Sample Excess Kurtosis
The Fat Ale microbrewery tracks bottle sales on a daily basis. After adjusting for several factors, the company developed a model to estimate the daily sales with some accuracy since sales are nearly always around 1,200 bottles per day and have been approximately normally distributed in the past year. The standard deviation of this distribution is 200.
On September 13, a social media stunt gave Fat Ale some instant popularity and 2,000 bottles were sold that day. What would you say this observation will do to the measures of skewness and kurtosis?
Incorrect.
Kurtosis will not decrease.
Incorrect.
Skewness will not decrease.
That's the simple way of saying it although the actual calculation for excess kurtosis looks pretty impressive:
$$\displaystyle K_E = \left[\frac{n(n+1)}{(n-1)(n-2)(n-3)} {{\sum_{i=1}^{n}(X_{i}-{\bar X})^{4}}\over{s^4}}\right] - \frac{3(n-1)^2}{(n-2)(n-3)} $$
Before you bang your head against a wall, look carefully at the _n_ terms and imagine that _n_ gets larger and larger. What would you say about the effect of those _n_ terms as the sample size grows?
Incorrect.
Notice that the algebra on the left side would leave an _n_2 on the top and an _n_3 on the bottom.
Now it becomes a little more clear where "excess kurtosis" is different from just kurtosis. Kurtosis is the main part of the calculation without the -3. So if someone tells you they calculated kurtosis as 2.5 what would you call that distribution?
Incorrect.
Notice that the algebra on the right side would leave an _n_2 on the top and an _n_2 on the bottom.
Incorrect.
A mesokurtic distribution has excess kurtosis of zero, but that's not the case here.
Absolutely!
Since kurtosis is 2.5, then excess kurtosis is 2.5 - 3 = -0.5. This means that the distribution is platykurtic. If excess kurtosis is zero, then it is mesokurtic (as in a normal distribution). If the distribution has positive excess kurtosis, then it is leptokurtic.
But be careful; some people like to say things like "positive kurtosis of 2.5" when they really mean "positive _excess_ kurtosis of 2.5."
Incorrect.
A leptokurtic distribution has positive excess kurtosis, but that's not the case here.
Below are some rounded return observations of a portfolio.
| Number of observations | Observation value |
|---|---|
| 50 | -10% |
| 100 | 0% |
| 250 | 5% |
| 100 | 10% |
| 50 | 20% |
You can probably tell that this is symmetric with a mean of 5%. The sample standard deviation is 7.0775%. What is the excess kurtosis of this sample?
Incorrect.
This is the total kurtosis measure, but not excess kurtosis.
Yes!
You can calculate this using the four deviations from the mean of 0.05 and taking each to the fourth power. Here are the complete results:
| __$$ n $$__ | __$$ Observation \, value $$__ | __$$ Deviation $$ (%)__ | __$$ Deviation^4 $$__ | __$$ n \cdot Deviation^4 $$__ |
|---|---|---|---|---|
| 50 | -10% | -15 | 50625 | 2,531,250 |
| 100 | 0% | -5 | 625 | 62500 |
| 250 | 5% | 0 | 0 | 0 |
| 100 | 10% | 5 | 625 | 62500 |
| 50 | 20% | 15 | 50625 | 2,531,250 |
| TOTAL | | | | 5,187,500 |
You can then place this total of the summed deviations to the fourth power into the simplified excess kurtosis calculation:
$$\displaystyle K_E = \frac{1}{550} \times \frac{5{,}187{,}500}{7.0775^4} - 3 = 0.759 $$
or close to 0.76.
Incorrect.
You might have forgotten to consider the number of observations for each value. Just assuming a single observation for each will lead to this result.
To summarize this discussion:
[[summary]]
Exactly!
This relatively large positive surprise will increase the measured skewness of the distribution, but also add to the measured kurtosis. This is because kurtosis is calculated using the fourth power of each deviation from the mean.
So this distribution has some excess kurtosis, and is slightly leptokurtic. This is typical for equity returns. There are "surprises" in equity markets more often than a normal distribution would allow, causing equity returns to be leptokurtic in their distribution.
That's right!
If you can assume that you'll have a sample of a decent size, then a really close approximation for __sample excess kurtosis__ can be found in this simplified version:
$$\displaystyle K_E = {1\over{n}}{{\sum_{i=1}^{n}(X_{i}-{\bar X})^{4}}\over{s^4}} - 3$$
Skewness will increase, kurtosis will decrease
Skewness will decrease, kurtosis will increase
Skewness and kurtosis will both increase
They should grow close to 1 on the left side and 3/_n_ on the right side
They should grow close to 1/_n_ on the left side and 3 on the right side
They should grow close to 1/_n_ on the left side and 3/_n_ on the right side.
Mesokurtic
Platykurtic
Leptokurtic
-2.93
0.04
0.76
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