Leptokurtic
Also written Leptokurtic distribution
A return distribution with kurtosis above 3 (excess kurtosis above zero): values bunch near the mean or sit far from it — a high peak with fat tails, which the workbook reads as riskier.
In plain language
Two funds can have similar average returns and still behave very differently. One delivers roughly the same return every year. The other delivers mostly good years punctuated by the occasional dreadful one.
Kurtosis measures that difference — how "peaked" a return distribution is compared with a normal distribution, and therefore how much of its variation comes from outliers.
A leptokurtic distribution has more kurtosis than a normal one. Its values are either close to the mean or a long way from it. That shape — a tall peak with fatter tails — is what a fund with occasional extreme returns produces. The workbook treats it as the riskier profile and as a sign of inconsistent performance.
Its opposite is platykurtic: kurtosis below 3, values spread with consistent frequencies around the mean and no significant deviations.
How it works
The benchmark is the normal distribution, which has a kurtosis of 3. The workbook defines excess kurtosis as kurtosis minus 3.
| Kurtosis | Excess kurtosis | Name | What the returns look like |
|---|---|---|---|
| Above 3 | Above 0 | Leptokurtic | Around the mean or substantially away from it; fat tails, high peak |
| Equal to 3 | 0 | Normal | — |
| Below 3 | Below 0 | Platykurtic | Consistent frequencies around the mean, no significant deviations |
In Example 9.8.2 kurtosis is computed in Excel with =KURT(range), and 3 is then subtracted to get the excess.
Kurtosis sits in the workbook next to skewness, and both come from the same source: outliers. Skewness says which side the outliers are on; kurtosis says how much of the distribution lives in the tails. Neither replaces the mean or the standard deviation — the chapter's conclusion is that sophisticated investors must not analyse any single risk measure in isolation.
The formula
Excess kurtosis = Kurtosis - 3
Excess kurtosis > 0 → leptokurtic
Excess kurtosis < 0 → platykurtic
A worked example
The workbook's Example 9.8.2 compares six years of net returns for two funds:
| Y1 | Y2 | Y3 | Y4 | Y5 | Y6 | Mean | SD | |
|---|---|---|---|---|---|---|---|---|
| Fund A | 12.50% | 11.25% | 13.65% | 17.30% | 14.65% | 13.85% | 13.87% | 1.88% |
| Fund B | 21.50% | −0.50% | 16.75% | 20.25% | 21.90% | 20.80% | 16.78% | 7.91% |
Check the means: Fund A sums to 83.20 over six years (13.87%); Fund B sums to 100.70 (16.78%).
| Skewness | Kurtosis | Excess kurtosis | Shape | |
|---|---|---|---|---|
| Fund A | 0.705 | 1.162 | 1.162 − 3 = −1.838 | Platykurtic |
| Fund B | −2.222 | 5.034 | 5.034 − 3 = +2.034 | Leptokurtic |
Fund B's five years around 17% to 22% form the tall peak; its one year at −0.50% is the fat tail. That single outlier is what makes Fund B both negatively skewed and leptokurtic.
The investor's view. Suppose Ravi (illustrative) has Rs 2 crore to place and a high risk appetite. On mean and standard deviation alone he might still choose Fund B for its extra 2.91 percentage points of average return. The workbook's point is that without skewness and kurtosis he would be ignoring the probability of outliers — and for Fund B the outlier is a loss year.
Why NISM asks about it
Chapter 9 (Fee Structure and Fund Performance), section 9.6 on risk measures defines kurtosis, leptokurtic and platykurtic distributions and works Example 9.8.2 on Funds A and B, directly after skewness in Example 9.8.1. Expect a definition question ("a distribution with kurtosis above 3 is…"), a question on the kurtosis of a normal distribution, and an interpretation question that gives two kurtosis values and asks which fund is riskier.
Common exam traps
- The normal distribution has kurtosis 3, not 0. Excess kurtosis of a normal distribution is 0. Keep the two numbers apart.
- Lepto = above 3 = fat tails = riskier. Platy = below 3 = consistent around the mean.
- A high peak does not mean low risk. A leptokurtic distribution has both a high peak and fat tails; the tails are the risk.
- The workbook calls Fund A "negative kurtosis" — it means negative excess kurtosis (1.162 − 3). Kurtosis itself is positive.
- Kurtosis is not skewness. Skewness is about asymmetry and the side of the outliers; kurtosis is about how much weight sits in the tails.
- Do not read any risk measure alone. Mean, standard deviation, skewness and kurtosis together told the workbook that Fund B was riskier.
Check yourself
1.Fund A has skewness of +0.705 and kurtosis of 1.162. Fund B has skewness of −2.222 and kurtosis of 5.034. Which is riskier and why?
- a)Fund A, because positive skewness means positive outliers which increase volatility
- b)Fund B — negative skewness means a long left tail with the Mean below the Mode, and kurtosis above 3 makes it leptokurtic with fat tails and inconsistent performance
- c)They are equally risky since both deviate from normal
- d)Fund B, but only because its kurtosis is higher; skewness is irrelevant to risk
Show the answer
Answer: (b) Fund B — negative skewness means a long left tail with the Mean below the Mode, and kurtosis above 3 makes it leptokurtic with fat tails and inconsistent performance
Both measures point the same way. On skewness: a distribution having Skewness of greater than zero indicates a positively skewed distribution and a distribution having Skewness of less than zero indicates a negatively skewed distribution, and in a negatively skewed distribution... the Mean is always lesser than the Mode. The preference is explicit: while the investors will not prefer the distribution of returns to be skewed, Positive Skewness is preferred over Negative Skewness. On kurtosis: a Normal Distribution has a Kurtosis of Three, so Fund A's excess kurtosis is 1.162 − 3.000 = −1.838 (platykurtic) and Fund B's is 5.034 − 3.000 = 2.034 (leptokurtic). Leptokurtic Distributions have a Kurtosis higher than 3... Such distributions have data values which are either around the mean or substantially deviating away from the Mean, and investors must be diligent about funds exhibiting return distributions with fatter tails, as it is an indicator of inconsistent performance. The workbook's verdict: on comparison of risk based on Kurtosis, Fund B is riskier as compared to Fund A.
Where this is taught
Free preparation for NISM Series XIX-CRelated terms
- Standard deviationA measure of how far returns typically stray from their own average — the standard statistic for total risk, counting company-specific and market-wide causes alike.
- Downside riskThe probability of a loss on account of falling asset prices in changing market conditions — the half of volatility investors actually mind, measured by maximum drawdown and value at risk.
- KurtosisA measure of how peaked a return distribution is and how fat its tails are. The workbook takes a normal distribution as kurtosis 3; above 3 (leptokurtic) signals outliers and higher risk.
- Maximum DrawdownThe largest peak-to-trough fall in a Category III AIF's AUM over a reporting period, quoted as a percentage of the peak — the worst loss an investor could have suffered in that window.
- SkewnessA measure of how far a fund's return distribution departs from symmetry around its mean. Zero is symmetrical; positive has a long right tail; negative has a long left tail and is riskier.
- OutliersExtreme returns in a fund's track record, whether highly positive or highly negative — the thing skewness and kurtosis are computed to detect.
- PlatykurticA return distribution with kurtosis below 3, so excess kurtosis is negative: values cluster at consistent frequencies around the mean with no significant deviations.