Download Instability of downside risk measures

Instability of Portfolio Optimization under
Coherent Risk Measures
Imre Kondor
Collegium Budapest and Eötvös
University, Budapest
Large Databases in Social and Economic
Complex System Research,
Jerusalem, September 17, 2008
Coauthor
István Varga-Haszonits (ELTE PhD student and
Analytics Department of Fixed Income Division,
Morgan Stanley, Budapest, Hungary)
Preliminaries
The proper choice of risk measures (basically the
fundamental objective function) is of central importance
in finance.
The widest spread risk measure today is VaR which, as a
quantile, has no reason to be convex. A non-convex risk
measure violates the principle of diversification, does
not allow the correct pricing and aggregation of risk, and
cannot form the basis of a consistent limit system.
As a remedy to VaR’s shortcomings the concept of
coherent risk measures were introduced by Artzner at al.
in 1999 (Ph. Artzner, F. Delbaen, J. M. Eber, and D.
Heath, Coherent measures of risk, Mathematical
Finance, 9, 203-228, (1999).
This work triggered a tremendous response: there are 489
papers citing it on Gloriamundi
Preliminaries II
In a recent paper (I. K., Sz. Pafka, G. Nagy: Noise
sensitivity of portfolio selection under various
risk measures, Journal of Banking and Finance,
31, 1545-1573 (2007)) we investigated the risk
sensitivity of various risk measures (variance,
mean absolute deviation, expected shortfall,
maximal loss) and found that the estimation error
diverges at a critical value of the ratio N/T,
where N is the number of securities in the
portfolio and T is the sample size (the length of
the time series per item).
Furthermore, we realised that for some risk
measures portfolio optimisation does not always
have a solution.
Of course, for small samples (T < N) the
optimisation task never has a solution for any of
the risk measures – this is a triviality.
However, for T > N it is always possible to find a
solution for the variance and MAD, but the
feasibility of the optimization under ES or ML is
not guaranteed, it is a probabilistic issue, the
existence of a finite solution depends on the
sample.
The case of Maximal Loss
Definition of the problem (for simplicity, we are looking
for the global minimum and allow unlimited short
selling):
 N

min max    wi xit 
w 1 t T
 i 1

N
w
i 1
i
1
where the w’s are the portfolio weights and the x’s the
returns.
Probability of finding a solution for
the minimax problem (for elliptic
underlying distributions):
 T  1

p  T 1  
2 k  N 1  k 
1
T 1
The calculation of the probability of a
solution is equivalent to some problems
in operations research or random
geometry: Todd, M.J., Probabilistic
models for linear programming, Math.
Oper. Res. 16, 671-693 (1991).
In the limit N,T → ∞, with N/T fixed, the
transition becomes sharp at N/T = ½.
Interpretation of the instability
For ML it is easy to see that the risk measure
becomes unbounded from below if and only if it
is possible to form a portfolio that dominates all
the others on the given sample.
We say that portfolio u dominates (strictly
dominates) portfolio v (notation u  v , resp.
u  v ) if the return on u is larger or equal
(strictly larger) than the return on v for each
time period in the sample.
Expected Shortfall
ES is the average loss above a high threshold defined in
probability, not in money (ES is sometimes called
CVaR).
Optimisation under ES can be reduced to linear
programming. (R.T. Rockafellar and S. Uryasev,
Optimization of Conditional Value-at-Risk, The Journal
of Risk, 2, 21-41 (2000)
Maximal Loss is a limiting case of ES, corresponding to
the threshold going to 1.
Both ML and ES are coherent measures (C. Acerbi and D.
Tasche, On the Coherence of Expected Shortfall,
Journal of Banking and Finance, 26, 1487-1503 (2002))
in the sense of Artzner & al.
Feasibility of optimization under ES
Probability of the existence of an optimum under CVaR. F is the
standard normal distribution. Note the scaling in N/√T.
For ES the critical value of N/T
depends on the threshold β
With increasing N, T ( N/T= fixed) the
transition becomes sharper and sharper…
…until in the limit N, T →∞ with N/T= fixed we get
a „phase boundary”. The exact phase boundary has
been obtained by A. Ciliberti, I. K., and M. Mézard:
On the Feasibility of Portfolio Optimization under
Expected Shortfall, Quantitative Finance, 7, 389-396
(2007), from replica theory.
For ES the presence of a dominating portfolio is sufficient (but not
necessary) for the nonexistence of a solution (a dominating
combination of items will do).
The observations made on ML and ES can be generalized to any
measure ̂ w X satisfying the coherence axioms:
u  v  ˆ u X  ˆ v X
ˆ u  v X  ˆ u X  ˆ v X
a  0  ˆ au X  aˆ u X
ˆ u X  a  ˆ u X  a
defined on the sample X
Theorem 1. If there exist two portfolios u and v
so that u  v then the portfolio optimisation
task has no solution under any coherent
measure.
Theorem 2. Optimisation under ML has no
solution, if and only if there exists a pair of
portfolios such that one of them strictly
dominates the other.
Neither of these theorems assumes anything
about the underlying distribution.
For elliptically distributed underlyings we can say more:
Corollary 1: For elliptically distributed items the
probability of the existence of a pair of portfolios such
that one of them dominates the other on a given sample
X is 1 - p(N,T). (Think of the minimax.)
Corollary 2: The probability of the unfeasibility of the
portfolio optimisation problem under any coherent
measure on the sample X is at least 1 - p(N,T) if the
underlying assets are elliptically distributed.
Corollary 3: If there is a sharp transition in the limit N,T
→ ∞, with N/T fixed also for coherent risk measures
other than ML or ES, then their critical N/T ratio is
smaller or equal to ½, for elliptical distributions again.
Summary and extension
The coherence axioms imply that for finite T
there will always be samples for which the
portfolio optimization cannot be carried out
under a given coherent risk measure, because
the measure becomes unbounded from below.
Paradoxically, this instability is related to a very
attractive feature of coherent measures: if one
of the assets dominates the rest for all times,
that is for infinitely large samples, then the
coherent measures signal this by going to
minus infinity.
It may happen, however, that in a given finite
sample a single asset dominates the others even
if there is no such dominance relationship
between them for infinitely long observation
periods, and the coherent measures become
unbounded from below also in this case,
thereby giving a false signal.
Constrained optimization
We have allowed unlimited short selling so far.
If short selling is excluded, or any other set of
constraints that limit the domain of the
problem are imposed, the instability shows up
by the solutions sticking to the walls of the
allowed region, and jumping around from
sample to sample.
Constraints do not solve the problem of
instability, just mask it.
Coherent measures grasp some of the
most important features of risk.
However, in addition to mathematical
consistency, robustness to sample to
sample fluctuations is also a desirable
property of risk measures.
The above findings can be generalized to the even
larger class of downside risk measures, including
VaR.
VaR is a quantile, the loss beyond which a given
percentage (like 5% or 1%) of the worst losses
reside.
Historical VaR is not convex. We can consider
parametric VaR, however, by fitting e.g. a
Gaussian to the row data, and using its variance as
a risk measure. This way we can deduce a new
phase diagram, somewhat similar to the ES case,
along which the instability sets in.
Portfolio optimization suffers from the lack of
sufficient data. We are trying to extract
more information from the data than what
they contain, and this leads to overfitting.
Some kind of complexity control is needed,
but not an artificial one like imposing a ban
on short selling.
Methods taken over from machine learning
theory are being considered in our group.