Financial Risk Management

Portfolio Risk, Analytical Methods Following P. Jorion, Value at Risk, McGraw-Hill, Chapter 7

Daniel HERLEMONT

Portfolio of Random Variables N

Y = ∑ wi X i = wT X i =1

N

E (Y ) = µ p = w E ( X ) = w µ X = ∑ wi µ i T

T

N

N

i =1

σ 2 (Y ) = wT Ωw = ∑∑ wiσ ij w j i =1 j =1

VAR p = αW wT Ωw Daniel HERLEMONT

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Diversified VAR

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VAR and correlations

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Exercise

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VAR Tools

Marginal VAR

Incremental VAR

Component VAR

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Marginal VaR

How risk sensitive is my portfolio to increase in size of each position? - calculate VaR for the entire portfolio VaRP=X - increase position A by one unit (say 1% of the portfolio) - calculate VaR of the new portfolio: VaRPa= Y - incremental risk contribution to the portfolio by A: Z = X-Y i.e. Marginal VaR of A is Z = X-Y

Marginal VaR can be Negative; what does this mean...?

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Relation to beta and CAPM

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Incremental VaR

Risk contribution of each position in my portfolio. - calculate VaR for the entire portfolio VaRP= X - remove A from the portfolio - calculate VaR of the portfolio without A: VaRP-A= Y - Risk contribution to the portfolio by A: Z = X-Y i.e. Incremental VaR of A is Z = X-Y Incremental VaR can be Negative; what does this mean...?

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Example

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Component VAR Objective: to get a risk decomposition of the portfolio Taking individual VAR is not useful since it ignore diversification Rather, the component VAR defined as

indicates how the portfolio would change (approximately) if the component is deleted from the portfolio

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VaR decomposition VaR

Incremental VaR Marginal VaR Portfolio VaR Component VaR 100

Position in asset A

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Summary

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Example of VaR decomposition

Currency Position Individual Marginal Component

Contribution

VaR

VaR

VaR

to VaR in %

CAD

$2M

$165,000

0.0528

$105,630

41%

EUR

$1M

$198,000

0.1521

$152,108

59%

Total

$3M

$257,738

100%

Undiversified Diversified

$363K

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Barings Example

Long $7.7B Nikkei futures Short of $16B JGB futures σNK=5.83%, σJGB=1.18%, ρ=11.4% σ P2 = 7.7 2 ⋅ 0.05832 + 16 2 ⋅ 0.01182 + 2 ⋅ 7.7 ⋅16 ⋅ 0.0583 ⋅ 0.114 ⋅ 0.0118

VaR95%=1.65⋅σP = $835M VaR99%=2.33 ⋅σP=$1.18B Actual loss was $1.3B Daniel HERLEMONT

Baring's Risk

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Simplifying the correlation Matrix

With 10 assets, the covariance matrix need to estimate 10*11/2=55 elements. With 100 assets, we need to estimate 5050 elements ...
lead to estimation errors
need for simplification and robustness

 One Factor (Market) Model - Sharpe / CAPM  Multi Factors Model - APT, Ross & Roll, BARRA, ...  Implicit Factors (Principal Component Analysis) Daniel HERLEMONT

PCA on US Bonds

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