Leture 6: Multivariate Volatility Modelling – Part
III – PCA and Orthogonal GARCH
FM321: Risk Management and Zhu
1 November 2022
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LSE Finance
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Previously, we wrote down several models for Σt EWMA: Σt = (1 − λ)rt−1rt′−1 + λΣt−1
BEKK: Σt = Ω′Ω + Art−1rt′−1A′ + BΣt−1B′
Correlation models: Σt = DtCtDt, where Dt = diag{σ1,t,…,σN,t}
and Ct correlation.
Constant conditional correlation models (CCC): Ct = C Dynamic conditional correlation models (DCC)
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What benefit does CCC or DCC give us relative to BEKK?
Why do we use qˆt instead of rt to estimate Ct ?
What is a common issue with the models above?
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The curse of dimensionality
Managers deal with large universes.
MSCI World All-Country Index has over 2400 stocks.
Around 2,800 stocks trade in the NYSE.
Financial institutions can have tens of thousands of assets, if not more.
None of the models we have discussed so far can be scaled to that number of securities.
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The idea of factor models
Correlations among asset returns arise from a small number of common sources of risk (called risk factors).
rt =βft +εt
r is an N × 1 vector of (excess) returns.
β is an N × K matrix of factor exposures (containing the exposure of
each security to each factor).
f is an K × 1 vector of factor returns.
ε is an N × 1 vector of residual returns for the securities.
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The idea of factor models
If we assume cov(ft,εt) = 0, we have
Σt = Vart−1(rt) = βΣf ,tβ′ + Σε,t
Then we can just model Σf ,t and Σε,t . The parameters to estimate are:
The N × K factor exposures in β (can be constant or time-varying); The 12K(K +1) elements of Σf,t; and
the N non-zero elements of Σε,t.
The total number of parameters is N(K +1)+ 12K(K +1) , which is linear in N.
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How to find the risk factors ft?
Examples motivated by economic or finance theory:
Market risk
Country risk
Sector risk
Macroeconomic risk factors
Risk factors derived from anomalies (value, size, momentum)
We are going to discuss one statistical way of constructing risk factors, i.e. principal component analysis (PCA)
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Principal component analysis
a method of dimensional reduction
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The idea of PCA: example with two assets
Figure: Scatterplot of daily returns of GE and JPM stocks
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The idea of PCA
By plotting the returns on this coordinate, a natural basis to express the data is a set of two directions given by vectors {(1, 0), (0, 1)}.
r1,t 1 0 r1,t rt=r =01 r
The naive basis reflects the way we gathered the data, but it may fail to uncover the simpler structure that underlie the data.
Is there another basis, which is a linear combination of the naive basis, that best re-expresses the data?
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The idea of PCA
Let’s define a new direction by vector (w1,w2) which yields
r1,t
r ̃=w w t12
as a projection of rt on to that direction.
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The idea of PCA
Let’s define a new direction by vector (w1,w2) which yields
r1,t
r ̃=w w t12
as a projection of rt on to that direction.
What do we mean by “best re-express”?
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The idea of PCA
Let’s define a new direction by vector (w1,w2) which yields
r1,t
r ̃=w w t12
as a projection of rt on to that direction.
What do we mean by “best re-express”?
We assume that the direction with largest variance of data contains the most interesting dynamics.
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The idea of PCA
Figure: Scatterplot of daily returns of GE and JPM stocks
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The idea of PCA
Figure: Scatterplot of daily returns of GE and JPM stocks
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More generally, we have N assets.
r1,1 … r1,T
r=… … …= r1 … rT
rN,1 … rN,T
First look for a direction in the N-dimensional space denoted with an N-by-1 vector p1 so that data has the largest variance in that direction.
Find another direction along which variance is maximized, however, restrict the search to all directions perpendicular to all previous selected directions. Save this vector as pi .
Repeat this procedure until K vectors (K ≤ N) are selected.
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Put all these vectors into a matrix P
p1 p1,1 . . . p1,N P = . . . = . . . . . . . . .
pK pK,1 … pK,N
P is orthonormal because pipj′ = 0 if i ̸= j and pipi′ = 1.
P transforms original return data into
r ̃p …pr
1,t 1,1 1,N 1,t …=… … … …
r ̃ p…p r K,t K,1 K,N N,t
so that r ̃ has a diagonal variance-covariance matrix and its diagonal t
elements decline in value from the top-left to the bottom-right.
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PCA: implementable by eigen decomposition
PCA can be implemented simply by eigen decomposition.
It turns out that to diagonalize Var(r ̃), we can simply set p to be ti
the eigenvector of Var(rt) associated with its i-th largest eigenvalue.
p1,…,pK arecalledtheprincipalcomponents.
The variance associated with pi , or the i-th eigenvalue, quantifies how important the direction pi is for capturing the dynamics of data.
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PCA: example
Consider a portfolio with four stocks: C, AAPL, MSFT, JPM.
What does each column mean?
What does Pct. Total Var. mean?
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Benefits of PCA
From the point of view of risk modeling, the procedure has a few benefits from a computational and conceptual perspective:
As mentioned above, it makes it possible to identify factor structures on the basis of historical data alone, so the informational requirements are low.
PCA achieves dimensional reduction and can greatly simplify many high-dimensional problems.
For example for multivariate volatility modelling, the principal components (factors in the model) are portfolios of the underlying securities, so their variance can be readily computed.
The factors are orthogonal by construction, so their covariance is zero, simplifying some of the estimation of covariance matrix.
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Challenges associated with PCA
One needs to decide how many principal components to include in the factor model, which is usually done on the basis of:
Relevance of the factor (does it capture meaningful amount of common variance across securities?)
Economic interpretation (does it seem to reflect a recognizable reason of an economic or financial nature that would cause comovement across securities?)
While there are statistical techniques to help with this decision, there are no definite rules, and judgement often plays a part in that process.
PCA requires large data, i.e. a large number of securities, relative to the number of factors in order to properly identify the factor structure.
The principal components and the factors can sometimes be hard to interpret.
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Orthogonal GARCH (O-GARCH) = PCA + univariate GARCH
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The Orthogonal GARCH procedure consists of the following steps:
Determine the first K factors from PCA, and compute their historical returns;
Σt =βΣf,tβ′+Σεi,t
Estimate a univariate volatility model for factor variances, which are
diagonal elements of Σf ,t (usually GARCH(1, 1));
Compute the model’s estimate of conditional variance for each of the
factors over time;
For each period t, use the estimated betas, residual variances (time-invariant or time-varying), and estimated factor variances to compute an estimate of the conditional variance matrix Σˆt
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Applying O-GARCH to our example with two stocks, GE and JPM), we obtain the following estimates for GE volatility:
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The following are the estimates for JPM volatility:
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As expected, correlations vary over time in a way that reflects the time-varying volatility of the common factor:
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Implication of factor models for correlations
In the market model,
which implies
Var(ri)=βi2σM2 +σε2 i
Cov(ri,rj) = βiβjσM2 βi βj σM2
Corr(ri,rj)= q(β2σ2 +σ2)(β2σ2 +σ2) iM εi jM εj
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Implication of factor models for correlations
The higher the ratio of idiosyncratic risk to market risk in either asset, the lower the correlation between them.
If we use a time-varying model for market risk (e.g., GARCH), this
will endogenously generate higher correlations between securities
when σ2 rises relatively to the σ2 , which is what typically happens M εi
in times of market crisis.
Therefore, time-varying correlations do not necessarily have to be modeled as a separate phenomenon in the context of factor models, if time-varying volatility is already incorporated in the model.
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