew support that claim.
With three options and no penalty, always answer. Eliminate options that contradict the mean-median order or the kurtosis comparison with 3, then choose among those left.

Practice questions from Statistical Characteristics of Asset Returns

Skewness and Kurtosis of Returns in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Skewness and Kurtosis of Returns: frequently asked questions

What is the order of mean, median and mode for positive and negative skew?

For positive skew, mean > median > mode. For negative skew, mean < median < mode. The mean moves toward the long tail.

What is excess kurtosis and why is it measured against 3?

Excess kurtosis is kurtosis minus 3. The normal distribution has kurtosis of 3, so excess kurtosis shows how far the tails depart from normal. Above 0 is leptokurtic, below 0 is platykurtic.

What does leptokurtic mean for investment risk?

Leptokurtic returns have fatter tails, so extreme gains and losses occur more often than a normal distribution predicts. A risk model that assumes normality will understate the chance of large losses.

Is negative skewness bad for investors?

Negative skew means large losses are more likely than large gains of similar size, which most risk-averse investors dislike. Standard deviation alone would not show this asymmetry.