8/20/2019 VaR vs CVaR CARISMA Conference 2010
1/75
Stan Uryasev
Joint presentation with
Sergey Sarykalin, Gaia Serraino and Konstantin Kalinchenko
Risk Management and Financial Engineering Lab, University of Floridaand
American Optimal Decisions
VaR vs CVaR in
Risk Management and Optimization
1
8/20/2019 VaR vs CVaR CARISMA Conference 2010
2/75
Agenda
Compare Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR)
definitions of VaR and CVaR
basic properties of VaR and CVaR
axiomatic definition of Risk and Deviation Measures
reasons affecting the choice between VaR and CVaR
risk management/optimization case studies conducted with
Portfolio Safeguard package by AORDA.com
2
8/20/2019 VaR vs CVaR CARISMA Conference 2010
3/75
Risk Management
Risk Management is a procedure for shaping a loss distribution
Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are
popular function for measuring risk The choice between VaR and CVaR is affected by:
differences in mathematical properties,
stability of statistical estimation, simplicity of optimization procedures,
acceptance by regulators
Conclusions from these properties are contradictive
3
8/20/2019 VaR vs CVaR CARISMA Conference 2010
4/75
Risk Management
Key observations:
CVaR has superior mathematical properties versus VaR
Risk management with CVaR functions can be done very efficiently
VaR does not control scenarios exceeding VaR
CVaR accounts for losses exceeding VaR
Deviation and Risk are different risk management concepts
CVaR Deviation is a strong competitor to the Standard Deviation
4
8/20/2019 VaR vs CVaR CARISMA Conference 2010
5/75
VaR and CVaR Representation
5
8/20/2019 VaR vs CVaR CARISMA Conference 2010
6/75
x
Risk
VaR
CVaR
CVaR+
CVaR-
VaR, CVaR, CVaR+ and CVaR-
6
8/20/2019 VaR vs CVaR CARISMA Conference 2010
7/75
Value-at-Risk
is non convex and discontinuous function of the
confidence level α for discrete distributions
is non-sub-additive
difficult to control/optimize for non-normal distributions:VaR has many extremums for discrete distributions
7
[1,0]∈α })(|min{)( α α ≥= zF z X VaR X for
)( X VaRα
X a loss random variable
)( X VaRα
8/20/2019 VaR vs CVaR CARISMA Conference 2010
8/75
Conditional Value-at-Risk
Rockafellar and Uryasev, “Optimization of Conditional Value-at-Risk”,
Journal of Risk, 2000 introduced the term Conditional Value-at-
Risk
For
where
8
∫+∞
∞−
= )()( z zdF X CVaR X α
α [1,0]∈α
⎪⎩
⎪
⎨
⎧
≥−
−
<
= )( when1
)(
)(nwhe 0
)( X VaR z zF
X VaR z
zF X X α
α α
α α
8/20/2019 VaR vs CVaR CARISMA Conference 2010
9/75
Conditional Value-at-Risk
CVaR+
(Upper CVaR): expected value of X strictly exceeding VaR(also called Mean Excess Loss and Expected Shortfall)
CVaR- (Lower CVaR): expected value of X weakly exceeding VaR(also called Tail VaR)
Property: is weighted average of and
zero for continuous distributions!!! 9
)](|[)( X VaR X X E X CVaR α α >=+
)( X C V a Rα ( )C V aR X α +
( )VaR X α
( ) ( ) (1 ( )) ( ) if ( ( )) 1( )
( ) if ( ( )) 1 X
X
X VaR X X CVaR X F VaR X CVaR X
VaR X F VaR X α α α α α α
α α
λ λ +⎧ + −
8/20/2019 VaR vs CVaR CARISMA Conference 2010
10/75
Conditional Value-at-Risk
Definition on previous page is a major innovation
and for general loss distributions are
discontinuous functions
CVaR is continuous with respect to α
CVaR is convex in X
VaR, CVaR- ,CVaR+ may be non-convex
VaR ≤ CVaR- ≤ CVaR ≤ CVaR+
10
)( X CVaR+
α )( X VaRα
8/20/2019 VaR vs CVaR CARISMA Conference 2010
11/75
x
Risk
VaR
CVaR
CVaR+
CVaR-
VaR, CVaR, CVaR+ and CVaR-
11
8/20/2019 VaR vs CVaR CARISMA Conference 2010
12/75
CVaR: Discrete Distributions
α does not “split” atoms: VaR < CVaR- < CVaR = CVaR+,λ = (Ψ- α)/(1- α) = 0
12
1 2 41 2 6 6 3 6
1 15 62 2
Six scenarios, ,
CVaR CVaR =
p p p
f f
α = = = = = =
= +
L
+
Probability CVaR
16
16
16
16
16
16
1 f
2 f
3 f 4 f 5 f 6 f
VaR --CVaR
+CVaR
Loss
8/20/2019 VaR vs CVaR CARISMA Conference 2010
13/75
CVaR: Discrete Distributions
• α “splits” the atom: VaR < CVaR- < CVaR < CVaR+,λ = (Ψ- α)/(1- α) > 0
13
711 2 6 6 12
1 4 1 2 24 5 65 5 5 5 5
Six scenarios, ,
CVaR VaR CVaR =
α = = = = =
= + + +
L p p p
f f f +
Probability CVaR
16
16
16
16
112
16
16
1 f 2 f 3 f 4 f 5 f 6 f
VaR
--
CVaR
+1 1
5 62 2
CVaR + = f f Loss
8/20/2019 VaR vs CVaR CARISMA Conference 2010
14/75
CVaR: Discrete Distributions
• α “splits” the last atom: VaR = CVaR- = CVaR ,CVaR+ is not defined, λ = (Ψ - α)/(1- α) > 0
14
8/20/2019 VaR vs CVaR CARISMA Conference 2010
15/75
CVaR: Equivalent Definitions
Pflug defines CVaR via an optimization problem, as in Rockafellar
and Uryasev (2000)
Acerbi showed that CVaR is equivalent to Expected Shortfalldefined by
15
Pflug, G.C., “Some Remarks on the Value-at-Risk and on the Conditional Value-at-Risk”, Probabilistic
Constrained Optimization: Methodology and Applications, (Uryasev ed), Kluwer, 2000
Acerbi, C., “Spectral Measures of Risk: a coherent representation of subjective risk aversion”, JBF, 2002
8/20/2019 VaR vs CVaR CARISMA Conference 2010
16/75
RISK MANAGEMENT: INSURANCE
Accident lost
Payment
Accident lost
Payment
Premium
Deductible
Payment
∞
Payment
∞ ∞
Premium
Deductible
8/20/2019 VaR vs CVaR CARISMA Conference 2010
17/75
TWO CONCEPTS OF RISK
Risk as a possible loss
Minimum amount of cash to be added to make a portfolio (orproject) sufficiently safe
Example 1. MaxLoss
-Three equally probable outcomes, { -4, 2, 5 };
MaxLoss = -4; Risk = 4
-Three equally probable outcomes, { 0, 6, 9 };
MaxLoss = 0; Risk = 0
Risk as an uncertainty in outcomes
Some measure of deviation in outcomes
Example 2. Standard Deviation
-Three equally probable outcomes, { 0, 6, 9 }; StandardDeviation > 0
8/20/2019 VaR vs CVaR CARISMA Conference 2010
18/75
Risk Measures: axiomatic definition
A functional is a coherent risk measure in theextended sense if:
R1: for all constant C R2: for (convexity)
R3: when (monotonicity)
R4: when with (closedness)
A functional is a coherent risk measure in the basic
sense if it satisfies axioms R1, R2, R3, R4 and R5:
R5: for (positive homogeneity)
18
]0,1[λ ∈' X X ≤
0>λ
8/20/2019 VaR vs CVaR CARISMA Conference 2010
19/75
A functional is an averse risk measure in theextended sense if it satisfies axioms R1, R2, R4 and R6:
R6: for all nonconstant X (aversity)
A functional is an averse risk measure in the basic
sense if it satisfies axioms R1, R2, R4, R6 and R5
Aversity has the interpretation that the risk of loss in a
nonconstant random variable X cannot be acceptable unless EX
8/20/2019 VaR vs CVaR CARISMA Conference 2010
20/75
Examples of coherent risk measures: A
Z
Examples of risk measures not coherent:
, λ >0, violates R3 (monotonicity)
violates subadditivity
is a coherent measure of risk
in the basic sense and it is an averse measure of risk !!! Averse measure of risk might not be coherent, a coherent
measure might not be averse
20
Risk Measures: axiomatic definition
8/20/2019 VaR vs CVaR CARISMA Conference 2010
21/75
A functional is called a deviation measure in the extendedsense if it satisfies:
D1: for constant C , but for nonconstant X
D2: for (convexity)
D3: when with (closedness)
A functional is called a deviation measure in the basic
sense if it satisfies axioms D1,D2, D3 and D4:
D4: (positive homogeneity)
A deviation measure in extended or basic sense is also coherent if itadditionally satisfies D5:
D5: (upper range boundedness)
21
Deviation Measures: axiomatic definition
[1,0]∈λ
8/20/2019 VaR vs CVaR CARISMA Conference 2010
22/75
Examples of deviation measures in the basic sense: Standard Deviation
Standard Semideviations
Mean Absolute Deviation α-Value-at-Risk Deviation measure:
α-VaR Dev does not satisfy convexity axiom D2 it is not a
deviation measure
α-Conditional Value-at-Risk Deviation measure:
22
Deviation Measures: axiomatic definition
Coherent deviation measure in basic sense !!!
8/20/2019 VaR vs CVaR CARISMA Conference 2010
23/75
23
Deviation Measures: axiomatic definition
CVaR Deviation Measure is a coherent deviation measure in the
basic sense
8/20/2019 VaR vs CVaR CARISMA Conference 2010
24/75
Rockafellar et al. (2006) showed the existence of a one-to-one
correspondence between deviation measures in the extended sense
and averse risk measures in the extended sense:
24
Risk vs Deviation Measures
Rockafellar, R.T., Uryasev, S., Zabarankin, M., “Optimality conditions in portfolio analysis
with general deviation measures”, Mathematical Programming, 2006
8/20/2019 VaR vs CVaR CARISMA Conference 2010
25/75
25
Risk vs Deviation Measures
Deviation
Measure
Counterpart Risk
Measure
where
8/20/2019 VaR vs CVaR CARISMA Conference 2010
26/75
26
Chance and VaR Constraints
Let be some random loss function. By definition:
Then the following holds:
In general is nonconvex w.r.t. x, (e.g., discretedistributions)
may be nonconvex constraints
mi x f i ,..1 ),,( =ω }}),(Pr{:min{)( α ε ω ε α ≥≤= x f xVaR
ε α ε ω α ≤↔≥≤ )(VaR }),(Pr{ X x f
)( xVaRα
}),(Pr{and)(VaR α ε ω ε α ≥≤≤ x f X
8/20/2019 VaR vs CVaR CARISMA Conference 2010
27/75
27
VaR vs CVaR in optimization
VaR is difficult to optimize numerically when losses are not
normally distributed
PSG package allows VaR optimization
In optimization modeling, CVaR is superior to VaR:
For elliptical distribution minimizing VaR, CVaR or Variance is
equivalent CVaR can be expressed as a minimization formula (Rockafellar
and Uryasev, 2000)
CVaR preserve convexity
8/20/2019 VaR vs CVaR CARISMA Conference 2010
28/75
28
CVaR optimization
Theorem 1:
1. is convex w.r.t. α
2. α -VaR is a minimizer of F with respect to ζ :
3. α - CVaR equals minimal value (w.r.t. ζ ) of function F :
),( ζ α xF
( ) ( )VaR ( , ) ( , ) arg min ( , ) f x f x F xα α α ζ
ξ ζ ξ ζ = =
( )CVaR ( , ) min ( , ) f x F xα α ζ
ξ ζ =
}]),({[1
1),( +−
−+= ζ ξ
α ζ ζ α x f E xF
8/20/2019 VaR vs CVaR CARISMA Conference 2010
29/75
29
CVaR optimization
Preservation of convexity: if f(x,ξ ) is convex in x thenis convex in x
If f(x,ξ ) is convex in x then is convex in x and ζ
If f(x*,ζ *) minimizes over then
is equivalent to
)( X CVaRα
),( ζ α xF
ℜ× X
8/20/2019 VaR vs CVaR CARISMA Conference 2010
30/75
30
CVaR optimization
In the case of discrete distributions:
The constraint can be replaced by a system ofinequalities introducing additional variables η k :
ω ζ α ≤),( xF
N1,..,k ,0),( ,0 =≤−−≥ k k k y x f η ζ η
ω η α
ζ ≤−
+ ∑=
N
k
k k p11
1
1
1
( , ) (1 ) [ ( , ) ]
max { , 0}
N k
k
k
F x p f x
z z
α ζ ζ α ξ ζ − +
=+
= + − −
=
∑
8/20/2019 VaR vs CVaR CARISMA Conference 2010
31/75
31
Generalized Regression Problem
Approximate random variable by random variables
Error measure satisfies axioms
Y 1 2, ,..., .n X X X
Rockafellar, R. T., Uryasev, S. and M. Zabarankin:
“Risk Tuning with Generalized Linear Regression”, accepted for publication in Mathematics of
Operations Research, 2008
8/20/2019 VaR vs CVaR CARISMA Conference 2010
32/75
Error, Deviation, Statistic
32
For an error measure :
the corresponding deviation measure is
the corresponding statistic
is
8/20/2019 VaR vs CVaR CARISMA Conference 2010
33/75
Theorem: Separation Principle
General regression problem
is equivalent to
33
8/20/2019 VaR vs CVaR CARISMA Conference 2010
34/75
Percentile Regression and CVaR Deviation
34
Koenker, R., Bassett, G.
Regression quantiles.
Econometrica 46, 33–50(1978)
( )1min [ ] ( 1) [ ] ( )C R
E X C E X C CVaR X EX α α −
+ −∈− + − − = −
8/20/2019 VaR vs CVaR CARISMA Conference 2010
35/75
35
Stability of Estimation
VaR and CVaR with same confidence level measure “differentparts” of the distribution
For a specific distribution the confidence levels α1 and α2 for
comparison of VaR and CVaR should be found from the equation
Yamai and Yoshiba (2002), for the same confidence level:
VaR estimators are more stable than CVaR estimators
The difference is more prominent for fat-tailed distributions
Larger sample sizes increase accuracy of CVaR estimation
More research needed to compare stability of estimators for the
same part of the distribution.
)()(21
X CVaR X VaR α α =
Decomposition According to Risk Factors
8/20/2019 VaR vs CVaR CARISMA Conference 2010
36/75
36
For continuous distributions the following decompositions ofVaR and CVaR hold:
When a distribution is modeled by scenarios it is easier to
estimate than
Estimators of are more stable than estimators
of
iii
n
i i
z X VaR X X E z z
X VaR X VaR )](|[
)()(
1 α
α
α
==∂
∂=
∑=
iii
n
i i
z X VaR X X E z z
X CVaR X CVaR )](|[
)()(
1
α α
α ≥=∂∂
= ∑=
)](|[ X VaR X X E i α ≥ )](|[ X VaR X X E i α =
i z X CVaR
∂∂)(α
i z
X VaR
∂∂ )(α
Decomposition According to Risk Factors
Contributions
8/20/2019 VaR vs CVaR CARISMA Conference 2010
37/75
Generalized Master Fund Theorem and CAPM
Assumptions:• Several groups of investors each with utility function
• Utility functions are concave w.r.t. mean and deviation
increasing w.r.t. mean
decreasing w.r.t. deviation
• Investors maximize utility functions s.t. budget constraint.
37
))(( , j j j j R ERU D
Rockafellar, R.T., Uryasev, S., Zabarankin, M. “Master Funds in Portfolio Analysis with
General Deviation Measures”, JBF, 2005
Rockafellar, R.T., Uryasev, S., Zabarankin, M. “Equilibrium with Investors using a Diversity of
Deviation Measures”, JBF, 2007
8/20/2019 VaR vs CVaR CARISMA Conference 2010
38/75
8/20/2019 VaR vs CVaR CARISMA Conference 2010
39/75
8/20/2019 VaR vs CVaR CARISMA Conference 2010
40/75
Generalized Master Fund Theorem and CAPM
Equilibrium exists w.r.t. Each investor has its own master fund and invests in its own master
fund and in the risk-free asset
Generalized CAPM holds:
is expected return of asset i in group jis risk-free rate
is expected return of market fund for investor group j
is the risk identifier for the market fund j
40
)])([()r ,Covar(G ij j ijij j j Er r EGG E −−=
8/20/2019 VaR vs CVaR CARISMA Conference 2010
41/75
41
When then
When then
When then
Generalized Master Fund Theorem and CAPM
Cl i l CAPM Di ti F l
8/20/2019 VaR vs CVaR CARISMA Conference 2010
42/75
Classical CAPM => Discounting Formula
( )
( )2
0
0
00
,cov
)(1
1
)(1
M
M ii
M iii
M i
ii
r r
r r E r
r r r
E
σ
β
β ζ π
β
ζ π
=
−−+=
−++=
All investors have the same risk preferences:
standard deviation
Discounting by risk-free rate with adjustment foruncertainties (derived in PhD dissertation of
Sarykalin)
G li d CAPM
8/20/2019 VaR vs CVaR CARISMA Conference 2010
43/75
Generalized CAPM
There are different groups of investors k=1,…,K
Risk attitude of each group of investors can be expressedthrough its deviation measure Dk
Consequently:
Each group of investors invests its own Master Fund M
G li d CAPM
8/20/2019 VaR vs CVaR CARISMA Conference 2010
44/75
Generalized CAPM
( )( )
( ))(1
1
)(1
,cov
0
0
00
r r E
r
r r r E
r D
Qr
M iii
M i
ii
M
D
M ii
−−
+
=
−++=
−=
β ζ π
β ζ π
β
= deviation measure
Q
M
= risk identifier for corresponding to M
Investors Buying Out-of-the-money
8/20/2019 VaR vs CVaR CARISMA Conference 2010
45/75
Investors Buying Out-of-the-money
S&P500 Put Options
Group of investors buys S&P500 options
Risk preferences are described by mixed CVaR deviation
Assume that S&P500 is their Master Fund
Out-of-the-money put option is an investments in low tail of
price distribution. CVaR deviations can capture the tail.
( )∑=
Δ=n
i
i X CVaR X D i1
)( α λ
Mixed CVaR Deviation Risk Envelope
8/20/2019 VaR vs CVaR CARISMA Conference 2010
46/75
Mixed CVaR Deviation Risk Envelope
EX XQ X DQ
−=∈Q
sup)(
( )
( ){ } X VaR X Q
X CVaR X D
X α
α
ω ω ≥=
= Δ
)()(
)(
1
( )
( ){ }∑∑
=
=
Δ
≥=
=n
i
i X
n
ii
X VaR X Q
X CVaR X D
i
i
1
1
)()(
)(
α
α
ω λ ω
λ
1
)(ω X Q
)(ω X
( ) X VaRα
)(ω X Q
)(ω X ( ) X VaR
1α
( ) X VaR2α
( ) X VaR3α
α 1
1
1
α λ
2
2
1
1
α
λ
α
λ +
3
3
2
2
1
1
α λ
α λ
α λ ++
Data
8/20/2019 VaR vs CVaR CARISMA Conference 2010
47/75
Data
Put options prices on Oct,20 2009 maturing on Nov, 20 2009
S&P500 daily prices for 2000-2009
2490 monthly returns (overlapping daily):
every trading day t: r t=lnSt-lnSt-21
Mean return adjusted to 6.4% annually.
2490 scenarios of S&P500 option payoffs on Nov, 20 2009
CVaR Deviation
8/20/2019 VaR vs CVaR CARISMA Conference 2010
48/75
CVaR Deviation
Risk preferences of Put Option buyers:
We want to estimate values for coefficients
( )
%50%75%85%95%99
)(
54321
5
1
=====
=
∑=Δ
α α α α α
λ α j
j X CVaR X D j
jλ
Prices Implied Volatilities
8/20/2019 VaR vs CVaR CARISMA Conference 2010
49/75
Prices Implied Volatilities
Option prices with different strike prices vary very significantly
Black-Scholes formula: prices implied volatilities
Graphs: ( )XCV RXD Δ)(
8/20/2019 VaR vs CVaR CARISMA Conference 2010
50/75
Graphs: ( ) X CVaR X D = %50)(
0%
5%
10%
15%
20%
25%
30%
35%
91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110
Market IV CAPM price IV No Risk IV Generalized CAPM IV alpha=50%
Graphs: ( )XCVaRXD Δ)(
8/20/2019 VaR vs CVaR CARISMA Conference 2010
51/75
Graphs: ( ) X CVaR X D = %75)(
0%
5%
10%
15%
20%
25%
30%
35%
91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110
Market IV CAPM price IV No Risk IV Generalized CAPM IV alpha=75%
Graphs: ( )XCVaRXD Δ=)(
8/20/2019 VaR vs CVaR CARISMA Conference 2010
52/75
Graphs: ( ) X CVaR X D = %85)(
0%
5%
10%
15%
20%
25%
30%
35%
91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110
Market IV CAPM price IV No Risk IV Generalized CAPM IV alpha=85%
Graphs: ( )XCVaRXD Δ=)(
8/20/2019 VaR vs CVaR CARISMA Conference 2010
53/75
Graphs: ( ) X CVaR X D = %95)(
Graphs: ( )∑ Δ=5
)( XCVaRXD λ
8/20/2019 VaR vs CVaR CARISMA Conference 2010
54/75
0%
5%
10%
15%
20%
25%
30%
35%
91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110
Market IV CAPM price IV No Risk IV Ceneralized CAPM Mixed
Graphs:
111.0219.0227.0239.0204.0
%50%75%85%95%99
54321
54321
==========
λ λ λ λ λ
α α α α α
( )∑=
=1
)( j
j X CVaR X D jα λ
8/20/2019 VaR vs CVaR CARISMA Conference 2010
55/75
VaR: Pros
1. VaR is a relatively simple risk management concept and has a clearinterpretation
2. Specifying VaR for all confidence levels completely defines thedistribution (superior to σ)
3. VaR focuses on the part of the distribution specified by theconfidence level
4. Estimation procedures are stable
5. VaR can be estimated with parametric models
55
8/20/2019 VaR vs CVaR CARISMA Conference 2010
56/75
VaR: Cons
1. VaR does not account for properties of the distribution beyond
the confidence level
2. Risk control using VaR may lead to undesirable results for skewed
distributions
3. VaR is a non-convex and discontinuous function for discrete
distributions
56
8/20/2019 VaR vs CVaR CARISMA Conference 2010
57/75
CVaR: Pros
1. VaR has a clear engineering interpretation
2. Specifying CVaR for all confidence levels completely defines the
distribution (superior to σ)3. CVaR is a coherent risk measure
4. CVaR is continuous w.r.t. α
5. is a convex function w.r.t.
6. CVaR optimization can be reduced to convex programming and insome cases to linear programming
57
)...( 11 nn X w X wCVaR ++α ),...,( 1 nww
8/20/2019 VaR vs CVaR CARISMA Conference 2010
58/75
CVaR: Cons
1. CVaR is more sensitive than VaR to estimation errors
2. CVaR accuracy is heavily affected by accuracy of tail modeling
58
8/20/2019 VaR vs CVaR CARISMA Conference 2010
59/75
VaR or CVaR in financial applications?
VaR is not restrictive as CVaR with the same confidence levela trader may prefer VaR
A company owner may prefer CVaR; a board of director may
prefer reporting VaR to shareholders VaR may be better for portfolio optimization when good models
for the tails are not available
CVaR should be used when good models for the tails are available CVaR has superior mathematical properties
CVaR can be easily handled in optimization and statistics
Avoid comparison of VaR and CVaR for the same level α
59
8/20/2019 VaR vs CVaR CARISMA Conference 2010
60/75
Adequate accounting for risks, classical and downside risk measures:
-Value-at-Risk (VaR)
- Conditional Value-at-Risk (CVaR)
- Drawdown
- Maximum Loss- Lower Partial Moment
- Probability (e.g. default probability)
- Variance
- St.Dev.- and many others
Various data inputs for risk functions: scenarios and covariances
- Historical observations of returns/prices- Monte-Carlo based simulations, e.g. from RiskMetrics or S&P CDO
evaluator
- Covariance matrices, e.g. from Barra factor models
PSG: Portfolio Safeguard
60
PSG P tf li S f d
8/20/2019 VaR vs CVaR CARISMA Conference 2010
61/75
Powerful and robust optimization tools:
four environments:
Shell (Windows-dialog)
MATLAB
C++
Run-file
simultaneous constraints on many functions at various times
(e.g., multiple constraints on standard deviations obtained by
resampling in combination with drawdown constraints)
PSG: Portfolio Safeguard
61
8/20/2019 VaR vs CVaR CARISMA Conference 2010
62/75
PSG: Risk Functions
orRisk function
(e.g., St.Dev, VaR, CVaR, Mean,…)covariance
matrix
matrix of
scenarios
matrix with
one rowLinear function
62
PSG E l
8/20/2019 VaR vs CVaR CARISMA Conference 2010
63/75
PSG: Example
63
PSG O ti ith F ti
8/20/2019 VaR vs CVaR CARISMA Conference 2010
64/75
PSG: Operations with Functions
Risk functions
Optimization Sensitivities Graphics
64
C St d Ri k C t l i V R
8/20/2019 VaR vs CVaR CARISMA Conference 2010
65/75
Case Study: Risk Control using VaR
Risk control using VaR may lead to paradoxical results for skeweddistributions
Undesirable feature of VaR optimization: VaR minimization mayincrease the extreme losses
Case Study main result: minimization of 99%-VaR Deviation leadsto 13% increase in 99%-CVaR compared to 99%-CVaR of optimal
99%-CVaR Deviation portfolio
Consistency with theoretical results: CVaR is coherent, VaR is not
coherent
65
Larsen, N., Mausser, H., Uryasev, S., “ Algorithms for Optimization of Value-at-Risk”,
Financial Engineering, E-commerce, Supply Chain, P. Pardalos , T.K. Tsitsiringos Ed.,
Kluwer, 2000
C St d Ri k C t l i V R
8/20/2019 VaR vs CVaR CARISMA Conference 2010
66/75
Case Study: Risk Control using VaR
66
Columns 2, 3 report value of risk functions at optimal point of Problem 1 and 2;
Column “Ratio” reports ratio of Column 3 to Column 2
P.1 Pb.2
C St d Li R i H d i
8/20/2019 VaR vs CVaR CARISMA Conference 2010
67/75
Case Study: Linear Regression-Hedging
67
Investigate performance of optimal hedging strategies based ondifferent deviation measures
Determining optimal hedging strategy is a linear regression problem
Benchmark portfolio value is the response variable, replicatingfinancial instruments values are predictors, portfolio weights arecoefficients of the predictors to be determined
Coefficients chosen to minimize a replication error
function depending upon the residual
I I x x θ θ θ ++= ...ˆ 11
θ θ ˆ0
− I x x ,...,1
C St d Li R i H d i
8/20/2019 VaR vs CVaR CARISMA Conference 2010
68/75
Case Study: Linear Regression-Hedging
68
Out-of-sample performance of hedging strategies significantly dependson the skewness of the distribution
Two-Tailed 90%-VaR has the best out-of-sample performance
Standard deviation has the worst out-of-sample performance
(1)
(2)
(3)
(4)
(5)
C St d Li R i H d i
8/20/2019 VaR vs CVaR CARISMA Conference 2010
69/75
Case Study: Linear Regression-Hedging
69
Each row reports value o different risk functions evaluated at optimal points of the 5
different hedging strategies (e.g., the first raw is for hedging strategy)
the best performance has TwoTailVar
Out-of-sample performance of different deviation measures evaluated at optimal points
of the 5 different hedging strategies (e.g., the first raw is for hedging strategy)0.9CVaRΔ
0.9CVaRΔ
0.9
Δ
Example:
h d i i l
8/20/2019 VaR vs CVaR CARISMA Conference 2010
70/75
Chance and VaR constraints equivalence
70
We illustrate numerically the equivalence:
At optimality the two
problems selected
the same portfoliowith the same
objective function
value
Case Study: Portfolio Rebalancing Strategies,
8/20/2019 VaR vs CVaR CARISMA Conference 2010
71/75
Risks and Deviations
71
We consider a portfolio rebalancing problem:
We used as risk functions VaR, CVaR, VaR Deviation, CVaRDeviation, Standard Deviation
We evaluated Sharpe ratio and mean value of each sequence ofportfolios
We found a good performance of VaR and VaR Deviationminimization
Standard Deviation minimization leads to inferior results
Case Study: Portfolio Rebalancing Strategies,
i k d i i ( d)
8/20/2019 VaR vs CVaR CARISMA Conference 2010
72/75
Risks and Deviations (Cont'd)
72
Results depend on the scenario dataset and on k In the presence of mean reversion the tails of historical distribution
are not good predictors of the tail in the future
VaR disregards the unstable part of the distribution thus may leadto good out-of-sample performance
Sharpe ratio for the rebalancing strategy when different risk functions are
used in the objective for different values of the parameter k
Conclusions: key observations
8/20/2019 VaR vs CVaR CARISMA Conference 2010
73/75
Conclusions: key observations
73
CVaR has superior mathematical properties: CVaR is coherent,CVaR of a portfolio is a continuous and convex function withrespect to optimization variables
CVaR can be optimized and constrained with convex and linear
programming methods; VaR is relatively difficult to optimize
VaR does not control scenarios exceeding VaR
VaR estimates are statistically more stable than CVaR estimates
VaR may lead to bearing high uncontrollable risk
CVaR is more sensitive than VaR to estimation errors
CVaR accuracy is heavily affected by accuracy of tail modeling
C l i k b i
8/20/2019 VaR vs CVaR CARISMA Conference 2010
74/75
Conclusions: key observations
74
There is a one-to-one correspondence between Risk Measuresand Deviation Measures
CVaR Deviation is a strong competitor of Standard Deviation
Mixed CVaR Deviation should be used when tails are not
modeled correctly. Mixed CVaR Deviation gives different weight
to different parts of the distribution
Master Fund Theorem and CAPM can be generalized with the for
different deviation measure.
C l i C St di
8/20/2019 VaR vs CVaR CARISMA Conference 2010
75/75
Conclusions: Case Studies
Case Study 1: risk control using VaR may lead to paradoxicalresults for skewed distribution. Minimization of VaR may lead to astretch of the tail of the distribution exceeding VaR
Case Study 2: determining optimal hedging strategy is a linear
regression problem. Out-of-sample performance based ondifferent Deviation Measures depends on the skewness of thedistribution, we found standard deviation have the worst
performance Case Study 3: chance constraints and percentiles of a distribution
are closely related, VaR and Chance constraints are equivalent
Case Study 4: the choice of the risk function to minimize in aportfolio rebalancing strategy depends on the scenario dataset. Inthe presence of mean reversion VaR neglecting tails may lead togood out-of-sample performance
Top Related