LO 7.3: Identify the best-fit distribution for equity, bond, and default correlations.

LO 7.3: Identify the best-fit distribution for equity, bond, and default correlations.
Seventy-seven percent of the correlations between stocks listed on the Dow from 1972 to 2012 were positive. Three distribution fitting tests were used to determine the best fit for equity correlations. Based on the results of the Kolmogorov-Smirnov, Anderson-Darling, and chi-squared distribution fitting tests, the Johnson SB distribution (which has two shape parameters, one location parameter, and one scale parameter) provided the best fit for equity correlations. The Johnson SB distribution best fit was also robust with respect to testing different economic states for the time period in question. The normal, lognormal, and beta distributions provided a poor fit for equity correlations.
There were three mild recessions and three severe recessions from 1972 to 2012. The time periods for the mild recessions occurred in 1980, 1990 to 1991, and 2001. More severe recessions occurred from 1973 to 1974 and from 1981 to 1982. Both of these severe recessions were caused by huge increases in oil prices. The most severe recession for this time period occurred from 2007 to 2009 following the global financial crisis. The percentage change in correlation volatility prior to a recession was negative in every case except for the 1990 to 1991 recession. This is consistent with the findings discussed earlier where correlation volatility is low during expansionary periods that often occur prior to a recession.
An empirical investigation of 7,643 bond correlations found average correlations for bonds of 42%. Correlation volatility for bond correlations was 64%. Bond correlations were also found to exhibit properties of mean reversion, but the mean reversion rate was only 26%. The best fit distribution for bond correlations was found to be the generalized extreme value (GEV) distribution. However, the normal distribution is also a good fit for bond correlations.
A study of 4,633 default probability correlations revealed an average default correlation of 30%. Correlation volatility for default probability correlations was 88%. The mean
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Topic 7 Cross Reference to GARP Assigned Reading – Meissner, Chapter 2
reversion rate for default probability correlations was 30%, which is closer to the 26% for bond correlations. However, the default probability correlation distribution was similar to equity distributions in that the Johnson SB distribution is the best fit for both distributions. Figure 1 summarizes the findings of the empirical correlation analysis.
Figure 1: Empirical Findings for Equity, Bond, and Default Correlations
Correlation Type
Average Correlation
Correlation Volatility
Reversion Rate
Equity
Bond
Default Probability
35%
42%
30%
80%
64%
88%
78%
26%
30%
Best Fit Distribution Johnson SB Generalized Extreme Value Johnson SB
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Topic 7 Cross Reference to GARP Assigned Reading – Meissner, Chapter 2
K e y C o n c e p t s
LO 7.1
Risk managers should be cognizant that historical correlation levels for common stocks in the Dow are highest during recessions. Correlation volatility for Dow stocks is high during recessions but highest during normal economic periods.
LO 7.2
W hen a regression is run where St St l (the ^variable) is regressed with respect to St l (the X variable), the (3 coefficient of the regression is equal to the negative mean reversion rate, a.
Equity correlations show high mean reversion rates (78%) and low autocorrelations (22%). These two rates must sum to 100%. Bond correlations and default probability correlations show much lower mean reversion rates and higher autocorrelation rates.
LO 7.3
Equity correlation distributions and default probability correlation distributions are best fit with the Johnson SB distribution. Bond correlation distributions are best fit with the generalized extreme value distribution, but the normal distribution is also a good fit.
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Topic 7 Cross Reference to GARP Assigned Reading – Meissner, Chapter 2
C o n c e p t C h e c k e r s
1.
2.
3.
Suppose a risk manager examines the correlations and correlation volatility of stocks in the Dow Jones Industrial Average (Dow) for the period beginning in 1972 and ending in 2012. Expansionary periods are defined as periods where the U.S. gross domestic product (GDP) growth rate is greater than 3.5%, periods are normal when the GDP growth rates are between 0 and 3.5%, and recessions are periods with two consecutive negative GDP growth rates. Which of the following statements characterizes correlation and correlation volatilities for this sample? The risk manager will most likely find that: A. correlations and correlation volatility are highest for recessions. B. correlations and correlation volatility are highest for expansionary periods. C. correlations are highest for normal periods, and correlation volatility is highest
for recessions.
D. correlations are highest for recessions, and correlation volatility is highest for
normal periods.
Suppose mean reversion exists for a variable with a value of 30 at time period t 1. Assume that the long-run mean value for this variable is 40 and ignore the stochastic term included in most regressions of financial data. What is the expected change in value of the variable for the next period if the mean reversion rate is 0.4? A. -10. B. -4. C. 4. D. 10.
A risk manager uses the past 480 months of correlation data from the Dow Jones Industrial Average (Dow) to estimate the long-run mean correlation of common stocks and the mean reversion rate. Based on historical data, the long-run mean correlation of Dow stocks was 32%, and the regression output estimates the following regression relationship: Y = 0.24 – 0.75X. Suppose that in April 2014, the average monthly correlation for all Dow stocks was 36%. What is the expected correlation for May 2014 assuming the mean reversion rate estimated in the regression analysis? A. 32%. B. 33%. C. 35%. D. 37%.
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4.
3.
A risk manager uses the past 480 months of correlation data from the Dow Jones Industrial Average (Dow) to estimate the long-run mean correlation of common stocks and the mean reversion rate. Based on this historical data, the long-run mean correlation of Dow stocks was 34%, and the regression output estimates the following regression relationship: Y = 0.262 – 0.77X. Suppose that in April 2014, the average monthly correlation for all Dow stocks was 33%. What is the estimated one-period autocorrelation for this time period based on the mean reversion rate estimated in the regression analysis? A. 23%. B. 26%. C. 30%. D. 33%.
In estimating correlation matrices, risk managers often assume an underlying distribution for the correlations. Which of the following statements most accurately describes the best fit distributions for equity correlation distributions, bond correlation distributions, and default probability correlation distributions? The best fit distribution for the equity, bond, and default probability correlation distributions, respectively are: A. B. Johnson SB, generalized extreme value, and Johnson SB. C. beta, normal, and beta. D. Johnson SB, normal, and beta.
lognormal, generalized extreme value, and normal.
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C o n c e p t C h e c k e r A n s w e r s
1. D Findings of an empirical study of monthly correlations of Dow stocks from 1972 to 2012
revealed the highest correlation levels for recessions and the highest correlation volatilities for normal periods. The correlation volatilities during a recession and normal period were 80.5% and 83.4%, respectively.
2. C The mean reversion rate, a, indicates the speed of the change or reversion back to the mean.
If the mean reversion rate is 0.4 and the difference between the last variable and long-run mean is 10 (= 40 – 30), the expected change for the next period is 4 (i.e., 0.4 x 10 = 4).
3. B There is a -4% difference from the long-run mean correlation and April 2014 correlation
(32% – 36% = -4%). The inverse of the (3 coefficient in the regression relationship implies a mean reversion rate of 75%. Thus, the expected correlation for May 2014 is 33.0%:
St = a(p, – St_j) + St l
St = 0.75(32% – 36%) + 0.36 = 0.33
4. A The autocorrelation for a one-period lag is 23% for the same sample. The sum of the mean reversion rate (77% given the beta coefficient of-0.77) and the one-period autocorrelation rate will always equal 100%.
5. B Equity correlation distributions and default probability correlation distributions are best fit with the Johnson SB distribution. Bond correlation distributions are best fit with the generalized extreme value distribution.
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The following is a review of the Market Risk Measurement and Management principles designed to address the learning objectives set forth by GARP. This topic is also covered in:
S t a t is t ic a l C o r r e l a t io n M o d e l s C a n W e A p p l y Th e m t o F in a n c e ?
Topic 8
E x a m F o c u s
This topic addresses the limitations of financial models and popular statistical correlation measures such as the Pearson correlation measure, the Spearman rank correlation, and the Kendall t . For the exam, understand that the major limitation of the Pearson correlation coefficient is that most financial variables have nonlinear relationships. Also, be able to discuss the limitations of ordinal correlation measures, such as Spearmans rank correlation and Kendalls t underlying joint distributions of variables; however, applications of ordinal risk measures are limited to ordinal variables where only the rankings are important instead of actual numerical values.
. These nonparametric measures do not require assumptions about the
L i m i t a t i o n s o f F i n a n c i a l M o d e l s

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