extreme risk analysis

7/30/2019 Extreme Risk Analysis

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Extreme Risk AnalysisMSCI Barra Research

Lisa R. GoldbergMichael Y. HayesJose MencheroIndrajit Mitra


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Summary

Timely: draft version published in February 2009, Lehman co

on Sep 15, 2008. Volatility is not an ideal risk measure, VaR has shortcomings

for the usage of expected shortfall (which they call shortfal

Shows how to use some intuitive volatility decompositions tand manage risk

Example-based, math in the Appendix


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Quantitative Risk Management

Risk measurement: quantifying the overall risk of a portfolio

Risk analysis: gaining insight into the sources of risk

Crisis in 2008-2009 provides motivation to think differently aband shows the importance of considering risk measures that well in all types of climates.


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Risk Measurement


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Risk measures - Volatility

Volatility is the classic risk measure, since Markowitz (1952)several positive features: Favors diversification over concentration

Possible analytic solutions

Easy to measure and possible to forecast

Can be traded in the open market (VIX) and indirectly by using de


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Volatility has some major problems

If returns were normal, mean and volatility would fully characterizreturns are non-normal, and in particular, have fatter tails. Lehman Brothers, Long-Term Capital Holdings, the not-so-long-term firm in

Scholes and Merton worked, Black Monday

Indifferent between loss and gain

Risk measures - Volatility


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Risk MeasuresJP Morgans VaR

As we are familiar from class, one possible definition of VaR

Where L is the loss distribution and q is the quantile function is one minus the probability of loss (authors call it confidence

Advantages: Accounts only for downside

Included in Basel II

Disadvantages: May favor concentration

Lower-bound estimate of the loss: VaR is not a measure ofextrem

VaR (L)q


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Example 1: VaR concentration

Imagine a bank has $1M to invest.

In their investment set, they have two bonds, get 10% intere99.3% of the time, lose everything 0.7% of the time.

If a financial institution holds one of the bonds, VaR 99% is z

If they put half of their money in each bond, their VaR 99% less than $500k.

Therefore, a bank with these investment opportunities mighdiversify when held to VaR targets.

Jn Danelson would probably rightfully argue that this examhand-crafted.


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Shortfall: a true measure of extreme

Expected loss given that the VaR has been breached

Always encourages diversification

[ | VaR]ES E L L

E l 2 Sh t P iti i C ll


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Example 2: Short Position in CallOptionsHow to print money:

Imagine a portfolio like one (N) shares of an ETF and, forconcreteness, imagine its for the MSCI US Broad Market, anspot price is $60. Average daily return is close to zero, and itunlikely to be something like 6.6%. In order to enhance retusell a out-of-the-money call at $64 with one day to expiratio

Maybe sell several options! Free money! Not so fast: theres unlimiteddownside if the ETFs price goe

$64 and if you sell more than one option.

Note that you can also lose money if the ETF price falls.


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Example 2: Shortfall vs. VaR

Example 2 illustranother problemVaR: its a lower the probability osomething badhappening.

Expected shortfahave captured thproblem.


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Risk Analysis


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Marginal Contribution to Risk

RC MCR m m mx

Any risk measure that is homogeneous of degree 1 (scale invariant) ca

decomposed using Eulers theorem:

p

p m

m m

xx

We can define risk contribution as:

risk exp re: osumx

: (marginal contributionMCR to risk)p

m

mx


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X-Sigma-Rho Risk Attribution (I)

Lets start by thinking about volatility. From the class slides:


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X-Sigma-Rho Risk Attribution (II)

p pp m p m p m

m m m

x x x MCx x

,p m m m px

The stand-alone volatility and the correlation of the asset with the por

contribute to the portfolios risk. Thinking in this way may help manag

portfolio risk.


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X-Sigma-Rho Risk Attribution (II)

p pp m p m p m

m m m

x x x MCx x

,p m m m px

The stand-alone volatility and the correlation of the asset with the por

contribute to the portfolios risk. Thinking in this way may help manag

portfolio risk.

Generalizing X Sigma Rho Risk


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Generalizing X-Sigma-Rho RiskAttribution

p m m

m

x MCR

MCRm m m

p m m m

m

x

Generalizing X Sigma Rho Risk


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Generalizing X-Sigma-Rho RiskAttribution

1MCR

m m

m

ES ES

ES

[ | VaR]

| VaR][

mm

E

P P

S P

E

E L L

LL

[ | VaR]| VaR[ ]

m P

P P

E L LE L L

1

if L is symme

m

ES


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The marginal contribution to risk canlot!


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Example 3: Portfolio Insurance

ETF on the MSCIBroad Market, spotprice $60

Put option to insureagainst large losses,strike $50

Volatility gives a verydifferent picture ofrisk than ES


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Example 3: Portfolio Insurance

On bad portfolio days, theoption has negative loss


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Example 4: Coincidental Losses

Two identical portfolios joined by a copula.

In the first case, the portfolios are joined by a Gaussian copula. In the second case, the portfolios are joined by a t-copula

The point is that in the second case, were modeling a higher probability

coincidental losses, and the x-sigma-rho decomposition is going to show


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extreme risk analysis

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