LO 8.2: Assess the Pearson correlation approach, Spearm ans rank correlation, and Kendall s t , and evaluate their lim itations and usefulness in finance.
Pearson Correlation
The Pearson correlation coefficient is commonly used to measure the linear relationship between two variables. The Pearson correlation is defined by dividing covariance (covXY) by the product of the two assets standard deviations (oxa Y).
PXY
covXY ox a Y
Covariance is a measure of how two assets move with each other over time. The Pearson correlation coefficient standardizes covariance by dividing it by the standard deviations of each asset. This is very convenient because the correlation coefficient is always between 1 and 1.
Covariance is calculated by finding the product of each assets deviation from their respective mean return for each period. The products of the deviations for each period are then added together and divided by the number of observations less one for degrees of freedom.
( X t – n x )(Yt – j i Y)
c o v x y
n 1
There is a second methodology that is used for calculating the Pearson correlation coefficient if the data is drawn from random processes with unknown outcomes (e.g., rolling a die). The following equation defines covariance with expectation values. If E(X) and E(Y) are the expected values of variables X and Y, respectively, then the expected product of deviations from these expected values is computed as follows:
E{ [X – E(X)1 [Y – E(Y)]} or E(XY) – E(X)E(Y)
When using random sets of data, the correlation coefficient can be rewritten as:
PXY
E(XY) – E(X)E( Y)
Ve (X 2) (E(X))2 x Ve (Y2)-(E (Y ))2
Because many financial variables have nonlinear relationships, the Pearson correlation coefficient is only an approximation of the nonlinear relationship between financial
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variables. Thus, when applying the Pearson correlation coefficient in financial models, risk managers and investors need to be aware of the following five limitations:
1. The Pearson correlation coefficient measures the linear relationship between two
variables, but financial relationships are often nonlinear.
2. A Pearson correlation of zero does not imply independence between the two variables. It simply means there is not a linear relationship between the variables. For example, the parabola relationship defined as Y = X2 has a correlation coefficient of zero. There is, however, an obvious nonlinear relationship between variables Y and X.
3. When the joint distribution between variables is not elliptical, linear correlation
measures do not have meaningful interpretations. Examples of common elliptical joint distributions are the multivariate normal distribution and the multivariate Students ^-distribution.
4. The Pearson correlation coefficient requires that the variance calculations of the
variables X and Y are finite. In cases where kurtosis is very high, such as the Students ^-distribution, the variance could be infinite, so the Pearson correlation coefficient would be undefined.
3. The Pearson correlation coefficient is not meaningful if the data is transformed. For
example, the correlation coefficient between two variables X and Ywill be different than the correlation coefficient between ln(X) and ln(Y).
Spearm ans Rank Correlation
Ordinal measures are based on the order of elements in data sets. Two examples of ordinal correlation measures are the Spearman rank correlation and the Kendall t . The Spearman rank correlation is a nonparametric approach because no knowledge of the joint distribution of the variables is necessary. The calculation is based on the relationship of the ranked variables. The following equation defines the Spearman rank correlation coefficient where n is the number of observations for each variable, and d- is the difference between the ranking for period i.
The Spearman rank correlation coefficient is determined in three steps:
Step 1: Order the set pairs of variables X and Y with respect to the set X. Step 2: Determine the ranks of Xy and Y- for each time period i. Step 3: Calculate the difference of the variable rankings and square the difference.
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Example: Spearmans rank correlation
Calculate the Spearman rank correlation for the returns of stocks X and Yprovided in Figure 1.
Figure 1: Returns for Stocks X and Y
X
25.0% 60.0% -20.0% 40.0% -10.0% 19.0%
Y
-20.0% 40.0% 10.0% 20.0% 30.0% 16.0%
Year 2010 2011 2012 2013 2014 Average
Answer:
The calculations for determining the Spearman rank correlation coefficient are shown in Figure 2. The first step involves ranking the returns for stock X from lowest to highest in the second column. The first column denotes the respective year for each return. The returns for stock Y are then listed for each respective year. The fourth and fifth columns rank the returns for variables X and Y The differences between the rankings for each year are listed in column six. Lastly, the sum of squared differences in rankings is determined in column 7.
Figure 2: Ranking Returns for Stocks X and Y
Year 2012 2014 2010 2013 2011
X
-20.0% -10.0% 25.0% 40.0% 60.0%
Y
10.0% 30.0% -20.0% 20.0% 40.0%
X Rank
Y Rank
1 2 3 4 5
2 4 1 3 5
d, -1 -2 2 1 0
Sum =
1 4 4 1 0 10
The Spearman rank correlation coefficient can then be determined as 0.3:
n
6Ed? i=l n(n2 – l )
Ps 1
1 –
6×10 5(23-1)
0.5
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Kendall S T
Topic 8 Cross Reference to GARP Assigned Reading – Meissner, Chapter 3 is also a
is another ordinal correlation measure that is becoming more widely applied Kendalls t in financial models for ordinal variables such as credit ratings. Kendalls t is also a nonparametric measure that does not require any assumptions regarding the joint probability distributions of variables. Both Spearmans rank correlation coefficient and Kendalls t perfectly correlated variables will have a coefficient of 1 . The Kendall t Y always increases with an increase in variable X. The numerical amount of the increase does not matter for two variables to be perfectly correlated. Therefore, for most cases, the Kendall t
are similar to the Pearson correlation coefficient for ranked variables because
and the Spearman rank correlation coefficients will be different.
will be 1 if variable
The mathematical definition of Kendalls t
is provided as follows:
nc ~ nd n(n 1)/ 2
In this equation, the number of concordant pairs is represented as nc, and the number of discordant pairs is represented as nd. A concordant pair of observations is when the rankings of two pairs are in agreement:
< Y and X ~ < Y* or X^ >
t*
t*
t
t
t
t
and X * > Y * and t ^ t
*
t*
t*
A discordant pair of observations is when the rankings of two pairs are not in agreement:
Xt < Yt and Xt* > Yt* or Xt > Yt and Xt* < Yt* and t ^ t*
A pair of rankings is neither concordant nor discordant if the rankings are equal:
xt = Yt or xt, = Yt, equation computes the total number of pair The denominator in the Kendall t equation computes the total number of pair combinations. For example, if there are six pairs of observations, there will be 13 combinations of pairs:
[n x (n - 1)] / 2 = (6 x 3) / 2 = 13
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Example: Kendalls T
Calculate the Kendall t Figure 3.
correlation coefficient for the stock returns of X and Klisted in
Figure 3: Ranked Returns for Stocks X and Y
X
-20.0% -10.0% 25.0% 40.0% 60.0%
Y
10.0% 30.0% -20.0% 20.0% 40.0%
X Rank
Y Rank
1 2 3 4 5
2 4 1 3 5
Year 2012 2014 2010 2013 2011
Answer:
Begin by comparing the rankings of X and Ystock returns in columns four and five of Figure 3. There are five pairs of observations, so there will be ten combinations. Figure 4 summarizes the pairs of rankings based on the stock returns for X and Y There are two concordant pairs, four discordant pairs, and four pairs that are neither concordant nor discordant.
Figure 4: Categorizing Pairs of Stock X and TReturns
Concordant Pairs {(1,2),(2,4)} {(3,1),(4,3)}
Discordant Pairs {(1,2),(3,1)} {(1,2),(4,3)} {(2,4),(3,1)} {(2,4),(4,3)}
Neither {(1,2),(3,3)} {(2,4),(5,5)} {(3,1),(5,5)} {(4,3),(5,5)}
Kendalls t
can then be determined as 0.2:
T
n c n d n(n 1) / 2
2 - 4
5(5 1)/2
- 0.2
Thus, the relationship between the stock returns of X and Kis slightly negative based on the Kendall t
correlation coefficient.
Lim itations o f Ordinal Risk Measures
Ordinal correlation measures based on ranking (i.e., Spearmans rank correlation and Kendalls t ) are implemented in copula correlation models to analyze the dependence of market prices and counterparty risk. Because ordinal numbers simply show the rank of observations, problems arise when ordinal measures are used for cardinal observations, which show the quantity, number, or value of observations.
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Example: Impact of outliers on ordinal measures
Suppose we triple the returns of X in the previous example to show the impact of outliers. If outliers are important sources of information and financial variables are cardinal, what are the implications for ordinal correlation measures?
Answer:
Notice from Figure 3 that Spearmans rank correlation and Kendalls t with an increased probability of outliers. Thus, ordinal correlation measures are less sensitive to outliers, which are extremely important in VaR and stress test models during extreme economic conditions. Numerical values are not important for ordinal correlation measures where only the rankings matter. Thus, since outliers do not change the rankings, ordinal measures underestimate risk by ignoring the impact of outliers.
do not change
Figure 5: Ranking Returns with Outliers
Year 2012 2014 2010 2013 2011
3X -60.0% -30.0% 75.0% 120.0% 180.0%
Y
3XRank
Y Rank
10.0% 30.0% -20.0% 20.0% 40.0%
1 2 3 4 5
2 4 1 3 5
d, -1 -2 2 1 0
Sum =
d ? 1 4 4 1 0 10 calculation can be
occurs when there are a large number of pairs that Another limitation of Kendalls t are neither concordant nor discordant. In other words, the Kendall t calculation can be distorted when there are only a few concordant and discordant pairs. For example, there were 4 out of 10 pairs that were neither concordant nor discordant in Figure 4. Thus, the Kendall t
calculation was based on only 6 out of 10, or 60%, of the observations.
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K e y C o n c e p t s
LO 8.1
Limitations of financial models arise due to inaccurate input values, erroneous underlying distribution assumptions, and mathematical inconsistencies.
Copula correlation models failed during the 20072009 financial crisis due to assumptions of a negative correlation between the equity and senior tranches in a collateralized debt obligation (CDO) structure and the calibration of correlation estimates with pre-crisis data.
LO 8.2
A major limitation of the Pearson correlation coefficient is that it measures linear relationships when most financial variables are nonlinear.
The Spearman rank correlation coefficient, where n is the number of observations for each variable and d- is the difference between the ranking for period /, is computed as follows:
i=l
The Kendall t as nc and the number of discordant pairs is represented as nd, is computed as follows:
correlation coefficient, where the number of concordant pairs is represented
nc ~ nd n(n 1) / 2
correlation coefficients should not be used with cardinal Spearmans rank and Kendalls t financial variables because ordinal measures underestimate risk by ignoring the impact of outliers.
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C o n c e p t C h e c k e r s
1.
2.
Kirk Rozenboom, FRM, uses the Black-Scholes-Merton (BSM) model to value options. Following the financial crisis of 20072009, he is more aware of the limitations of the BSM option pricing model. Which of the following statements best characterizes a major limitation of the BSM option pricing model? A. The BSM model assumes strike prices have nonconstant volatility. B. Option traders often use a volatility smile with lower volatilities for out-of-the-
money call and put options when applying the BSM model.
C. For up-and-out calls and puts, the BSM model is insensitive to changes in
implied volatility when the knock-out strike price is equal to the strike price and the interest rate equals the underlying asset return.
D. For down-and-out calls and puts, the BSM model is insensitive to changes in
option maturity when the knock-out strike price is greater than the strike price and the interest rate is greater than the underlying asset return.
New copula correlation models were used by traders and risk managers during the 2007-2009 global financial crisis. This led to miscalculations in the underlying risk for structured products such as collateralized debt obligation (CDO) models. Which of the following statements least likely explains the failure of these new copula correlation models during the financial crisis? A. The copula correlation models assumed a negative correlation between the
equity and senior tranches of CDOs.
B. Correlations for equity tranches of CDOs increased during the financial crisis. C. The correlation copula models were calibrated with data from time periods that
had low risk.
D. Correlations for senior tranches of CDOs decreased during the financial crisis.
3.
A risk manager gathers five years of historical returns to calculate the Spearman rank correlation coefficient for stocks X and Y The stock returns for X and Y from 2010 to 2014 are as follows:
Year 2010 2011 2012 2013 2014
X 5.0% 50.0% -10.0% -20.0% 30.0%
Y
-10.0% -5.0% 20.0% 40.0% 15.0%
What is the Spearman rank correlation coefficient for the stock returns of X and F? A. -0.7. B. -0.3. C. 0.3. D. 0.7.
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4.
A risk manager gathers five years of historical returns to calculate the Kendall t correlation coefficient for stocks X and Y The stock returns for X and Yfrom 2010 to 2014 are as follows:
Year 2010 2011 2012 2013 2014
X 5.0% 50.0% -10.0% -20.0% 30.0%
Y
-10.0% -5.0% 20.0% 40.0% 15.0%
correlation coefficient for the stock returns of X and Y?
What is the Kendall t A. -0.3. B. -0.2. C. 0.4. D. 0.7.
3.
A risk manager is using a copula correlation model to perform stress tests of financial risk during systemic economic crises. If the risk manager is concerned about extreme outliers, which of the following correlation coefficient measures should be used? A. Kendalls t B. Ordinal correlation. C. Pearson correlation. D. Spearmans rank correlation.
correlation.
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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. C For up-and-out calls and puts and for down-and-out calls and puts, the BSM option pricing model is insensitive to changes in implied volatility when the knock-out strike price is equal to the strike price and the interest rate equals the underlying asset return. The BSM model assumes strike prices have a constant volatility, and option traders often use a volatility smile with higher volatilities for out-of-the-money call and put options.
2. D During the crisis, the correlations for both the equity and senior tranches of CDOs
significantly increased causing losses in value for both.
3. A The following table illustrates the calculations used to determine the sum of squared ranking
deviations:
Year 2013 2012 2010 2014 2011
Y
X Rank
Y Rank
-20.0% -10.0% 5.0% 30.0% 50.0%
40.0% 20.0% -10.0% 15.0% -5.0%
1 2 3 4 5
5 4 1 3 2
4 -A -2 2 1 3
Sum =
d f
16 4 4 1 9 34
Thus, the Spearman rank correlation coefficient is -0.7:
n
Ps 1 i=l
n(n2 -1 )
= 1
6x34 5(25-1)
= -0 .7
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4. B The following table provides the ranking of pairs with respect to X.
Year 2013 2012 2010 2014 2011
X
-20.0% -10.0% 5.0% 30.0% 50.0%
Y
40.0% 20.0% -10.0% 15.0% -5.0%
X Rank
Y Rank
1 2 3 4 5
5 4 1 3 2
There are four concordant pairs and six discordant pairs shown as follows:
Concordant Pairs 1(1,5),(2,4)} {(3,1),(4,3)} {(3,1),(5,2)} {(4,3),(5,2)}
Discordant Pairs {(1,5), (3,1)} {(1,5),(4,3)} {(1,5),(5,2)} {(2,4),(3,1)} {(2,4),(4,3)} {(2,4),(5,2)}
Thus, the Kendall t
correlation coefficient is -0.2:
nc nd n(n 1)/ 2
4 - 6
5(5 1)/2
5. C The Pearson correlation coefficient is preferred to ordinal measures when outliers are a
concern. Spearmans rank correlation and Kendalls t that should not be used with cardinal financial variables because they underestimate risk by ignoring the impact of outliers.
are ordinal correlation coefficients
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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:
F in a n c ia l C o r r e l a t io n M o d e l i n g B o t t o m -U p A p p r o a c h e s
Topic 9
E x a m F o c u s
A copula is a joint multivariate distribution that describes how variables from marginal distributions come together. Copulas provide an alternative measure of dependence between random variables that is not subject to the same limitations as correlation in applications such as risk measurement. For the exam, understand how a correlation copula is created by mapping two or more unknown distributions to a known distribution that has well-defined properties. Also, know how the Gaussian copula is used to estimate joint probabilities of default for specific time periods and the default time for multiple assets. The material in this topic is relatively complex, so your focus here should be on gaining a general understanding of how a copula function is applied.
C o p u l a F u n c t i o n s
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