pvar
TRANSCRIPT
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Portfolio Problems and
Copulas
FIN285a: Lecture 4.3
Fall 2008
Reading: Jorion, 7, and 8.3
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Outline
Portfolio VaR definitions
Portfolio VaR global equity example
Analytic tools:
MVaR, IVaR, CVaR
Copulas and dependence
Partial hedging and nonnormality
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Software
gport.m
mcgport.m
bgport.m
bsensgport.m
optdist.musoptchoice.m
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Portfolio VaR
VaR on portfolio of assets
Similar to standard VaR with new
complications Covariance
Dependence
Portfolio weights
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Global Portfolio Example
Data
wldeqp.dat, wldeqp.info
Column 1: date (mm/dd/yy)
92-2002
Column 2-6, MSCI equity indices (US $)
World
Japan
US Germany
UK
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Historical VaR
Matlab
gport.m
Notes: Portfolio weights:
Equal weighted over US, Japan, Germany, UK
Compares delta normal with historical
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Monte-Carlo VaR
Matlab
mcgport.m
Critical issue: Variance covariance matrix
See normal.m
Similar patterns to univariate VaR
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Bootstrap VaR
Matlab:
bgport.m
Note: Bootstrap modeling of dependence
(sample)
No need to estimate variances andcovariances
Results are sensitive to doing this
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Correlations and VaR
Simple example
2 Assets
Multivariate normal
Mean returns = 0
Constant variance and correlation
Wealth fraction in each asset = 0.5
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Portfolio Formulasrp w1r1 w2r2 p w11 w22
p w121
2w2
22
22w1w212
w1 w2 0.5,1 2
p (1)
2
VaRp Pt(p CLp ) Pt(p CL(1)
2)
n assets
VaRp
Pt
(p
CL
1
n(n1)
n)
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Component Issues
Sensitivity to portfolio changes
Analytic tools
Bootstrap and monte-carlo methods
Try sweeping through different portfolios
Applications
US to Global change
bsensgport.m
US to Japan change
bsensgport2.m
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Summary
Portfolio choice adds differentdimensions
Covariances Joint bootstrapping
Often critical
May be most important part of modellingrisk factors
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Outline
Portfolio VaR definitions
Portfolio VaR global equity example
Analytic tools:
MVaR, IVaR, CVaR
Copulas and dependence
Partial hedging and nonnormality
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General Approaches
Marginal VaR: MVaR
Incremental VaR: IVaR
Component VaR: CVaR
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MVaR: Marginal VaR(Change in VaR from $1change in investment i.)
VaR
VaR
xi
xi wiW
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IVaR: Incremental VaR
IVaR = VaR(p+a) - VaR(p)
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Analytic Approximations
wi (pi ai ) pi ai
IVaR(a) VaR(p a) VaR(p) VaR
wiwi
i1
n
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Weakness
Local linear approximation
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CVaR: Component VaR(VaR linear homogeneous
function)aVaR(w1,w2 ,K ,wn ) VaR(aw1,aw2 ,K ,awn )
VaR wi
VaR
wii1
n
CVaRi wiVaR
wi
VaR CVaRii1
n
%Contribution CVaR
iVaR
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Component VaR
Compare risk components
Pockets of risk
Break down firm wide risk to components Business units
Drill down capability
Best hedges Most effective position changes to reduce
risk
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Outline
Portfolio VaR definitions
Portfolio VaR global equity example
Analytic tools:
MVaR, IVaR, CVaR
Copulas and dependence
Partial hedging and nonnormality
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Copulas
Generate dependence between x and y
Fix distribution of x alone, and y alone
(Marginal distributions)
Example
Fit returns to student-t
Adjust dependence
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Copulas: How to guideFirst lesson: Generate a
distribution
Let F(x) be a cumulative density
function for some distributionGenerate a uniform random number z =
[0,1]
Generate y = F-1(z)Y is follows F distribution
Matlab: copulaex1.m
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Second LessonPair of Random Variables
Gaussian Copula, Student-t Marginal
Generate (w,u) normal
Get F(w), F(u) (F is normal CDF)
Generate x = G-1(w), y = G-1(u)
G(x), student-t cdf
Matlab: copulaex2.m
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Third Lesson:Gaussian Copula
Empirical Marginal: Use actual dataGenerate (w,u) normal
Get F(w), F(u) (F is normal CDF)
Generate x = G-1
(w), y = G-1
(u) G(x), empirical CDF from data
Like a kind of bootstrap
Matlab: copulaex3.m See code for how to do G(x)
Spearman rank correlation Correlate quantile(w) with quantile(u)
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Copula Advantages
Separates problems
Fitting marginal distribution
Modeling dependence Many copulas for dependence
Key problem
Determining correct model
May beat bootstrap for small samples Generating joint (x,y) extreme observations
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Puzzles In Multivariate
ReturnsSee Jorion 9.3.4
Do correlations change with volatility
and overall risk?
Some think yes.
When volatility is high, correlations are
high? Ability to diversity risk is low.
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Outline
Portfolio VaR definitions
Portfolio VaR global equity example
Analytic tools: MVaR, IVaR, CVaR
Copulas and dependence
Options, partial hedging and nonnormality
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Adding Options to Equity
Portfolios Problem:
50/50 US/UK equity portfolio
Cover the US portion by purchasing a putoption
Do this at the money
20 day (1 month European option)
First, what does the eventual portfoliodistribution look like?
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Part 1
What does an option do to thedistribution?
optdist.m
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Part 2
Evaluating option purchases
usoptchoice.m
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Outline
Portfolio VaR definitions
Portfolio VaR global equity example
Analytic tools:
MVaR, IVaR, CVaR
Copulas and dependence
Options, partial hedging and nonnormality