Covariance · 2020. 2. 7. · Covariance and correlation If X;Y 2RN is a random variables with means X; Y, we know that: Cov(X;Y) = E((X X)(Y Y)0) Notation: Cov(X;Y) = ˙ XY = ˙

Covariance

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Remember this chart:

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What are the ellipses drawn there?

Contour lines:

Ic = {x : (x − µ)′Σ−1(x − µ) = c}

From the definition of the normal density it is clear thatφµ,Σ(x) is constant on the set Ic .

Therefore the contours of normal densities are ellipsoids.

The same is true for Student-t densities.

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A random variable, which has a density, is called elliptical if itsdensity is constant on ellipsoids.

Elliptical distributions like the normal and Student t distributionplay a pivotal role in modeling financial data and at their heart isthe covariance matrix Σ.

We will now discuss some key properties, empirical observations,and current research work.

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Covariance and correlation

If X ,Y ∈ RN is a random variables with means µX , µY , we knowthat:

Cov(X ,Y ) = E ((X − µX )(Y − µY )′)

Notation:Cov(X ,Y ) = σXY = σXσY ρXY

The correlation ρXY is a number between −1 and 1 (we will provethis).

It is not observable in the market. Just with the variance of thereturns we can estimate it.

Also, like with the variance, the market prices the so-called impliedcorrelation.

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The covariance defines an inner product on RN :

(·, ·) : RN × RN → R

Inner products satisfy:

Symmetry

Bilinearity

Non-negativity

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Example of inner product:

Consider A ∈ RN×N symmetric, positive definite.

Define (x , y)A = x ′Ay .

Proof:

Symmetry:

(y , x)A = y ′Ax

= y ′A′x , by symmetry of A

= (y ′A′x)′

= x ′Ay

= (x , y)A

The other two properties follow similarly.

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If we consider the space of random variables on R.

Cov(X ,Y ) satisfies almost all three conditions.

Cov(X ,X ) = 0 in the case X is a constane random variable (andnot necessarily as the definition of inner product requires).

One thing one can do is to identify al the constants, computingthe quotient space.

For example, in the quotient space two random variables that differby a constant are identified.

You might have seen quotient spaces in arithmetic: given a naturalnumber n all the remainders of dividing by n (0, 1, . . . , n − 1) forma quotient space.

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For a given inner product the Cauchy- Schwarz inequality statesthat:

|(u, u)| ≤ ‖u‖2‖v‖2

where the norm ‖ · ‖ is defined as ‖ · ‖ =√

(·, ·) (the norm inducedby the inner product).

But...what is a norm anyway...?

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A norm on a space V is a function ‖ · ‖ : V → R+ sothat forX ,Y ∈ V and a a scalar:

Absolute scalability

‖aX‖ = |a| · ‖X‖

Triangle inequality

‖X + Y ‖ ≤ ‖X‖+ ‖Y ‖

Non-negativity

‖X‖ ≥ 0

with equality only in the case X ≡ 0.

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Now,

|Cov(X ,Y )| ≤√Cov(X ,X )

√Cov(Y ,Y

=√Var(X )

√Var(Y )

Then

|Cov(X ,Y )|√Var(X )

√Var(Y )

≤ 1

|σX ,Y |σXσY

≤ 1

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Then by defining

ρX ,Y :=σX ,YσXσY

we see that it is a number between −1 and 1.

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Example: r = βm + ε, where m, ε are rv’s s.t.:

Cov(m, ε) = 0,Var(m) = σ2m,Var(ε) = σ2

ε

Then Cov(r ,m) = βCov(m,m) =⇒ β = σεσmρ

Interpretation: This is CAPM.

r represents a particular stock’s returns, m the market’s (whateverthat may be) returns, and ε the so-called idiosyncratic componentof that stock.

The model says that stock returns have a linear relationship withthe market’s returns, with some variation defined by ε.

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CAPM

The parameters need to be estimated.

For example: ρ =σX ,YσX σY

.

What kind of estimator is this?

Method of moments.

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Method of moments

Example:The method of moments estimator of the variance is:

1

N

N∑i=1

(xi − x)2

We know this estimator is biased.

Is it good in any way?

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Consistency, Unbiasedness

Statisticians use different ways to measure whether or not a certainestimator is good or not.

One of them which we have mentioned before is the concept of”biasedness” (E(θ) = θ).

Another measure is consistency.

Roughly speaking an estimator is consistent if it converges inprobability to the true value.

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Biasedness 6=⇒ Consistency

Consistency 6=⇒ Biasedness

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Result relating biasedness and consistency:

If an estimator θn of θ is unbiased and it converges to a value,then it is consistent.

Actually, more generally it is consistent if it is unbiased and itsvariance goes to 0.

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Proof:We want to prove that it is consistent, so we need to prove that

P(|θn − θ| > ε)n→∞−−−→ 0

But, Markov says that:

P(|θn − θ| > ε) ≤ E(|θn − θ|2)

ε2

=E(|θn − E(θn) + E(θn)− θ|2)

ε2

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Now,

E(|θn − E(θn) + E(θn)− θ|2) ≤ E(|θn − E(θn)|2)

+ 2E(|θn − E(θn)|)E(|E(θn)− θ|)+ E(|E(θn)− θ|2)

Since θn converges its variance goes to 0 so the first term goes to0.

The second and third terms are zero due to unbiasedness.

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CAPM

We may now calculate β and ρ using the whole sample. Weupdate the original figure in Figure 3.1 to show the regression linegiven by the estimated values below:

β = 1.03

σIBM = 0.0787

σSPX = 0.0443

ρ = .59

Notice that the slope of the regression line is not parallel to themajor axis of the ellipses defined by the covariance matrix.While this is visually jarring, it is by design as we shall see that β isthe result of minimizing squared errors in the y-axis dimension.

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CAPM

Other questions that arise:

If we look at rolling time windows, will the results besubstantially different?

Does our estimate of β change through time?

Is it correlation or the ratio of vols that drives the variation inβ?

We see in Figure 3.1 what a rolling window using 252 trading days(approximately one year) of data yields for our estimate of β.

We clearly identify that our estimate varies significantly throughtime.

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CAPM

Interestingly, the bursting of the tech bubble in early 2000 is visiblein the dramatic uptick of β. This is less pronounced, however, inthe financial crisis in 2008. A reasonable observation is that theremay be sector-specific exposures that impact a company inaddition to the market as such.

Focusing on these time periods a bit more, we have yet todistinguish whether the driver of β dynamics is correlation orvolatility.

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CAPM

From the chart showing both the ratio of estimated volatilities ofIBM and the S&P and the estimated correlation between the twowe see that the two crises exhibit different behavior.

The tech bubble shows an increase in the volatility ratio as well asan increase in correlation, while the financial crisis shows adecrease in the former and increase in the latter.

In both cases, we see that in the event of a crisis, there is someevidence to expect that correlations increase between securities.This is yet another stylized feature of equity returns.

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Correlation and independence

Correlation and independence are related but distinct propertiesbetween random variables.

It is easy to show that if two univariate random variables, X and Y, are independent, then they are uncorrelated

Cov(X ,Y ) = E((X − µX )(Y − µY ))

= E(X − µX )E(Y − µY )

= 0

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The converse is not always true. Examples:

X ∼ N(0, 1),Y = X 2.Clearly, they are not independent and:

Cov(X ,Y ) = E(X (Y − 1))

= E(XY )− E(X )

= 0

X ,Y discrete with:

P(X = −1) = P(X = 0) = P(X = 1) = 13 .

Y = 1 if X = 0 and Y = 0 otherwise.

Again, clearly not independent and:

Cov(X ,Y ) = E((X − µX )(Y − µY ))

= E((X )(Y − 1

3))

= E(XY )

= 0

Take U to be uniformly distributed on [0, 2π] andX = cos(U),Y = sin(U).

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One more example

Take U to be uniformly distributed on [0, 2π] andX = cos(U),Y = sin(U).

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Correlation and independence.

One exception is in the case of jointly normal random variables.

If X ∼ N(µX ,ΣX ) and Y ∼ N(µY ,ΣY ) are each multivariatenormal random variables which are jointly normal and uncorrelated,then X and Y are independent.Proof:Let us define:

Z =

(XY

)Then Z ∼ N(µ,Σ) with:

µ =

(µXµY

)Σ =

(ΣX 00 ΣY

)

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The idea of the proof is to write the density of Z and to noticethat it factorizes as the densities of X and Y .

As we know:

φµ,Σ(z) =1

(2π)N2

1

det(Σ)12

e−12

(z−µ)′Σ−1(z−µ)

By the structure of Σ we have that det(Σ) = det(ΣX )det(ΣY )and:

Σ−1 =

(Σ−1X 0

0 Σ−1Y

)

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Now, the exponent can be decomposed:

(z − µ)′Σ−1(z − µ) =

(x − µXy − µY

)′(Σ−1X 0

0 Σ−1Y

)(x − µXy − µY

)= (x − µX )′Σ−1

X (x − µX )′

+ (y − µY )′Σ−1Y (y − µY )′

And this proves that the density factorizes (which is the definitionof independence).

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In the above example we have to explicitly say ”jointly normalrandom variables”.

We could easily have two random variables X1 and X2 which areeach normally distributed but whose joint distribution is notnormal.

In fact, due to a result by Sklar, we know that for a set ofunivariate random variables, {Xi}Ni=1, with marginal distributionfunctions, Fi (·), any joint distribution may be constructed whichrespects the marginal distributions prescribed.

So, for instance, it is possible to have normal marginals with aStudent t joint distribution. Further, this is a constructiveprocedure which we establish in the next section on copulas.

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Copulas

A copula is the joint distribution of random variables, {Ui}Ni=1,each of which is uniformly distributed on [0, 1].

In the previous slide we mentioned Sklar’s Theorem. It states thatfor any random variables, {Xi}Ni=1 with marginals Fi (·) and jointdistribution F (·), there exists a copula, C , such thatF (x1, . . . , xN) = C (F1(x1), . . . ,FN(xN)) and that if the Fi ’s areunique, then so is C .

This is a powerful result. In the continuous case it can be provedfairly easily.

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F (x1, . . . , xN) = P(X1 ≤ x1, . . . ,XN ≤ xN)

= P(F1(X1) ≤ F1(x1), . . . ,FN(XN) ≤ FN(xN))

= P(U1 ≤ F1(x1), . . . ,UN ≤ FN(xN))

= C (F1(x1), . . . ,FN(xN))

This says that, for a specified F (·) and a continuous set of Fi (·)’swe can define a copula by:

C (u1, . . . , uN) = F (F−1i (u1), . . . ,F−1

N (uN))

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So, we can divorce the joint and marginal densities because thelinking may be done entirely through some copula.

Or, as so happens in practice, we may specify marginaldistributions and a joint distribution separately.

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Example: Normal marginals, joint Student t.

Let X1 and X2 be distributed as standard normal random variables,and let Stµ,Σ;ν(·) be the joint distribution of a two dimensionalStudent t distribution with ν degrees of freedom. Then

C (u1, u2) = Stµ,Σ;ν(Φ−1(u1),Φ−1(u2)),

exhibits a copula jointly Student t pair of random variables withmarginals that are standard normal.

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Rank-invariant property

For elliptical distributions, we may focus solely on correlation.

This is due to the fact that copulas have a so-called rank-invariantproperty:

If gi (·) are each strictly increasing functions gi : R→ R, fori = 1, ...,N, and C is the copula of {Xi} then C is also the copulaof {gi (Xi )}.

Proof:

Let F (·) be the joint distribution function of X , a multivariaterandom variable and gi (·) strictly increasing functions from RtoR.We know by the change of variable theorem that the CDF ofgi (Xi ) is

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Rank-invariant property

Fi (·) = Fi ◦ g−1(·) = Fi (g−1)(·)

whose inverse is:

F−1i (·) = gi ◦ F−1

i (·) = gi (F−1i )(·)

Now:

C (u1, . . . , uN) = F (F−11 (u1), . . . ,F−1

N (uN)

= P(X1 ≤ F−11 (u1), . . . ,XN)

= P(g1(X1) ≤ g1(F−11 )(u1),

. . . , gN(XN) ≤ gN(F−1N )(uN))

= Fg (g1(F−11 )(u1), . . . , gN(F−1

N )(uN))

where Fg is the CDF corresponding to (g1(X1), . . . , gN(XN)).

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Example: Student t marginals, joint normal.

In practice, F (·) and marginals, Fi (·) are determined a priori.

Oftentimes, copulas are used to simulate data with theseprescribed distributions.

Here, we look at a simple case of simulating jointly normal datawith Student t marginals.

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Specifically, let

X1 ∼ St(µ1, σ21; 5),X2 ∼ St(µ2, σ

22; 5),

and let F (·) = Φµ,Σ.

By the rank-invariant property we can choose µ = 0 and Σ acorrelation matrix.

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Remember: how do we create a normals with correlation Σ givennormals with correlation I?

Cholesky factorization: if Σ is positive definite there is a uniquelower triangular Λ so that ΛΛ′ = Σ.

Now, if X ∼ N(0, I ) then define Y = ΛX .

Cov(Y ,Y ) = Cov(SX ,SX )

= E((SX )(SX )′)

= E(SXX ′S ′)

= SE(XX ′)S ′

= SIS ′

= Σ

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Then, starting with independent Z 01 ,Z

02 we obtain Z1,Z2 with the

right covariance.

Now, using the standard normal cdf we obtainU1 = Φ(Z1),U2 = Φ(Z2),

and X1 = St(µ1, σ21; 5)−1(U1),X2 = St(µ2, σ

22; 5)−1(U2).

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The copula approach found wide appeal in credit derivativesmarkets due to a paper published by David Li in the Journal ofFixed Income.

On Default Correlation: A Copula Function Approach modeleddefault correlation in a novel way, linking marginal default risksobtained from credit default swap (CDS) pricing through a copulawith a very simple structure to imply a joint distribution of creditdefaults.

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The copula that became widely used and whose parametereventually became a quoted market price was a multivariatenormal copula with a covariance (correlation) matrix given by:

1 ρ . . . ρρ 1 . . . ρ...

. . . ρρ ρ . . . 1

Much like the versions of the Capital Asset Pricing Model we haveseen already, the above model does two things: it provides asimplification of market relationships via market pricing andnormative relationships, and produces an interpretable parameter.

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The model above (here told in generalities, but a rigoroustreatment is not too much more involved) was used to estimateprobabilities of joint defaults within pools of hundreds or eventhousands of loans.

The constant pairwise correlation is concerning, but the use of thenormal distribution is even more so.

Our previous analysis of the inability of the normal distribution tocapture market extremes applies here as well.

And yet, the copula-based model here was used to mint hugenumbers of triple-A rated bonds (made up from tiered levels ofpools of bonds). The pooled bonds were known as collateralizeddebt obligations, or CDOs.

Concurrent with the acceptance of the modeling above, the CDOmarket grew from $275 billion in 2000 to $4.7 trillion in 2006.

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Not only are correlations unstable (as we have already seen) andextreme events terribly likely, the CDO market (because of suchgreat ratings by the ratings agencies) saw massive leverage.

This was a recipe for disaster and culminated in the financial crisisof 2008.

There were many people who could see this trainwreck coming farbefore it occurred, but in large part, the market did not.

In effect, the market wasn’t efficient at pricing pairwisecorrelations; or, even worse, systemic crashes.

Even with the above stain against it, we maintain that thepowerful capability of modeling joint and marginal distributionsseparately is incomparable.

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Eigenvalues and eigenvectors

We may further analyze the structure of the covariance matrix bystudying its eigenvalues and eigenvectors.

Recall that for a square matrix, A ∈ RN×N , the scalar λ is aneigenvalue if

Av = λv

We say that the nonzero vector v is the eigenvector associatedwith λ.

Notice that if v is an eigenvector, then a scalar multiple, cv ,satisfies A(cv) = cAv = cv = (cv), and hence we may assumewithout loss of generality that ‖v‖ = 1.

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Eigenvalues may be determined by considering that if then

Av = λv ⇒ (A− λI )v = 0

So, v is in the ker(A− λI )) which means, in particular, that(A− λI ) is singular.

Characteristic equation: det(A− λI ) = 0.

It turns out to be a polynomial of degree N. We know that, in C ithas N roots.

However, we are interested in the case where all of the eigenvaluesare real, and as it happens, positive definiteness (andsemidefiniteness) is a sufficient condition for just such a result.

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We will denote positive definitness as:

A � 0

and positive semidefinitness as:

A � 0

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Theorem: The eigenvalues of a positive semidefinite real matrix,A � 0, are real and nonnegative. If A � 0, then the eigenvalues arestrictly positive.

Idea of the proof:

Av = λv =⇒ 0 ≤ v ′Av = λ‖v‖2 = λ

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In a financial context, we may interpret an eigenvector, ei , as avector of portfolio weights or positions.

Consider,

Var(e ′iX ) = e ′iΣei

= e ′iλiei

= λi (1)

So that ei is exactly the variance of the portfolio with positions ei .

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For a covariance matrix, Σ ∈ RN×N with eigenvalues {λi}Ni=1 andassociated eigenvectors {ei}Ni=1 with 0 ≤ λ1 ≤ · · · ≤ λN we havethat the eigenvectors of distinct eigenvalues are orthogonal.Proof:

e ′jΣei = e ′iΣej ( Σ is symmetric)

e ′jλiei = e ′iλjej

λie′jei = λje

′i ej

Since λi 6= λj we necessarily have that e ′jei = e ′i ej = 0

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Fact: Assuming that the eigenvalues of Σ are distinct, we maydecompose the covariance matrix as:

Σ =N∑i=1

λieie′i

Proof:

By orthogonality: e1...eN

e1 . . . eN

= I

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Now, suposse that we have square matrices A and B so that:BA = I . Then: AB = I .Proof:

The fact that BA = I means that B is the right inverse of A (or Aleft inverse of B). In particular both of them are invertible (so,B−1 and A−1 both exist).Now, we would know that M = AB = I if ABC = C for any C .

AB = M

ABC = MC

BABC = BMC

BABC = BMC

BC = BMC

C = MC left multiply by B−1

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Therefore: e1 . . . eN

e1

...eN

= I

By looking at entry-by-entry we see that:

e1e′1 + · · ·+ eNe

′N = I

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Then:

Σ = ΣI

= Σ∑i

eie′i

=∑i

Σeie′i

=∑i

λieie′i

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Using this:

tr(Σ) = tr(∑i

λieie′i )

=∑i

λi tr(eie′i )

=∑i

λi tr(e ′i ei )

=∑i

λi

But, remember that the trace of Σ is the sum of the variances.

This fact tells us that the sum of the variances coincides with thesum of the eigenvalues.

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We call∑

i λi the total variance of Σ. In addition to relating theeigenvalues of Σ to the sum of variances, this gives us a methodfor dimension reduction.

Let X be an N-dimensional random vector representing the returnsof N assets. For a threshold, τ , with, 0 ≤ τ ≤ 1 we may choose Meigenportfolios explaining τ% of the total variance by choosing thesmallest M satisfying: ∑M

i=1 λi∑Ni=1 λi

≥ τ

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An estimated covariance matrix was calculated monthly for across-section of the 50 largest stocks at the time by market cap.The covariance was calculated using 121 trailing weeks of returns.The largest N eigenvalues were chosen with τ = 80%. A smoothedapproximation, looking at the mean Nt for the trailing 18 monthsis shown as well.Throughout, no more than 18 eigenportfolios were needed toexplain more than 80% of the total variance.This is a significant decrease from the original dimension of 50.

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We may also ask how much of the total variance is explained bythe eigenportfolio related to the largest eigenvalue.

The next figure shows the time variation of the explanatory powerof this eigenportfolio.

We see again the significant upswing after the Financial Crisis,achieving levels of market coordination not seen in the precedingtwenty years.

While the adage that in a crisis correlations go to one is evidencedhere.

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Finally, we may look at the distribution of eigenvalues in a mannersimilar to our previous analysis of daily log returns for variousstocks.

The next figure shows the empirical density of the eigenvalues ofthe covariance matrix as before available on 12/31/2007. As withour discussion of the distribution of daily log returns, certainstylized features emerge.

Particularly, even with observations that are linearly independent,we see a peak of near-zero eigenvalues.

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In recent years, the tools of Random Matrix Theory (RMT) havebeen implemented in math finance to study this phenomenon.Authors like Bouchaud and Potters present a methodology basedon RMT to identify random, and hence noisy, eigenportfolios.

Doing so seeks to modify the covariance (correlation) matrix toeliminate eigenportfolios with erroneously low contributions to risk.This effect is particularly important when consideringmean-variance optimization.In addition to a large bulk of eigenvalues clustering around zero,we also note one very large eigenvalue, in this case, 2, 745 timeslarger than the smallest eigenvalue, and 3.80 times larger than thesecond largest eigenvalue.

The eigenportfolio for this eigenvalue very often has all positiveentries.

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Litterman-Scheinkman (1991)

Looked at the treasury yield curve.

Found that just a few eigenvectors are the important ones.

Three of them explain most of the moves.

Level-Slope-Curvature

Very Intuitive.Curve trades.

Cortazar-Schwartz (2004) found the same in copper

Loads (or lots?) of other people report the same kind ofresults in many other markets.

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Very Intuitive.

Curve trades.



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Table: Correlation Matrix for Changes of the First 12 Crude Oil Futures Prices

1.000 0.992 0.980 0.966 0.951 0.936 0.922 0.08 0.892 0.877 0.860 0.8480.992 1.000 0.996 0.988 0.978 0.966 0.954 0.941 0.927 0.913 0.898 0.8860.980 0.996 1.000 0.997 0.991 0.982 0.973 0.963 0.951 0.939 0.925 0.9140.966 0.988 0.997 1.000 0.998 0.993 0.986 0.978 0.968 0.958 0.946 0.9360.951 0.978 0.991 0.998 1.000 0.998 0.994 0.989 0.981 0.972 0.963 0.9540.936 0.966 0.982 0.993 0.998 1.000 0.999 0.995 0.90 0.983 0.975 0.9670.922 0.954 0.973 0.986 0.994 0.999 1.000 0.999 0.996 0.991 0.984 0.9780.08 0.941 0.963 0.978 0.989 0.995 0.999 1.000 0.999 0.996 0.991 0.985

0.892 0.927 0.951 0.968 0.981 0.90 0.996 0.999 1.000 0.999 0.995 0.9910.877 0.913 0.939 0.958 0.972 0.983 0.991 0.996 0.999 1.000 0.998 0.9960.860 0.898 0.925 0.946 0.963 0.975 0.984 0.991 0.995 0.998 1.000 0.9980.848 0.886 0.914 0.936 0.954 0.967 0.978 0.985 0.991 0.996 0.998 1.000

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First four eigenvectors for oil

2 4 6 8 10 12

−0.

6−

0.4

−0.

20.

00.

20.

40.

6

Contract

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2 4 6 8 10 12

−0.

6−

0.4

−0.

20.

00.

20.

40.

6

Contract

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2 4 6 8 10 12

−0.

6−

0.4

−0.

20.

00.

20.

40.

6

Contract

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2 4 6 8 10 12

−0.

6−

0.4

−0.

20.

00.

20.

40.

6

Contract

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Random Matrix Theory

Financial data is subject to a substantial amount of noise.

Filtering the true signal from the noise becomes of paramountimportance.

We can even doubt that a true, stationary, signal exists.

If we take, say, stocks data and compute covariance matrices wefind a well determined structure.

However, this is also true if we generate random data.

Therefore, we need to be able to distinguish structure from noisefrom structure from signal.

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Consider a matrix A ∈ RN×N formed by elements aij ∼ N(0, σ2).

To make it symmetric let us define A = A+A′√2

Notice that A does not need to be positive definite.

Let us compute the eigenvalues of A.

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Semicircle Law:

As N →∞ the density of the eigenvalues of A stisfies:

ρN(λ) =

{1

2πσ2

√4σ2 − λ2 if |λ| ≤ 2σ

0 otherwise

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Gatheral

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The matrix A can’t be a covariance matrix.

To generate a (random) covariance matrix suppose that we haveM stock return series with T elements each.

Let us assume that the means of the series are zero and thevariances have been normalized to 1.

An estimate of the covariance between series i and series j is thengiven by the matrix E :

Eij =1

T

T∑t=1

xitxjt

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Suppose now that, instead of getting stocks data, we get iid,N(0, σ2) data.

In the previous case we got a clear structure for the density of theeigenvalues.

Is that true now? If so, what is that structure?

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Marcenko-Pastur:

Then, in the limit T ,M →∞ keeping the ratio Q := TM ≥ 1

constant, the density of eigenvalues of E is given by:

ρ(λ) =Q

2πσ2

√(λ+ − λ)(λ− − λ)

λ

where the maximum and minimum eigenvalues are given by:

λ± = σ2

(√1± 1

Q

)2

ρ(λ) is known as the Marcenko-Pastur density.

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Laloux, Cizeau, Potters, Bouchaud

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High correlations in crisis.

A well acknowledged phenomenon.

Similar to ”leverage effect” in volatility.

It makes intuitive sense.

There are many references in academic papers describing,explaining, modeling it.

I have just picked one example:

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Application of copulas:

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A critique of assuming joint distributions to be elliptical:

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Documents

Covariance · 2020. 2. 7. · Covariance and correlation If X;Y 2RN is a random variables with means X; Y, we know that: Cov(X;Y) = E((X X)(Y Y)0) Notation: Cov(X;Y) = ˙ XY = ˙