UBI Pramerica SGR
Implementation of portfolio optimization
with spectral measures of risk
Roberto Strepparava
January 2009
Summary
 Introduction and motivation
 Optimization of Spectral Measures of risk
 α-ES efficient frontier via parametric method
 Efficient frontier with general Spectral Measures
 Backtest on real portfolios
 Conclusions and future work
1
Introduction and motivation
 Efficient frontier in the Markowitz plane using different risk measures: using
VaR the problem is impossible to solve: problem non convex, plagued by local
fake minima, due to non-subadditivity of VaR.
 Portfolio optimization issue for portfolio managers (stable weights, i.e. wellposedness of the problem).
 Need to solve efficient frontier with more general risk measures ρ: all advantages
of VaR and none of the shortcomings:
1.
ρ universal measure (= applies to any kind of risk)
2.
ρ global measure (= “sums” different risks into a single number)
3.
ρ probabilistic (= provides probabilistic info on the risk)
4.
ρ expressed in units of “lost money”
2
Summary
 Introduction and motivation
 Optimization of Spectral Measures of risk
 α-ES efficient frontier via parametric method
 Efficient frontier with general Spectral Measures
 Backtest on real portfolios
 Conclusions and future work
3
Optimization of Spectral Measures of Risk
 Coherent Risk Measures (CRM) if for all X,Y portfolio’s P&L r.v.’s
X Y

ρ( X )  ρ(Y )
monotonicity
X , Y
ρ( X  Y )  ρ( X )  ρ(Y )
subadditivity
X , h  0
ρ(hX )  hρ( X )
homogeneity
X , a  
ρ( X  a)  ρ( X )  a
tr. invariance
 Subadditivity related to the “risk diversification principle”. Hedging benefit:
H(X, Y; ρ)  ρ(X )  ρ(Y )  ρ(X  Y )
 Given a prob. measure P, a CMR is said to be law-invariant (LI) if (X) is in
fact a functional of the distribution function FX(x) only.
 A measure of risk  is said to be Comonotonic Additive (CA) if adding
together two comonotonic risks X and Y provides NO HEDGING AT ALL
4
Optimization of Spectral Measures of Risk
Kusuoka (2001) showed the class of LI CA CRMs is given by all
convex combinations of possible ES’s at different conf. levels:
  ( X )   d ( p) ES p ( X )
d any measure on [0,1]
Acerbi (2001) introduced the same class of CMRs calling it
“spectral measures of risk” (SM)
1
M  ( X )    ( p) FX ( p) dp
0
which is a CMR if the “risk spectrum” :[0,1] satisfies
1.
0
2.
 =1
3.
 weakly decreasing
5
Optimization of Spectral Measures of Risk
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.
To this class belongs the Expected Shortfall 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 ):
ES ( X ) 

1


0
1

VaR p ( X ) dp


0
FX ( p) dp
 " the average loss in the  worst cases"
6
Optimization of Spectral Measures of Risk
Optimization of ES: Uryasev et al. (2000, 2001) develop an
efficient procedure for the minimization of ES, avoiding to deal
with ordered statistics (read: sorting operations).


Theorem: let X (w) a portfolio with weights w . Define


1
 ( X ( w), )    E[  X ( w)]
Then
1.
a
2.
b
3.
c



ES ( X (w))  min  ( X (w), )
 

arg min  ( X (w), )  VaR ( X (w))



min
 ES ( X ( w))  min
  ( X ( w), )
w
 ,w
7
Optimization of Spectral Measures of Risk
In an N-scenarios pdf this problem is a nonlinear convex
optimization of the form:
min


(N )
w,

1

( X ( w), )  min





w,
N

 
(


X
(
w
)) 

i
i 1

N

w
 
Notice the absence of sorting operations (read: ordered statistics).
The objective function is piecewise linear in  and w
8
Optimization of Spectral Measures of Risk
But the problem can be mapped again into a linear programming
(LP) optimization problem of the form :
1

min

   
w , , z
N


zi    X i ( w)

N
 z 
i 1
i
i  1,  , N

w
 
zi  0
Where linearity has been bought at the price of introducing N new
variables z
9
Optimization of Spectral Measures of Risk
Optimization of general Spectral Measures: Acerbi and Simonetti
(2002) extend the method above to a general SM.
The objective function in this case takes the form:

 

d
 [ X ( w), ]   dt
t (t )  E[ (t )  X ( w)]   (1) E[ X ( w)]
dt
0
1


and therefore, in the general case the additional parameter is a whole
function (t).
10
Optimization of Spectral Measures of Risk
The N-scenarios optimization problem can be cast again into a
nonlinear convex program:
min
  
(N)
w,
N
N


 


( X ( w), )  min
    j  j j   ( j  X i )    N  X i 
w,
i 1
i 1


 jJ


w

  J
where we have discretized the risk spectrum φ to a piecwiseconstant function with J jumps
11
Optimization of Spectral Measures of Risk
And again the problem can be mapped into a linear program in a
(generally) huge number of variables:
N
N




X


z


j


min
   

i
N
ij 
j
j
w, , z
i 1
i 1



 jJ

zij   j  X i ( w)
i  1,..., N ;
jJ

w

  J
zij  0
12
Optimization of Spectral Measures of Risk
Theorem (risk-reward optimization for Spectral Measures):
The optimal portfolios of the (M  X  ,E(X)) risk-reward
constrained optimization problem are the solutions of the
unconstrained minimization problem of the SMs:
Mhat( X )  E ( X )  (1   ) M ( X )
Defined for all  in [0,1] , where
hat ( )    1   
Most useful for implementation:
•
we get all and only optimal portfolios and no dominated frontier
•
the range of the parameter to vary is exactly known [0,1]
13
Summary
 Introduction and motivation
 Optimization of Spectral Measures of risk
 α-ES efficient frontier via parametric method
 Efficient frontier with general Spectral Measures
 Backtest on real portfolios
 Conclusions and future work
14
α-ES efficient frontier via parametric method
Constrained α-ES minimization drawbacks:
• chosen constraint value μ incompatible with efficient frontier
• even if all compatible constraints, portion of dominated frontier
As
Parametric α-ES minimization advantages:
 all and only optimal portfolios retrieved
 range of the Lagrange parameter λ exactly known
15
Summary
 Introduction and motivation
 Optimization of Spectral Measures of risk
 α-ES efficient frontier via parametric method
 Efficient frontier with general Spectral Measures
 Backtest on real portfolios
 Conclusions and future work
16
Efficient frontier with general Spectral Measures
Two-percentile SM: piecewise constant SM with parametric method
(tipically α=1%, β=5%) interesting features
•
Measures interpolate extrema α-ES and β-ES, belonging to class
of weighted V@R measures -Cherny (2006)-
•
Well-possess of the optimization problem: how smoothly
portfolio weights depend on risk spectrum φ
•
Solve practical issue: risk manager that finds 1%-ES too loose
and 5%-ES too strict can calibrate the measure .
Especially useful in the present
context of financial crisis
17
Efficient frontier with general Spectral Measures
Well-posedness of optimization: smooth dependence of portfolio
weights on shape of the spectrum, within a certain range
M
For minimum risk portfolio new weights come into play.
18
Efficient frontier with general Spectral Measures
Open problems (suitable theorems needed?):
1. Optimization fails (problem unbounded) as soon as measure
becomes slightly incoherent.
2. Well-posedness for ptf with derivatives –Alexander et al.
(2006)
3. Success of LP optimization with simplex method, failure
with interior point method of the same problem
19
Summary
 Introduction and motivation
 Optimization of Spectral Measures of risk
 α-ES efficient frontier via parametric method
 Efficient frontier with general Spectral Measures
 Backtest on real portfolios
 Conclusions and future work
20
Backtest on real portfolios
Case study: portfolio of an Asset Management company
•
63 assets in portfolio (Italian stocks)
•
Medium-depth HS (764 daily observations)
•
3 months backtest
•
Simplest SM: 5% Expected Shortfall
•
Inclusion of transaction costs (1 BP per transaction) +
management fees
Results: even including costs + fees, high risk portfolio beats NAV
of the fund, while minimum risk portfolio stays very close to
NAV
21
Summary
 Introduction and motivation
 Optimization of Spectral Measures of risk
 α-ES efficient frontier via parametric method
 Efficient frontier with general Spectral Measures
 Backtest on real portfolios
 Conclusions and future work
22
Conclusions and future work
Feasibility of more general Spectral Measures than simple ES: now that ES is
being widely used (thereby slowly replacing VaR), SMs await their turn.
SMs both interesting theoretically and useful to risk managers
Parametric method of efficient frontier construction very efficient and fast (all
simulations on laptop, Pentium processor 1.7GHz with Matlab™ R2007b on)
Numerical simulations hint at theorems that need to be properly stated
Agenda:
• Analysis of SMs with more general risk aversion functions φ (discrete
exponential spectrum).
• Analysis on real portfolio requiring major changes in constraints: leveraged
portfolios, portfolios short of derivatives.
• Heavy MC simulations to see computational burden on the optimizer.
• Rolling analysis on assets’ HS to see well-posedness of the problem w.r.t.
changes in empirical distribution of the portfolio P&L random variable
23