cours econometrie finance r1 part 1

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Arthur CHARPENTIER - ´ econom´ etrie de la finance (2008/2009) ´ Econom´ etrie de la finance Partie 1 Mesurer les risques Arthur Charpentier http ://perso.univ-rennes1.fr/arthur.charpentier/ Master 1, Universit´ e Rennes 1 1

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Page 1: Cours Econometrie Finance R1 Part 1

Arthur CHARPENTIER - econometrie de la finance (2008/2009)

Econometrie de la finance

Partie 1

Mesurer les risques

Arthur Charpentier

http ://perso.univ-rennes1.fr/arthur.charpentier/

Master 1, Universite Rennes 1

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Page 2: Cours Econometrie Finance R1 Part 1

Arthur CHARPENTIER - econometrie de la finance (2008/2009)

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Page 3: Cours Econometrie Finance R1 Part 1

Arthur CHARPENTIER - econometrie de la finance (2008/2009)

Premier fil rouge du cours : la VaR

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

Premier fil rouge du cours : la VaR

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

Premier fil rouge du cours : la VaR

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

Second fil rouge du cours : RiskMetrics

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

Second fil rouge du cours : RiskMetrics

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

Plan du cours

Introduction generale• Mesurer les risques, une introduction au Risk Manageemnt

Mesurer les “risques” ? Value-at-Risk Contexte et cadre regelementaire, Bale II Un (tout petit) peu d’economie de l’incertain• Modeliser des rendements boursiers Que cherche-t-on a modeliser ? Processus ARCH et GARCH Processus a volatilite stochastique Du rendement d’un titre au rendement d’un portefeuille• Retour a la VaR : les problemes d’estimation Estimation de la Value-at-Risk, un mot de theorie des extremes Estimation de la Value-at-Risk pour des processus GARCH

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

La gestion des risques

A la fin des annees 90, les reglementations prudentielles convergent versl’adoption de la VaR comme mesure de risque de reference.

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

History of risk measures

The evolution of (analytical) Risk Management Tools (from Jorion (2007))1938 bond duration1952 Markowitz mean-variance framework1963 Sharpe’s single beta model1973 Black & Scholes option pricing formula1983 RAROC, Risk Adjusted Return1992 Stress testing1993 Value-at-Risk (VaR)1994 RiskMetrics1997 CreditMetrics1998 integration of credit and market risk1999 coherent risk measures

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

Market risks

Classical models for stock prices,• dynamic models (Bachelier (1900), Black & Scholes (1973)), Brownian

geometricdSt = µStdt︸ ︷︷ ︸

drift

+√V StdWt︸ ︷︷ ︸

random part

,

where (Wt)t≥0 is a standard brownian motion,• more advanced dynamic models (Heston (1993)) have stochastic volatility dSt = µStdt+

√VtdW

St

dVt = κ(θ − Vt)dt+ ξ√VtdW

Vt ,

where (WSt )t≥0 and (WV

t )t≥0 are two standard brownian motions (possiblycorrelated).

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

0.0 0.2 0.4 0.6 0.8 1.0

5010

015

020

0Stock price over 1 year, large volatility

Time

0.0 0.2 0.4 0.6 0.8 1.0

5010

015

020

0

Stock price over 1 year, large volatility

Time

Fig. 1 – Random generation of a stock price, dSt = µStdt+ σStdWt.

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

How to quantify market risks : volatility

All the information about uncertainty is summarized by the volatiliy - orvariance - parameter.

Note that this is one of the reason for the use of the Gaussian distribution, i.e.X ∼ N (µ, σ2) having density

f(x) =1

σ√

2πexp

(−1

2

(x− µσ

)2)

Then µ = E(X) and σ2 = V ar(X).

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

Market risks

• the capital asset pricing model (Markowitz (1970) or the Sharpe index arebased on the mean-variance framework,

0 5 #0 #5

!#.0

!0.5

0.0

0.5

#.0

#.5

%.0

%.5

&ca)t!type

&sp/)ance

0 " #0 #"!#.0

!0."

0.0

0."

#.0

#."

%.0

%."

&ca)t!t+pe

&sp/)ance

Fig. 2 – Capital asset pricing model, the mean-variance framework.

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

A (very) short word on diversification

Naturally, in higher dimension (when dealing with multiple stocks), Gaussianvectors are considered

X =

X1

X2

...

Xd

∼ N

µ1

µ2

...

µd

,

σ2

1 ρ1,2σ1σ2 · · · ρ1,dσ1σd

ρ2,1σ2σ1 σ22 · · · ρ2,dσ2σd

......

...

ρd,1σdσ1 ρd,2σdσ2 · · · σ2d

All the information about marginal risks is in the variances (σ2

i ) while all theinformation on the dependence is in the correlation coefficients (ρi,j).

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

On the Gaussian distribution

The Gaussian distribution is very important for many reasons,• it is a stable distribution, i.e. it appears as a limiting distribution in the central

limit theorem : for i.i.d. Xi’s with finite variance,

√nX − E(X)√

V X

L→ N (0, 1).

• it is an elliptic distribution, i.e. X = µ+AX0 where A′A = Σ, and whereX0 has a spheric distribution, i.e. f(x0) is a function of x′0x0 (spherical levelcurves),

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

−3 −2 −1 0 1 2 3

−3

−2

−1

01

23

Level curves of a spherical distribution

−3 −2 −1 0 1 2 3

−3

−2

−1

01

23

Level curves of a elliptical distribution

Fig. 3 – The Gaussian distribution.

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

On the Gaussian distribution

As a consequence, if X ∼ N (µ,Σ), and if

X =

X1

X2

∼ N µ1

µ2

,

Σ11 Σ12

Σ21 Σ22

• Xi ∼ N (µi,Σi), for all i = 1, · · · , d,• α′X = α1X1 + · · ·+ αdXd ∼ N (α′µ,α′Σα),• X1|X2 = x2 ∼ N (µ1 + Σ12Σ−1

2,2(x2 − µ2),Σ1,1 −Σ12Σ−12,2Σ21)

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

−3−2

−10

12

3

−3

−2

−1

0

1

23

0.00

0.05

0.10

0.15

0.20

Density of the Gaussian distribution

−3 −2 −1 0 1 2 3

−3

−2

−1

01

23

Level curves of a elliptical distribution

Fig. 4 – The Gaussian distribution.

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

Value-at-Risk

The expression is quite recent and its origin is uncertain : in the 80’s, somepapers introduced dollars-at-risk, capital-at-risk, income-at-risk, earning-at-riskand finally value-at-risk

Denomination has been stabilized after the publication of RiskMetrics TechnicalDocument in 1994, by JPMorgan. Note that the work accomplished by JPMorganwas more a pulic relation campaign than an advanced technical study : VaR ismore a practice than a theory.

“VaR summarizes the worst loss ever on a target horizon that will not be exceededwith a given level of confidence”, i.e. formaly it is a quantile of the projecteddistibution of gains and losses over the target horizon

Till Guldimann (1992) created the term value-at-risk while head of globalresearch at JP Morgan in the late 80’s. It appeared in the G30 report (group ofthirty) in July 1993.

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The Basel II accord (2004)

June 2004, the Basel Committee finalized the Basel Accords, based on threepillars• minimum regulatory requierements, i.e. some risk-based capital requirements :

set capital charges against credit risk (internal rating based), market risk(internal model approach) and operational risk. the goal is to keep constantthe level of capital in the global banking syste : 8% of risk weighted assets,

• supervisorv review, i.e. expanded role for bank regulartors, to ensure thatbanks operate above the minimum regulatory capital ratios, that banks haveappropriate processes for assessing their risks, and appropriate correctiveactions

• market discipline, i.e. set of disclosure recommendations, encouraging topublish informations about exposures, risk profiles, capital cushion...

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

The Basel II accord (2004)

From the first pillar, there should be a credit risk charge (CRC), a market riskcharge (MRC) and an operationnal risk charge (ORC), and the bank’s totalcapital must exceed the total-risk charge (TRC)

Capital > TRC = CRC + MRC + ORC.

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

Why using VaR as a risk measure ?

Markowitz (1952) claimed that standard deviation should be an intuitive andappropriate risk measure (leading to the mean-variance trade-off).

The same year, Roy (1952) claimed that “the optimal bundle of assets(investment) for investors who employ the safety first principle is the portfoliothat minimizes the probability of disaster”.

Roy A. D. (1952), Safety first and the holding of assets, Econometrica, 20,431-449.

Markowitz H. M. (1952), Portfolio selection, Journal of Finance, 7, 77-91.

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Edgeworth, et l’hypothese normale

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Edgeworth, et l’hypothese normale

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L’hypothese normale, depassee ?

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Markowitz et l’approche moyenne-variance

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

Markowitz et l’approche moyenne-variance

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Roy et la notion de safety first

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Roy et la notion de safety first

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Introduction the risk measures, and risk perception

“S. Clam [...] once said : I define a coward as someone who will not bet whenyou offer him two-to-one odds and let him choose his side ”.

“With the centuries old St. Petersburg paradox in my mind, I pedanticallycorrected him : You mean will not make a sufficiently small bet (so that thechange in the marginal utility of money will not contaminate his choice).”.

“Recalling this conversation, a few years ago I offered some lunch colleagues tobet each $200 to $100 that the side of a coin they specified would not appear atthe first tom. One distinguished scholar - who lays no claim to advancedmathematical skills - gave the following answer :”

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

Introduction the risk measures, and risk perception

“ I won’t bet because I would fell the $100 loss more than the $200 gain. ButI’ll take you on if you promise to let me make 100 such bets .”

“What was behind this interesting answer ? He, and many others, have givensomething liko tho following explanation. One toss is not enough to make itreasonably sure that the law of averages will turn out in my favor. But in ahundred tosses of a coin, the law of large numbers will make a dam good bet. Iam, so to speak, virtually sure to come out ahead in such a sequence, and that iswhy I accept the sequence while rejecting the single toss. ”.

One can check that P(gain > 0) = P(at least 34 odds) ∼ 99.91%.

However, with one toss, the maximal loss is $100 but it becomes $10,000 with100 tosses.

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

Notations

Let X be a real valued random variable, interpreted as a (net) loss.

Definition 1. A risk measure is a function R : X → R, interpreted as the capitalnecessary.

Example 2. R(X) = supX(ω), ω ∈ Ω, R(X) = supEQ(X),Q ∈ Q where Qis a set of probabilities (called scenarios), R(X) = F−1

X (α) where α ∈ (0, 1) ...etc.

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

Risk measures and price of a risk

Pascal, Fermat, Condorcet, Huygens, dAlembert in the XVIIIth centuryproposed to evaluate the “produit scalaire des probabilites et des gains”,

< p,x >=n∑i=1

pixi = EP(X),

based on the “regle des parties”.

For Quetelet, the expected value was, in the context of insurance, the price thatguarantees a financial equilibrium.

From this idea, we consider in insurance the pure premium as EP(X). As inCournot (1843), “l’esperance mathematique est donc le juste prix des chances”(or the “fair price” mentioned in Feller (1953)).

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

What is probability P ?

“my dwelling is insured for $ 250,000. My additional premium for earthquakeinsurance is $ 768 (per year). My earthquake deductible is $ 43,750... The more Ilook to this, the more it seems that my chances of having a covered loss are aboutzero. I’m paying $ 768 for this ?” (Business Insurance, 2001).

• Estimated annualized proability in Seatle 1/250 = 0.4%,• Actuarial probability 768/(250, 000− 43, 750) ∼ 0.37%

The probability for an actuary is 0.37% (closed to the actual estimatedprobability), but it is much smaller for anyone else.

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Saint Petersbourg’s paradox

Problem proposed by Bernoulli (1713),

“Une piece de monnaie est lancee jusqu’a ce que pile apparaisse. Le joueur Areoit alors de la banque B la somme de 2n francs, ou n est le nombre total delancers. Quelle mise doit disposer A avant le premier jet pour que la partie soitequitable ? ”

It is a paradox since the expected value is infinite∞∑i=1

P( stop after n draw) · 2n =∞∑i=1

12n· 2n =

∞∑i=1

1 =∞.

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

Saint Petersbourg’s paradox

Many answers have been investigated

• the bank does not have infinite liabilities, and thus, the player can play only afinite time (Buffon (1777), Poisson (1837), Borel (1949)),

• the player has a “moral utility” of money (Cramer(1728), Bernoulli (1738),von Neumann & Morgenstern (1956)) where a concave utility function isconsidered,

• the player bets using “subjective probabilities”, were rare events are assumed tobe impossible (D’Alembert (1754), Menger (1934), Yaari (1987))

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Risk measures : the expected utility approach

Ru(X) =∫u(x)dP =

∫P(u(X) > x))dx

where u : [0,∞)→ [0,∞) is a utility function.

Example with an exponential utility, u(x) = [1− e−αx]/α,

Ru(X) =1α

log(EP(eαX)

).

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Risk measures : Yarri’s dual approach

Rg(X) =∫xdg P =

∫g(P(X > x))dx

where g : [0, 1]→ [0, 1] is a distorted function.

Example if g(x) = I(X ≥ α) Rg(X) = V aR(X,α), and if g(x) = minx/α, 1Rg(X) = TV aR(X,α) (also called expected shortfall),Rg(X) = EP(X|X > V aR(X,α)).

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

Yarri’s dual approach : capacities and Choquet’s integral

Here Rg(X) =∫g(P(X > x))dx =

∫g(FX(x))dx with g : [0, 1]→ [0, 1]

increasing. Thus, g FX is a decreasing function taking values in [0, 1] on [0,∞) :g FX is a survival function.

Can Rg(X) be seen as an expected value of X with a change of measure ?

Yes if there exists a probability measure Q such that g FX(x) = Q(X > x). If itis possible to define such a measure Q, generally Q is not a probability measure.In fact, Q satisfies• Q(∅) = 0 (since FX(∞) = 0 and g(0) = 0),• Q(Ω) = 1 (since FX(0) = 1 and g(1) = 1),• Q(A) ≤ Q(B) if A ⊂ B (since FX(·) is decreasing and g(·) is increasing).

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

Yarri’s dual approach : capacities and Choquet’s integral

Such a measure Q satisfies only Q(A) ≤ Q(B) if A ⊂ B : Q is a capacity.

With this notation,

Rg(X) =∫xdg P =

∫g(P(X > x))dx =

∫Q(X > x)dx,

but since Q is not a probability measure, Rg(·) is not an expected value : it is theso-called Choquet’s integral.

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

Distortion of values versus distortion of probabilities

0 1 2 3 4 5 6

0.00.2

0.40.6

0.81.0

Calcul de l’esperance mathématique

Fig. 5 – Expected value∫xdFX(x) =

∫P(X > x)dx.

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

Distortion of values versus distortion of probabilities

0 1 2 3 4 5 6

0.00.2

0.40.6

0.81.0

Calcul de l’esperance d’utilité

Fig. 6 – Expected utility∫u(x)dFX(x).

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

Distortion of values versus distortion of probabilities

0 1 2 3 4 5 6

0.00.2

0.40.6

0.81.0

Calcul de l’intégrale de Choquet

Fig. 7 – Distorted probabilities∫g(P(X > x))dx.

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Axiomatic approach for risk measures

There are three way to describe risk measure : characterizing natural propertiesthat should satisfy• the risk measure R(·), e.g. R(·) is subadditive (R(X + Y ) ≤ R(X) +R(Y )),• induced stochastic ordering , i.e. X Y (“Y is more risky than X”) if and

only if R(X) ≤ R(Y ) [Economics],• induced set of acceptable risks A, i.e. X ∈ A (“X is is acceptable”) if and only

if R(X) ≤ 0 [Financial Mathematics].

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

Ordering and comparing risks

Assume that “risks” are positive random variables.

The higher R(X), the risker X is. Y will be said to be more risky than X will bedenoted X Y .

In Pascal’s approach FX(x) = P(X ≤ x)

X Y ⇐⇒ R(X) ≤ R(Y ) where R(X) = EP(X) =∫xdFX(x).

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

More difficult to quantify than to compare

Definition 3. is an ordering relationship if it is reflexive (FX FX),transitive (if FX FY and FY FZ then FX FZ) and antisymmetric (ifFX FY and FY FX then FX = FY ).

Note that the ordering on the set of distribution functions will be extended tothe set of positive random variables (with X ∼ Y if FX = FY , i.e. X L= Y ).

Definition 4. satisfies the additivity axiom if for any risks X, Y and Z suchthat X Y , then X + Z Y + Z.

It denotes the invariance of perception in case of a common variation. It mightalso be called the linearity axiom.

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

More difficult to quantify than to compare

Definition 5. satisfies the continuity axiom (or Archimedean axiom) if forany FX , FY and FZ such that FX FY FZ , then for all α, β ∈ (0, 1)

αFX + [1− α]FZ FY βFX + [1− β]FZ .

Proposition 6. If satisfies the continuity and associativity axioms,

X Y ⇐⇒ R(X) ≤ R(Y )

where

R(X) = EP(X) =∫xdFX(x) =

∫ ∞0

P(X > x)dx.

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

The expected utility approach

In order to answer Saint Petersbourg’ paradox, one solution, proposed byBernoulli was to introduce a “moral value of money”, i.e. a nonlinear perceptionof gains : he suggests to consider log(1 +X) instead of X. The price of the gamewas then EP(log(1 +X)). Analogously, Cramer suggested to consider

√X, so

that the price was EP(√X).

Hence, the idea was to consider a “utility function of gains”, u(·), which canchange for all players.

“Several mathematicians, for example Laplace, discussed the Bernoulli principlein the following century, and its relevance to insurance systems seems to havebeen generally recognized. In 1832, Barrois presented a fairly complete theory offire insurance based on Laplace’s work on the Bernoulli principle.” (Borch

(1974)).

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

The expected utility approach

Definition 7. satisfies the independence axiom if for any distribution functionFX , FY and FZ such that FX FY , then for all λ ∈ [0, 1]

λFX + [1− λ]FZ λFX + [1− λ]FZ .

or equivalently(λX)⊕ ([1− λ]Z) (λY )⊕ ([1− λ]Z),

where ⊕ denotes a mixture.

Hence, ordering are not modified when mixing risks with a third one. Recall that

(λX)⊕ ([1− λ]Z) 6= (λX) + ([1− λ]Z).

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Example 8. If X,Y are two Bernoulli variables B(2/3) and B(1/3) respectively, independent,

X =

0 p = 1/3

1 p = 2/3and Y =

0 p = 2/3

1 p = 1/3

X ⊕ Y =

X p = 1/2

Y p = 1/2=

0 p = 1/3× 1/2

1 p = 2/3× 1/2 0 p = 2/3× 1/2

1 p = 1/3× 1/2

=

0 p = 1/2

1 p = 1/2

X + Y =

0 + 0 p = 1/3× 2/3

0 + 1 p = 1/3× 1/3

1 + 0 p = 2/3× 2/3

1 + 1 p = 2/3× 1/3

=

0 p = 2/9

1 p = 5/9

2 p = 2/9

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

Proposition 9. If satisfies the continuity and independence axioms, thereexists a function u with values in R, continuous, strictly increasing, unique up toan affine transformation, such that

X Y ⇐⇒ Ru(X) ≤ Ru(Y )

where

Ru(X) = EP(u(X)) =∫u(x)dFX(x).

Demonstration. von Neumann & Morgenstern (1944) or Fishburn

(1970).

The continuity of u comes from the continuity assumption of the ordering.

If u is concave, the risk taker is said to be risk adverse since (Jensen’s inequality)

EP(u(X)) ≤ u(EP(X)).

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The expected utility approach

The insurance premium is then obtained by the null utility principle : π(X)satisfies

EP(u(π(X)−X)) = 0.

Example 10. With an exponential utility, u(x) = [1− e−αx]/α, alors

π(X) =1α

log(EP(eαX)

).

Note that the exponential utility does not exist for heavy tailed risks.

Example 11. With a quadratic utility, u(x) = x− x2/2s where x < s, then

π(X) ∼ EP(X) +κ

2V arP(X).

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From expected utility approach to variance

The use of the standard deviation (or variance) as a risk measure can be relatedto the theory of utility functions.

Consider Taylor’s approximation,

E(u(X)) = u(E(X)) +u′′[E(X)] · V ar(X)

2+ higher-order terms.

In the Gaussian case, those higher terms are identically zero. But fornon-Gaussian random variable X, they can be extremely large, especially ifTaylor’s expansion can not be used (it is valid only in the neighborhood of themean).

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Risk aversion and finance

One can introduce, Arrow-Pratt coefficient of absolute risk aversion,

RA(x) = −u′′(x)u′(x)

, and the coefficient of relative risk aversion, RR(x) = −xu′′(x)

u′(x)CARA (Constant Absolute Risk Aversion) means that RA(·) is constant, i.e.

u(x) = − 1α

exp(−αx).

CRRA (Constant Relative Risk Aversion) means that RA(·) is constant, i.e.

u(x) = − x1−α

1− α, for α > 0, including the limiting case u(x) = log(x) (when α→ 1.

• modeling portfolios with Gaussian returns and CARA utility

Assume that X ∼ N (µ, σ2) and u(x) = − 1α

exp(−αx), for some α > 0.

By solving u(EP(X)− π) = EP(u(X)), using the expression of the Laplacetransform of the Gaussian distribution,

EP(u(X)) = − 1α

EP(exp(−αX)) = − 1α

exp(−αµ+

α2

2σ2

),

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and

u(EP(X)− π) = − 1α

exp(−α(µ− π)),

thus, one gets that π =α

2σ2.

• modeling portfolios with lognormal risks and CRRA utility

Assume that logX ∼ N (µ, σ2) and u(x) = − x1−α

1− α, for some α > 0.

By solving u(EP(X)− π) = EP(u(X)), one gets that π =ασ2

2× EP(X).

• quadratic utility principle

If u is quadratic u(x) = x− x2/2s, then

EP(u(X)) = EP(X)(

1− 12s

EP(X))− 1

2sV arP(X),

hence only the mean and the variance matter.

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Changing probabilities ?

Question : what is probability P ?

“Il y a donc une aversion particuliere pour l’incertitude liee a l’ignorance. Onprefere avoir un modele probabiliste que pas de modele du tout, on prefere evaluerraisonnablement ses chances de succes, fussent-elles minces, que de n’en avoiraucune idee.” (Ekeland (1991)).

Idee de Ramsey (1931), formalisee par Savage (1972) : les individus ne raisonnepas sous P, la probabilite reelle (inconnue), mais sous une probabilite subjectiveQ.

Probleme : difficile d’estimer une probabilite d’evenement rare.

Travaux de Selvige (1975) : importance des evenements rares aux consequencesimportantes. Approche psychologique du risque : besoin de comparer a desevenements rares quantifiables (taux de mortalite infantile, quinte flush aupocker...).

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Changing probabilities ?

Definition 12. est une relation verifiant l’axiome de monotonie siP(X + ε ≤ Y ) = 1 implique X Y , pour tout ε > 0.

Proposition 13. Si est une relation d’ordre verifiant les axiomes decontinuite, d’additivite et de monotonie, alors il existe une probabilite Q telle que

X Y ⇐⇒ RQ(X) ≤ RQ(Y )

ou

RQ(X) = EQ(X) =∫ ∞

0

Q(X > x)dx.

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Using subjective probabilities

Considerons un call europeen, dont le payoff actualise est e−rT (ST −K)+. Leprix n’est pas EP

(e−rT (ST −K)+

).

Le prix d’un call europeen proposant de toucher (ST −K)+ a maturite. Lavalorisation de l’option, a la date d’aujourd’hui est basee sur la notion deportefeuille de replication : deux portefeuilles offrant le meme payoff a une dateT ont necessairement le meme prix aujourd’hui (sinon il serait possible deconstituer une opportunite d’arbitrage).

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Changing probabilities ?

Considerons le modele de Cox, Ross & Rubinstein (1979), avec un actif sansrisque valant 1 aujourd’hui, et 1 + r dans un an, et un actif risque valant S0

aujourd’hui, et, dans un an S1, valant soit Su, soit Sd, avec d < 1 + r < u,suivant l’etat de la nature. Considerons un call europeen donnant le droitd’acheter le sous-jacent a maturite (dans un an) a la valeur K. Le payoff dans unan est alors (S1 −K)+. Construisons un portefeuille α+ βS0 permettant derepliquer la valeur de l’option dans un an :

• si le marche monte, le portefeuille vaudra α (1 + r) + βSu, et l’optionCu = (Su−K)+

• si le marche baisse, le portefeuille vaudra α (1 + r) + βSd, et l’optionCd = (Sd−K)+

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Changing probabilities ?

Dans un marche avec absence d’opportunite d’arbitrage, si ces deux produits ontla meme valeur dans un an, c’est donc qu’ils ont le meme prix aujourd’hui. Leportefeuille qui permet de repliquer le payoff de l’option est obtenu en resolvant α (1 + r) + βSu = Cu

α (1 + r) + βSd = Cd

c’est a dire que

α =Cu − CdS0u− S0d

et β =1

1 + r

(Cu − S0u

Cu − CdS0u− S0d

).

Notons au passage qu’il est ainsi toujours possible de constituer un uniqueportefeuille de replication.

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Changing probabilities ?

Le prix de l’option aujourd’hui s’ecrit

α+ βS0 =1

1 + r

(1 + r − du− d

Cu +u− (1 + r)u− d

Cd

),

qui peut s’ecrire

π =1

1 + r(qCu + (1− q)Cd) , ou q =

1 + r − du− d

.

Notons que q ∈ [0, 1], c’est a dire que le prix de l’option est l’esperancemathematique, sous une probabilite Q appelee probabilite risque neutre du payoffa echeance : π = EQ (payoff). Notons que Q n’a rien n’a voir avec la probabilitedite historique P qu’a le sous-jacent de monter ou de descendre : le prix d’unpayoff aleatoire X ne s’ecrit pas EP (X).

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

Les paradoxes de Allais et Ellsberg

Allais (1953), “l’effet de certitude” ou “l’effet de securite”

Choisir entre les deux lotteries suivantesLoterie A 100% de chance de recevoir 1 million ,

Loterie B

10% de chance de recevoir 5 millions

89% de chance de recevoir 1 million

1% de chance de ne rien recevoir,

puis entre Loterie C

11% de chance de recevoir 1 million

89% de chance de ne rien recevoir,

Loterie D

10% de chance de recevoir 5 millions

90% de chance de ne rien recevoir,

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Preferer A a B, et D a C (observe empiriquement) viole l’hypothesed’independance (et meme le principe de la chose sre).

Ellsberg (1961), “l’effet d’ambiguite”

Choisir entre les deux lotteries suivantes Loterie A win 1000 si la boule tiree est rouge

Loterie B win 1000 si la boule tiree est bleue,

puis entre Loterie C win 1000 si la boule tiree n’est pas rouge

Loterie D win 1000 si la boule tiree n’est pas bleue,

Preferer A a B, et C a D (observe empiriquement) viole l’hypothesed’independance (et meme le principe de la chose sre).

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La distortion de probabilites, Yaari (1987)

Example 14. Un exemple de relation d’ordre est la dominance stochastique al’ordre 1. X 1 Y si et seulement si une des conditions suivantes (equivalentes)sont satisfaites,• E(g(X)) ≤ E(g(Y )) pour g croissante,• pour tout x ∈ R, P(X ≤ x) ≥ P(Y ≤ x),• pour tout x ∈ R, P(X > x) ≤ P(Y > x),• pour tout x ∈]0, 1[, V aR(X,α) ≤ V aR(Y, α).Cette relation d’ordre est notee V aR dans Denuit & Charpentier (2004).

Example 15. Un exemple de relation d’ordre est la dominance stochastique al’ordre 2. X 2 Y si et seulement si une des conditions suivantes (equivalentes)sont satisfaites,• E(g(X)) ≤ E(g(Y )) pour g croissante et convexe,• E((X − t)+) ≤ E((Y − t)+) pour t ∈ R,

• pour tout x ∈]0, 1[,∫ α

0

V aR(X, p)dp ≥∫ α

0

V aR(Y, p)dp,

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• pour tout x ∈]0, 1[,∫ ∞α

V aR(X, p)dp ≤∫ ∞α

V aR(Y, p)dp,

• pour tout x ∈ [0, 1[, TV aR(X,α) ≤ TV aR(Y, α).Cette relation d’ordre est notee TV aR dans Denuit & Charpentier (2004).

Definition 16. est une relation verifiant l’axiome d’independance comonotonesi X Y implique X + Z Y + Z pour tout Z tel que les couples (X,Z) et(Y, Z) soient comonotones.

Remark 17. X et Z sont comonotones s’il n’existe pas ω, ω′ tels que

X(ω) > X(ω′) et Y (ω) < Y (ω′).

Definition 18. est une relation verifiant l’axiome de coherence si pour desvariables X et Y constantes (P(X = x) = P(Y = y) = 1), FX FY impliquex ≤ y.

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Proposition 19. Si est une relation d’ordre verifiant les axiomes decontinuite, d’independance comonotone, monotonie, et est compatible avec ladominance stochastique a l’ordre 1, alors il existe une unique fonction dedistortion croissante g : [0, 1]→ [0, 1] telle que

X Y ⇐⇒ Rg(X) ≤ Rg(Y )

ou

Rg(X) =∫xdg P =

∫g(P(X > x))dx =

∫ 1

0

F−1X (1− p)dg(p).

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Coherent risk measures and the axiomatic approach

A risk measure is said to be coherent (from Artzner, Delbaen, Eber &

Heath (1999)) if• R(·) is monotonic, i.e. X ≤ Y implies R(X) ≤ R(Y ),• R(·) is positively homogeneous, i.e. for any λ ≤ 0, R(λX) = λR(X),• R(·) is invariant by translation, i.e. for any κ, R(X + κ) = R(X) + κ,• R(·) is subadditive, i.e. R(X + Y ) ≤ R(X) +R(Y ).“subadditivity” can be interpreted as “diversification does not increase risk”.

Example : the Expected-Shortfall is coherent, the Value-at-Risk is not.

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Convex risk measures

A risk measure is said to be convex (from Artzner, Delbaen, Eber & Heath

(1999)) if• R(·) is monotonic, i.e. X ≤ Y implies R(X) ≤ R(Y ),• R(·) is invariant by translation, i.e. for any κ, R(X + κ) = R(X) + κ,• R(·) is convex, i.e. R(λX + (1− λ)Y ) ≤ λR(X) + (1− λ)R(Y ), for anyλ ∈ [0, 1].

Hence, if a convex measure satisfies the homogeneity condition, it is coherent.

Remark A natural way to define a convex measure satisfying the small sizecoherent condition is adding a coherent measure a liquidity charge,

Rconvex(X) = Rcoherent(X) + Cliquidity(X).

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Sets of acceptable risksDefinition 20. Given a risk measure R, a risk X is acceptable if X ∈ AR whereAR = Y ∈ X such that R(Y ) ≤ 0.

Conversely,Theorem 21. Given a set of acceptable risks A, the associated risk measure isthe smallest capital amont m such that X −m is acceptable, i.e.

RA(X) = infm ∈ R such that X −m ∈ A.

Then RAR(·) = R(·) and ARA = A.Proposition 22. If R is a convex risk measure, then AR is convex. Conversely,if A is convex, then RA is a convex risk measure.Proposition 23. If R is a positively homogeneous risk measure, then AR is apositive cone. Conversely, if A is a positive cone, then RA is a positivelyhomogeneous risk measure.Example 24. If R(X) = supX(ω), ω ∈ Ω, then AR = Y, Y ≤ 0. IfR(X) = F−1

X (α) where α ∈ (0, 1), then AR = Y,P(Y ≤ 0) ≥ α.

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Characterizations of coherent risk measures

Proposition 25. If R is a coherent risk measure, then there exists a set ofprobability measures Q such that

R(X) = supQ∈QEQ(X).

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Value-at-Risk : going furtherProposition 26. The Value-at-Risk is (generally) not a coherent risk measure.

If X,Y ∼ B(92.5%), independent, then

V ar(X, 90%) + V ar(Y, 90%) = 0 + 0 ≤ V ar(X + Y, 90%) = 1.

0.80 0.85 0.90 0.95 1.00

0.0

0.5

1.0

1.5

2.0

Proposition 27. The Value-at-Risk is a coherent risk measure for ellipticalrisks.Proposition 28. For all X, note that

V aR(X,α) = infR(X) such that R is coherent and V aR(X,α) ≤ R(X).

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Tail Value-at-Risk (or Expected Shortfall)

Define

TV aR(X,α) = E(X|X > V aR(X,α)) =1

1− α

∫ 1

α

FX−1(u)du.

In some sense, the TailVaR is the average of worst cases, while V aR was the bestworst case.

Proposition 29. Tail Value-at-Risk is a coherent risk measure.

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La Value-at-Risk, ou VaR

Definition 30. La Value-at-Risk est le montant des pertes qui ne doivent pasetre depassees pour un niveau de confiance donne sur un horizon temporel donne.

On se donne un horizon temporel T et un niveau de confiance α ∈]0, 1[. Soit (Xt)la valeur d’un portefeuille a horizon t, alors V aR(XT , α) verifie

P(XT > V aR(XT , α)) = 1− α.

Cette mesure n’est pas (trop) technique et peut meme etre comprise par lesdirigeants d’une compagnie bancaire.

Definition 31. La Value-at-Risk pour un niveau de confiance α ∈]0, 1[ et pourun horizon temporel donne T est le quantile d’ordre α de la distribution de profitset pertes (profit and loss distribution) a horizon T .

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La Value-at-Risk, ou VaR

Si on suppose que XT ∼ N (0, 1), alors V aR(XT , α) = −1.645.

−6 −4 −2 0 2 4

0.0

0.1

0.2

0.3

0.4

Distribution P&L, à horizon T

Den

sité

de

prob

abili

seuil 5%

Quantile à 95% = −1.645

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Arthur CHARPENTIER - econometrie de la finance (2008/2009)

La Value-at-Risk, ou VaR

Par exemple, pour les risques les plus classiques• Market Risk (MR) : α = 99%, T = 10 jours• Trading desk limits (MR) : α = 95%, T = 1 jour• Credit Risk (CR) : α = 99.9%, T = 1 an• Operational Risk (OR) : α = 99.9%, T = 1 an• Economic Capital (EC) : α = 99.97%, T = 1 an=⇒ Les Value-at-Risk mesurent des quantiles pour des seuils α (tres) eleves.

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Calcul de VaR, approche reglementaire

Les accords de Bale II donne une formule pour calculer le capital economiquepour “couvrir” son portefeuille actions. Le capital necessaire est alors

CRt = max

V aRt(99%) +

k

60

60∑i=1

V art−i+1(99%)

+ CR?t ,

ou V aRt(99%) est la Value-at-Risk du portefeuille, sur un horizon de 10 jours,avec un seuil de 99%, CR?t est un montant de capital pour des “risquesspecifiques”, et k designe un “facteur de stress” compris entre 2 et 3 (voire 5, quidoit dependre du backtesting statistique et de la qualite de la methodologiestatistique utilisee).

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Un “facteur de stress” ?

Pour mieux comprendre l’origine de ce facteur de stress, rappelons qu’on peuttrouver une borne superieure a la VaR en utilisant l’ingalite de BienaymeTchebychev : si X est une variable de variance finie (σ2 <∞) et d’esperance µ,alors pour tout α

P (|X − µ| ≥ α) ≤ σ2

α2.

Si X a une distribution symmetrique (ce qui implique entre autres µ = 0), enposant α = −cσ,

P (X ≥ −cσ) ≤ 12c2

.

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Un “facteur de stress” ?

Aussi, si c =√

1/2α, i.e. α = 1/2c2, on obtient une borne pour la VaR a un seuil1− α,

P(X ≥ σ√

)= P

(X ≥ V aR(1− α)

)≤ α.

Pour α = 1%, on obtient c = 7.07 soit V aR(99%) = 7.07σ. Rappelons que dans lecas N (0, σ2), V aR(99%) = 2.33σ, i.e.

V aR(99%) = 3.04 · V aRN (0,σ2)(99%).

Notons que V aR(95%) = 1.92 · V aRN (0,σ2)(95%).

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