Week 7 Quantitative Analysis of Financial Markets ...€¦ · C V 1;V 2 = E V 1V 2 E V 1 E V 2: R...

Introduction Correlation & Dependence Multivariate Distributions Vasicek’s Model Summary

Week 7Quantitative Analysis of Financial Markets

Correlations and Copulas

Christopher Ting

Christopher Ting

http://www.mysmu.edu/faculty/christophert/

k: [email protected]: 6828 0364

ÿ: LKCSB 5036

November 21, 2017

Christopher Ting QF 603 November 21, 2017 1/47

http://www.mysmu.edu/faculty/christophert/

mailto:[email protected]


Lesson Plan

1 Introduction

2 Correlation & Dependence

3 Multivariate Distributions

4 Vasicek’s Model

5 Summary



Introduction

A The objective is to examine how two random variables(investments) interact to produce an overall risk exposure of theportfolio.

A It is important to use tools such as copulas to quantify thecorrelation structure between two or more variables, regardless ofthe shapes of their probability distributions.

A A copula can be used to create a model of default correlation for aportfolio of loans. In fact, the copula model is used in the Basel IIcapital requirements.



Learning Outcomes of QA15

Chapter 11, John Hull, Risk Management and Financial Institutions, 4th Edition(Hoboken, NJ: John Wiley & Sons, 2015).

A Define correlation and covariance, differentiate betweencorrelation and dependence.

A Calculate covariance using the EWMA and GARCH (1,1) models.A Apply the consistency condition to covariance.A Describe the procedure of generating samples from a bivariate

normal distribution.A Describe properties of correlations between normally distributed

variables when using a one-factor model.A Define copula, describe the key properties of copula and copula

correlation.A Explain tail dependence.A Describe Gaussian copula, Student t-copula, multivariate copula

and one factor copula.Christopher Ting QF 603 November 21, 2017 4/47


Covariance and Correlation

R The covariance between two random variables V 1 and V 2 isdefined as

C(V 1, V 2

)= E

(V 1V 2

)− E

(V 1

)E(V 2

).

R The coefficient of correlation, ρ, between two random variables V 1

and V 2 is defined as

ρ :=C(V 1, V 2)

S(V 1)S(V 2).

R Here, S(V a) is the standard deviations of V a, where a = 1, 2.



Correlation vs. Dependence

R Two variables are statistically independent if knowledge about oneof them does not affect the probability distribution for the other, i.e.,

f(V 2

∣∣V 1 = x)= f(V 2)

for all x where f(·) denotes the probability density function.

R Otherwise, one variable may depend on the other.

R Question: If the coefficient of correlation between two variables iszero, does it mean that there is no dependence between thevariables?

R Answer: No!



Independence 6= Zero Correlation

R Suppose V 1 = -1, 0, or +1, equally likely. Also, V 2 = 0, 1, equallylikely.

R We construct a dependence structure between V 1 and V 2 asfollows:

If V 1 6= 0, then V 2 = 1.If V 1 = 0, then V 2 = 0.

R The coefficient of correlation is zero because the covariance iszero.

The means are E(V 1) = 0 and E(V 2) =1

2.

The covariance C(V 1, V 2) is zero:

E(V 1V 2)−E(V 1)E(V 2) =1

3(−1× 1)+

1

3(1× 1)+

1

3(0× 0)−0×1

2= 0.

R But by construction, V 2 is dependent on V 1 and vice versa.



Estimates of Variance and Covariance Rates

R Suppose that Xi and Yi are the values of two variables, X and Y ,at the end of day i. The returns on the variables on day i are

xi =Xi −Xi−1Xi−1

, yi =Yi − Yi−1Yi−1

,

R Expected daily returns are assumed to be zero.

R Using equal weights for the last m observations on X, theestimate of variance rate per day (denoted by VX;n) of X for day nis calculated on day n− 1 by

VX;n =1

m

m∑i=1

x2n−i.



Estimates of Variance and Covariance Rates (cont’d)

R The variance rate for Y is estimated with the same formula and isdenoted by VY ;n.

R Analogously, the estimate of covariance rate (denoted by Cn) isgiven by:

Cn =1

m

m∑i=1

xn−iyn−i.

Cn is the covariance calculated on day n− 1 for day n.



Predictive Updates by Two Models

R EWMACn = λCn−1 + (1− λ)xn−1yn−1.

R GARCH(1, 1)

Cn = ω + αxn−1yn−1 + βCn−1.



EWMA Example

R Suppose λ = 0.95 and the estimate of the correlation between twovariables X and Y on day n− 1 is 0.6.

R Suppose further that the estimate of the volatilities for X and Y onday n− 1 are 1% and 2%, respectively.

R Suppose that the percentage changes in X and Y on day n− 1are 0.5% and 2.5%, respectively.

R From the relationship between correlation and covariance, theestimate of the covariance rate between X and Y on day n− 1 is

0.6× 0.01× 0.02 = 0.00012.

R The variance rates and covariance rate for day n would beupdated as follows:σ2X,n = 0.95× 0.012 + 0.05× 0.0052 = 0.00009625

σ2Y ,n = 0.95× 0.022 + 0.05× 0.0252 = 0.00041125

Cn = 0.95× 0.00012 + 0.05× 0.005× 0.025 = 0.00012025Christopher Ting QF 603 November 21, 2017 11/47


Consistency Condition

R The condition for an N ×N variance-covariance matrix, ΩΩΩ, to beinternally consistent is

www>ΩΩΩwww ≥ 0

for all N × 1 vectors www. A matrix that satisfies this property isknown as positive-semi-definite.

R The expression www>ΩΩΩwww is the variance rate of a portfolio where anamount www is invested in market variable i. As such, it cannot benegative.

R To ensure that a positive semi-definite matrix is produced,variances and covariances should be calculated consistently.



Consistency Condition (cont’d)

R For example, if variance rates are calculated by giving equalweight to the last m data items, the same should be done forcovariance rates. If variance rates are updated using an EWMAmodel with λ = 0.94, the same should be done for covariancerates.

R We know that ΩΩΩ is a symmetric matrix.

R How to check whether a symmetric matrix is semi-definite?

R For a start, suppose there are only two assets and

ΩΩΩ =

[a bb c

]

We write www =

[w1

w2

]and www> =

[w1 w2

].



Consistency Condition (cont’d)

R Thenwww>ΩΩΩwww = aw2

1 + 2bw1w2 + cw22.

R Completion of the squares:

www>ΩΩΩwww = a

(w1 +

b

aw2

)2

+

(ac− b2

a

)w22

.R To have www>ΩΩΩwww > 0, it must be that a > 0 and

ac− b2 = det(ΩΩΩ)> 0.

R For variance-covariance matrix, the diagonal elements a and c arealways positive because they are variances. Therefore, thenecessary and sufficient condition for the 2× 2 matrix ΩΩΩ is thatdet(ΩΩΩ)> 0.



Positive Semi-Definite

R The condition for an N ×N variance-covariance matrix, Ω, to beinternally consistent is

www>ΩΩΩwww ≥ 0

for all N × 1 vectors www. A matrix that satisfies this property isknown as positive-semi-definite.

R The expression www>ΩΩΩwww is the variance rate of a portfolio where anamount www is invested in market variable i. As such, it cannot benegative.

R To ensure that a positive semi-definite matrix is produced,variances and covariances should be calculated consistently.



Positive Semi-Definite (cont’d)

R For example, if variance rates are calculated by giving equalweight to the last m data items, the same should be done forcovariance rates. If variance rates are updated using an EWMAmodel with λ = 0.95, the same should be done for covariancerates.

R We know that ΩΩΩ is a symmetric matrix.

R How to check whether a symmetric matrix is semi-definite?

R For a start, suppose there are only two assets and

ΩΩΩ =

[a bb c

].

If detΩΩΩ = ac− b2 > 0, then ΩΩΩ is positive semi-dfinite.



Consistency Condition for N ×N Symmetric Matrix

R For N ×N symmetric ΩΩΩ to be positive definite, all its eigenvaluesmust be positive.

R For example, the matrix ΩΩΩ =

1 0 0.90 1 0.90.9 0.9 1

is not internally

consistent.R Proof: Note that the characteristic equation is

det(ΩΩΩ − λIII

)= (1− λ)3 − 0.92(1− λ)− 0.92(1− λ) = 0.

Solving the equation finds that λ = 3.25, 1, and −1.15.R If instead of 0.9, suppose the covariance is 0.7, which is quite.

Then we find that the eigenvalue is positive.R When ΩΩΩ is large, such as 100× 100, accuracy of the estimates is

crucial. A small error is quite likely to lead to it no longer beingpositive semi-definite.



Bivariate Conditional Mean and StandardDeviation

® We start by considering a bivariate normal distribution where thereare only two variables, V 1 and V 2.

® Suppose that we know that V 1 has some value. Conditional onthis, the value of V 2 is normal with mean

µ2 + ρσ2V 1 − µ1σ1

and standard deviationσ2√

1− ρ2.

® Here µ1 and µ2 are the unconditional means of V 1 and V 2, σ1 andσ2 are their unconditional standard deviations, and ρ is thecoefficient of correlation between V 1 and V 2.

® Note that the expected value of V 2 conditional on V 1 is linearlydependent on the value of V 1.



Generating Random Samples from NormalDistributions

® Obtain independent samples z1 and z2 from a univariatestandardized normal distribution.

® The samples are

ε1 = z1, ε2 = ρz1 +√1− ρ2z2,

where ρ is the coefficient of correlation in the bivariate normaldistribution.

® In the case of multivariate normal distribution, the procedures areSample n independent variables zi from univariate standardizednormal distributions.The required samples are, for 1 ≤ j < i,

εi =

i∑k=1

αikzk, subject toi∑

k=1

α2ik = 1 and

j∑k=1

αikαjk = ρij .



Factor Models

® Suppose that U1, U2, . . . , UN have standard normal distributions.

® In a one-factor model, each U i has a component dependent on acommon factor, F , and a component that is uncorrelated with theother variables. Formally,

U i = aiF +√

1− a2i Zi,

where F and the Zi have standard normal distributions and ai is aconstant between -1 and +1. The Zi are uncorrelated with eachother and uncorrelated with F .

® In the one-factor model, all the correlation between U i and U j

arises from their dependence on the common factor, F , and thecoefficient of correlation is aiaj .



Factor Models (cont’d)

® Without assuming a factor model, the number of correlations thathave to be estimated for N variables is N(N − 1)/2. But with onefactor, we need only estimate N parameters.

® The one-factor model can be extended to a two-factor,three-factor, or M -factor model. In the M -factor model,

U i = ai1F 1 + · · ·+ aiMFM +√

1− a21 − a2i2 − · · · − a2iM Zi.

® The M factors F j have uncorrelated standard normal distributionsand the Zi are uncorrelated both with each other and with thefactors.

® In this case, the correlation between U i and U j isM∑

m=1

aimajm.



What is Copula?

Ò In probability theory and statistics, a copula is a multivariateprobability distribution for which the marginal probabilitydistribution of each variable is uniform.

Ò A copula describes the dependence between random variables.

Ò In probabilistic terms, C : [0, 1]d → [0, 1] is a d-dimensional copulaif C is a joint cumulative distribution function of a d-dimensionalrandom vector on the unit cube [0, 1]d with uniform marginals.

Ò Sklar’s theorem provides the theoretical foundation for theapplication of copulas. It states that every multivariate cumulativedistribution function of a random vector can be expressed byinvolving only the marginals and a copula.



Copula Construction Illustrated

-0.2 0 0.2 0.6 0.8 1 1.2 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

-6 -4 -2 0 2 4 6 2 4 6

U1

V1 V2

One-to-One Mapping

Correlation Assumption

-6 -4 -2 0 2 4 6 2 4 6

U2



Two Marginal Probability Density Functions

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

PDF of V 1

0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

PDF of V 2



Mapping to Standard Normal Distribution

Ò Mapping of V 1 and V 2 to standard normal U1 and U2, respectively.V 1 or V 2 Percentile of U1 Percentile of U2

Value V 1 Distribution Value V 2 Distribution Value0.1 5.00 -1.64 2.00 -2.050.2 20.00 -0.84 8.00 -1.410.3 38.75 -0.29 18.00 -0.920.4 55.00 0.13 32.00 -0.470.5 68.75 0.49 50.00 0.000.6 80.00 0.84 68.00 0.470.7 88.75 1.21 82.00 0.920.8 95.00 1.64 92.00 1.410.9 98.75 2.24 98.00 2.05

Ò The correlation between U1 and U2 is referred to as the copulacorrelation.



Constructing a Copula

Ò The essence of copula is therefore that, instead of defining acorrelation structure between V 1 and V 2 directly, we do soindirectly.

Ò We map V 1 and V 2 into other variables that have well-behaveddistributions and for which it is easy to define a correlationstructure.

Ò We have assumed that U1 and U2 are jointly bivariate normal.

Ò Further, for illustration, we assume that the correlation betweenU1 and U2 is 0.5.

Ò We can now look at the joint cumulative probability distributionbetween V 1 and V 2.



CDF for V 1 and V 2

V 2

V 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.1 0.006 0.017 0.028 0.037 0.044 0.048 0.049 0.050 0.0500.2 0.013 0.043 0.081 0.120 0.156 0.181 0.193 0.198 0.2000.3 0.017 0.061 0.124 0.197 0.273 0.331 0.364 0.381 0.3870.4 0.019 0.071 0.149 0.248 0.358 0.449 0.505 0.535 0.5480.5 0.019 0.076 0.164 0.281 0.417 0.537 0.616 0.663 0.6830.6 0.020 0.078 0.173 0.301 0.456 0.600 0.701 0.763 0.7930.7 0.020 0.079 0.177 0.312 0.481 0.642 0.760 0.837 0.8770.8 0.020 0.080 0.179 0.318 0.494 0.667 0.798 0.887 0.9360.9 0.020 0.080 0.180 0.320 0.499 0.678 0.816 0.913 0.970

Ò The table is the cumulative joint probability distribution for V 1 andV 2 in the Gaussian Copula Model (correlation parameter = 0.5).

Ò The table shows the joint probability that V 1 and V 2 are less thanthe specified values.



Calculation of Joint Cumulative Distribution

Ò Probability that V 1 and V 2 are both less than 0.2 is the probabilitythat U1 < −0.84 and U2 < −1.41.

Ò When copula correlation is 0.5 this joint probability is

M(−0.84,−1.41, 0.5) = 0.043,

where M is the cumulative distribution function for the bivariatenormal distribution.



3-D Graphs of Bivariate Normal Distribution



Expressing the Approach Algebraically

Ò Suppose that G1 and G2 are the cumulative marginal (i.e.,unconditional) probability distributions of V 1 and V 2.

Ò Map V 1 = v1 to U1 = u1 and V 2 = v2 to U2 = u2 so that

G1(v1) = N(u1), G2(v2) = N(u2),

where N(·) is the cumulative normal distribution function.Ò It follows that

u1 = N−1(G1(v1)

), u2 = N−1

(G2(v2)

)v1 = G−11

(N(u1)

), v2 = G−12

(N(u2)

)Ò The variables U1 and U2 are then assumed to be bivariate normal.Ò The key property of a copula model: Preserving the marginal

distributions of V 1 and V 2 while defining a correlation structurebetween them.



Other Copulas

Ò The Gaussian copula is just one copula that can be used to definea correlation structure.

Ò There are many other copulas leading to many other correlationstructures.

Ò One that is sometimes used is the Student’s t-copula. It works inthe same way as the Gaussian copula except that the variablesU1 and U2 are assumed to have a bivariate Student’s t-distributionwith f degrees of freedom instead of a bivariate normaldistribution.

Sample from the inverse chi-square distribution to get a value χ.Sample from a bivariate normal distribution with correlation ρ.Multiply the bivariate normally distributed samples by

√f/χ.



Simulated Copula



Tail Dependence

Ò A tail value of a distribution is a value in the left or right 1% tail ofthe distribution.

Ò There is a tail value for the normal distribution when the variable isgreater than 2.33 or less than -2.33.

Ò Similarly, there is a tail value in the Student’s t-distribution whenthe value of the variable is greater than 3.75 or less than -3.75.

Ò It is much more common for the two variables to have tail valuesat the same time in the bivariate Student’s t-distribution than in thebivariate normal distribution. That is, the tail dependence is higherin a bivariate Student’s t-distribution than in a bivariate normaldistribution.

Ò Correlations between market variables tend to increase in extrememarket conditions.



Multivariate Copulas

Ò Copulas can be used to define a correlation structure betweenmore than two variables.

Ò The simplest example of this is the multivariate Gaussian copula.Suppose that there are N variables, V 1, V 2, . . . , V N and that weknow the marginal distribution of each variable.

Ò For each i (1 ≤ i < N), we transform V i into U i where U i has astandard normal distribution.

Ò The transformation is accomplished on a percentile-to-percentilebasis.

Ò We then assume that the U i have a multivariate normaldistribution.



Credit Default Correlation

F The credit default correlation between two companies is ameasure of their tendency to default at about the same time.

F Default correlation is important in risk management whenanalyzing the benefits of credit risk diversification.

F It is also important in the valuation of some credit derivatives.



Annual Percentage Default Rates

Year Default Year Default Year DefaultRate Rate Rate

1970 2.621 1985 0.960 2000 2.8521971 0.285 1986 1.875 2001 4.3451972 0.451 1987 1.588 2002 3.3191973 0.453 1988 1.372 2003 2.0181974 0.274 1989 2.386 2004 0.9391975 0.359 1990 3.750 2005 0.7601976 0.175 1991 3.091 2006 0.7211977 0.352 1992 1.500 2007 0.4011978 0.352 1993 0.890 2008 2.2521979 0.088 1994 0.663 2009 6.0021980 0.342 1995 1.031 2010 1.4081981 0.162 1996 0.588 2011 0.8901982 1.032 1997 0.765 2012 1.3811983 0.964 1998 1.317 2013 1.3811984 0.934 1999 2.409

Source: Moody’s.



Probability of Default

F To model the defaults of the loans in a portfolio, we define T i asthe time when company i defaults.

F We make the simplifying assumption that all loans have the samecumulative probability distribution for the time to default. Wedefine PD as the probability of default by time T :

PD = P(T i < T ).

F The Gaussian copula model can be used to define a correlationstructure between the times to default of the loans. Each time todefault T i is mapped to a variable U i that has a standard normaldistribution on a percentile-to-percentile basis.

U i = aF +√1− a2iZi



Probability of Default (cont’d)

F As before, the variables F and Z have independent standardnormal distributions. The copula correlation between each pair ofloans is in this case the same. It is

ρ = a2.

F The expression for U i reduces to

U i =√ρF +

√1− ρZi.



Worst Case Default Rate

F What is the default rate WCDR(T,X) (i.e., percentage of loansdefaulting) during time T (e.g. one year) that will not be exceededwith probability X%?

F Vasicek’s Model (1987)

WCDR(T,X) = N

(N−1

(PD)+√ρN−1

(X)

√1− ρ

)

F Note that if ρ = 0, the loans default independently of each otherand WCDR = PD. As ρ increases, WCDR increases.



Numerical Example of Worst Case Default Rate

F Suppose that a bank has a large number of loans to retailcustomers.

F The one-year probability of default for each loan is 2% and thecopula correlation parameter, ρ, in Vasicek’s model is estimatedas 0.1. What is the 99.9% worst case one-year default rate?

F Hence,

WCDR(1, 0.999) = N

(N−1

(0.02

)+√0.1N−1

(0.999

)√1− 0.1

)= 0.128

F So the 99.9% worst case one-year default rate is 12.8%.

F Analyst: “I’m 99.9% sure that, for a period of one year, thepercentage of loans defaulting is at most 12.8%. But there is a0.1% chance that it will be worst than 12.8%.”



Proof of Vasicek’s One-Factor Model (1987)

F From the properties of the Gaussian copula model

PD = P (T i < T ) = P (U i < u) ,

whereU i = N−1(PD).

F In the one-factor model, the standard normal random variable isbacked out as

Zi =Ui −

√ρF

√1− ρ

.

F The probability that U i < u conditional on the factor value, F , is

P (U i < u|F ) = P(Zi <

Ui −√ρF

√1− ρ

)= N

(Ui −

√ρF

√1− ρ

).



Proof of Vasicek’s One-Factor Model (1987) (cont’d)

F This is also the probability of conditional default rate. So

P (T i < T |F ) = N

(Ui −

√ρF

√1− ρ

).

F Because F has a normal distribution, the probability that F will beless than N−1(y) is y, i.e.,

P(F < N−1(y)

)= y.

F Note that as F decreases, the default rate increases. There istherefore a probability of y that the default rate will be greater than

N

(Ui −

√ρN−1(y)

√1− ρ

).



Proof of Vasicek’s One-Factor Model (1987) (cont’d)

F The default rate that we are X% certain will not be exceeded isobtained by substituting y = 1−X into the preceding expression.

F Based on the left-right symmetry of normal distribution’s tails,

N−1(X) = −N−1(1−X),

we then obtain the default rate conditional on factor F .

P (T i < T |F ) = N

(Ui +

√ρN−1(X)√1− ρ

).



Estimating PD and ρ

F The maximum likelihood methods can be used to estimate PD andρ from historical data on default rates.

F If DR is the default rate and G(DR) is the cumulative probabilitydistribution function for DR, the Vasicek (1987) model shows that

DR = N

(N−1

(PD)+√ρN−1

(G(DR)

)√1− ρ

).

F Rearranging this equation,

G(DR) = N

(√1− ρN−1

(DR)−N−1

(PD)

√ρ

).

F Differentiating, we obtain the probability density function for thedefault rate:

g(DR) =

√1− ρρ

e

12

((N−1(DR)

)2−(√

1−ρN−1(DR)−N−1(PD)√ρ

)2).



Probability Density Function of Default Rate

0

10

20

30

40

50

60

70

0.00 0.01 0.02 0.03 0.04 0.05 0.06

Default Rate Christopher Ting QF 603 November 21, 2017 45/47


Maximum Likelihood Estimates

F Procedures1 Choose trial values for PD and ρ, e.g., 0.01 and 0.02.2 Calculate the logarithm of the probability density for each of the

observations on DR.3 Use Solver to search for the values of PD and ρ that maximize the

sum of the values in 2.

F The estimates for ρ and PD from the data are 0.108 and 1.41%,respectively.

F Worst case default rate is

WCDR(1, 0.999) = N

(N−1

(0.0141

)+√0.108N−1

(0.999

)√1− 0.108

)= 0.106.



Takeawayskkk Dependence vs. correlation

kkk Correlations are notoriously unstable in financial time series!When the financial crisis occurs, correlations become very closeto either 1, -1 or 0.

kkk Risk managers need to keep track of a variance-covariance matrixfor all the variables to which the portfolio is exposed.

kkk Analysts often assume that a one-factor copula model relates theprobability distributions of the times to default for different loans.The percentiles of the distribution of the number of defaults on alarge portfolio can then be calculated from the percentiles of theprobability distribution of the factor.

kkk This approach is used in determining credit risk capitalrequirements for banks under Basel II.


Week 7 Quantitative Analysis of Financial Markets ...€¦ · C V 1;V 2 = E V 1V 2 E V 1 E V 2: R...

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Transcript of Week 7 Quantitative Analysis of Financial Markets ...€¦ · C V 1;V 2 = E V 1V 2 E V 1 E V 2: R...