1. Portfolio Optimization with Spectral Measures of Risk
Carlo Acerbi and Prospero Simonetti
Torino – January 30, 2003

2. Outline of the talk What is a Spectral Measure of Risk ?
Coherency and Convexity
Minimization of Expected Shortfall (ES)
Minimization of Spectral Measures
Risks and Rewards: are they really two orthogonal dimensions ?

3. Part 1:
What is a Spectral Measure of Risk ?

4. 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
(Positive Homogeneity) if then
(Translational Invariance)
(Subadditivity)

5. VaR = the best of worst cases ES = the average of worst cases
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 not-Coherent Measure of Risk
VaR = the best of worst cases
ES = the average of worst cases

6. Spectral Measures of Risk:
Let’s consider the class of Spectral Measures of Risk defined as
where the ”Risk Spectrum” is an arbitrary real function on [0,1]
They include in particular:
ES with: Heavyside Step Function
VaR with: Dirac Delta Function

7. Spectral Measures of Risk
Theorem: the Spectral Measure of Risk
Is coherent if and only if its Risk Spectrum satisfies
is positive
is decreasing

8. 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

9. Risk Aversion Function (p) for ES and VaR
Expected Shortfall:
Step function
positive
decreasing
not coherent
Value at Risk:
Spike function
positive
not decreasing

10. 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)
Discretized  function

11. Part 2:
Coherency and Convexity

12. Coherency of the Risk Measure
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

13. Bond payoff = Nominal (or 0 with probability 2%)
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.

14. “risk” versus number of bonds in the portfolio
The surface of risk of ES has a single global minimum at n= and no fake local minima.
ES just tell us: “buy more bonds you can”
VaR and TCE suggest us NOT TO BUY the 6th, 36thor 83rd bond because it would increase the risk of the portfolio .... (?)
Are things better for large portfolios ???...

15. 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

16. ...maybe there’s really some tricky risk in the 36th bond !
If we use a 3% VaR instead of a 5% VaR, the “dangerous bond” is not the 36th anymore, but the 28th.... (!?)

17. 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.

18. Part 3:
Minimization of Expected Shortfall

19. 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
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.
Notice: also in the case of non parametric VaR a SORTING operation is needed in the estimator and the same problem appears

20. 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 
In the discrete case with N scenarios it becomes
Notice: the SORTING operator on data has disappeared. The dependence on data is manifestly analytic.

21. (w) and ES(w) coincide but just in the minimum !
Properties of : the Pflug-Uryasev-Rockafellar theorem
Minimizing  in its arguments (w,) amounts to minimizing ES in (w) only
Moreover the  parameter in the extremum takes the value of VaR(X(w)).
(w) and ES(w) coincide but just in the minimum !
The auxiliary parameter in the minimum becomes the VaR

22. 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

23.  The role of the auxiliary variable  .......
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
XN:N
In the minimization process  places on the specified quantile and gets the value of VaR.

24. Application – unconstrained ES minimization

25. Part 4:
Minimization of Spectral Measures of Risk

26. Minimizing a general Spectral Measure M
The “SORTING” problem appears in the minimization of any Spectral Measure

27. 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
In the discrete case it becomes

28. Properties of the generalized 
Minimizing  in all parameters (w,) amounts to minimizing M in (w)
Moreover, in the extremal, k takes the value of VaR(X(w)) associated to the quantile k/N.

29. The role of 
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) k 1 2 3 N
X1:N
X2:N
X3:N
XN:N
Xk:N
In the minimization any k goes to the quantile k/N. The  vector separates all scenarios X.

30. Part 5:
Risks and Rewards: are they really two orthogonal axis ?

31. - Σneg.Φ(i) X(i) - Σpos. Φ(i) X(i) = + 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.
% -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.

32. The simplest example ...
Take for instance the case of a Spectral Mesure obtained as convex combination of ES(X) and ES100%(X)= - average(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.

33. Optimal portfolios for different  values

34. with no constraints, for any λ
Integrated Markowitz problem
One shows in fact that (Acerbi, Simonetti, 2002):
minimizing
with no constraints, for any λ
amounts to minimizing
with constrains, for any specified return value

35. constrained optimal prtf for M unconstrained optimal prtf for M,
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
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,.
Expected return
constrained optimal prtf for M
unconstrained optimal prtf for M,
Where for some 

36. 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.

37. 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