Download - OptiRisk Systems

Coherent Measures of Risk
in everyday market practice
Carlo Acerbi
4th Annual CARISMA conference, London, July 2 2008
Motivations
The present talk is a guided tour on the recent (sometimes very technical)
literature on Coherent Measures of Risk (CMRs)
Our purpose is to overview the theory of coherent measures in the perspective of
practical risk management applications. We will therefore focus on those results
of the theory which help understanding what coherent measures can today be
considered as realistic candidate alternatives to VaR in the market practice.
This has also been the spirit of some research done in these years together with
some collaborators of mine (Tasche, Simonetti)
Part 1:
Coherency and VaR shortcomings
Coherence axioms
Artzner et al (1997) established an axiomatic framework to define what are the
characteristics defining financial risk and the requirements a (coherent) measure
of risk has to satisfy
X,Y : portfolios' profit & loss r.v.' s
ρ is said to be ' coherent' if :
X Y
X , Y
 ρ( X )  ρ(Y )
ρ( X  Y )  ρ( X )  ρ(Y )
X , h  0
X , a  
ρ(hX )  hρ( X )
ρ( X  a)  ρ( X )  a
monotonicity
subadditivity
homogeneity
tr. invariance
VaR’s lack of coherence
VaR is a statistics that does not satisfy the axiom of subadditivity.
Recall that subadditivity is related to the “risk diversification principle”. The quantity
H(X, Y; ρ)  ρ(X )  ρ(Y )  ρ(X  Y )
represents the ‘hedging benefit’ (or equivalently the ‘capital relief’) in term of risk
, which comes from merging portfolios X and Y together
The problem of VaR is that for some portfolios X and Y one may have a negative H,
which is nonsensical
 X,Y |
H(X , Y;VaR)  0
Examples of VaR subadditivity violations can easily be found in literature.
Example 1: two bonds with mutual exclusive defaults
Initial Value
Bond A
Bond B
Bond A + Bond B
104.6
104.6
209.2
Final Payoff
Final Event
Hard default B
Soft Default B
Hard default A
Soft Default A
No default
Probability
3%
2%
3%
2%
90%
Bond A
108
108
0
100
108
Bond B
0
100
108
108
108
Bond A + Bond B
108
208
108
208
216
Bond A + Bond B
101.2
101.2
Subadditivity
violated
not violated
Risk Measurement
Risk Variable
5% VaR
5% ES
Bond A
4.6
64.6
Bond B
4.6
64.6
Example 2: a subadditivity violation of VaR with two normal pdfs
A somehow puzzling example is the following. Take two standard normal r.v.’s:
X N(0,1), Y N(0,1)
Suppose they are dependent through a copula with density
δ(q  x  y )

C(x, y )   δ(x  y )

0

0  x, y  q
q  x, y  1
else
for some q(0,1), then it is easy to show that choosing q=p+
(0   1)
VaRp(X+Y)  VaRp(X) + VaRp(Y)
For example, choosing p=.01, =.0001 we obtain
VaRp(X) = VaRp(Y)  2.4
VaRp(X+Y)  5.2 > 2.4 + 2.4 = 4.8
But VaR should be subadditive with normals: where’s the catch ?
Convex risk surfaces
The ‘risk surface’ of a coherent risk measure is always convex.

Π(w) 
portfolio
ρ - risk surface
ρ
coherent 

i
wi X i


w  ρ(Π(w))

ρ(Π(w))

convex in w
CRMs minimization is convex optimization.
VaR minimization, being not convex is often plagued by the presence of local
nonsensical minima.
Risk surfaces of Example 1
Coherent risk measures display always
convex risk surfaces with a unique global
minimum and no local minima
70
ES
60
50
40
30
VaR
20
10
0
0
100% A
0.1
0.2
0.3
0.4
0.5
0.6
50%-50%
Non-coherent measures display in general
risk surfaces affected by multiple (local)
minima
0.7
0.8
0.9
1
100% B
Example 3: a simple prototype portfolio
Consider a portfolio made of n risky bonds all of which have a 3% default
probability and suppose for simplicity that all the default probabilities are
independent of one another.
Portfolio = { 100 Euro invested in n i.i.d. Bonds }
Bond payoff = Nominal (or 0 with probability 3%)
Question: let’s choose n in such a way as to minimize the risk of the portfolio
Let’s try to answer this question with a 5% VaR, ES and TCE on a time horizon
equal to the maturity of the bond.
VaR, ES and TCE
VaRq ( X )   sup s | FX (s)  q 
not coherent
q
1
ESq ( X )    FX (s) ds
q0
TCEq ( X )   E X | X  VaRq ( X )
In the example, q=5%
coherent
‘almost’ coherent
coincides with ES on continuous pdf
“risk” versus number of bonds in the portfolio
ES
TCE
VaR
The surface of risk of ES has a single global
minimum at n= and no fake local minima.
0.5
VaR and TCE suggest us NOT TO
BUY the 13th, 28thor 47th bond
because it would increase the
0.4
risk of the portfolio .... (!?!?!?)
ES just tells us: “buy as many bonds as you can !”
0.3
0.2
0.1
0
0
5
10
15
20
25
30
35
40
Are things better for large portfolios ???...
45
50
...but for large n the pdf should be normal and VaR coherent ... (!!!)
Notice that the pdf really becomes
normal-shaped for large N
... or not ?
0.1
ES
TCE
VaR
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
50
100
150
200
250
300
350
But convexity problems still remain !!!
400
The importance of convex optimization
Risk minimization is intrinsically a convex program, because convexity derives from
the diversification principle.
Minimizing risks through VaR means
• dealing with immensely more complicated computational task, often unsolvable
for sufficiently complex and large portfolios
• having to do with a structure of local minima whose complexity does not contain
any risk information, but only distortions induced by the adoption of a statistics
which is unfit to measure risk.
Sacrificing the huge computational advantage of convex optimization for
the sake of VaR fanatism only is pure masochism
As a matter of fact no bank in the world has probably ever really performed a VaR
minimization in their real portfolios unless in gaussian models (where VaR  ),
while nowadays with coherent measures, we know algorithms efficient enough
for optimizing large and complex portfolios.
General (robust) representation of CMRs
The most general CMR can be expressed as (Artzner et al. 97, Delbaen 2000)
ρ( X )  supQQ EQ [ X ]
Where Q is a family of probability measures.
This representation gives enormous freedom. For a practitioner it is impossible to
find his way in this huge class of measures.
Moreover, this representation introduces a philosophical impasse, because it is
written in a language which is manifestly subjectivist, whereas the vast majority
of practitioners behave as objectivists when they manage risk …
Only a subjectivist can in principle exploit all the freedom left in this representation.
The objectivist paradigm
The main heritage of the advent of VaR is probably the objectivist’s attitude a risk
manager has learned to take
1.
Model and fit the (real) probability distribution of your portfolio
2.
Compute the quantile (VaR)
Accepting this paradigm, we want to replace step 2 of this assembly line with
suitable CMRs instead of VaR
Notice that the (overwhelmingly larger) work done in the setup of 1 can be left as
it is when replacing VaR with any other measure.
A change of paradigm (from obj. to subj.) is very difficult to happen for the time
being in risk mgmt practice.
The “advantages of VaR”
The main advantages introduced in risk management by VaR are
1.
VaR is a universal measure (= applies to any kind of risk)
2.
VaR is a global measure of risk (= “sums” different risks into a single number)
3.
VaR is probabilistic (= provides probabilistic info on the risk)
4.
VaR is expressed in units of “lost money”
A glance is sufficient to realize that these features have in fact nothing to do with VaR
itself, but rather with the objectivist paradigm, irrespectively of the (monetary)
measure eventually computed
Part 2:
The importance of estimability
Law-Invariant CMRs
The property that makes a CRM suitable for objectivists is law-invariance (LI)
(Kusuoka, 2001)
Given a prob. measure P, a CMR is said to be law-invariant if (X) is in fact a
functional of the distribution function FX(x) only
ρ(X )  ρ[FX ()]
Or in other words if it has the same value on variables with the same pdf
FX (t )  FY (t )
t

ρ(X )  ρ(Y )
This property must not be taken for granted: the class of CMRs contains a lot of
non-LI measures (WCE for instance ...).
Law-Invariance means “estimability from data”
LI is a necessary and sufficient condition for a CRM to be estimable from data
Proof (necessity):
Suppose X and Y have the same pdf
Given two samples of N iid realizations {xi} and {yi} and an estimator ρ̂
N 
ˆ(xi )  ρ( X )
ρ
N 
ˆ(y i )  ρ(Y )
ρ
But for N we also have
ˆ(xi )  ρ
ˆ(yi )
ρ
because for large N the two samples are indistinguishable. So, (X)=(Y)
Proof (sufficiency): just define
ˆ(xi )  ρ[Fˆ(xi )]
ρ
The first choice to make
CMRs
LI CRMs
objectivists
subjectivists
Popular examples of LI CMRs
• Expected Shortfall (Acerbi and Tasche 2001, Uryasev and Rockafellar 2001)(ES
aka CVaR aka AV@R aka ...) with (0,1)
ESα ( X ) 
1 α
VaRp ( X ) dp
α 0

1 α 
FX (p) dp
α 0
 " the average loss in the α worst cases"
• CMRs based on one-sided moments (Fisher, 2001)
ρp, a( X )  E[X ]  a ( X  E[X ])
p
For a=1, p=2, one recognizes a coherent definition of “semivariance”
Part 3:
Spectral Measures of Risk
Comonotonic Additivity
Comonotony between two r.v.’s is the strongest form of positive dependence (the
‘copula maxima’)
Adding together two comonotonic risks X and Y provides NO HEDGING AT ALL in
any circumstance, because the two risks can be expressed as two monotonic
functions of a common risk driver Z.
It is therefore very natural to impose that in this particular circumstance the
hedging benefit reaches its limiting case H(X,Y;)=0
A measure of risk  is said to be Comonotonic Additive (CA) if
X , Y comonotonic 
ρ(X  Y )  ρ(X )  ρ(Y )
Also this condition is not generally enforced by the axioms of coherence.
Diversification principle revisited – cheating strategies
In a context of capital adequacy the diversification principle is fully encoded in the
couple of requirements
1.
Subadditivity
2.
Comonotonic Additivity (as a limiting case of subadditivity)
To support this point of view it suffices to observe that if any of the two conditions is
violated by a measure of risk used for capital adequacy, there exists a “cheating
strategy” able to reduce the capital requirement without any real risk reduction.
1.
“Split your portfolio into suitable subportfolios and compute capital adequacy on
each one …”
2.
“Merge your portfolio with the one of new ‘comonotone’ partners and compute
capital adequacy on the global portfolio”
Law-invariant, Comonotonic Additive CRMs
Kusuoka (2001) (see also Follmer-Schied theorem 4.87) showed that the class of
LI CA CRMs is given by all convex combinations of possible ES’s at different
confidence levels
ρμ ( X )   dμ(p) ESp ( X )
dμ any measure on [0,1]
Acerbi (2001) introduced the same class of CMRs calling them
“spectral measures of risk”
1
ρφ ( X )    φ(p) FX (p) dp
0
which was shown to be a CMR iff the “risk spectrum” :[0,1] satisfies
1.
0
2.
 =1
3.
 weakly decreasing
Law-invariant, Comonotonic Additive CRMs, (cont’d)
It was soon realized that the class of LI CA CRMs = ‘Spectral Measures’ essentially
coincided with the “distortion measures” dating back to the works of
Denneberg (1990) and Wang (1996) in actuarial math, and in the context of
premium principles (instead of coherence axioms)
Spectral Measures
CMRs
LI CRMs
spectral
measures
CA CRMs
Understanding spectral measures
1.
0
2.
 =1
3.
 weakly decreasing
1
ρφ ( X )    φ(p) FX (p) dp
0
The spectral measure with spectrum  is the (p)-weighted average loss in
all cases (p-quantiles p[0,1]) of the portfolio.
Subadditivity (through condition 3.) imposes to give larger weights to worse cases.
In this class it is easy to notice ES (flat spectrum with domain [0,]) and even
VaR which however is not a CMR because it fails to satisfy 3. (Dirac-
spectrum peaked on ).
Part 4:
Coherent Estimators
Coherent estimators for LI CMRs


Given a market of risks X  X (1) ,  , X (W ) and N iid realizations xi distributed
according their joint pdf FX() one can study the estimators of a LI CMR 

 N 
ˆ(xi )  ρ( X )
ρ
and ask whether coherency is a property that is valid also for any finite N.
A sufficient condition for having ‘canonical estimators’ which are CMRs for any finite
N is very simple: a coherent estimator for finite N is obtained if we apply the
CMR  to the empirical distribution defined by the realizations (or ‘joint empirical
pdf’)
(N )
ˆ
X
F

1
(t ) 
N

ˆ
ρ(xi )  ρ[FX(N)()]
N
W
  θ (t
i 1 K 1

(k )
 X i(k ) )
ˆ coherent N
ρ
Canonical estimators
Applying ES to the empirical pdf one obtains the canonical estimator
(N )
α
ES

 1  [Nα]
  X i:N  Nα  [Nα] X[Nα]1:N 
(X ) 
Nα  i 1

Where Xi:N is the ith “ordered statistics”, namely the ith worst scenario in the sample.
This formula is a literal translation of “average of losses in the worst N cases”
For a general spectral measure the canonical estimator is
N
ρφ(N )( X )   X i:N φi
i 1
i/N
with
φi 
 φ(p) dp
(i-1 )/N
Part 5:
Optimization: exploiting convexity
Optimization of ES – the method of Pflug, Rockafellar, Uryasev
Uryasev et al. (2000, 2001) develop a formidably efficient procedure for the
minimization of ES, circumventing the problem of dealing with ordered
statistics (read: sorting operations).


Theorem: let X (w) a prtf with weights w . Define
Then


1
α (X (w), ψ)  ψ  E[ψ  X (w)]
α
1.


ESα (X(w))  min α (X (w), ψ)
2.


arg min Γα (X(w), ψ)  VaRα (X(w))
3.


min
ESα (X(w))  min

 Γα ( X (w), ψ)
ψ
ψ
w
ψ,w
the method of Pflug, Rockafellar, Uryasev (cont’d)
In a N-scenarios pdf this problem is a nonlinear convex optimization of the
form

 

1 N
(N )
min
Γ
(
X
(
w
),
ψ
)

min

ψ

(ψ  X i (w)) 




α
w ,ψ
w ,ψ
N α i 1



wΩ
ψ
Notice the absence of sorting operations (read: ordered statistics).
The objective function is piecewise linear in  and w
the method of Pflug, Rockafellar, Uryasev (cont’d)
But the problem can be mapped again into a linear optimization of the form

1 N 
min

  ψ 
 zi 
w ,ψ , z
Nα i 1 


z i  ψ  X i (w)
i  1,  , N

wΩ
ψ
zi  0
Where linearity has been bought at the price of introducing n new variables z
Extension to general spectral measures
Acerbi and Simonetti (2002) extended the PRU method to a general spectral
measure.
The objective function in this case takes the form
1



dφ
Γφ[X (w), ψ]   dt
t ψ(t )  E[ψ(t )  X (w)]  φ(1) E[X (w)]
dt
0


and therefore, in the general case the additional parameter is a whole function
(t), leading in general to a variational problem
Nonlinear convex program
The N-scenarios optimization problem can be cast again into a nonlinear convex
program
(N )
φ
min
  Γ
w ,ψ

w Ω

ψ  J
N
N
 




( X (w), ψ)  min
   Δφ j  jψ j   (ψ j  X i )   φN  X i 
w ,ψ
i 1
i 1


 jJ

Linear program
...or into a linear program in a (generally) huge number of variables
N
N




min
    Δφ j  jψ j   z ij   φN  X i 
w ,ψ , z
i 1
i 1


 jJ


z ij  ψ j  X i (w)

wΩ

ψ  J
z ij  0
i  1,..., N;
jJ
Conclusions
The class of CMRs is too large. Unless an unlikely “subjectivistic revolution” takes
place among risk practitioners, they will concentrate only on the Law-Invariant
subclass
All LI CRMs can replace VaR in a risk management system, leaving unaltered most
of it. They perfectly fit the objectivist paradigm and can share the same
probabilistic hypotheses made for VaR
Among this subclass the further natural requirement of comonotonic additivity,
identifies the family of ‘spectral measures of risk’ or ‘distorsion measures’.
Spectral measures of risk, and in particular ES are gaining popularity because for
these measures there exist consistent, coherent estimators and efficient convex
optimization algorithms.
As a matter of fact, optimization of large-complex portfolios is nowadays a realistic
task to perform, under any probability measure modeling, within the class of
spectral measures. Similar tasks were simply impossible with VaR.
Suggested readings
• Follmer and Schied: Stochastic Finance, 2nd ed., chapter 4 (2004)
• Acerbi: “Coherent representations of subjective risk aversion”, in “Risk measures for
the 21st century”, (2004)
And reference therein, in particular of the works of the following groups of researchers
Artzner, Delbaen, Eber, Heath,
Kusuoka
Uryasev, Rockafellar, Pflug et al.
Follmer, Schied,
Acerbi, Tasche, Simonetti et al.
Wang, Denneberg, et al
Fisher
others …