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P1.T2.21.4. Non-normal distributions and rank correlations

Nicole Seaman

Director of FRM Operations
Staff member
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Learning objectives: Explain how the Jarque-Bera test is used to determine whether returns are normally distributed. Describe the power law and its use for non-normal distributions. Define correlation and covariance and differentiate between correlation and dependence. Describe properties of correlations between normally distributed variables when using a one-factor model.

Questions:

21.4.1. For three assets (Gold, S&P5000, and the USDJPY exchange rate), Elizabeth collected monthly returns over the last five years (T = 60 months) and computed the skewness (aka, skew) and kurtosis for each return series, as displayed below. Given the skew and kurtosis, she also computed the corresponding Jarque-Bera test statistics. The Jarque-Bera test will help her decide if the returns are normal because it is a test of the null hypothesis that (jointly) skewness and excess kurtosis are zero.

Elizabeth knows it will be easier to work with the dataset if the returns are normally distributed. She would like to reach a decision with the typical confidence levels: either 95.0% or 99.0%. Are these returns normal?

a. No, none of the return sets is plausibly normal: null hypothesis is rejected at all reasonably high confidence levels
b. Only the USDJPY exchange rate return set might be normal: we fail to reject the null with 99.0% confidence
c. Only the S&P500 index return series is normal: we fail to reject the null with 95.0% confidence
d. Yes, all three return sets are normal: we fail to reject the null at all reasonably low confidence levels

21.4.2. For his firm's equities portfolio, Luke estimates the one-month 95.0% confident relative value at risk (VaR) is $12.0 million. He is assuming the loss and profit (L/P) has a normal distribution. Based on this normality assumption, he re-scales this VaR in order to retrieve the corresponding 99.0% VaR. However, he worries this will underestimate the VaR given the tails are likely to be heavy. Consequently, using the same assumption of a one-month 95.0% VaR of$12.0 million, he instead assumes a power law distribution with an alpha constant, α = 1.90. Recall the power law distribution says that P(X > x) = k*x^(-α) where α and k are constants. Which are nearest to the one-month 99.0% VaR under, respectively, the normal and power law assumptions?

a. $17.0 and$28.0 million
b. $25.0 and$33.3 million
c. $49.0 and$39.6 million
d. $57.7 and$74.4 million

21.4.3. Emily listed the daily returns last week for two cryptocurrencies, bitcoin (BTC) and Ethereum (ETH). These are very small samples of only five returns for each currency. The returns are displayed below along with their ranks; e.g., Wednesday saw the worst performance for both cryptocurrencies, while Tuesday saw the best performance. Albeit a very small sample, the ranks do match on three of the days (Monday, Tuesday, and Wednesday) and differ on only two days (Thursday and Friday).

What are, respectively, the Spearman's Rank and Kendall's tau?

a. -0.70 (rank) and -0.55 (Kendall's)
b. 0.33 (rank) and 0.67 (Kendall's)
c. 0.90 (rank) and 0.80 (Kendall's)
d. 1.40 (rank) and 1.60 (Kendall's)