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Conditional Value-at-Risk (CVaR):
Algorithms and Applications
Stan Uryasev
Risk Management and Financial Engineering Lab
University of Florida
e-mail: uryasev@ise.ufl.edu
http://www.ise.ufl.edu/uryasev
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OUTLINE OF PRESENTATION
Background: percentile and probabilistic functions inoptimization
Definition of Conditional Value-at-Risk (CVaR) and basicproperties
Optimization and risk management with CVaR functions
Case studies:
Definition of Conditional Drawdown-at-Risk (CDaR)
Conclusion
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PAPERS ON MINIMUM CVAR APPROACH
Presentation is based on the following papers:
[1] Rockafellar R.T. and S. Uryasev (2001): Conditional Value-at-Risk for General Loss Distributions. Research Report 2001-5. ISE
Dept., University of Florida, April 2001.(download: www.ise.ufl.edu/uryasev/cvar2.pdf)
[2] Rockafellar R.T. and S. Uryasev (2000): Optimization of
Conditional Value-at-Risk. The Journal o f Risk. Vol. 2, No. 3, 2000,21-41 (download: www.ise.ufl.edu/uryasev/cvar.pdf)
Several more papers on applications of Conditional Value-at-Risk
and the related risk measure, Conditional Drawdown-at-Risk, canbe downloaded from www.ise.ufl.edu/rmfe
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ABSTRACT OF PAPER1
Fundamental properties of Conditional Value-at-Risk (CVaR), as a
measure of risk with significant advantages over Value-at-Risk,
are derived for loss distributions in finance that can involve
discreetness. Such distributions are of particular importance inapplications because of the prevalence of models based on
scenarios and finite sampling. Conditional Value-at-Risk is able to
quantify dangers beyond Value-at-Risk, and moreover it is
coherent. It provides optimization shortcuts which, through linearprogramming techniques, make practical many large-scale
calculations that could otherwise be out of reach. The numerical
efficiency and stability of such calculations, shown in several
case studies, are illustrated further with an example of indextracking.
1Rockafellar R.T. and S. Uryasev (2001): Conditional Value-at-Risk for General Loss Distributions.
Research Report 2001-5. ISE Dept., University of Florida, April 2001.
(download: www.ise.ufl.edu/uryasev/cvar2.pdf)
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Let f(x ,y) be a loss functions depending upon a decision vector
x = ( x1,, x
n) and a random vector y = ( y
1,, y
m)
VaR= percentile of loss distribution (a smallest value such thatprobability that losses exceed or equal to this value isgreater or equal to ))))
CVaR+ ( upper CVaR ) = expected losses strictly exceeding VaR(also called Mean Excess Loss and Expected Shortfall)
CVaR- ( lower CVaR ) = expected losses weakly exceeding VaR,i.e., expected losses which are equal to or exceed VaR
(also called Tail VaR)
CVaR is a weighted average of VaR and CVaR+
CVaR = VaR + (1- ) CVaR+ , 0 1
PERCENTILE MEASURES OF LOSS (OR REWARD)
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VaR, CVaR, CVaR+ and CVaR-
Loss
Freq
uency
1111
VaR
CVaR
Probability
Maximum
loss
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CVaR: NICE CONVEX FUNCTION
x
Risk
VaR
CVaR
CVaR+
CVaR-
CVaR is convex, but VaR, CVaR- ,CVaR+ may be non-convex,
inequalities are valid: VaR CVaR- CVaR CVaR+
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Value-at-Risk (VaR) is a popular measure of risk:current standard in finance industry
various resources can be found at http://www.gloriamundi.org
Informally VaR can be defined as a maximum loss in aspecified period with some confidence level (e.g.,confidence level = 95%, period = 1 week)
Formally, VaR is the percentile of the lossdistribution:
VaR is a smallest value such that probability that loss exceedsor equals to this value is bigger or equals to
VaR IS A STANDARD IN FINANCE
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FORMAL DEFINITION OF CVaR
Notations:! = cumulative distribution of losses,
! = -tail distribution, which equals to zero for losses below VaR,and equals to (!- )/(1)/(1)/(1)/(1 )))) for losses exceeding or equal to VaR
Definition: CVaR is mean of -tail distribution !
Cumulative Distribution of Losses, !
0 VaR
+
1
!()
0
1
VaR
-Tail Distribution, !
1
+
!()
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CVaR: WEIGHTED AVERAGE
Notations:
VaR= percentile of loss distribution (a smallest value such thatprobability that losses exceed or equal to this value is greater or
equal to ))))CVaR+ ( upper CVaR ) = expected losses strictly exceeding VaR
(also called Mean Excess Loss and Expected Shortfall)
!(VaR) = probability that losses do not exceed VaR or equal to VaR
= (!(VaR) - )/ (1)/(1)/(1)/(1 ) , ( 0) , ( 0) , ( 0) , ( 0 1 )1 )1 )1 )
CVaR is weighted average of VaR and CVaR+
CVaR = VaR + (1- ) CVaR+
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CVaR: DISCRETE DISTRIBUTION, EXAMPLE 1
does not split atoms: VaR < CVaR- < CVaR = CVaR+,
= (!- )/(1)/(1)/(1)/(1 ) =) =) =) = 0
1 2 41 2 6 6 3 6
1 15 62 2
Six scenarios, ,
CVaR CVaR =
p p p
f f
= = = = = =
= +
+
Probability CVaR
1
6
1
6
1
6
1
6
1
6
1
6
1f
2f
3f 4f 5f 6f
VaR --
CVaR
+CVaR
Loss
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CVaR: DISCRETE DISTRIBUTION, EXAMPLE 2
splits the atom: VaR < CVaR- < CVaR < CVaR+ ,
= (!- )/(1)/(1)/(1)/(1 ) >) >) >) > 0
711 2 6 6 12
1 4 1 2 24 5 65 5 5 5 5
Six scenarios, ,
CVaR VaR CVaR =
= = = = =
= + + +
p p p
f f f+
Probability CVaR
16
16
16
16
112
16
16
1f 2f 3f 4f 5f 6f
VaR
--
CVaR
+1 1
5 62 2
CVaR+ =f fLoss
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CVaR: DISCRETE DISTRIBUTION, EXAMPLE 3
splits the last atom: VaR = CVaR- = CVaR,
CVaR+ is not defined, = (! )/(1)/(1)/(1)/(1 ) >) >) >) > 0
711 2 3 4 4 8
4
Four scenarios, ,
CVaR VaR =
= = = = ==
p p p p
f
Probability CVaR
14
14
14
14
18
1f 2f 3f 4f
VaR
Loss
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CVaR: NICE CONVEX FUNCTION
Position
Risk
VaR
CVaR
CVaR+
CVaR-
CVaR is convex, but VaR, CVaR- ,CVaR+ may be non-convex,
inequalities are valid: VaR CVaR- CVaR CVaR+
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CVaR FEATURES1,2
- simple convenient representation of risks (one number) - measures downside risk
- applicable to non-symmetric loss distributions
- CVaR accounts for risks beyond VaR (more conservative than VaR) - CVaR is convex with respect to portfolio positions
- VaR CVaR- CVaR CVaR+
- coherent in the sense of Artzner, Delbaen, Eber and Heath3:
(translation invariant, sub-additive, positively homogeneous, monotonic w.r.t. Stochastic Dominance1)
1Rockafellar R.T. and S. Uryasev (2001): Conditional Value-at-Risk for General Loss Distributions. Research Report 2001-5. ISE Dept., University of Florida, April 2001. (Can be downloaded:
www.ise.ufl.edu/uryasev/cvar2.pdf)
2 Pflug, G. Some Remarks on the Value-at-Risk and the Conditional Value-at-Risk, in ``Probabilistic
Constrained Optimization: Methodology and Applications'' (S. Uryasev ed.), Kluwer Academic
Publishers, 2001.
3Artzner, P., Delbaen, F., Eber, J.-M. Heath D. Coherent Measures of Risk, Mathematical Finance, 9 (1999), 203--228.
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CVaR FEATURES (Contd)
- stable statistical estimates (CVaR has integral characteristicscompared to VaR which may be significantly impacted by one scenario)
- CVaR is continuous with respect to confidence level ,,,,
consistent at different confidence levels compared to VaR (((( VaR, CVaR-, CVaR+ may be discontinuous in ))))
- consistency with mean-variance approach: for normal loss distributions optimal variance and CVaR portfolios coincide
- easy to control/optimize for non-normal distributions;
linear programming (LP): can be used for optimization of very large problems (over 1,000,000 instruments and
scenarios); fast, stable algorithms
- loss distribution can be shaped using CVaR constraints (many LP constraints with various confidence levels in different intervals)
- can be used in fast online procedures
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CVaR versus EXPECTED SHORTFALL
CVaR for continuous distributions usually coincides withconditional expected loss exceeding VaR (also called Mean ExcessLoss or Expected Shortfall).
However, for non-continuous (as well as for continuous)distributions CVaR may differ from conditional expected loss
exceeding VaR. Acerbi et al.1,2 recently redefined Expected Shortfall to be consistent
with CVaR definition.
Acerbi et al.2 proved several nice mathematical results on properties
of CVaR, including asymptotic convergence of sample estimates toCVaR.
1Acerbi, C., Nordio, C., Sirtori, C. Expect ed Shortf al l as a Tool for Financ ial Risk
Management, Working Paper, can be downloaded: www.gloriamundi.org/var/wps.html
2Acerbi, C., and Tasche, D. On the Coherence of Expected Sho rt fal l.Working Paper, can be downloaded: www.gloriamundi.org/var/wps.html
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CVaR OPTIMIZATION
Notations:x = (x
1,...x
n) = decision vector (e.g., portfolio weights)
X = a convex set of feasible decisions
y = (y1,...y
n) = random vector
yj = scenario of random vector y , (j=1,...J )
f(x,y) = loss functions
Example: Two Instrum ent Port fo l io A portfolio consists of two instruments (e.g., options). Let x=(x
1,x
2) be a
vector of positions, m=(m1,m
2) be a vector of initial prices, and y=(y
1,y2) be
a vector of uncertain prices in the next day. The loss function equals thedifference between the current value of the portfolio, (x
1m
1+x
2m
2), and an
uncertain value of the portfolio at the next day (x1y1+x
2y2), i.e.,
f(x,y) = (x1m
1+x
2m
2)(x
1y1+x
2y2) = x
1(m
1y
1)+x
2(m
2y
2) .
If we do not allow short positions, the feasible set of portfolios is a two-dimensional set of non-negative numbers
X = {(x
1,x
2), x1 0, x2 0} .
Scenarios yj= (yj1,yj
2),j=1,...J , are sample daily prices (e.g., historical
data for Jtrading days).
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CVaR OPTIMIZATION (Contd)
CVaR minimizationmin{ xX } CVaR
can be reduced to the following linear programming (LP) problem
min{ xX , R, z RJ} + {j =1,...,J} zj
subject to
zj
f(x ,yj) - , zj 0 , j=1,...J ( = (( 1- )J)-1= const )
By solving LP we find an optimal portfolio x* , corresponding VaR,which equals to the lowest optimal *, and minimal CVaR, whichequals to the optimal value of the linear performance function
Constraints, x X , may account for various trading constraints,including mean return constraint (e.g., expected return shouldexceed 10%)
Similar to return - variance analysis, we can construct an efficient
frontier and find a tangent portfolio
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RISK MANAGEMENT WITH CVaR CONSTRAINTS
CVaR constraints in optimization problems can be replaced by aset of linear constraints. E.g., the following CVaR constraint
CVaR C
can be replaced by linear constraints
+ {j =1,...,J} zj C
zj f(x ,y
j) - , zj 0 , j=1,...J ( = (( 1- )J)-1= const )
Loss distribution can be shaped using multiple CVaR constraintsat different confidence levels in different times
The reduction of the CVaR risk management problems to LP is a
relatively simple fact following from possibility to replace CVaR bysome function F(x, ) , which is convex and piece-wise linear withrespect to x and . A simple explanation of CVaR optimizationapproach can be found in paper1 .
1Uryasev, S. Condi t ional Value-at-Risk: Opt imizat ion Algor i thms and A ppl icat ions.Financial Engineering News, No. 14, February, 2000.
(can be downloaded: www.ise.ufl.edu/uryasev/pubs.html#t).
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CVaR OPTIMIZATION: MATHEMATICAL BACKGROUND
Defini t ion
F(x , ) = + j=1,J ( f(x,yj)- )+, = (( 1- )J)-1= const
Theorem 1.
CVaR((((x) = minR F(x , ) and (x) is a smallest minimizer
Remark. This equality can be used as a definition of CVaR ( Pflug ).
Theorem 2.
min xX CVaR((((x) = minR, xX F(x , ) (1)
Minimizing of F(x , ) simultaneously calculates VaR= (x),optimal decision x, and optimal CVaR
Problem (1) can be reduces to LP using additional variables
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Propos i t ion 1.
Let f(x ,y) be a loss functions and (x) be - percentile (-VaR) then
(x) Pr{ f(x ,y) }
Proof follows from the definition of - percentile (x)
(x) = min { : Pr{ f(x,y)
} } } } }
Generally, (x) is nonconvex (e.g., discrete distributions),
therefore (x) as well as Pr( f(x ,y) ) may be
nonconvex constraints
Probabilistic constraints were considered by Prekopa, Raik,
Szantai, Kibzun, Uryasev, Lepp, Mayer, Ermoliev, Kall, Pflug,
Gaivoronski,
PERCENTILE V.S. PROBABILISTIC CONSTRAINTS
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Low partial moment constraint (considered in finance literaturefrom 70-th)
E{ ((f(x ,y) - )+)a } b, a 0, g+=max{0,g}
special cases
a =0 => Pr{ f(x ,y) - }a =1 => E{ (f(x ,y) - )+ }a =2 , = E f(x ,y) => semi-variance E{ ((f(x ,y) - )+)2 }
Regret (King, Dembo) is a variant of low partial moment with=0 and f(x ,y) = performance-benchmark
Various variants of low partial moment were successfully applied
in stochastic optimization by Ziemba, Mulvey, Zenios,Konno,King, Dembo,Mausser,Rosen,
Haneveld and Prekopa considered a special case of low partial
moment with a =1, = 0: in tegrated chance cons traints
NON-PERCENTILE RISK MEASURES
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Low partial moment with a>0 does not control percentiles. It isapplied when loss can be hedged at additional cost
total expected value = expected co st w i thout high los ses+ expected cos t of h igh losses
expec ted cos t o f h igh losses= p E{ (f(x ,y) - )+ }
Percentiles constraints control risks explicitly in percentile terms.
Testury and Uryasev1 established equivalence between CVaR
approach (percentile measure) and low partial moment, a =1 (non-percentile measure) in the following sense:
a) Suppose that a decision is optimal in an optimization problemwith a CVaR constraint, then the same decision is optimal with a low
partial moment constraint with some >0;b) Suppose that a decision is optimal in an optimization problemwith a low partial moment constraint, then the same decision isoptimal with a CVaR constraint at some confidence level ....
1Testuri, C.E. and S. Uryasev. On Relation between Expected Regret and Conditional Value-At-Risk.Research Report 2000-9. ISE Dept., University of Florida, August 2000. Submitted to Decis ions inEconom ics and Financejournal. (www.ise.ufl.edu/uryasev/Testuri.pdf)
PERCENTILE V.S. LOW PARTIAL MOMENT
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CVaR AND MEAN VARIANCE: NORMAL RETURNS
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CVaR AND MEAN VARIANCE: NORMAL RETURNS
If returns are normally distributed, and return constraint is active,the following portfolio optimization problems have the samesolution:
1. Minimize CVaR
subject to return and other constraints
2. Minimize VaR
subject to return and other constraints
3. Minimize variance
subject to return and other constraints
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EXAMPLE 1: PORTFOLIO MEAN RETURN AND COVARIANCE
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OPTIMAL PORTFOLIO (MIN VARIANCE APPROACH)
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PORTFOLIO, VaR and CVaR (CVaR APPROACH)
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EXAMPLE 2: NIKKEI PORTFOLIO
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NIKKEI PORTFOLIO
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HEDGING: NIKKEI PORTFOLIO
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HEDGING: MINIM CVaR APPROACH
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ONE INSTRUMENT HEDGING
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ONE - INSTRUMENT HEDGING
MU TIP E INSTRUMENT HEDGING CV R APROACH
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MULTIPLE INSTRUMENT HEDGING: CVaR APROACH
MULTIPLE INSTRUMENT HEDGING
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MULTIPLE - INSTRUMENT HEDGING
MULTIPLE HEDGING MODEL DESCRIPTION
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MULTIPLE-HEDGING: MODEL DESCRIPTION
EXAMPLE 3 PORTFOLIO REPLICATION USING CVaR
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EXAMPLE 3: PORTFOLIO REPLICATION USING CVaR
Problem Statement: Replicate an index using instruments. Considerimpact of CVaR constraints on characteristics of the replicating portfolio.
Daily Data: SP100 index, 30 stocks (tickers: GD, UIS, NSM, ORCL, CSCO, HET, BS,TXN, HM,INTC, RAL, NT, MER, KM, BHI, CEN, HAL, DK, HWP, LTD, BAC, AVP, AXP, AA, BA, AGC, BAX, AIG, AN, AEP)
Notations= price of SP100 index at times
= prices of stocks at times= amount of money to be on hand at the final time
= = number of units of the index at the final time
= number of units ofj-th stock in the replicating portfolio
Definitions (similar to paper1 )
= value of the portfolio at time
= absolute relative deviation of the portfolio from the target
= relative portfolio underperformance compared to target at time
1Konno H. and A. Wijayanayake. Minimal Cost Index Tracking under Nonl inear Transact ion Costs and
Minimal Transact ion Unit Constraints,Tokyo Institute of Technology, CRAFT Working paper 00-07,(2000).
1, ,t T= tI
j tp 1, ,j n= 1, ,t T= T
TI T
jx
t1
n
j t jj
p x=
1
| ( ) /( ) |n
t j t j t
j
I p x I =
tI
1, ,j n=
t1
( , ) ( ) /( )n
t t j t j t
j
f x p I p x I =
=
PORTFOLIO REPLICATION (Contd)
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PORTFOLIO REPLICATION (Contd)
Index and optimal portfolio values in in-sample region, CVaRconstraint is inactive (w= 0.02)
0
2000
4000
6000
8000
10000
12000
1 51 101 151 201 251 301 351 401 451 501 551
Day number: in-sample region
Po
rtfoliovalue(U
SD)
portfolio
index
PORTFOLIO REPLICATION (Contd)
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PORTFOLIO REPLICATION (Contd)
Index and optimal portfolio values in out-of-sample region,CVaR constraint is inactive (w= 0.02)
8500
9000
9500
10000
10500
11000
11500
12000
1 713
19
25
31
37
43
49
55
61
67
73
79
85
91
97
Day number in out-of-sample region
Po
rtfoliovalue(U
SD)
portfolioindex
PORTFOLIO REPLICATION (Contd)
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PORTFOLIO REPLICATION (Cont d)
Index and optimal portfolio values in in-sample region,CVaR constraint is active (w= 0.005).
0
2000
4000
6000
8000
10000
12000
1 51
101
151
201
251
301
351
401
451
501
551
Day number: in-sample region
Portfoliovalue(USD)
portfolio
index
PORTFOLIO REPLICATION (Contd)
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PORTFOLIO REPLICATION (Cont d)
Index and optimal portfolio values in out-of-sample region,CVaR constraint is active (w= 0.005).
8500
9000
9500
10000
10500
11000
11500
12000
1 713
19
25
31
37
43
49
55
61
67
73
79
85
91
97
Day number in out-of-sample region
Por
tfoliovalue(U
SD)
portfolio
index
PORTFOLIO REPLICATION (Contd)
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PORTFOLIO REPLICATION (Cont d)
Relative underperformance in in-sample region, CVaRconstraint is active (w = 0.005) and inactive (w = 0.02).
-6
-5
-4-3
-2
-1
0
1
2
3
1 51 101 151 201 251 301 351 401 451 501 551
Day number: in-sample region
D
iscrepancy(%
)
active
inactive
PORTFOLIO REPLICATION (Contd)
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PORTFOLIO REPLICATION (Cont d)
Relative underperformance in out-of-sample region, CVaRconstraint is active (w= 0.005) and inactive (w= 0.02)
-2
-1
0
1
2
3
4
5
6
1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96
Day number in out-of-sample region
Discrepancy(%)
active
inactive
PORTFOLIO REPLICATION (Contd)
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PORTFOLIO REPLICATION (Cont d)
In-sample objective function (mean absolute relative deviation),out-of-sample objective function, out-of-sample CVaR for various
risk levels w in CVaR constraint.
0
1
2
3
4
5
6
0 .0 2 0 .0 1 0 .0 0 5 0 .0 0 3 0 .0 0 1
o m e g a
Value(%)
in - s a m p le o b je c t iv efu n c t i o n
o u t -o f - s a m p le o b je c t iv efu n c t i o n
o u t -o f -s a m p le C V A R
PORTFOLIO REPLICATION (Contd)
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PORTFOLIO REPLICATION (Cont d)
Calculation results
CVaR constraint reduced underperformance of the portfolio versus the index both in
the in-sample region (Column 1 of table) and in the out-of-sample region (Column 4) .
For w=0.02, the CVaR constraint is inactive, for w0.01, CVaR constraint is active.
Decreasing of CVaR causes an increase of objective function (mean absolutedeviation) in the in-sample region (Column 2).
Decreasing of CVaR causes a decrease of objective function in the out-of-sampleregion (Column 3). However, this reduction is data specific, it was not observed for
some other datasets.
C VaR in -sam ple (600 d ays) ou t -of-sam p le ( 100 days) ou t -of-sam ple C VaR
level w ob ject ive fu n ct ion , in % ob ject ive fun ct ion , in % in %
0.02 0.71778 2.73131 4.886540.01 0.82502 1.64654 3.88691
0.005 1.11391 0.85858 2.62559
0.003 1.28004 0.78896 2.16996
0.001 1.48124 0.80078 1.88564
PORTFOLIO REPLICATION (Contd)
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PORTFOLIO REPLICATION (Cont d)
In-sample-calculat ions: w=0.005
Calculations were conducted using custom developed software (C++) in combination
with CPLEX linear programming solver
For optimal portfolio, CVaR= 0.005. Optimal *= 0.001538627671 gives VaR. Probabilityof the VaR point is 14/600 (i.e.14 days have the same deviation= 0.001538627671). The
losses of 54 scenarios exceed VaR. The probability of exceeding VaR equals
54/600 < 1- , and
= (!(VaR) - ) / (1) / (1) / (1) / (1 - ) =) =) =) = [546/600 - 0.9]/[1 - 0.9] = 0.1
Since splits VaR probability atom, i.e., !(VaR) - >0, CVaR is bigger than CVaR-(lower CVaR)and smaller than CVaR+ ( upper CVaR, also called expected shortfall)
CVaR- = 0.004592779726 < CVaR = 0.005 < CVaR+=0.005384596925
CVaR is the weighted average of VaR and CVaR+
CVaR = VaR + (1- ) CVaR+= 0.1 * 0.001538627671 + 0.9 * 0.005384596925= 0.005
In several runs, * overestimated VaR because of the nonuniqueness of the optimalsolution. VaR equals the smallest optimal *.
EXAMPLE 4: CREDIT RISK (Related Papers)
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( p )
Andersson, Uryasev, Rosen and Mausser applied the CVaRapproach to a credit portfolio of bonds
Andersson, F., Mausser, Rosen, D. and S. Uryasev (2000),
Credit risk optimization with Conditional Value-at-Risk criterion,
Mathematical Programm ing, series B, December)
Uryasev and Rockafellar developed the approach to minimizeConditional Value-at-Risk
Rockafellar, R.T. and S. Uryasev (2000), Optimization of Conditional
Value-at-Risk, The Jou rnal of Risk, Vol. 2 No. 3 Bucay and Rosen applied the CreditMetrics methodology to
estimate the credit risk of an international bond portfolio
Bucay, N. and D. Rosen, (1999)Credit risk of an international bondportfolio: A case study, Algo Research Quar ter ly , Vol. 2 No. 1, 9-29
Mausser and Rosen applied a similar approach based on theexpected regret risk measure
Mausser, H. and D. Rosen (1999), Applying scenario optimization toportfolio credit risk, Algo Research Quar ter ly, Vol. 2, No. 2, 19-33
Basic Definitions
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Basic Definitions
Credit risk
The potential that a bank borrower or counterpart will fail to
meet its obligations in accordance with agreed terms
Credit loss Losses due to credit events, including both default and creditmigration
Credit Risk Measures
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Credit Risk Measures
Credit loss
Frequ
ency
Target insolvency rate = 5%
Unexpected loss (95%)
(Value-at-Risk (95%))
Conditional Value-at-Risk (95%)
CVaR
Allocated economic capital
Expected loss
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Portfolio Loss Distribution
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Generated by a Monte Carlo
simulation based on 20000
scenarios
Skewed with a long fat tail Expected loss of 95 M USD
Only credit losses, no
interest income
Standard deviation of 232 MUSD
VaR (99%) equal 1026 M USD
CVaR (99%) equal 1320 M
USD 0
500
1000
1500
2000
2500
-502
-239
24
286
549
812
1075
1337
1600
1863
2125
2388
Portfolio loss (millions of USD)
F
requency
Model Parameters
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Definitions
1) Obligor weights expressed as multiples of current holdings
2) Future values without credit migration, i.e. the benchmarkscenario
3) Future scenario dependent values with credit migration
4) Portfolio loss due to credit migration
1 2
1 2
1 2
(Instrument positions) ( , , ..., ) (1)
(Future values without credit migration) ( , , ..., ) (2)
(Future values with credit migration) ( , , ..., ) (3)
(Portfolio loss) ( , ) ( ) (4)
n
n
n
T
x x x
b b b
y y y
f
=
=
=
=
x
b
y
x y b y x
OPTIMIZATION PROBLEM
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niqqx
xRrq
qxq
niuxl
Jjz
Jjxybz
zJ
n
iiii
ii
n
ii
n
i
n
iiii
iii
j
n
iiijij
J
jj
,...,1,20,0position)Long(
,0)(return)(Expected
,value)(Portfolio
,,...,1,bound)er(Upper/low
,,...,1,0
,,...,1,))((loss)(Excess
:Subject to
)1(
1
Minimize
1
1
1 1
1,
1
=
=
=
=
=
+
=
=
= =
=
=
SINGLE - INSTRUMENT OPTIMIZATION
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2993729724-294.23Romania
1130911015-3.24Philippines
212923998-3.75Mexico
2994129727-610.14MoscowTel
2598924777-21.25RussiaIan
22103523792-88.29Morocco
21104021808-45.07Colombia
2698028740-7.35Peru
2599027751-10.30Argentina
3388033683-4.29Venezuela
3586335667-9.55Russia
4276740612-5.72Brazil
CVaR (%)CVaR (M USD)VaR (%)VaR (M USD)Best HedgeObligor
MULTIPLE - INSTRUMENT OPTIMIZATION
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0
500
1000
1500
2000
2500
90 95 99 99.9
Percentile level (%)
Va
Ran
dC
Va
R(millionso
fUSD)
Original VaR
No Short VaR
Long and Short VaR
Original CVaR
No Short CVaR
Long and Short CVaR
RISK CONTRIBUTION (original portfolio)
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RISK CONTRIBUTION (original portfolio)
Russia
Venezuela
Argentina
Colombia
Morocco
RussioIan
Moscow Tel
Turkey
Panama
Romania
Mexico
Croatia
PhillippinesPolandChina
Bulgaria
TeveCap
Jordan
Brazil
Rossiysky
Multicanal
Globo
BNDES
Slovakia
Peru
0
5
10
15
20
25
30
35
40
45
50
0 100 200 300 400 500 600 700 800 900 1000
Credit exposure (millions of USD)
MarginalCVaR(%)
RISK CONTRIBUTION (optimized portfolio)
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( p p )
PolandMexico
Philippines
BulgariaArgentina
KazakhstanJordan
Croatia
Moscow Tel
Israel
Brazil
South Af ricaKorea
TurkeyPanama
Romania
Globo
BNDES
Columbia
Slovakia
Multicanal
ThailandTelefarg
Moscow
China
0
5
10
15
20
25
30
35
40
45
50
0 100 200 300 400 500 600 700 800
Credit exposure (millions of USD)
Marg
inalrisk(%)
EFFICIENT FRONTIER
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1026 1320
4
6
8
10
12
14
16
0 200 400 600 800 1000 1200 1400
Conditional Value-at-Risk (99%) (millions of USD)
Port
folior
eturn
(%)
VaR
CVaR
Original VaR
Original CVaR
Original Return
7,62
CONDITIONAL DRAWDOWN-AT-RISK (CDaR)
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CDaR1 is a new risk measure closely related to CVaR
Drawdown is defined as a drop in the portfolio value compared tothe previous maximum
CDaR is the average of the worst z% portfolio drawdowns observedin the past (e.q., 5% of worst drawdowns). Similar to CVaR,
averaging is done using -tail distribution. Notations:
w(x,t) = uncompounded portfolio value
t = time
x = (x1,...xn) = portfolio weights
f(x,t) = max{ 0 t } [w(x, )] - w(x,t) = drawdown
Formal definition:CDaR is CVaR with drawdown loss function f(x,t) .
CDaR can be controlled and optimized using linear programmingsimilar to CVaR
Detail discussion of CDaR is beyond the scope of this presentation
1
Chekhlov, A., Uryasev, S., and M. Zabarankin. Port fol io Opt imizat ion with DrawdownConstraints. Research Report 2000-5. ISE Dept., University of Florida, April 2000.
CDaR: EXAMPLE GRAPH
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Return and Drawdown
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
1 3 5 7 911
13
15
17
19
21
23
25
27
29
31
33
35
time (working days)
Rate of Return DrawDown function
CONCLUSION
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CVaR is a new risk measure with significant advantages comparedto VaR
- can quantify risks beyond VaR- coherent risk measure
- consistent for various confidence levels (((( smooth w.r.t )
- relatively stable statistical estimates (integral characteristics)
CVaR is an excellent tool for risk management and portfoliooptimization
- optimization with linear programming: very large dimensions and stable
numerical implementations- shaping distributions: multiple risk constraints with different
confidence levels at different times
- fast algorithms which can be used in online applications, such as activeportfolio management
CVaR methodology is consistent with mean-variancemethodology under normality assumption
- CVaR minimal portfolio (with return constraint) is also variance minimalfor normal loss distributions
CONCLUSION (Contd)
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Various case studies demonstrated high efficiency andstability of of the approach (papers can be downloaded:www.ise.ufl.edu/uryasev)
- optimization of a portfolio of stocks
- hedging of a portfolio of options
- credit risk management (bond portfolio optimization)- asset and liability modeling
- portfolio replication
- optimal position closing strategies
CVaR has a great potential for further development. It stimulatedseveral areas of applied research, such as Conditional Drawdown-at-Risk and specialized optimization algorithms for riskmanagement
Risk Management and Financial Engineering Lab at UF(www.ise.ufl.edu/rmfe) leads research in CVaR methodology andis interested in applied collaborative projects
APPENDIX: RELEVANT PUBLICATIONS
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[1] Bogentoft, E. Romeijn, H.E. and S. Uryasev (2001): Asset/Liability Management forPension Funds Using Cvar Constraints. Submitted to The Journal of Risk Finance(download: www.ise.ufl.edu/uryasev/multi_JRB.pdf)
[2] Larsen, N., Mausser H., and S. Uryasev (2001): Algorithms For Optimization OfValue-At-Risk . Research Report 2001-9, ISE Dept., University of Florida, August, 2001(www.ise.ufl.edu/uryasev/wp_VaR_minimization.pdf)
[3] Rockafellar R.T. and S. Uryasev (2001): Conditional Value-at-Risk for General LossDistributions. Submitted to The Journal of Banking and Finance (relevant
Research Report 2001-5. ISE Dept., University of Florida, April 2001,
www.ise.ufl.edu/uryasev/cvar2.pdf)
[4] Uryasev, S. Conditional Value-at-Risk (2000): Optimization Algorithms andApplications. Financial Eng ineering News, No. 14, February, 2000.(www.ise.ufl.edu/uryasev/finnews.pdf )
[5] Rockafellar R.T. and S. Uryasev (2000): Optimization of Conditional Value-at-Risk.The Journal of Risk. Vol. 2, No. 3, 2000, 21-41.
( www.ise.ufl.edu/uryasev/cvar.pdf).
[6] Andersson, F., Mausser, H., Rosen, D., and S. Uryasev (2000): Credit RiskOptimization With Conditional Value-At-Risk Criterion. Mathematical Programming,Series B, December, 2000.(/www.ise.ufl.edu/uryasev/Credit_risk_optimization.pdf)
APPENDIX: RELEVANT PUBLICATIONS (Contd)
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[7] Palmquist, J., Uryasev, S., and P. Krokhmal (1999): Portfolio Optimization withConditional Value-At-Risk Objective and Constraints. Submitted to The Journal o fRisk (www.ise.ufl.edu/uryasev/pal.pdf)
[8] Chekhlov, A., Uryasev, S., and M. Zabarankin (2000): Portfolio Optimization WithDrawdown Constraints. Submitted to Appl ied Mathematical Financejournal.
(www.ise.ufl.edu/uryasev/drd_2000-5.pdf)
[9] Testuri, C.E. and S. Uryasev. On Relation between Expected Regret andConditional Value-At-Risk. Research Report 2000-9. ISE Dept., University of Florida,August 2000. Submitted to Decisions in Econom ics and Financejournal.(www.ise.ufl.edu/uryasev/Testuri.pdf)
[10] Krawczyk, J.B. and S. Uryasev. Relaxation Algorithms to Find Nash Equilibria
with Economic Applications. Environmental Modeling and Assessment, 5, 2000, 63-73. (www.baltzer.nl/emass/articlesfree/2000/5-1/ema505.pdf)
[11] Uryasev, S. Introduction to the Theory of Probabilistic Functions and Percentiles(Value-at-Risk). In Probabilistic Constrained Optimization: Methodology andApplications, Ed. S. Uryasev, Kluwer Academic Publishers, 2000.
(www.ise.ufl.edu/uryasev/intr.pdf).
APPENDIX: BOOKS
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[1] Uryasev, S. Ed. Probabilistic Constrained Optimization:Methodology and Applications, Kluwer AcademicPublishers, 2000.
[2] Uryasev, S. and P. Pardalos, Eds. StochasticOptimization: Algorithms and Applications, KluwerAcademic Publishers, 2001 (proceedings of the conference
on Stochastic Optimization, Gainesville, FL, 2000).