Transcript No Slide Title

Portfolio Optimization with Spectral Measures of Risk
Carlo Acerbi and Prospero Simonetti
Torino – January 30, 2003
Outline of the talk
1.
What is a Spectral Measure of Risk ?
2.
Coherency and Convexity
3.
Minimization of Expected Shortfall (ES)
4.
Minimization of Spectral Measures
5.
Risks and Rewards: are they really two orthogonal dimensions ?
Part 1:
What is a Spectral Measure of Risk ?
Coherent Measures of Risk
In “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
   then
 ()   ()
(Translational Invariance)
 ()   ( )
 (  a)   ()  a
(Subadditivity)
 (  )   ()   ()
(Positive Homogeneity) if
then
0
The VaR vs ES debate
Value at Risk (for a chosen x% confidence level and time horizon) is defined as
“The VaR of a portfolio is the minimum loss that a portfolio can suffer in the x% worst cases”
Expected Shortfall is defined as:
“The ES of a portfolio is the average loss that a portfolio can suffer in the x% worst cases”
The debate arises since ES, turns out to be a Coherent Measure of Risk while VaR is well known to be a notCoherent Measure of Risk
VaR = the best of worst cases
ES = the average of worst cases
Spectral Measures of Risk:
Let’s consider the class of Spectral Measures of Risk defined as
1
M  ( X )     ( p) FX ( p) dp
0
where the ”Risk Spectrum”
is an arbitrary
 ( preal
) function on [0,1]
They include in particular:
ES with:
VaR with:
 ES ( p) 
1

 (  p )
Heavyside Step Function
VaR ( p)   ( p   )Dirac Delta Function
Spectral Measures of Risk
Theorem: the Spectral Measure of Risk
1
M  ( X )     ( p) FX ( p) dp
Is coherent if and only if its Risk Spectrum
0
satisfies
1.
 ( p)
is positive
1.
 ( p)
is decreasing
1
1.
  ( p )dp  1
0
 ( p)
The “Risk Aversion Function” (p)
Any admissible (p) represents a possible legitimate rational attitude toward risk
A rational investor may express her own subjective risk aversion through her own subjective (p) which in turns give her
own spectral measure M
(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
• 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)
Part 2:
Coherency and Convexity
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
An interesting prototype portfolio
Consider a portfolio made of n risky bonds all of which have a 2% 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 2%)
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 vs VaR vs TCE
0.25
The surface of risk of ES has a single global minimum at n= and no
fake local minima.
0.20
ES just tell us: “buy more bonds you can”
0.15
0.10
0.05
0.00
0
20
40
60
80
100
120
-0.05
VaR and TCE suggest us NOT TO BUY the 6 th, 36thor
83rd bond because it would increase the risk of the
portfolio .... (?)
Number of Bonds
ES
VaR
Are things better for large portfolios ???...
TCE
On large portfolios the same messy pattern occurs on and on ...
Notice that a n=320 portfolio has a smaller VaR than a n=400.
Big n ... same pattern
ES vs VaR vs TCE
0.020000
0.018000
0.016000
0.014000
0.012000
0.010000
0.008000
0.006000
0.004000
0.002000
0.000000
200
250
300
350
400
450
500
550
Number of Bonds
ES
VaR
TCE
...maybe there’s really some tricky risk in the 36th bond !
ES vs VaR vs TCE
0,35
If we use a 3% VaR instead of a 5% VaR, the “dangerous bond” is not
the 36th anymore, but the 28th.... (!?)
0,30
0,25
0,20
0,15
0,10
0,05
0,00
-0,05
0
20
40
ES
60
VaR
80
TCE
100
120
Coherency and convexity
The lack of coherence of VaR and TCE is the reason why their risk surfaces are wrinkled displaying meaningless local minima.
In such a situation, an optimization process always selects a wrong local solution, irrespectively of the starting point.
In real life finance, on large and complex portfolios, such local minima for VaR surfaces are the rule rather than the exception. In
other words, the local minima are not due to the simplicity of our chosen portfolio. They always surround a diversified global
optimal solution.
Even though manifest VaR subadditivity violations are very rare to happen when adding large (quasi-gaussian) portfolios, it is
nevertheless true that on the same large portfolios marginal VaR systematically fails to properly assess the change of risk
associated to buying or selling a single asset.
Part 3:
Minimization of Expected Shortfall
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
The role of the auxiliary variable 

(N)
1
( X ( w), )    
N
N





X
(
w
)

i
i 1
The auxiliary variable is introduced to SPLIT the “5%” worst scenarios from the remaining “95%”.
It is thanks to this variable that the data SORTING disappears.

X1:N
X2:N
X3:N
.......
XN:N
.......
In the minimization process  places on the specified quantile and gets the value of VaR.
XN:N
Application – unconstrained ES minimization
Asset
Asset 1
Asset 2
Asset 3
Portfolio
Weight
-0.53%
47.31%
43.87%
8.82%
100.0%
Return
10.23%
20.78%
23.34%
16.01604%
Standard
Deviation
16%
18%
29%
11.4%
Expected
Shortfall
1.31%
1.39%
2.46%
0.86222%
Part 4:
Minimization of Spectral Measures of Risk
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
The role of 

(N)

1 N

( X (w), )    j j  j    j  X i (w) 
j i 1
j 1



N
The N auxiliary variables are needed to separate completely from one another all the ordered scenarios Xi:N
In the case of a general spectral measures in fact, splitting the data sample into TWO SUBSET is not enough (as in the case of
ES)
1
X1:N
2
X2:N
3
X3:N
.......
k
Xk:N
N
.......
In the minimization any k goes to the quantile k/N. The  vector separates all scenarios X.
XN:N
Part 5:
Risks and Rewards: are they really two orthogonal axis ?
An elementary observation ...
A generic Spectral Measure weights all scenarios of a portfolio from the worst to the best with decreasing weights (decreasing
risk aversion function).
It therefore weights at the same time RISKS and REWARDS in an integrated way.
M
(N)
N
( X ( w))  i X i:N ( w) 
i 1
....
-5.72%
-4.94%
-3.21%
+1.23%
+2.34%
+3.03%
+4.92%
....
....
Φ(n)
Φ(n+1)
Φ(n+2)
Φ(n+3)
Φ(n+4)
Φ(n+5)
Φ(n+6)
....
=
- Σneg.Φ(i) X(i)
+
- Σpos. Φ(i) X(i)
Minimizing a Spectral Measure amounts to a certain MINIMIZATION of RISKS captured by the negative contribute and a certain
MAXIMIZATION of REWARDS captured by the positive contribute.
The simplest example ...
Take for instance the case of a Spectral Mesure obtained as convex combination of ES(X) and ES100%(X)= - average(X)
M  ( X )   ES ( X )  (1   ) ES1 ( X )
• This is a particular family of Spectral Measures with parameter λ between 0 and 1:
• for λ=1 it is ES with confidence level 
• for λ=0 it reduces to (- return)
•Minimizing this spectral measure already amounts to minimizing Expected Shortfall at confidence level  and maximizing the
return at the same time.
Optimal portfolios for different  values
Efficient Frontier
26%
Expecte d Return
24%
22%
20%
=1
=0
18%
16%
14%
12%
10%
0.0%
0.5%
1.0%
1.5%
Expected Shortfall
2.0%
2.5%
3.0%
Integrated Markowitz problem
One shows in fact that (Acerbi, Simonetti, 2002):
minimizing
M  ( X (w))
with no constraints, for any λ
amounts to minimizing
ES (X )
with constrains, for any specified return
value
Risk Minimization and Return Maximization cannot be disentangled. Given an optimal portfolio there always exist a Spectral Measure for which
that portfolio is a “minimal risk portfolio”.
General result
Expected return
More generally one can show that any point in the efficient frontier in the “Markowitz plane” of abscissa M represents also the
unconstrained minimal solution for another spectral measure M,.
• constrained optimal prtf for M
• unconstrained optimal prtf for M,
M ( X (w))
Where for some 
M , ( X (w))   M ( X (w))  (1   )  X (w) 
Conclusions
• The minimization problem of a Spectral Measure of Risk is a convex problem, but dramatic computational problems are
encountered if a straightforward approach is adopted.
• An extension of the P.R.U. methodology however allows to exactly convert the minimization problem into the minimization of
an analytic, piecewise linear and convex functional.
• Complexity can be further reduced by an exact linearization of the problem.
• Standard linear optimizators (say CPLEX) allow to face in an efficient way the optimization problem of any Spectral Measure,
under any probability distribution function for large size portfolios.
• Splitting an optimization problem in “Risk Minimization” and “Returns Maximization” is arbitrary. All “optimal portfolios” in
Markowitz-like efficient frontiers are in fact absolute unconstrained minima of other Spectral Measures. The trade-off between
risks and reward is already taken into account in the choice of the risk measure itself.
References
P. Artzner, F. Delbaen, J.M. Eber and D. Heath, 1999, “Coherent Measures of Risk”
R.T Rockafellar and S.Uryasev, 2000, “Optimization of Conditional Value-at-Risk”
C. Acerbi, 2001, “Risk Aversion and Coherent Risk Measures: a Spectral
Representation Theorem”
C. Acerbi, P. Simonetti, 2002: Portfolio Optimization with Spectral Measures of Risk