Sample Excess Kurtosis
An analyst estimates the following statistics for a sample of portfolio returns ($$X$$).
| Statistics | Value |
|----------------------------|-------|
| Sample size ($$n$$) | 100 |
| Variance ($$s^2$$) | 0.025 |
| $$\sum(X_i-{\bar X})^{4}$$ | 0.335 |
Which of the following is the sample excess kurtosis _closest_ to?
Incorrect.
This answer choice incorrectly divides $$\sum(X_i-{\bar X})^{4}$$ by $$n$$ and $$s^2$$ to compute the sample kurtosis. The formula is mistaken.
Correct.
Kurtosis measures how a distribution is more or less peaked than a normal distribution.
The formula for sample kurtosis $$K_E$$ is:
$$\displaystyle K_E = {1\over{n}}{{\sum(X_{i}-{\bar X})^{4}}\over{s^4}} $$.
Please note for the second denominator, $$s^4 = (s^2)^2 $$.
$$\displaystyle K_E = {1 \over {100}} \times {{0.335} \over {0.025^2}} $$
$$\displaystyle = {0.335 \over{100 \times 0.000625}} = 5.36$$
The sample excess kurtosis is then $$K_E - 3 = 5.36 - 3 = 2.36$$.
Incorrect.
This is the kurtosis for the sample, but not the sample excess kurtosis.
-2.87
2.36
5.36
