Transcript No Slide Title

Spectral Measures of Risk
Coherence in theory and practice
Budapest – September 11, 2003
Subject of the talk: only finance (and a bit of statistic)
Risk Management Questions
Financial
What do I
measure ?
Statistical
How do I
estimate it ?
Our investigation will be completely devoted to
financial and statistical
questions.
Probabilistic
Computational
What
How can I
hypotheses
The resultscarry
will beout the
should I
computation
however absolutely
general
make ?
(in time) ?
Part 1:
Defining a Risk Measure
The qualitative concept of “risk” and “risk premium”
Everybody has an innate feeling for financial risk ....
... more or less this ......
How to define risk in a quantitative fashion ? ...
Concept of Risk
?
Risk Measure
test
fundamental shared
principles
requirements
(axioms) on the risk
measure
The risk diversification principle
Portfolio A
The aggregation of portfolios has always the effect of reducing or at most leaving unchanged the overall risk.
+
Portfolio B
is less or equal to
=
Portfolio A + B
Risk of ( A + B )
Risk of (A) + Risk of (B)
The diversification principle goes here
Coherent Measures of Risk
In the paper “Coherent measures of Risk” (Artzner et al. Mathematical Finance, July 1999) a set of axioms was proposed as the
key properties to be satisfied by any “coherent measure of risk”.
(Monotonicity) if
(Positive Homogeneity) if
(Translational Invariance)
(Subadditivity)
   then
 ()   ()
 ()   ( )
 (  a)   ()  a
then
0
 (  )   ()   ()
Strange
Value at Risk (VaR): how
it works as
it may seem, this is the question
To compute VaR, we need to specify

most frequently asked to risk managers
A time horizon: for instance one day. It represents the future period over which we measure the risks of a
portfolio

worldwide today
A confidence level: for instance a 5% probability. It represents the fraction of future worst case scenarios
of the portfolio that we want to single out.
The definition of VaR is then:
“The VaR of a portfolio is the minimum loss that a portfolio can suffer in one day in the 5% worst cases”
Or equivalently:
“The VaR of a portfolio is the maximum loss that a portfolio can suffer in one day in the 95% best cases”
Value at Risk (VaR): how it works
The formidable advantages introduced by VaR
Since its appearance, VaR turned out to be a more flexible instrument w.r.t. more traditional measures of risk such as the “greeks”
or “sensitivities”, because VaR is
1.
Universal: VaR can be measured on portfolios of any type (greeks on the contrary are designed “ad hoc” for specific risks)
2.
Global: VaR summarize in a single number all the risks of a portfolio (IR, FX, Equity, Credit, …) (while we need many
greeks to detect them all)
3.
Probabilistic: VaR provides a loss and a probability occurrence (while greeks are “what if” measures, which tell us nothing
on the probabilities of the “if”)
4.
Expressed in Lost Money: VaR is expressed in the best of possible units of measures: LOST MONEY. Greeks have
peculiar and less transparent u.o.m.
A VaR-based portfolio risk report is exceedingly clearer than a greeks-based one
No practitioner in 2003 would ever give up to these advantages anymore
The deadly sin of VaR
Unfortunately however VaR
1.
Violates the subadditivity axiom and so is not coherent
Or equivalently
2.
Violates the diversification principle and so for us it is not a risk measure at all
In other words it may happen that …
VaR = 2
+
VaR = 3
=
VaR = 10
The source of all VaR’s troubles: neglecting the tail
VaR doesn’t care what’s beyond the
threshold.
I do care !
Subadditivity and capital allocation
Due to the lack of subadditivity, VaR appears to be unfit for determining the capital adequacy of a bank.
In a financial institution made of several branches, it is common (or it might be unavoidable for practical reasons) to
perform the risk measurements in each branch separately, reporting the results to a central Risk Management dept.
Capital reserves as if VaR = 10 ?
BANK
VaR = 5
business unit: Equities
VaR = 3
VaR = 2
business unit: Fixed
Income
business unit: Forex
What is the concept of risk of VaR ?
From an epistemologic point of view however,
the main problem of VaR is not its lack of subadditivity
but the very lack of any associated consistent set of axioms
We still wonder what concept of risk Value at Risk has in mind !
A natural question
Is it possible to find coherent measures which are as versatile and flexible as VaR ?
The answer is fortunately YES
(… and they are also infinitely many …)
Expected Shortfall as an improvement of VaR
Definition of Expected Shortfall:
“The ES of a portfolio is the average loss that a portfolio can suffer in one day in the 5% worst cases”
Remember that
“The VaR of a portfolio is the minimum loss that a portfolio can suffer in one day in the 5% worst cases”
ES = the average of worst cases
VaR = the best of worst cases
Expected Shortfall: how it works
...does it make
such a big
difference ?
Is the Expected Shortfall coherent ?
The original definition of Expected Shortfall (also known as Tail Conditional Expectation TCE) is
TCE( X )  ES(OLD ) ( X )  E X X  VaR ( X )
This measure is also NON - SUBADDITIVE in general and so NON - COHERENT.
2001 : new definition of Expected Shortfall
ES  
1



F
u
 ( X ) du
0
This measure is SUBADDITIVE and in fact COHERENT with no hypotheses on the pdf
In this case ES is the average of the pdf defined by the
darkened area only (5% worst)
When TCE and ES differ
0.2
1
0.18
0.9
0.16
0.8
0.14
0.7
0.12
0.6
0.1
0.5
0.08
0.4
0.06
0.3
0.04
0.2
0.02
0.1
0
0
10
20
0
0
10
while TCE is the average of the pdf defined by all the
first two columns (>5% worst)
20
Estimating Expected Shortfall
One can show that ES is indeed estimable in a consistent way as the “Average of 100% worst cases”.
Ordered statistics
(= sorted data from worst to best)
[ N ]
1
ES( N ) ( X )  
X i:N

[ N ] i 1
ES ( X ) 


ES
(
X
)

N 
(N)
Example 1: a subadditivity violation of VaR
Consider a Bond A and suppose that, at maturity, there are three possible cases:
1) No default: it redeems the nominal (100 Euro) and the coupon (8 Euro)
or
2) Soft default: it redeems only the nominal (100 Euro) but not the coupon
or
3) Hard Default: it pays nothing
A subadditivity violation of VaR
Consider another Bond B perfectly identical to A, but issued by a different issuer
Suppose now that the risks of the two bonds happen to be mutually exclusive, in the sense that if issuer A defaults, B does not, and
vice-versa.
Typical case:
ANTICORRELATED RISKS =
RISK REDUCTION IN CASE OF DIVERSIFICATION
VaR dissuades from diversification !
ES advises diversification
Risk Measurement
Initial Value
Bond A
Bond B
Bond A + Bond B
104.6
104.6
209.2
Final Redeem
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
Risk Measurement on a fixed size portfolio (1000 Euro)
Risk Variable
1000 Euro VaR
1000 Euro ES
Bond A
44
618
Bond B
44
618
Bond A + Bond B
484
484
Convexity
violated
not violated
Coherent risk measures display always convex risk surfaces with a unique
global minimum and no local minima
Risk surfaces
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 2: 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 independent identical distributed Bonds }
Bond payoff = Nominal (or 0 with probability 3%)
Question: let’s choose n in such a way to minimize the risk of the portfolio
Let’s try to answer this question with a 5% VaR, ES and TCE (= ES (old)) with a time horizon equal to the maturity of the bond.
“risk” versus number of bonds in the portfolio
ES
TCE
VaR
13 th,
VaR and TCE suggest us NOT TO BUY the
47th bond because it would increase the risk of the
portfolio .... (?)
0.5
The surface of risk of ES has a single global minimum at n= and no
fake local minima.
28thor
ES just tell us: “buy more bonds you can”
0.4
0.3
0.2
0.1
0
0
5
10
15
20
25
Are things better for large portfolios ???...
30
35
40
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
But convexity problems still remain !!!
300
350
400
Part 2:
Defining a space of risk measures
A natural question: ... other coherent measures ?
Is the Expected Shortfall an “isolated exception” or does it belong to a large class of coherent measures ?
Is it possible to create new coherent measures
starting from some given known ones?
The answer is simple and allows to create a wide CLASS of coherent measures.
Given n coherent measuresof risk 1, 2,... n
any convex linear combination
 = 1 1 + 2 2 + ...+ n n ( with k k = 1 and k>0 )
is another coherent measure of risk
Geometrical interpretation
Given n coherent measures, their most general convex combination is any of the points contained in the generated “convex
hull”
If any point represents a given
coherent measure ...
... Then any other point in the
generated “convex hull” is a new
coherent measure of risk
Our strategy ....
We already know infinitely many coherent measures of risk, namely all the possible -Expected Shortfalls for any value 
between 0 e 1
In this way we can generate a new class of coherent measures.
This class is defined
“Spectral Measures of Risk”
Set of all Expected
Shortfalls with (0,1]
Convex hull =
New space of coherent measures
Spectral measures of risk: explicit characterization
 ( p)
Definition: Spectral measure of risk with spectrum
1
M  ( X )     ( p) FX ( p) dp
0
Theorem: the measure M(X) is coherent if and only if
1.
 ( p ) is positive
2.
 ( p ) is not increasing
1
1.
  ( p )dp  1
0
The “Risk Aversion Function” (p)
“(p) decreasing” explains the essence of coherence
Any admissible (p) represents a possible legitimate rational attitude toward risk
coherent
if itsubjective
maps(p) which in turns give her
A rational investor may express her...a
ownmeasure
subjective riskisaversion
throughonly
her own
own spectral measure M
“bigger weights to worse cases”
(p): Risk Aversion Function
It may thought of as a function which
“weights” all cases from the worst to the best
Worst cases
Best cases
Risk Aversion Function (p) for ES and VaR
Expected Shortfall:
• positive
Step function
• decreasing
•
1
  ( p )dp  1
0
Value at Risk:
Spike function
(Dirac delta)
• positive
• not decreasing
•
1
  ( p )dp  1
0
Estimating Spectral Measures of Risk
It can be shown that any spectral measure has the following consistent estimator:
Ordered statistics
(= data sorted from worst to best)
N
M ( N ) ( X )    X i:N i
i 1
Discretized  function
M ( X ) 


M
(
X
)

N 
(N)
Tailoring Risks !
The Expected Shortfall is just one out of infinitely many possible Spectral Measures
ES expresses just a specific risk aversion
But is there a spectral measure which is optimal for all portfolios ?
NO
DIFFERENT
PORTFOLIOS
DIFFERENT
SPECTRAL MEASURES
What are the distinguishing properties of Spectral measures ?
A characterization of spectral measures among coherent measures via additional properties (axioms) would give us not only more
information on Spectral Measures, but also useful information on NON-spectral measures.
Coherent but not Spectral
Spectral Measures
Coherent Measures
A fifth axiom ? A sixth
If “X is worse than Y in probability”, then its risk must be bigger
The measure of risk depends ONLY from the probability distribution of X and it is therefore estimable from
empirical
X.
If X and Ydata
are of“perfectly
correlated”, then the risk of X+Y must be the sum of the risks of X and Y.
(X+Y) = (X) + (Y)
one ?
One can show that the Spectral Measures M are all the coherent measures which satisfy two additional conditions: (Kusuoka 2001,
Tasche 2002)
The first condition may be expressed alternatively as
a.
(Monotonicity w.r.t. “First Stochastic Dominance”)
If Prob(X a)  Prob(Y a), aR then (Y)  (X)
b.
A coherent measure of risk which is NOT
estimable is WCE (Artzner et al. 1997)
(“Estimability from empirical data” or “law invariance”)
It must be possible to estimate (X) from empirical data of X
The second condition is:
c.
(“Comonotonic additivity”)
If X and Y are comonotonic risks, then (X+Y) = (X) + (Y)
A coherent measure of risk which is NOT
Comonotonic Additive is semivariance (Fischer
2001)
Part 3:
A wider class: Convex Measures of Risk
Handle with care
Liquidity Risk: when coherency violations make sense
When an asset position in a portfolio has a size which is comparable with the capacity of the market (market depth) of
absorbing a sudden sell off, we are in presence of liquidity risk. Selling large asset amounts moves market bids downwards.
In this case clearly
Risk (A+A) = Risk (2 A) > 2 Risk (A) = Risk(A) + Risk (A)
Subadditivity fails
Positive Homogeneity fails
 (  )   ()   ()
0
 ()   ( )
Convex Measures of Risk (or Weakly Coherent Measures of Risk)
Heath, Follmer et al., Frittelli et al. define a larger class of measures which allow for possible coherency violations due to
liquidity risk.
(Monotonicity) if
(Translational Invariance)
   then
 ()   ()
 (  a)   ()  a
(Convexity)
(Positive Homogeneity) if
then
, 0  0
 (1)   ( )
(Subadditivity)
 (
(
) 
())  
 (
)
(() )
Weaker Condition
Our point of view: some care is needed
We believe however that in absence of liquidity risk, coherency violations are completely undesired for a measure of risk. The
“small size limit” of a measure of risk should therefore be a (strongly) coherent measure of risk.
lim  ( X )  strongly coherent
X 0
This observation
•
Rules out measures of risk which are intrinsically non coherent in their analytical dependence from pdf’s.
•
Forces a convex measure to carry possible coherency violations only through dimensional constants (typically
the market depth di of each market’s asset Ai)
When each asset’s position is much smaller than its market depth we want the measure to be strongly coherent
Convex measures: a step forward ?
We are persuaded that convex measures of risk may represent a significant step forward in risk market practice provided that
they respect the “small size coherent limit”. Otherwise, trying to take liquidity into account we may jeopardize the properties of
coherency where it should hold in a strong sense.
A convex measure “beyond coherency” is therefore typically NOT a smarter formula which allows coherency violations,
because it should be sensitive to positions sizes.
A convex measure “beyond coherency” is more probably a measure with a coherent analytical structure PLUS a database of
each assets’ market depths to which the position sizes have to be compared in the search for illiquidities.
A natural solution
A natural way to define a convex measure satisfying the small size coherent condition is adding a coherent measure a liquidity
charge
convex ( X )  coherent ( X )  Cliquidity( X )
The liquidity charge C
•
Apply to illiquid assets only and contain their dimensional market depths.
•
Goes to zero in the liquid limit when all position becomes much smaller of its market depth.
We do not propose any specific modelling of the liquidity charge
Part 4:
Coherency and convexity: optimizing spectral measures
Coherency and Convexity in short
Coherency of the Risk Measure
Convexity of the “Risk Surface”
Absence of local minima / Existence of a unique global minimum
Minimizing the Expected Shortfall
Let a portfolio of M assets be a function of their “weights” wj=1....M and let X=X(wi ) be its Profit & Loss. We want to find
optimal weights by minimizing its Expected Shortfall
 1 

min ES ( X ( w))  min  FX ( w) ( p) dp
w
w
 0



In the case of a N scenarios estimator we have
PROBLEM ! A SORTING operation on data makes the dependence NOT EXPLICITLY ANALYTIC. Serious problems for any common
optimizator.


[ N ]


1
(N)
min ES ( X (w))  min
X i:N (w) 

w
w
 [ N ] i 1

Notice: also in the case of non parametric VaR a SORTING operation is needed in the
estimator and the same problem appears
The Pflug-Uryasev-Rockafellar solution
Pflug, Uryasev & Rockafellar (2000, 2001) introduce a function which is analytic, convex and piecewise linear in all its
arguments. It depends on X(w) but also on an auxiliary variable 
 ( X ( w),  )    
1

E  X ( w)   

In the discrete case with N scenarios it becomes
1
(N)
 ( X ( w), )    
N
N
   X (w)
i 1
Notice: the SORTING operator on data has disappeared. The dependence on data is
manifestly analytic.

i
Properties of : the Pflug-Uryasev-Rockafellar theorem
Minimizing  in its arguments (w,) amounts to minimizing ES in (w) only
(w) and ES(w) coincide but just in the minimum !
min  ( X ( w),  )  min ES ( X ( w))
w,
w
Moreover the  parameter in the extremum takes the value of VaR(X(w)).
arg min  ( X (w), ) VaR ( X (w))

The auxiliary parameter in the minimum becomes the VaR
Properties of  - linearizability of the optimization problem
A convex, piecewise linear function is the easiest kind of function to minimize for any optimizator. Its optimization problem can
also be reformulated as a linear progamming problem
It is a multidimensional faceted surface ... some kind of multidimensional diamond with a unique global minimum
Minimizing a general Spectral Measure M
The “SORTING” problem appears in the minimization of any Spectral Measure
 1


min M  ( X ( w))  min   ( p) FX ( w) ( p) dp
w
w
 0





N


(N)
min M  ( X ( w))  min i X i:N (w) 
w
w
 i 1

Generalization of the solution of Pflug-Uryasev-Rockafellar
Acerbi, Simonetti (2002) generalize the function of P-U-R to any spectral measure. Also in this case it is analytic, convex and
piecewise linear in all arguments. In general it depends however on N auxiliary variables i
d
 ( X ( w), ( ))   d

d
0
In the discrete case it becomes
1
1


  ( )  E ( )  X  



N

1

(N)
 ( X (w), )    j j  j    j  X i (w) 
j i 1
j 1



N
Properties of the generalized 
Minimizing  in all parameters (w,) amounts to minimizing M in (w)

min
 ( X ( w), )  min M  ( X ( w))

w,
w
Moreover, in the extremal, k takes the value of VaR(X(w)) associated to the quantile k/N.
arg min  ( X (w), ) VaR k ( X (w))
k
N
Part 5:
Some hints for an internal model ?
Coherent Measures – based internal models ?
Spectral measures are
1.
Universal
2.
Global
3.
Probabilistic
4.
Expressed in lost money
5.
Coherent (namely real risk measures)
But also at last
Are there still good reasons to use VaR ?
Let us see some issues more related to practical risk management aspects
Spectral Measures: worse statistical properties ?
Definitely not
1.
The statistical error of Spectral Measures is easily computable and is by no means worse in general than
VaR’s (Acerbi, Meucci, Tasche: in preparation).
2.
It is not true that “ES is Extreme Value Theory and VaR not”. They are probably both EVT in most cases,
and this depend mostly on the size of the chosen confidence levels.
More difficult probability modelling ?
No, identical situation to VaR’s.
One uses for spectral measures exactly the same models used in a VaR engine (parametric techniques, montecarlo,
historical bootstrap, subjective scenarios, stress tests, … )
The risk measure computation is just the very last step of a long assembly line and requires the same input data.
VaR
Probabilistic
hypotheses
Portfolios distributions
Spectral Measures
More serious computational problems ?
No, essentially the same situation as with VaR, however
subadditivity allows to split the “firmwide” capital allocation problem into subpreblems in separate subportfolios, which
can drastically reduce the computational complexity but giving nevertheless prudential extimates.
BANK
Spectral Measure
Spectral Measure
Spectral Measure
business unit: Equities
business unit: Forex
business unit: Fixed
Income
Optimization is more difficult ?
The contrary is certainly true !
Coherence in optimization problems implies convexity of risk measures.
Optimizing VaR on a portfolio of 100 assets may turn out to be a formidable NON-CONVEX problem of huge
complexity.
Optimizing a Spectral Measure on a portfolio of thousands of assets is a problem that can always been faced since it is
1.
CONVEX
2.
LINEARIZABLE
Pflug, Rockafellar, Uryasev, (2000 –2001), Acerbi, Simonetti (2002)
However, we personally don’t believe that VaR has been a source for systemic risk in recent crises. Panic selling had
been invented long before !!
Different Risk Measures in any bank ?
Regulating a bank industry where any bank could in principle use a different risk measure would prompt complications that
are beyond the scope of our talk.
We note however that
1.
No measure of risk fits any portfolio of a generic financial institution. The risk aversion function is a precision
instrument which allows to better detect the risks of portfolios in different business styles (e.g.: insurance portfolios
from bank portfolios …).
2.
The possibility of providing market players with different risk measures could cancel the suspect that a unique
measure of risk could prompt “flock effects” in the case of market crises.
Conclusions
 The space of Spectral Measures M provides the representation of a huge class of coherent measures suitable for
applications. Coherent measures which are not spectral display properties which are undesirable in real life finance.
 Any coherent measure of this space is in one-to-one correspondence with any possible rational aversion to risk of an
investor.
 For any spectral measure M a consistent estimator is available based on empirical data.
 The practical application of any spectral measure is elementary and the conversion of a VaR engine straightforward.
 Optimization of spectral measures is a convex and linearizable problem.
 There’s nothing special with ES. It is just one out of infinitely many other spectral measures.