1. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Gestion des risques bancaires et nanciers
mesures de risques
et cadre règlementaire
Arthur Charpentier
EdF, formation continue
arthur.charpentier@univ-rennes1.fr
1
2. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Overview of the two sessions
1. Notions de base en probabilité
2. Processus stochastiques en temps discret
3. Marchés nanciers en temps discret
4. Intégration stochastique en temps continu
5. Formule d'Itô et formule de Feynman-Kac
6. Changement de probabilité
7. Introduction aux processus stochastiques discontinus
8. Introduction aux modèles de taux d'intérêt
9. Mesures de risques, aspect règlementaires et principes de base
10. Corrélations, copules
11. Modèles statiques
12. Modèles dynamiques
2
3. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Overview of the two sessions
Consider a set of risks, denoted by a random vector X = (X1, . . . , Xd)
The interest is an agregation function of those risks g(X), where g : Rd
→ R, and
we wish to measure the risk of this quantity R(g(X)), for some risk measure R.
• Séance 9 : Mesures de risques, aspect règlementaires et principes
de base
what is a risk measure?
what are academic or pragmatic risks measures, and what about
accounting standards?
what are desirable properties of R?
how to estimate R(Z) given a sample {Z1, . . . , Zn}?
• Séance 10 : Corrélations, copules
3
4. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Overview of the two sessions
Consider a set of risks, denoted by a random vector X = (X1, . . . , Xd)
The interest is an agregation function of those risks g(X), where g : Rd
→ R, and
we wish to measure the risk of this quantity R(g(X)), for some risk measure R.
• Séance 9 : Mesures de risques, aspect règlementaires et principes
de base
• Séance 10 : Corrélations, copules
how to model X?
what about diversication eects?
what is the correlation of risks in X?
can we compare R(g(X)) and R(g(X⊥
)) (i.e. under independence)?
what is the contribution of Xi in the overall risk?
4
5. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Some possible motivations... in nance
Consider a set of stock prices at time T denoted X = (X1, . . . , Xd) and
Y = (Y1, . . . , Yd) the ratio of the price at time T divided by the price at time 0,
and and let g(X) denote the payo at time T of some nancial derivative,
• e.g. spread derivatives, g(x1, x2) = (x1 − x2 − K)+ based on the spread
between two assets, or more generally any extreme spread options, dual
spread options, correlation options or ratio spread options,
• e.g. buttery derivatives, g(x) = (a x − K)+, i.e. call option on a portfolio
of d assets,
• e.g. min-max derivatives or rainbow, g(x) = (min{x} − K)+,
g(x) = (max{x} − K)+, i.e. call option on the minimum or maximum of d
assets,
• e.g. Atlas derivatives, g(x) = (
i=i+
i=i−
Yi − K)+, where the sum is considered
skipping the i− lowest and the d − i+ largest returns, or Himalaya
derivatives, g(x) = (
i=d
i=i+
Yi − K)+,
5
6. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Some possible motivations... in environmental risks
6
7. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Some possible motivations... in credit risk
Applications with a high number of risks can also be considered, in credit risk for
instance. Let X = (X1, ..., Xd) denote the vector of indicator variables,
indicating if the i-th contract defaulted during a given period of time. If a credit
derivative is based on the occurrence of k defaults among d companies, and thus,
the pricing is related to the distribution of the number of defaults, N, dened as
N = X1 + ... + Xd. Under the assumption of possible contagious risks, the
distribution of N should integrate dependencies.
CreditMetrics in 1995 suggested a Gaussian model for credit changes, based on a
probit approach, Xi = 1(X∗
i ui), where X∗
i ∼ N(0, 1).
7
8. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
!6 !4 !2 0 2 4 6
0.00.10.20.30.40.50.6
Value of the company
Probit model in dimension 1
DEFAULT
!4 !2 0 2 4
!4!2024
Value of company (1)
Valueofcompany(2)
(1) DEFAULTS
(2)DEFAULTS
Probit model in dimension 2
Figure 1: Modeling defaults based on a probit approach.
8
9. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Some possible motivations... in risk management
Consider a set a risks X = (X1, ..., Xd) (returns in a portfolio, losses per line of
business, positions of nancial desks)
A classical risk measure (as in Markowitz (1959)) is the standard deviation,
σ(Xi) = V ar(Xi) = E((Xi − E(X))
2
). The risk of the portfolio
S = X1 + . . . + Xd is
σ(S) = σ(X1)2 + . . . + σ(Xd)2 + 2
ij
r(Xi, Xj)σ(Xi)σ(Xj).
Risks are now measured using Value-at-Risk (i.e. quantiles), and there is no
relationship between q(X1 + . . . + Xd) and q(X1) + . . . + q(Xd).
9
12. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Agenda
• General introduction
Financial risks
• Market risks
• Credit risk
• Operational risk
Risk measures and capital allocation
• Risk measures: an axiomatic introduction
• Risk measures: convexity and coherence
• Capital allocation: an axiomatic introduction
Risk measures and statistical inference
12
13. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Agenda
• General introduction
Financial risks
• Market risks
• Credit risk
• From variance to Value-at-Risk
Risk measures and capital allocation
• Risk measures: an axiomatic introduction
• Risk measures: convexity and coherence
• Capital allocation: an axiomatic introduction
Risk measures and statistical inference
13
14. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Some references on risk management
Föllmer, H. Schied, A. (2004). Stochastic nance. An introduction in
discrete time. Gruyter Studies in Mathematics,
Jorian, P. (2007). Value-at-Risk,
McNeil, A. Frey, R., Embrechts, P. (2005). Quantitative Risk
Management: Concepts, Techniques, and Tools. Princeton University Press,
Roncalli, T. (2004). La gestion des risques nanciers. Economica.
Basel Committee on Banking Supervision. International convergence of capital
measurement and capital standards. http://www.bis.org/publ/bcbs128.pdf
14
15. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Introduction to risk management: crisis
a statistical model is a probability distribution constructed to enable inferences to
be drawn or decisions made from data (Jorion (2007)).
1974 Herstatt Bank, 620 million USD (⇒ Real Time Gross Settlement Systems)
1994 Metallgesellschaft, 1340 million USD (oil futures)
1994 Orange County, 1810 million USD (derivaties)
1994 Procter Gamble, 102 million USD (derivaties)
1995 Barings, 1330 million USD (fraudulent manipulations)
1997 Natwest, 127 million USD (fraudulent manipulations)
1998 LTCM (Long Term Capital Management), 2000 million USD (liquidity crisis)
15
17. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
History of risk measures
The evolution of (analytical) Risk Management Tools (from Jorion (2007))
1938 bond duration
1952 Markowitz mean-variance framework
1963 Sharpe's single beta model
1973 Black Scholes option pricing formula
1983 RAROC, Risk Adjusted Return
1992 Stress testing
1993 Value-at-Risk (VaR)
1994 RiskMetrics
1997 CreditMetrics
1998 integration of credit and market risk
1999 coherent risk measures
17
18. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Agenda
• General introduction
Financial risks
• Market risks
• Credit risk
• From variance to Value-at-Risk
Risk measures and capital allocation
• Risk measures: an axiomatic introduction
• Risk measures: convexity and coherence
• Capital allocation: an axiomatic introduction
Risk measures and statistical inference
18
19. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Market risks
Classical models for stock prices,
• dynamic models (Bachelier (1900), Black Scholes (1973)), Brownian
geometric
dSt = 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 +
√
VtdWS
t
dVt = κ(θ − Vt)dt + ξ
√
VtdWV
t ,
where (WS
t )t≥0 and (WV
t )t≥0 are two standard brownian motions (possibly
correlated).
19
20. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0.0 0.2 0.4 0.6 0.8 1.0
50100150200 Stock price over 1 year, large volatility
Time
0.0 0.2 0.4 0.6 0.8 1.0
50100150200
Stock price over 1 year, large volatility
Time
Figure 4: Random generation of a stock price, dSt = µStdt + σStdWt.
20
21. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Market risks
Note that continuous GARCH processes can also be considered
dSt = µStdt +
√
VtdWS
t
dVt = κ(θ − Vt)dt + ξVtdWV
t ,
21
22. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Market risks
• the capital asset pricing model (Markowitz (1970) or the Sharpe index are
based on the mean-variance framework,
0 5 #0 #5
!#.0!0.50.00.5#.0#.5%.0%.5
ca)t!type
sp/)ance
0 #0 #
!#.0!0.0.00.#.0#.%.0%.
ca)t!t+pe
sp/)ance
Figure 5: Capital asset pricing model, the mean-variance framework.
22
23. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
How to quantify market risks : volatility
All the information about uncertainty is summarized by the volatiliy - or
variance - parameter.
Note that this is one of the drawback of the use of the Gaussian distribution.
23
24. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
A (very) short word on diversication
Naturally, in higher dimension (when dealing with multiple stocks), Gaussian
vectors are considered
X =
X1
X2
.
.
.
Xd
∼ N
µ1
µ2
.
.
.
µd
,
σ2
1 ρ1,2σ1σ2 · · · ρ1,dσ1σd
ρ2,1σ2σ1 σ2
2 · · · ρ2,dσ2σd
.
.
.
.
.
.
.
.
.
ρd,1σdσ1 ρd,2σdσ2 · · · σ2
d
All the information about marginal risks is in the variances (σ2
i ) while all the
information on the dependence is in the correlation coecients (ρi,j).
24
25. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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 nite variance,
√
n
X − E(X)
√
V X
L
→ N(0, 1).
• it is an elliptic distribution, i.e. X = µ + AX0 where A A = Σ, and where
X0 has a spheric distribution, i.e. f(x0) is a function of x0x0 (spherical level
curves),
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26. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
−3 −2 −1 0 1 2 3
−3−2−10123 Level curves of a spherical distribution
−3 −2 −1 0 1 2 3
−3−2−10123
Level curves of a elliptical distribution
Figure 6: The Gaussian distribution.
26
27. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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−1
2,2Σ21)
27
28. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
0.00
0.05
0.10
0.15
0.20
Density of the Gaussian distribution
−3 −2 −1 0 1 2 3
−3−2−10123
Level curves of a elliptical distribution
Figure 7: The Gaussian distribution.
28
29. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Agenda
• General introduction
Financial risks
• Market risks
• Credit risk
• From variance to Value-at-Risk
Risk measures and capital allocation
• Risk measures: an axiomatic introduction
• Risk measures: convexity and coherence
• Capital allocation: an axiomatic introduction
Risk measures and statistical inference
29
30. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Credit risk
Credit risk is the risk that an obligor does not full his payment obligations,
either wholly or in part.
For single obligors, one should model
• risk arrival (given a time horizon, it there a default ?)
• risk timing (when does the default occur ?)
• risk magnitude (how high is the loss ?)
and on a portfolio level
• risk correlation (will there be any contagion ?)
Here, loss distributions have low probabilities but high losses (high downside risk,
strongly skewed).
Hence, for credit risk, volatiliy has no meaning.
Main problem: no much historical data.
30
31. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Classical credit risk models
• credit scoring, logit/probit models to model default occurence
• models for rating transition based on Markov chains
• models for bond prices, intensity based
31
32. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
The binomial model for credit default
Consider defaults in a portfolio, until a xed time-horizon T, with no interest
rate.
The exposures are of identical size L, identical recovery rate c. Assume also that
the portfolio is homogeneous, i.e. each obligor defaults with a probability p
before time-horizon T.
Assume that defauts happen independenty of each other.
Let X denote the number of default in the portfolio, so that the loss is X(1 − c)L.
If defaults are independent,
P(X = k) =
d
k
pk
(1 − p)d−k
=
d!
k!(d − k)!
pk
(1 − p)d−k
= (k; d, p)
and
P(X ≤ k) =
k
i=0
d
k
pk
(1 − p)d−k
= B(k; d, p)
32
33. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0 5 10 15 20 25 30
0.000.050.100.150.20
Distribution of number of defaults, n=30, p=20%
Number of defaults
Probability
0 5 10 15 20 25 30
0.000.050.100.150.200.250.300.35
Distribution of number of defaults, n=30, p= 5%
Number of defaults
Probability
0 5 10 15 20 25 30
0.20.40.60.81.0
Distribution of number of defaults, n=30, p=20%
Number of defaults
Cumulativeprobability
0 5 10 15 20 25 30
0.20.40.60.81.0
Distribution of number of defaults, n=30, p= 5%
Number of defaults
Cumulativeprobability
Figure 8: Distribution of the number of defaults, independent case.
33
34. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Extending the binomial model for credit default
A simplied rm's value model can be considered, where the default of each
obligor k is triggered by the change of the value Vk(t) of the assets of its rm.
Assume that Vk(T) is normally distributed, and standardized, i.e.
Vk(T) ∼ N(0, 1). Assume more generally that
V (T) = (V1(T), · · · , Vd(T)) ∼ N(0, Σ), i.e. the asset values of dierent obligors
might be correleted with each other.
Obligor k defaults if its rm's value falls below a barrier Vk(T) ≤ Bk.
This can be seen as a multivariate probit model.
34
35. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
A one-factor model for credit default
Assume that the values of the assets of the obligors are driven by one common
factor Y , and an idiosyncratic standard normal noise component εk, so that
Vk(T) =
√
ρY + 1 − ρεk, k = 1, · · · , d,
where Y and the εk's are independent and N(0, 1) distributed.
This is an exchangeable model: conditional on the realisation of the systematic
factor Y , the rm's values and the defaults rae independent.
Assuming that Bk = B and that the exposure is Lk = 1, then
P(X = k) = P(X = k|Y = y)φ(y)dy,
and conditional on Y = y, the probability to have k defaults is
P(X = k|Y = y) =
d
k
p(y)k
(1 − p(y))d−k
35
36. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
A one-factor model for credit default
where p(y) is the probability that Vi(T) falls below B, i.e.
p(y) = P(Vi(T) ≤ B|Y = y) = P εi ≤
B −
√
Y
√
1 − ρ
≤ B Y = y = Φ
K −
√
ρy
√
1 − ρ
.
Substituting yields
P(X = k) =
d
k
Φ
K −
√
ρy
√
1 − ρ
k
1 − Φ
K −
√
ρy
√
1 − ρ
d−k
φ(y)dy
36
37. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
q
q
q
q
q
q
q
q
q
q
q
q
q
q q q q q q q q q q q q q q q q q q
0 5 10 15 20 25 30
0.000.050.100.15
Distribution of number of defaults, n=30, p=20%, r=0.1
Number of defaults
Probability
q
q
q
q
q
q
q
q
q
q
q
q
q
q q q q q q q q q q q q q q q q q q
0 5 10 15 20 25 30
0.000.050.100.15
Distribution of number of defaults, n=30, p=20%, r=0.2
Number of defaults
Probability
q
q
q
q
q
q
q
q
q
q
q
q
q
q q q q q q q q q q q q q q q q q q
0 5 10 15 20 25 30
0.000.050.100.15
Distribution of number of defaults, n=30, p=20%, r=0.3
Number of defaults
Probability
0 5 10 15 20 25 30
0.00.20.40.60.81.0
Distribution of number of defaults, n=30, p=20%, r=0.1
Number of defaults
Cumulativeprobability
0 5 10 15 20 25 30
0.00.20.40.60.81.0
Distribution of number of defaults, n=30, p=20%, r=0.2
Number of defaults
Cumulativeprobability
0 5 10 15 20 25 30
0.20.40.60.81.0
Distribution of number of defaults, n=30, p=20%, r=0.3
Number of defaults
Cumulativeprobability
Figure 9: Distribution of the number of defaults, nonindependent case.
37
38. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Using variance for credit risk ?
Is variance relevant to measure risk ?
No since it is not a downside risk.
38
39. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0.0 0.2 0.4 0.6 0.8
020406080100
Evolution of the variance of X as a function of rho, p=20%
Evolution of parameter rho
Variance
Figure 10: Evolution of the variance of X as a function of ρ.
39
40. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Using variance for credit risk ?
One has to nd a coecient which measures properly downside risk. An idea is
to use a quantile
Denition 1. The α-quantile of a distribution FX is qX(α), such that
qX(α) = F−1
X (α) where
F−1
X (u) = inf{x, FX(x) ≥ u}, where u ∈ (0, 1).
Then F−1
X (·) is increasing on (0, 1), continuous from the left, with limits from the
right, and further
F−1
X ◦ FX(x) ≤ x for any x and FX ◦ F−1
X (u) ≥ u for any p.
Remark Almost equivalently, it is possible to dene
F−1
X (u) = sup{x, FX(x) ≤ u}, where u ∈ (0, 1).
40
41. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0.0 0.2 0.4 0.6 0.8
20406080100
Evolution of the variance of X as a function of rho, p= 5%
Evolution of parameter rho
Quantile
90% quantile
95% quantile
99% quantile
0.0 0.2 0.4 0.6 0.8
406080100
Evolution of the variance of X as a function of rho, p=20%
Evolution of parameter rho
Quantile
90% quantile
95% quantile
99% quantile
Figure 11: Evolution of the quantile of X as a function of ρ.
41
42. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Agenda
• General introduction
Financial risks
• Market risks
• Credit risk
• From variance to Value-at-Risk
Risk measures and capital allocation
• Risk measures: an axiomatic introduction
• Risk measures: convexity and coherence
• Capital allocation: an axiomatic introduction
Risk measures and statistical inference
42
43. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Value-at-Risk
The expression is quite recent and its origin is uncertain: in the 80's, some
papers introduced dollars-at-risk, capital-at-risk, income-at-risk, earning-at-risk
and nally value-at-risk
Denomination has been stabilized after the publication of RiskMetrics Technical
Document in 1994, by JPMorgan. Note that the work accomplished by
JPMorgan was more a pulic relation campaign than an advanced technical study:
VaR is more a practice than a theory.
VaR summarizes the worst loss ever on a target horizon that will not be exceeded
with a given level of condence, i.e. formaly it is a quantile of the projected
distibution of gains and losses over the target horizon
Till Guldimann (1992) created the term value-at-risk while head of global
research at JP Morgan in the late 80's. It appeared in the G30 report (group of
thirty) in July 1993.
43
44. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
A technical denition for the Value-at-Risk
Denition 2. Let X denote the loss distribution, then
V aR(X, α) = qX(α), for all α ∈ (0, 1).
44
45. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
The Basel II accord (2004)
June 2004, the Basel Committee nalized the Basel Accords, based on three
pillars
• 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 constant
the level of capital in the global banking syste: 8% of risk weighted assets,
• supervisorv review, i.e. expanded role for bank regulartors, to ensure that
banks operate above the minimum regulatory capital ratios, that banks have
appropriate processes for assessing their risks, and appropriate corrective
actions
• market discipline, i.e. set of disclosure recommendations, encouraging to
publish informations about exposures, risk proles, capital cushion...
From the rst pillar, there should be a credit risk charge (CRC), a market risk
charge (MRC) and an operationnal risk charge (ORC), and the bank's total
45
46. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
capital must exceed the total-risk charge (TRC)
Capital TRC = CRC + MRC + ORC.
Why using VaR as a risk measure ?
Markowitz (1952) claimed that standard deviation should be an intuitive and
appropriate risk measure (leading to the mean-variance trade-o).
The same year, Roy (1952) claimed that the optimal bundle of assets
(investment) for investors who employ the safety rst principle is the portfolio
that minimizes the probability of disaster.
Roy A. D. (1952), Safety rst and the holding of assets, Econometrica, 20,
431-449.
Markowitz H. M. (1952), Portfolio selection, Journal of Finance, 7, 77-91.
46
47. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Agenda
• General introduction
Financial risks
• Market risks
• Credit risk
• From variance to Value-at-Risk
Risk measures and capital allocation
• Risk measures: an axiomatic introduction
• Risk measures: convexity and coherence
• Capital allocation: an axiomatic introduction
Risk measures and statistical inference
47
48. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Introduction the risk measures, and risk perception
S. Clam [...] once said: I dene a coward as someone who will not bet when
you oer him two-to-one odds and let him choose his side .
With the centuries old St. Petersburg paradox in my mind, I pedantically
corrected him: You mean will not make a suciently small bet (so that the
change in the marginal utility of money will not contaminate his choice). .
Recalling this conversation, a few years ago I oered some lunch colleagues to
bet each $200 to $100 that the side of a coin they specied would not appear at
the rst tom. One distinguished scholar - who lays no claim to advanced
mathematical skills - gave the following answer:
48
49. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Introduction the risk measures, and risk perception
I won't bet because I would fell the $100 loss more than the $200 gain. But
I'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 given
something liko tho following explanation. One toss is not enough to make it
reasonably sure that the law of averages will turn out in my favor. But in a
hundred tosses of a coin, the law of large numbers will make a dam good bet. I
am, so to speak, virtually sure to come out ahead in such a sequence, and that is
why 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 with
100 tosses.
49
50. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Notations
Let X be a real valued random variable, interpreted as a (net) loss.
Denition 3. A risk measure is a function R : X → R, interpreted as the capital
necessary.
Example 4. R(X) = sup{X(ω), ω ∈ Ω}, R(X) = sup{EQ(X), Q ∈ Q} where Q
is a set of probabilities (called scenarios), R(X) = F−1
X (α) where α ∈ (0, 1) ...etc.
50
51. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Risk measures and price of a risk
Pascal, Fermat, Condorcet, Huygens, d’Alembert in the XVIIIth century
proposed to evaluate the produit scalaire des probabilités et des gains,
p, x =
n
i=1
pixi = EP(X),
based on the règle des parties.
For Quételet, the expected value was, in the context of insurance, the price that
guarantees a nancial equilibrium.
From this idea, we consider in insurance the pure premium as EP(X). As in
Cournot (1843), l'espérance mathématique est donc le juste prix des chances
(or the fair price mentioned in Feller (1953)).
51
52. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
What is probability P ?
my dwelling is insured for $ 250,000. My additional premium for earthquake
insurance is $ 768 (per year). My earthquake deductible is $ 43,750... The more I
look to this, the more it seems that my chances of having a covered loss are about
zero. 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 estimated
probability), but it is much smaller for anyone else.
52
53. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Saint Pétersbourg's paradox
Problem proposed by Bernoulli (1713),
Une pièce de monnaie est lancée jusqu'à ce que pile apparaisse. Le joueur A
reçoit alors de la banque B la somme de 2n
francs, ou n est le nombre total de
lancers. Quelle mise doit disposer A avant le premier jet pour que la partie soit
équitable ?
It is a paradox since the expected value is innite
∞
i=1
P( stop after n draw) · 2n
=
∞
i=1
1
2n
· 2n
=
∞
i=1
1 = ∞.
53
54. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Saint Pétersbourg's paradox
Many answers have been investigated
• the bank does not have innite liabilities, and thus, the player can play only
a nite 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 is considered,
• the player bets using subjective probabilities, were rare events are assumed
to be impossible (D'Alembert (1754), Menger (1934), Yaari (1987))
54
55. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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) =
1
α
log EP(eαX
) .
55
56. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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) = min{x/α, 1}
Rg(X) = TV aR(X, α) (also called expected shortfall),
Rg(X) = EP(X|X V aR(X, α)).
56
57. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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 it
is possible to dene such a measure Q, generally Q is not a probability measure.
In fact, Q satises
• 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).
57
58. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Yarri's dual approach: capacities and Choquet's integral
Such a measure Q satises 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 the
so-called Choquet's integral.
58
59. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Distortion of values versus distortion of probabilities
0 1 2 3 4 5 6
0.00.20.40.60.81.0
Calcul de l’esperance mathématique
Figure 12: Expected value xdFX(x) = P(X x)dx.
59
60. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Distortion of values versus distortion of probabilities
0 1 2 3 4 5 6
0.00.20.40.60.81.0
Calcul de l’esperance d’utilité
Figure 13: Expected utility u(x)dFX(x).
60
61. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Distortion of values versus distortion of probabilities
0 1 2 3 4 5 6
0.00.20.40.60.81.0
Calcul de l’intégrale de Choquet
Figure 14: Distorted probabilities g(P(X x))dx.
61
62. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Axiomatic approach for risk measures
There are three way to describe risk measure: characterizing natural properties
that 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].
62
63. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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 be
denoted X Y .
In Pascal's approach FX(x) = P(X ≤ x)
X Y ⇐⇒ R(X) ≤ R(Y ) where R(X) = EP(X) = xdFX(x).
63
64. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
More dicult to quantify than to compare
Denition 5. is an ordering relationship if it is reexive (FX FX),
transitive (if FX FY and FY FZ then FX FZ) and antisymmetric (if
FX FY and FY FX then FX = FY ).
Note that the ordering on the set of distribution functions will be extended to the
set of positive random variables (with X ∼ Y if FX = FY , i.e. X
L
= Y ).
Denition 6. satises the additivity axiom if for any risks X, Y and Z such
that X Y , then X + Z Y + Z.
It denotes the invariance of perception in case of a common variation. It might
also be called the linearity axiom.
64
65. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
More dicult to quantify than to compare
Denition 7. satises the continuity axiom (or Archimedean axiom) if for
any 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 8. If satises the continuity and associativity axioms,
X Y ⇐⇒ R(X) ≤ R(Y )
where
R(X) = EP(X) = xdFX(x) =
∞
0
P(X x)dx.
65
66. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
The expected utility approach
In order to answer Saint Petersbourg' paradox, one solution, proposed by
Bernoulli was to introduce a moral value of money, i.e. a nonlinear perception
of gains: he suggests to consider log(1 + X) instead of X. The price of the game
was then EP(log(1 + X)). Analogously, Cramèr suggested to consider
√
X, so
that the price was EP(
√
X).
Hence, the idea was to consider a utility function of gains, u(·), which can
change for all players.
Several mathematicians, for example Laplace, discussed the Bernoulli principle
in the following century, and its relevance to insurance systems seems to have
been generally recognized. In 1832, Barrois presented a fairly complete theory of
re insurance based on Laplace's work on the Bernoulli principle. (Borch
(1974)).
66
67. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
The expected utility approach
Denition 9. satises the independence axiom if for any distribution function
FX, 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 modied when mixing risks with a third one. Recall that
(λX) ⊕ ([1 − λ]Z) = (λX) + ([1 − λ]Z).
67
68. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Example 10. If X, Y are two Bernoulli variables B(2/3) and B(1/3) respectively
, independent,
X =
0 p = 1/3
1 p = 2/3
and 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
68
69. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Proposition 11. If satises the continuity and independence axioms, there
exists a function u with values in R, continuous, strictly increasing, unique up to
an ane transformation, such that
X Y ⇐⇒ Ru(X) ≤ Ru(Y )
where
Ru(X) = EP(u(X)) = u(x)dFX(x).
Proof. 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)).
69
70. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
The expected utility approach
The insurance premium is then obtained by the null utility principle: π(X)
satises
EP(u(π(X) − X)) = 0.
Example 12. With an exponential utility, u(x) = [1 − e−αx
]/α, alors
π(X) =
1
α
log EP(eαX
) .
Note that the exponential utility does not exist for heavy tailed risks.
Example 13. With a quadratic utility, u(x) = x − x2
/2s where x s, then
π(X) ∼ EP(X) +
κ
2
V arP(X).
70
71. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Risk aversion and nance
One can introduce, Arrow-Pratt coecient of absolute risk aversion,
RA(x) = −
u (x)
u (x)
, and the coecient of relative risk aversion, RR(x) = −
xu (x)
u (x)
CARA (Constant Absolute Risk Aversion) means that RA(·) is constant, i.e.
u(x) = −
1
α
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) = −
1
α
exp(−αx), for some α 0.
By solving u(EP(X) − π) = EP(u(X)), using the expression of the Laplace
transform of the Gaussian distribution,
EP(u(X)) = −
1
α
EP(exp(−αX)) = −
1
α
exp −αµ +
α2
2
σ2
,
71
72. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
and
u(EP(X) − π) = −
1
α
exp(−α(µ − π)),
thus, one gets that π =
α
2
σ2
.
• modeling portfolios with lognormal risks and CRRA utility
Assume that log X ∼ 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 −
1
2s
EP(X) −
1
2s
V arP(X),
hence only the mean and the variance matter.
72
73. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Changing probabilities ?
Question: what is probability P ?
Il y a donc une aversion particulière pour l'incertitude liée à l'ignorance. On
préfère avoir un modèle probabiliste que pas de modèle du tout, on préfère évaluer
raisonnablement ses chances de succès, fussent-elles minces, que de n'en avoir
aucune idée. (Ekeland (1991)).
Idée de Ramsey (1931), formalisée par Savage (1972): les individus ne raisonne
pas sous P, la probabilité réelle (inconnue), mais sous une probabilité subjective
Q.
Problème: dicile d'estimer une probabilité d'évènement rare.
Travaux de Selvige (1975): importance des évènements rares aux conséquences
importantes. Approche psychologique du risque: besoin de comparer à des
évènements rares quantiables (taux de mortalité infantile, quinte ush au
pocker...).
73
74. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Changing probabilities ?
Denition 14. est une relation vériant l'axiome de monotonie si
P(X + ε ≤ Y ) = 1 implique X Y , pour tout ε 0.
Proposition 15. Si est une relation d'ordre vériant les axiomes de
continuité, d'additivité et de monotonie, alors il existe une probabilité Q telle que
X Y ⇐⇒ RQ(X) ≤ RQ(Y )
où
RQ(X) = EQ(X) =
∞
0
Q(X x)dx.
74
75. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Using subjective probabilities
Considérons un call européen, dont le payo actualisé est e−rT
(ST − K)+. Le
prix n'est pas EP e−rT
(ST − K)+ .
Le prix d'un call européen proposant de toucher (ST − K)+ à maturité. La
valorisation de l'option, à la date d'aujourd'hui est basée sur la notion de
portefeuille de réplication: deux portefeuilles orant le même payo à une date T
ont nécessairement le même prix aujourd'hui (sinon il serait possible de
constituer une opportunité d'arbitrage).
75
76. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Changing probabilities ?
Considérons le modèle de Cox, Ross Rubinstein (1979), avec un actif sans
risque valant 1 aujourd'hui, et 1 + r dans un an, et un actif risqué valant S0
aujourd'hui, et, dans un an S1, valant soit Su, soit Sd, avec d 1 + r u,
suivant l'état de la nature. Considérons un call européen donnant le droit
d'acheter le sous-jacent à maturité (dans un an) à la valeur K. Le payo dans un
an est alors (S1 − K)+. Construisons un portefeuille α + βS0 permettant de
répliquer la valeur de l'option dans un an:
• si le marché monte, le portefeuille vaudra α (1 + r) + βSu, et l'option
Cu = (Su − K)+
• si le marché baisse, le portefeuille vaudra α (1 + r) + βSd, et l'option
Cd = (Sd − K)+
76
77. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Changing probabilities ?
Dans un marché avec absence d'opportunité d'arbitrage, si ces deux produits ont
la même valeur dans un an, c'est donc qu'ils ont le même prix aujourd'hui. Le
portefeuille qui permet de répliquer le payo de l'option est obtenu en résolvant
α (1 + r) + βSu = Cu
α (1 + r) + βSd = Cd
c'est à dire que
α =
Cu − Cd
S0u − S0d
et β =
1
1 + r
Cu − S0u
Cu − Cd
S0u − S0d
.
Notons au passage qu'il est ainsi toujours possible de constituer un unique
portefeuille de réplication.
77
78. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Changing probabilities ?
Le prix de l'option aujourd'hui s'écrit
α + βS0 =
1
1 + r
1 + r − d
u − d
Cu +
u − (1 + r)
u − d
Cd ,
qui peut s'écrire
π =
1
1 + r
(qCu + (1 − q)Cd) , où q =
1 + r − d
u − d
.
Notons que q ∈ [0, 1], c'est à dire que le prix de l'option est l'espérance
mathématique, sous une probabilité Q appelée probabilité risque neutre du payo
à échéance: π = EQ (payo). Notons que Q n'a rien n'a voir avec la probabilité
dite historique P qu'a le sous-jacent de monter ou de descendre: le prix d'un
payo aléatoire X ne s'écrit pas EP (X).
78
79. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Les paradoxes de Allais et Ellsberg
Allais (1953), l'eet de certitude ou l'eet de sécurité
Choisir entre les deux lotteries suivantes
Loterie 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,
79
80. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Préférer A à B, et D à C (observé empiriquement) viole l'hypothèse
d'indépendance (et même le principe de la chose sûre).
Ellsberg (1961), l'eet d'ambiguité
Choisir entre les deux lotteries suivantes
Loterie A win 1000 si la boule tirée est rouge
Loterie B win 1000 si la boule tirée est bleue,
puis entre
Loterie C win 1000 si la boule tirée n'est pas rouge
Loterie D win 1000 si la boule tirée n'est pas bleue,
Préférer A à B, et C à D (observé empiriquement) viole l'hypothèse
d'indépendance (et même le principe de la chose sûre).
80
81. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
La distortion de probabilités, Yaari (1987)
Example 16. Un exemple de relation d'ordre est la dominance stochastique à
l'ordre 1. X 1 Y si et seulement si une des conditions suivantes (équivalentes)
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 notée V aR dans Denuit Charpentier (2004).
Example 17. Un exemple de relation d'ordre est la dominance stochastique à
l'ordre 2. X 2 Y si et seulement si une des conditions suivantes (équivalentes)
sont satisfaites,
• E(g(X)) ≤ E(g(Y )) pour g croissante et convexe,
• E((X − t)+) ≤ E((Y − t)+) pour t ∈ R,
81
82. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
• pour tout x ∈]0, 1[,
α
0
V aR(X, p)dp ≥
α
0
V aR(Y, p)dp,
• 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 notée T V aR dans Denuit Charpentier (2004).
Denition 18. est une relation vériant l'axiome d'indépendance comonotone
si X Y implique X + Z Y + Z pour tout Z tel que les couples (X, Z) et
(Y, Z) soient comonotones.
Remark 19. X et Z sont comonotones s'il n'existe pas ω, ω tels que
X(ω) X(ω ) et Y (ω) Y (ω ).
Denition 20. est une relation vériant l'axiome de cohérence si pour des
variables X et Y constantes (P(X = x) = P(Y = y) = 1), FX FY implique
x ≤ y.
82
83. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Proposition 21. Si est une relation d'ordre vériant les axiomes de
continuité, d'indépendance comonotone, monotonie, et est compatible avec la
dominance stochastique à l'ordre 1, alors il existe une unique fonction de
distortion croissante g : [0, 1] → [0, 1] telle que
X Y ⇐⇒ Rg(X) ≤ Rg(Y )
où
Rg(X) = xdg ◦ P = g(P(X x))dx =
1
0
F−1
X (1 − p)dg(p).
83
84. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Agenda
• General introduction
Financial risks
• Market risks
• Credit risk
• From variance to Value-at-Risk
Risk measures and capital allocation
• Risk measures: an axiomatic introduction
• Risk measures: convexity and coherence
• Capital allocation: an axiomatic introduction
Risk measures and statistical inference
84
85. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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 diversication does not increase risk.
Example: the Expected-Shortfall is coherent, the Value-at-Risk is not.
85
86. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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 any
λ ∈ [0, 1].
Hence, if a convex measure satises the homogeneity condition, it is coherent.
Remark A natural way to dene a convex measure satisfying the small size
coherent condition is adding a coherent measure a liquidity charge,
Rconvex(X) = Rcoherent(X) + Cliquidity(X).
86
87. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Sets of acceptable risks
Denition 22. Given a risk measure R, a risk X is acceptable if X ∈ AR where
AR = {Y ∈ X such that R(Y ) ≤ 0}.
Conversely,
Theorem 23. Given a set of acceptable risks A, the associated risk measure is
the smallest capital amont m such that X − m is acceptable, i.e.
RA(X) = inf{m ∈ R such that X − m ∈ A}.
Then RAR
(·) = R(·) and ARA
= A.
Proposition 24. If R is a convex risk measure, then AR is convex. Conversely,
if A is convex, then RA is a convex risk measure.
Proposition 25. If R is a positively homogeneous risk measure, then AR is a
positive cone. Conversely, if A is a positive cone, then RA is a positively
homogeneous risk measure.
Example 26. If R(X) = sup{X(ω), ω ∈ Ω}, then AR = {Y, Y ≤ 0}. If
R(X) = F−1
X (α) where α ∈ (0, 1), then AR = {Y, P(Y ≤ 0) ≥ α}.
87
88. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Characterizations of coherent risk measures
Proposition 27. If R is a coherent risk measure, then there exists a set of
probability measures Q such that
R(X) = sup
Q∈Q
{EQ(X)}.
88
89. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Value-at-Risk: going further
Proposition 28. 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.
q
0.80 0.85 0.90 0.95 1.00
0.00.51.01.52.0
Proposition 29. The Value-at-Risk is a coherent risk measure for elliptical
risks.
Proposition 30. For all X, note that
V aR(X, α) = inf{R(X) such that R is coherent and V aR(X, α) ≤ R(X)}.
89
90. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Tail Value-at-Risk (or Expected Shortfall)
Dene
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 best
worst case.
Proposition 31. Tail Value-at-Risk is a coherent risk measure.
90
91. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Agenda
• General introduction
Financial risks
• Market risks
• Credit risk
• From variance to Value-at-Risk
Risk measures and capital allocation
• Risk measures: an axiomatic introduction
• Risk measures: convexity and coherence
• Capital allocation: an axiomatic introduction
Risk measures and statistical inference
91
92. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Some simple example of capital allocation
Let X1, ..., Xn denote a set of risks, and dene the associated portfolio
X = X1 + ... + Xn.
Consider a risk measure R.
The objective is to distribute (or allocate) risk capital k = R(X) of the portfolio
to its components, i.e. compute k1, ..., kn such that k1 + ... + kn = k.
A rst approach would be to consider a proportional capital allocation, i.e.
ki =
R(Xi)
R(X1) + . . . + R(Xn)
· R(X).
92
93. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Capital allocation based on marginal capital cost
Idea from cooperative game theory.
Let S denote a subset of {1, 2, ..., n} (a subportfolio). Dene the associated cost
function,
Γ(S) = R
j∈S
Xj
,
satisfying Γ({1, ..., n}) = R(X) = k.
The marginal cost of risk i is
Mi = Γ({1, ..., n}) − Γ({1, ..., n} {i}) = R
j
Xj
− R
j=i
Xj
,
Problem is that M1 + ... + Mn = k.
93
94. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Capital allocation based on Shapley value
One idea is to use the Shapley value.
Let Si denote all the subsets of {1, 2, ..., n} that contain i.
The Shapley-value of risk i is
ki =
S∈Si
(|s| − 1)!)(n − |s|)!
n!
· [Γ(S) − Γ(S {i})],
where | · | denote the size of the subset (cardinality).
It is simply a weighted average of all possible congurations.
Note that here k1 + ... + kn = k.
This capital allocation has been used by Swiss Re.
94
95. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Capital allocation based on Shapley value
Note that if n = 2,
R(X1 + X2) =
1
2
(R(X1 + X2) − R(X2) + R(X1))
allocation for X1
+
1
2
(R(X1 + X2) − R(X1) + R(X2))
allocation for X2
.
95
96. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Capital allocation based on Aumann-Shapley value
An alternative is to use the Aumann-Shapley value.
Here the idea is to consider sensitivity of marginal cost. Dene the following cost
function, C : [0, 1]n
→ R,
C(ω) = R(ωt
X) = R
n
i=1
ωiXi .
Note that C(1) = R(X) = k.
Dene Ci(ω) =
∂C(ω)
∂ωi
, and dene the Aumann-Shapley value as
ki =
1
0
Ci(u1)du.
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97. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Aumann-Shapley value and capital allocation
Example 32. Consider the variance risk measure, R(X) = E(X) + θvar(X),
and assume that X ∼ N(µ, Σ), then
ki = E(Xi) + θcov(Xi, X) .
Example 33. Consider the standard deviation risk measure,
R(X) = E(X) + θ var(X), and assume that X ∼ N(µ, Σ), then
ki = E(Xi) + θ
cov(Xi, X)
var(X)
.
97
98. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Aumann-Shapley value and capital allocation
Assume that R is (positively) homogeneous , then so is C, i.e. C(λω) = λC(ω),
and thus Ci(λω)Ci(ω), i.e. the Aumann-Shapley value reduces to marginal costs,
ki =
1
0
Ci(u1)du =
1
0
Ci(1) = Ci(1).
Note that this allocation can also be derived directly using Euler's theorem for
homogeneous functions,
C(ω) = ω1
∂C(ω)
∂ω1
+ ... + ωn
∂C(ω)
∂ωn
.
98
99. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Capital allocation using distortion risk measures
If R is a distortion risk measure, R(X) =
∞
0
g(1 − Fx)dx.
Writing R as R(X) = EQX
(X), where dQX/dP = g(1 − FX(X)) then
k = R(X) = EQX
(X) and ki = EQX
(Xi).
99
100. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Capital allocation and dependence
Since E(XY ) = E(X)E(Y ) + cov(X, Y ), we can write for distortion risk measures
ki = EQX
(Xi) = ki = E(Xi · g(1 − FX(x))) = E(Xi) + cov(Xi, g(1 − FX(X))),
since E(g(1 − FX(X))) = 1.
Hence, the capital is maximum if Xi and X are comonotonic, i.e.
ki = E(Xi · g(FX(X))) ≤ E(X+
i · g(FX(X+
))) = R(X+
i ) = R(Xi).
The allocated capital is always lower than the stand-alone risk, i.e. the
allocation belongs to the core (in game theory terms).
Further, one can obtain ki 0 even if R(Xi) 0.
100
101. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Aumann-Shapley value for TVaR
TVaR is a distortion risk measure, R(X) = E(X|X F−1
X (α)) and thus,
ki = E(Xi|X F−1
X (α)).
Aumann-Shapley value for VaR
VaR is a positively homogenous, and marginal costs works, R(X) = F−1
X (α) and
thus,
ki = E(Xi|X = F−1
X (α)).
101
102. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Capital allocation based on expected utility
Consider for instance exponential risk measures, R(X) =
1
θ
log E(eθX
).
The Aumann-Shapley allocation is then
ki =
1
0
E(XieθuX
)
E(eθuX)
du = E Xi ·
1
0
eθuX
E(eθuX)
du
here again, change of measure representation.
Remark 34. If the Xi's are independent, then ki = R(Xi).
102
103. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Capital allocation and decision loss function
Capital allocation can also be seen as the solution of distance minimization
problem.
Consider some distance d(ki, Xi|R, k, X). Then
ki = argminκ(κ, Xi|R, k, X).
Example 35. Consider the following expected value distance,
d(ki, Xi) = E([Xi − ki]2
).
Then ki =
k
n
+ E(Xi) −
E(X)
n
.
Example 36. Consider the following tail VaR distance,
d(ki, Xi|α, k, X) = E([Xi − ki]2
|X F−1
X (α)).
Then ki =
k
n
+ E(Xi|X F−1
X (α)) −
E(X|X F−1
X (α))
n
.
103
104. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Example 37. Consider the following stop loss distance,
d(ki, Xi) = E([Xi − ki]+).
Then ki = F−1
Xi
(FX+ (k)).
Example 38. Consider the following distance on risk measures,
d(ki, Xi) = R(Xi) −
ki
k
(R(X1) + ... + R(Xn))
2
.
Then ki =
R(Xi)
R(X1) + ... + R(Xn)
· k.
104
105. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Axiomatic approach for capital allocation
Based on previous results, a standard assumption is to assume that capital
allocated
• depends on Xi and X,
• but not on the decomposition of the rest X − Xi = j=i Xj.
Denition 39. A capital allocation is a function Λ : R × R → R of two
arguments (x, y) where x is risk Xi and y is the portfolio X.
Denition 40. A capital allocation Λ is called capital allocation with respect to
risk measure R if Λ(X, X) = R(X).
The capital allocated to X is then the risk capital R(X) of X.
105
106. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
The three axioms for capital allocation
The rst axiom is linear aggregation: the risk capital of the portfolio equals the
sum of the risk capital of risks. Hence,
Λ(X1, X) + Λ(X2, X) + . . . + Λ(Xn, X) = Λ(X, X) = R(X).
Denition 41. A capital allocation Λ satises the linear aggregation axiom if
Λ(x, x + y) + Λ(y, x + y) = Λ(x + y, x + y).
106
107. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
The three axioms for capital allocation
The second axiom is diversication: the risk capital for a component should not
exceed the overall risk capital of the risk considered as a stand-alone portfolio.
Hence,
Λ(Xi, X) ≤ Λ(Xi, Xi) for all i = 1, ..., n.
Denition 42. A capital allocation Λ saties the diversication axiom if
Λ(x, y) ≤ Λ(x, x).
107
108. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
The three axioms for capital allocation
The third axiom is continuity: small changes to the portfolio have a limited
eect on the risk capital. Hence,
Λ(Xi, X + εXi) ∼ Λ(Xi, X) if εissmall.
Denition 43. A capital allocation Λ saties the continuity axiom if
lim
ε→0
Λ(x, y + εy) = Λ(x, y).
108
109. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Questions about capital allocation axioms
The existence issue: given a risk measure R, is it possible to derive capital
allocations satisfying those axioms ?
The completeness issue: if the allocation exists, is it unique ? If not, is it possible
to add more axioms ?
The explicit formulation issue: given a risk measure R, is it possible to specify
capital allocation?
109
110. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Existence of capital allocation
Recall that a risk measure is
• (positively) homogeneous if R(λX) = λR(X), λ ≥ 0,
• subadditive if R(X + Y ) ≤ R(X) + R(Y ).
If R is a positively homogeneous and subadditive risk measure such that the
following functional derivative
lim
ε→0
R(X + εZ) − R(X)
ε
exists for every Z, then
Λ(x, y = lim
ε→0
R(y + εx) − R(y)
ε
.
110
111. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Unicity of capital allocation based on R
Based on the three axioms, the risk capital allocation Λ is uniquely determined
as the derivative of the risk measure R at X in direction of risk Xi.
111
112. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Examples of capital allocation: standard deviation
For a risk measure based on standard deviation,
R(X) = E(X) + λ · var(X),
the derivative can be derived simply as
Λ(X, Y ) = E(X) + λ ·
cov(X, Y )
var(X)
.
112
113. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Examples of capital allocation: expected shortfall
Here also an explicit expression can be derived. If
R(X) =
1
1 − α
1
α
VaR(X, u)du =
E(X · I(X VaR(X, α)))
1 − α
(continuous random variable), then
Λ(Xi, X) =
1
1 − α
xi · I(X VaR(X, α))dFXi
(xi).
If P(X = V aR(X, α)) 0, set β =
P(X ≤ V aR(X, α)) − α
P(X = V aR(X, α))
, then
Λ(Xi, X) =
1
1 − α
xi · I(X VaR(X, α))dFXi
(xi) + β xi · I(X = VaR(X, α))dFXi
113
114. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Examples of capital allocation: Value-at-Risk
The VaR is not subadditive, and thus there does not exist a linear, diversifying
capital allocation with respect to VaR. Nevertheless, the directional derivative
lim
ε→0
V aR(X + εXi, α) − V aR(X, α)
ε
might exist for certain portfolios X. This was suggested by Hallerbach (1999),
and works well - under continuity assumptions (Tasche (1999) or Gouriéroux,
Laurent Scaillet (2000)).
Alternative is to write V aR(X, α) = E(X) + λ(X) · var(X), and to use a
covariance allocation:
if R(X)) = E(X) + λ · var(X), then Λ(X, Y ) = E(X) + λ ·
cov(X, Y )
var(X)
.
This technique is very popular in credit risk applications (see Kalkbrener et al.
(2004)).
114
115. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
And nally a similar idea is to write V aR(X, α) = TV aR(X, a(X)) for some a(·)
with values in [0, 1], and to consider capital allocation based on TVaR (see
Overbeck (2000)).
Numerical example based on a simulated portfolio
Consider three lines of business, with rather dierent behaviors,
• an exponential distribution X1 ∼ E(0.03), E(X1) ∼ 33,
• a log-normal distribution X2 ∼ LN(3, 1), E(X2) ∼ 33,
• a Pareto distribution X3 ∼ P(2, 30), E(X3) ∼ 33,
Assume further that the three risks are independent.
115
116. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0 50 100 150 200
0.0000.0100.0200.030 Densities of the three risks
0 50 100 150 200
0.00.20.40.60.81.0
Cumulative distribution function of the three risks
Figure 15: The three lines of business.
116
117. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
risk 1
risk 2
risk 3
Allocation of the pure premium
risk 1risk 2
risk 3
Allocation based on 90%!TVaR
risk 1
risk 2
risk 3
Allocation based on 95%!TVaR
risk 1
risk 2
risk 3
Allocation based on stop!loss distance
risk 1
risk 2
risk 3
Relative (proportional) capital allocation ! VaR 90%
risk 1
risk 2
risk 3
Relative (proportional) capital allocation ! VaR 99%
risk 1
risk 2
risk 3
Relative (proportional) capital allocation ! Tail VaR 90%
risk 1
risk 2
risk 3
Relative (proportional) capital allocation ! Tail VaR 99%
Figure 16: Comparing capital allocations for the three independent lines of business
117
118. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Agenda
• General introduction
Financial risks
• Market risks
• Credit risk
• From variance to Value-at-Risk
Risk measures and capital allocation
• Risk measures: an axiomatic introduction
• Risk measures: convexity and coherence
• Capital allocation: an axiomatic introduction
Risk measures and statistical inference
118
119. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Using a parametric models
A natural idea (that can be found in classical nancial models) is to assume
Gaussian distributions: if X ∼ N(µ, σ), then the α-quantile is simply
q(α) = µ + Φ−1
(α)σ,
where Φ−1
(α) is obtained in statistical tables (or any statistical software), e.g.
u = −1.64 if α = 90%, or u = −1.96 if α = 95%.
Denition 44. Given a n sample {X1, · · · , Xn}, the (Gaussian) parametric
estimation of the α-quantile is
qn(α) = µ + Φ−1
(α)σ, where µ =
1
n
n
i=1
Xi and σ =
1
n − 1
n
i=1
(Xi − µ)
2
119
120. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Using a parametric models
Actually, is the Gaussian model does not t very well, it is still possible to use
Gaussian approximation
If the variance is nite, (X − E(X))/σ might be closer to the Gaussian
distribution, and thus, consider the so-called Cornish-Fisher approximation (see
[?] or [?]), i.e.
Q(X, α) ∼ E(X) + zα V (X), (1)
where
zα = Φ−1
(α)+
ζ1
6
[Φ−1
(α)2
−1]+
ζ2
24
[Φ−1
(α)3
−3Φ−1
(α)]−
ζ2
1
36
[2Φ−1
(α)3
−5Φ−1
(α)],
(2)
where ζ1istheskewnessofX, andζ2
is the excess kurtosis, i.e. i.e.
ζ1 =
E([X − E(X)]3
)
E([X − E(X)]2)3/2
and ζ1 =
E([X − E(X)]4
)
E([X − E(X)]2)2
− 3. (3)
120
