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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



Analytic tools:

MVaR, IVaR, CVaR



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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



Analytic tools:

MVaR, IVaR, CVaR



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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



Analytic tools: MVaR, IVaR, CVaR


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



Analytic tools:

MVaR, IVaR, CVaR


Options, partial hedging and nonnormality

Documents

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