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Chapter 13

Market Risk VaR: Model-Building Approach

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Market Risk VaR: Model Building Approach

13:1 The Basic Methodology

13:2 Generalization

13:3 Correlation and Covariance Matrices

13:4 Handling the Interest Rate13: !pplications o" the #inear Model

13:$ #inear Model and %ptions

13:& '(adratic Model

13:) Monte Carlo *i+(lation13:, -on.nor+al /istri0(tion

13:1 Model B(ilding vs Historical *i+(lation

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Risk Management and Financial Institutions 3e, Chapter 15, Copyright John C. ull !"1!

The Model-Building Approach

The +ain alternative to historical si+(lation is to+ae ass(+ptions a0o(t the pro0a0ilitydistri0(tions o" the ret(rns on the +aret

varia0les This is non as the +odel 0(ilding approach 5or

so+eti+es the variance.covariance approach6

!ss(+e a 7oint distri0(tion o" changes in +aretvaria0les and (sing historical data to esti+ate+odel para+eters

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Risk Management and Financial Institutions 3e, Chapter 15, Copyright John C. ull !"1!

The Methodology

The port"olio consists o" one sec(rity:5Microso"t stoc6

8e have a position orth 91 +illion inMicroso"t shares

The volatility o" Microso"t is 2 per day

5a0o(t 32 per year68e (seN;1 andX;,,

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Risk Management and Financial Institutions 3e, Chapter 15, Copyright John C. ull !"1!

Microsoft Example continued

The standard deviation o" the change inthe port"olio in 1 day is 92

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Risk Management and Financial Institutions 3e, Chapter 15, Copyright John C. ull !"1!

Microsoft Example continued

8e ass(+e that the e=pected change inthe val(e o" the port"olio is zero

8e ass(+e that the change in the val(eo" the port"olio is nor+ally distri0(ted

*inceN5>2336;1< the ?aR is

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Risk Management and Financial Institutions 3e, Chapter 15, Copyright John C. ull !"1!

AT&T Example

Consider a position o" 9 +illion in !T@T

The daily volatility o" !T@T is 1 5appro=

1$ per year6The */ per 1 days is

The ?aR is

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Risk Management and Financial Institutions 3e, Chapter 15, Copyright John C. ull !"1!

ortfolio

-o consider a port"olio consisting o" 0othMicroso"t and !T@T

*(ppose that the correlation 0eteen theret(rns is 3

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Risk Management and Financial Institutions 3e, Chapter 15, Copyright John C. ull !"1!

!"#" of ortfolio

! standard res(lt in statistics states that

In this case X; 2

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Risk Management and Financial Institutions 3e, Chapter 15, Copyright John C. ull !"1!

VaR for ortfolio

The 1.day ,, ?aR "or the port"olio is

The 0ene"its o" diversi"ication are

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/aily changes in the val(e o" a port"olio e(al the total daily

changes in the val(es o" individ(al assets:

I" daily changes o" the val(es o" individ(al assets are nor+allydistri0(ted then daily changes in the val(e o" the port"olio are

nor+ally distri0(ted The variance o" the daily changes o" port"olio val(e is given 0y:

11

$%: 'enerali(ation:

=

=n

i

ii xP1

=