Return distributions · RiskAtoms Academy

By the end you will be able to

1Describe a return series with a histogram, mean, standard deviation, skewness and kurtosis, and compare it with the normal distribution.
2Explain what the signs and sizes of skewness and excess kurtosis say about a series.
3Count extreme days and compare the count with what a normal distribution predicts.
4Explain why fat tails matter when measuring risk.

Why it matters

If the equity index fund's daily returns were normal, a fall of more than 4% in a day would come about once every 37 years. In 500 days we saw 3.

Core explanation

The whole shape, not just the spread

Volatility (L2.5.01: Measuring volatility) is one number: the typical size of a daily move. But two series with the same volatility can behave very differently on their worst days. To see that, look at the whole distribution of returns.

Illustrative dataThe Module 2.5 data set

500 trading days of daily returns for an equity index fund, a bond fund and a bank share, generated for this module by a seeded model with a calm stretch, a turbulent spell and a few extreme days. It is not real market history. The same series runs through every lesson in the module, so the numbers line up. The fund we follow is worth $10 million: $6 million in the equity index fund and $4 million in the bond fund.

The histogram. Sort the 500 daily returns into bins of equal width, say 0.25% each, and draw a bar for each bin as tall as the number of days in it. That is a histogram. For the equity index fund it shows a tall peak near zero, most days within about ±2%, and a few lonely bars far out on the left.

The benchmark: the normal. The normal distribution is the bell curve fixed by just two numbers, the mean and the standard deviation (M-C06: The normal distribution). Draw a normal with the same mean and SD as the data over the histogram, and the differences jump out.

Four numbers describe the shape:

Statistic	What it measures	Normal	Equity index fund
Mean	The centre	any	0.066%
Standard deviation	The spread	any	1.10%
Skewness	Asymmetry	0	−0.34
Kurtosis	Tail weight	3	6.78

Skewness is negative here: the left tail (losses) reaches further than the right. Kurtosis of 6.78, an excess kurtosis of 3.78, means fat tails (M-C08: Skewness and fat tails). Fat-tailed data also have a taller peak and thinner “shoulders” than the normal: more quiet days, fewer moderate days, more extreme days.

Three shapes, each with mean 0 and standard deviation 1

Normal

Skew 0, kurtosis 3

Negative skew

A longer left tail: rare big losses

Fat tails

Taller peak, thinner shoulders, fatter tails

Dashed: the normal, for comparison.

Counting extreme days. A normal puts only about 0.27% of days beyond 3 standard deviations from the mean: 1.35 days in 500. The equity index fund has 7, of which 4 are falls. Beyond 4 standard deviations a normal expects 0.03 days; the data have 3, all of them falls.

Beyond	Normal expects	Equity index fund	The 60/40 fund
±2 SD	22.75	26 (9 falls)	24 (7 falls)
±3 SD	1.35	7 (4 falls)	8 (4 falls)
±4 SD	0.03	3 (3 falls)	2 (2 falls)

Why it matters for risk. Risk managers care most about the worst days. If you measure the tail with a normal curve fitted to the whole series, you will understate how often big losses come and how big they are. That is the gap between parametric and historical value at risk in L2.5.06: Value at risk: the idea and L2.5.07: Historical VaR, and the reason L2.5.08: Expected shortfall looks beyond VaR altogether.

Read the shape

Your callWhich shape would you expect?

0 of 3

Buying lottery-like assets: small losses most of the time, an occasional very large gain.
A portfolio of corporate loans: interest earned steadily, with rare defaults that wipe out years of income.
A major exchange rate in calm times, with no event risk on the horizon, measured over months rather than days.

The formal version

You built the mean and the sample standard deviation $s$ in L2.5.01: Measuring volatility. Skewness and kurtosis extend the same idea. First standardise each return into a z-score, its distance from the mean in standard deviations:

z_t = \frac{r_t - \bar r}{s}

Then average the third and fourth powers. Cubing keeps the sign, so big falls pull skewness down. The fourth power is always positive and makes the largest z-scores dominate, so kurtosis measures the tails:

\text{skewness} = \frac{1}{n}\sum_{t=1}^{n} z_t^{3} \qquad \text{kurtosis} = \frac{1}{n}\sum_{t=1}^{n} z_t^{4} \qquad \text{excess kurtosis} = \text{kurtosis} - 3

For the normal, skewness is 0 and kurtosis is 3. The normal's density with mean $\mu$ and standard deviation $\sigma$ , shown in full:

f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}

To lay it over a histogram, turn the density into counts: a bin of width $w$ centred on $x$ should hold about $n \times w \times f(x)$ of the $n$ days (M-D02: Histograms and empirical distributions).

Conventions

The z-scores use the sample SD (n − 1) and the averages divide by n. Spreadsheets apply a slightly different small-sample correction: Excel's SKEW and KURT, which the RiskAtoms calculator follows, give −0.340 and an excess kurtosis of 3.85 for the equity index fund, against our −0.338 and 3.78. With 500 days the choice hardly matters; with 20 it would. Some software reports excess kurtosis under the name “kurtosis”, so check which you are reading.

Try it

Start with the equity index fund. Switch on the log scale: the normal curve dives towards zero in the tails, while the real bars carry on. Then set the counter to ±4 SD and compare the actual count with what a normal expects. Finally, try the 60/40 fund and the bond fund.

Try itHistogram against the normal

Descriptive statistics calculator →

Series (illustrative data, 500 days)

Number of bins39

Count days beyond±3.0 SD

Days beyond ±3.0 SD

7

A normal expects1.35

Of which falls4

Mean (daily)0.066%

Standard deviation1.099%

Skewness−0.34

Kurtosis (normal 3)6.78

Excess kurtosis3.78

days in each bin beyond ±3.0 SD normal, same mean and SD

Horizontal axis: daily return, from the mean − 6.5 SD to the mean + 6.5 SD. Days outside sit in the end bins.

Worked example

Where the tail statistics come from

Describe the equity index fund's 500 days, and find which days drive its skewness and kurtosis. Try each step before revealing it.

Shape statistics, step by step

Step 1 of 5

1
Centre and spread
From L2.5.01: Measuring volatility: mean $\bar r = 0.0658\%$ , sample SD $s = 1.0987\%$ .

Regulatory lens

Fat tails are why supervisors test risk models against reality. Since 1996 the Basel framework has asked banks using their own value-at-risk models to backtest them: count the days in the last 250 when the trading loss exceeded the one-day 99% VaR. About 2.5 such days are expected. Zero to four keeps a model in the green zone, five to nine puts it in the yellow zone with a higher capital multiplier, and ten or more is red. A model that assumes normal returns for fat-tailed data tends to drift into yellow. The detail is in M4.3: Market risk capital (FRTB) and CVA, part of Level 4: Regulation, Capital & Accounting.

Sparks

Spark25 standard deviations, several days in a row

In August 2007, as quantitative hedge funds suffered sudden losses, Goldman Sachs's chief financial officer David Viniar told the Financial Times: “We were seeing things that were 25-standard deviation moves, several days in a row.” Under a normal distribution, a single 25-standard-deviation fall has a probability of about 3 × 10⁻¹³⁸. The lesson most people drew was not that the market was unlucky, but that the normal was the wrong model.

SparkDaily returns are less normal than monthly ones

Add up daily returns into weekly or monthly ones and the distribution looks more and more like a normal: the extremes get diluted by ordinary days. The economist Rama Cont listed this “aggregational Gaussianity” among the stylised facts of asset returns in 2001, next to fat tails and volatility clustering. So a monthly risk figure can look reassuringly normal while the daily path is anything but.

Key takeaways

A histogram shows the whole shape of a return series. Compare it with a normal of the same mean and standard deviation.
Skewness = average of z³ measures asymmetry; negative skew means a longer loss tail.
Kurtosis = average of z⁴ measures tail weight; the normal's is 3. Excess kurtosis above 0 means fat tails, with a taller peak and thinner shoulders.
Real returns have far more extreme days than a normal predicts: 7 days beyond 3 SD in our 500, against 1.35 expected.
Tail statistics rest on a few days, so they need long samples and a careful eye. Normal-based risk measures understate the tail.

Reaction

Check what stuck

Question 1 · objective 3

Daily returns are exactly normal. Over 1,000 days, how many days would you expect beyond 3 standard deviations from the mean, in either direction? (Two decimal places.)

days

Question 2 · objective 2

Five days have z-scores of −2, 0, 0, +1 and +1. What is their skewness, the average of z³?

Question 3 · objective 2

A strategy earns a small, steady income most days but occasionally loses a large amount (selling insurance, for example). Its returns most likely show:

APositive skewBNegative skewCZero skew and kurtosis of 3DKurtosis below 3

Question 4 · objective 1

A return series has excess kurtosis of 4. Compared with a normal with the same mean and SD, it has:

AFewer days near the mean and fewer extreme daysBMore days near the mean, fewer moderate days and more extreme daysCThe same tails but a shifted meanDExactly the same shape, because the SD is the same

Question 5 · objective 4

Under a normal distribution, how many of 500 days would you expect to fall more than 2 SD below the mean? (One decimal place.)

days

Glossary

Distribution: The full pattern of possible values and how often each occurs.
Histogram: A bar chart that sorts data into equal-width bins and shows how many values fall in each.
Normal distribution: The bell-shaped distribution fixed by its mean and standard deviation. About 68% of values lie within 1 SD of the mean, 95% within 2.
z-score: How many standard deviations a value lies from the mean: (x − mean) ÷ SD.
Skewness: A measure of asymmetry. Negative skew: a longer left tail, with rare large losses. Positive skew: a longer right tail.
Kurtosis: A measure of how heavy the tails are. The normal's kurtosis is 3; above 3 means more extreme values than a normal would produce.
Excess kurtosis: Kurtosis minus 3, so that the normal scores 0.
Fat tails: Tails that hold more probability than the normal's: extreme moves happen more often than a bell curve says.

Deep Dives & Origins

Deep Dives

R. Cont, “Empirical properties of asset returns: stylized facts and statistical issues”, Quantitative Finance, 1(2), 2001, pp. 223–236. Advanced.
A. J. McNeil, R. Frey and P. Embrechts, Quantitative Risk Management, revised edition, Princeton University Press, 2015, the chapter on empirical properties of financial data. Advanced.

Origins

D. N. Joanes and C. A. Gill, “Comparing measures of sample skewness and kurtosis”, Journal of the Royal Statistical Society, Series D, 47(1), 1998, pp. 183–189.
Basel Committee on Banking Supervision, Supervisory Framework for the Use of “Backtesting” in Conjunction with the Internal Models Approach to Market Risk Capital Requirements, January 1996.
Financial Times, 13 August 2007: David Viniar, chief financial officer of Goldman Sachs, on the losses at its quantitative funds.

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