Stochastic Optimization ESI 6912 Instructor: Prof. S. Uryasev NOTES 7: Algorithms and Applications
Jan 18, 2016
Stochastic OptimizationESI 6912
Instructor: Prof. S. Uryasev
NOTES 7:
Algorithms and Applications
Content
1. Value-at-Risk (VaR) a. Definition
b. Features
c. Examples
2. Conditional Value-at-Risk (CVaR)
a. Definition: Continuous and Discrete distribution
b. Features
c. Examples
3. Formulation of optimization problem
a. Definition of a loss function
b. Examples: CVaR in performance function and CVaR in constraints
4. Optimization techniques a. CVaR as an optimization problem: Theorem 1 and Theorem 2
b. Reduction to LP
5. Case Studies
Value-at-Risk Definition
Value-at-Risk (VaR) = - percentile of distribution of random variable (a smallest value such that probability that random variable
is smaller or equals to this value is greater than or equal to )
Definition:
Random variable,
Fre
qu
en
cy
VaR
Probability
Maximal value
Value-at-Risk Definition (cont’d)
Mathematical Definition:
- random variable
VaR ( ) inf ( )P
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 value in a specified period with some confidence level (e.g., confidence level = 95%, period = 1 week)
Remarks:
Value-at-Risk Features
• simple convenient representation of risks (one number)
• measures downside risk (compared to variance which is impacted by high returns)
• applicable to nonlinear instruments, such as options, with non- symmetric (non-normal) loss distributions
• may provide inadequate picture of risks: does not measure losses exceeding VaR (e.g., excluding or doubling of big losses in November 1987 may not impact VaR historical estimates)
• reduction of VaR may lead to stretch of tail exceeding VaR: risk control with VaR may lead to increase of losses exceeding VaR. E.g, numerical experiments1 show that for a credit risk portfolio, optimization of VaR leads to 16% increase of average losses exceeding VaR. Similar numerical experiments conducted at IMES2 .
1 Larsen, N., Mausser, H. and S. Uryasev. Algorithms for Optimization of Value-At-Risk. Research Report, ISE Dept., University of Florida, forthcoming.
2 Yamai, Y. and T. Yoshiba. On the Validity of Value-at-Risk: Comparative Analyses with Expected Shortfall. Institute for Monetary and Economic Studies. Bank of Japan. IMES Discussion Paper 2001-E-4, 2001.
(Can be downloaded: www.imes.boj.or.jp/english/publication/edps/fedps_2001index.html)
Value-at-Risk Features (cont’d)
• since VaR does not take into account risks exceeding VaR, it may provide conflicting results at different confidence levels:
e.g., at 95% confidence level, foreign stocks may be dominant risk contributors, and at 99% confidence level, domestic stocks may be dominant risk contributors to the portfolio risk
• non-sub-additive and non-convex: non-sub-additivity implies that portfolio diversification may increase the risk
• incoherent in the sense of Artzner, Delbaen, Eber, and Heath1
• difficult to control/optimize for non-normal distributions: VaR has many extremums
1Artzner, P., Delbaen, F., Eber, J.-M. Heath D. Coherent Measures of Risk,
Mathematical Finance, 9 (1999), 203--228.
Value-at-Risk Example
- normally distributed random variable with mean and standard deviation
-2 2 4 6 8
0.05
0.1
0.15
0.2
2
p( )
VaR
area = 1-
)()()(VaR 1 k
Value-at-Risk Example (cont'd)
)()()(VaR 1 k
)12(erf2)( 1 k
tdtzz
0
2 )exp(2
)erf(
65.1)95.0( k
33.2)99.0( k
Conditional Value-at-Risk Definition
Notations:
Ψ = cumulative distribution of random variable ,
Ψ = -tail distribution, which equals to zero for below VaR, and equals to (Ψ- )/(1- ) for exceeding or equal to VaR
Definition: CVaR is mean of -tail distribution Ψ
Cumulative Distribution of , Ψ
0 VaR
1
Ψ()
1
0
1
VaR
-Tail Distribution, Ψ
Ψ()
Conditional Value-at-Risk Definition (cont’d)
Notations:
CVaR+ ( “upper CVaR” ) = expected value of strictly exceeding
VaR (also called Mean Excess Loss and Expected Shortfall)
CVaR- ( “lower CVaR” ) = expected value of weakly exceeding VaR, i.e., value of which is equal to or exceed VaR (also called Tail VaR)
Ψ (VaR) = probability that does not exceed VaR or equal to VaR
Property: CVaR is weighted average of VaR and CVaR+)(CVaR)1(VaR)(CVaR
10,)1()(
Conditional Value-at-Risk Definition (cont’d)
Random variable,
Fr
eq
ue
nc
y
VaR
CVaR
Probability
Maximal value
Conditional Value-at-Risk Features
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 control variables
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.
Conditional Value-at-Risk Features (cont'd)
x
Risk
VaR
CVaR
CVaR+
CVaR-
CVaR is convex, but VaR, CVaR- ,CVaR+ may be non-convex,
inequalities are valid: VaR CVaR- CVaR CVaR+
Conditional Value-at-Risk Features (cont'd)
stable statistical estimates (CVaR has integral characteristics compared 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
Conditional Value-at-Risk Features (cont'd)
qdq
1
)()(CVaR1
1
CVaR for continuous distributions usually coincides with conditional expected loss exceeding VaR (also called Mean Excess Loss 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 to CVaR.
1Acerbi, C., Nordio, C., Sirtori, C. Expected Shortfall as a Tool for Financial Risk Management, Working Paper, can be downloaded: www.gloriamundi.org/var/wps.html
2Acerbi, C., and Tasche, D. On the Coherence of Expected Shortfall. Working Paper, can be downloaded: www.gloriamundi.org/var/wps.html
CVaR: Continuous Distribution, Example 1
- normally distributed random variable with mean and standard deviation
2
)()](|[CVaR 1kE
-2 2 4 6 8
0.05
0.1
0.15
0.2
p( )
VaR
area = 1-
CVaR
CVaR: Continuous Distribution, Example 1
- normally distributed random variable with mean and st. dev.
1211 )1()12(erfexp2)(
k
tdtzz
0
2 )exp(2
)erf(
67.2)99.0(1 k
)()](|[CVaR 1kE
06.2)95.0(1 k
CVaR: Discrete Distribution, Example 2
does not “split” atoms: VaR < CVaR- < CVaR = CVaR+, = (Ψ- )/(1- ) = 0
1 2 41 2 6 6 3 6
1 15 62 2
Six scenarios, ,
CVaR CVaR =
p p p
f f
L+
Probability CVaR
16
16
16
16
16
16
1f 2f 3f 4f 5f 6f
VaR --CVaR
+CVaRLoss
CVaR: Discrete Distribution, Example 3
“splits” the atom: VaR < CVaR- < CVaR < CVaR+, = (Ψ- )/(1- ) > 0
711 2 6 6 1 2
1 4 1 2 24 5 65 5 5 5 5
S i x s c e n a r i o s , ,
C V a R V a R C V a R =
Lp p p
f f f+
P r o b a b i l i t y C V a R
16
16
16
16
11 2
16
16
1f 2f 3f 4f 5f 6f
V a R - -C V a R
+1 15 62 2 C V a R f f
L o s s
CVaR: Discrete Distribution, Example 4
“splits” the last atom: VaR = CVaR- = CVaR,
CVaR+ is not defined, = (Ψ - )/(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
VaRLoss
Formulation of Optimization Problem
- random variable, x - vector of control variables
),( xf = loss or reward
Stochastic functions:
Notations:
0min ( ( , ))
. .
( ( , )) 0, 1, ,
,
j
F f x
s t
F f x j n
x X
(.)CVaR(.)jαjF
Optimization problem:
where for some of j :
Examples
1. 2.
3.
.0
,),(
..
),(CVaRmin
x
xfE
ts
xfx
.0
,),(CVaR,),(CVaR
..
),(max
21 21
x
xfxf
ts
xfEx
),( xf = loss function),( xf = reward function,
.0
,),(CVaR
..
),(max
x
xf
ts
xfEx
Examples (cont'd)
4.
.0,),(
..
),(CVaR),(CVaRmin21 21
xxfE
ts
xfxfx
5.
.0,),(CVaR)(..
),(max
1
0
xdxfts
xfEx
6.
.0,),(..
),(CVaR)(min1
0
xxfEts
dxfx
Optimization Techniques
Notations:1( , ) (1 ) [ ( , ) ] ( ) ,
max { , 0}
F x f x p d
z z
Theorem 1
a) -VaR is a minimizer of F with respect to :
VaR ( , ) ( , ) arg min ( , )f x f x F x
b) - CVaR equals minimal value (w.r.t. ) of function F :
CVaR ( , ) min ( , )f x F x
Remark. This equality can be used as a definition of CVaR ( Pflug ).
Optimization Techniques (cont'd)Proof:
1
1
( , ) 1 (1 ) ( , ) ( )
1 (1 ) ( , ) 0,
F x I f x p d
P f x
where
),(,0
),(,1),(
xf
xfxfI
Consequently,
).(1),( * xfP
b) * * 1 *min ( , ) ( , ) (1 ) [ ( , ) ] ( ) .F x F x f x p d
a) (in the case of continuous distribution)
Integral of losses exceeding VaR
Optimization Techniques (cont'd)
Theorem 2
,min CVaR ( , ) min ( , )
x xf x F x
Proof:
,
min ( , ) min min ( , ) min CVaR ( , )x x x
F x F x f x
• Minimization of G (x, ) simultaneously calculates VaR = a(x), optimal decision x and optimal CVaR
• CVaR minimization can be reduced to LP using dummy variables
Reduction to LP
[ ( , ) ] ( , ) , 0, 1,k kk kf x z f x z k N
1
1
( , ) (1 ) [ ( , ) ]
max { , 0}
Nk
kk
F x p f x
z z
In the case of discrete distribution:
Discrete distribution (continuous distribution can be approximated using scenarios).
),(),(,1,)(,1
kN
kkk
k xfxfppp
Reduction to LP by expanding the problem dimension:
Reduction to LP (cont'd) CVaR minimization
min{ xX } CVaR
is reduced to the following linear programming (LP) problem
By solving LP we find an optimal x* , corresponding VaR, which equals to the lowest optimal *, and minimal CVaR, which equals to the optimal value of the linear performance function
Constraints, x X , may account for various constraints, including constraint on expected value
Similar to return - variance analysis, we can construct an efficient frontier and find a tangent portfolio
.,,1,0
,,1,),(..
)1(min1
1
,
XxNkz
Nkxfzts
zp
k
kk
N
kkk
x
Reduction to LP (cont'd)
CVaR constraints in optimization problems can be replaced by a set of linear constraints. E.g., the following CVaR constraint CVaR C
is replaced by linear constraints
Loss distribution can be shaped using multiple CVaR constraints at different confidence levels in different times
.,1,0,),(..
)1(1
1
Nkzxfzts
Czp
kk
k
N
kkk
Financial Engineering Applications
Notations:x = (x1,…,xn) = decision vector (e.g., portfolio weights)X = a convex set of feasible decisions = (1,…,n) = random vector j = scenario of random vector , ( j=1,...J )f(x,) = T x = reward function
- f(x,) = - T x = loss function
Example: Two Instrument Portfolio A portfolio consists of two instruments (e.g., options). Let x = (x1, x2) be a vector of
positions, m = (m1, m2) be a vector of initial prices, and y = (1, 2) be a vector of uncertain prices in the next day. The loss function equals the difference between the current value of the portfolio, (x1m1+x2m2), and an uncertain value of the portfolio at the next day (x11+x22), i.e.,
- f(x,) = (x1m1+x2m2) – (x11+x22) = x1(m1–1) + x2(m2–2) .
If we do not allow short positions, the feasible set of portfolios is a two-dimensional set of non-negative numbers
X = {(x1,x2), x1 0, x2 0} . Scenarios j = ( j
1,j2), j=1,...J , are sample daily prices (e.g., historical data for J
trading days).
Portfolio optimization
Financial Engineering Applications (cont'd)CVaR and Mean Variance: normal returns
If returns are normally distributed, and return constraint is active, the following portfolio optimization problems have the same solution:
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
2( , ) , ( )T TE f x m x x x V x
RxmT
1,01
n
iii xxxX
Mean return and variance:
No-shorts and budget constraints:
Return constraint:
Financial Engineering Applications (cont'd)Optimization problem formulations
1min ( ) ( )min CVaR ( , )
. . ,. . , .xx
TT
k x Rf x
s t m x R x Xs t m x R x X
2min ( )
. . ,
x
T
x
s t m x R x X
CVaR minimization (assumption: risk constraint is active):
VaR minimization (assumption: risk constraint is active):
Variance minimization:
min ( ) ( )min VaR ( , )
. . ,. . , .xx
TT
k x Rf x
s t m x R x Xs t m x R x X
Financial Engineering Applications (cont'd)Optimization problem formulations (cont'd):
)12(erf2)( 1 k
1211 )1()12(erfexp2)(
k
tdtzz
0
2 )exp(2
)erf(
06.2)95.0(1 k
67.2)99.0(1 k
65.1)95.0( k
33.2)99.0( k
Example Data:
INSTRUMENT MEAN RETURNS&P 0.0101110
Gov Bond 0.0043532Small Cap 0.0137058
S&P GOV BOND SMALL CAPS&P 0.00324625 0.00022983 0.00420395
GOV BOND 0.00022983 0.00049937 0.00019247SMALL CAP 0.00420395 0.00019247 0.00764097
Portfolio Mean return
Portfolio Covariance Matrix
Example (cont'd)Results:
Optimal Portfolio with the Minimum Variance Approach
Optimal VaR and CVaR with the Minimum Variance Approach
S&P GOV BOND SMALL CAP0.452013 0.115573 0.432414
9.0 95.0 99.0VaR 0.067848 0.090200 0.132128
CVaR 0.096975 0.115908 0.152977
Example (cont'd)
The Portfolio, VaR and CVaR with the Minimum CVaR Approach
STOCHASTIC WTA PROBLEM
Weapon-Target Assignment (WTA) problem: find an optimal assignment of I weapons to K targets
I weapons with different munitions capacities
K targets to be destroyed
STOCHASTIC WTA PROBLEM
Define:
vik = 1, if weapon i fires at target k, vik = 0 otherwise
xik is a number of munitions to be fired by weapon i at target k
cik is the cost of firing 1 unit of munitions i at target k
mi is the munitions capacity of weapon i
ti is the maximum number of targets that weapon i can attack
pik is the probability of destroying target k by firing 1 unit of munitions by weapon i
dk is the minimum required probability of destroying target k
Minimize total cost of the mission subject to the destruction of all targets with prescribed probabilities and constraints on munitions
WTA problem is formulated as a Mixed Integer Linear Programming Problem (MILP)
WTA WITH UNCERTAIN PROBABILITIES
WTA with known probabilities of destroying:
minimize the cost of the mission
constraint on the munitionscapacities of the weapons
constraint on how many targets a
weapon can attack
destroy all targets with prescribed probabilities
(can be linearized!)
1 1
1
1
1
min
s.t. , 1,..., ,
, 1,..., , 1,..., ,
, 1,..., ,
1 (1 ) , 1,..., ,ik
K I
ik ikk i
K
ik ik
ik i ik
K
ik ik
Ix
ik ki
c x
x m i I
x m v i I k K
v t i I
p d k K
WTA WITH UNCERTAIN PROBABILITIES
Introduce uncertainty in the model by making probabilities pik dependent on the random parameter : pik = pik()
Assume a scenario model for stochastic parameters
WTA with uncertain probabilities:
CVaR constraint on risk of failure to destroy a target
1
( , ) ln(1 ( )) ln(1 )I
k ik ik ki
L x p x d
1 1
1
1
min
s.t. , 1,..., ,
, 1,..., , 1,..., ,
, 1,..., ,
CVaR[ ( , ] 1,..., ,
K I
ik ikx
k i
K
ik ik
ik i ik
K
ik ik
k k
c x
x m i I
x m v i I k K
v t i I
L x C k K
Loss function Lk(x,) quantifies the risk of not destroying target k
EXAMPLE: STOCHASTIC WTA PROBLEM 5 targets (K = 5)
5 aircraft, each with 4 missiles (I = 5, mi = 4)
probabilities and costs do not depend on targets to be attacked
20 scenarios for pik() = pi(s)
any aircraft can attack any target (ti = 5)
destroy targets with 95% confidence (dk = 0.95)
90% confidence level in CVaR constraints ( = 0.90)
T 1 T 2 T 3 T 4 T 5
Weapon 1 0 2 1 0 0
Weapon 2 0 1 2 0 0
Weapon 3 1 0 0 1 1
Weapon 4 1 0 0 1 1
Weapon 5 0 0 0 0 0
T 1 T 2 T 3 T 4 T 5
Weapon 1 0 1 1 0 1
Weapon 2 0 0 1 1 1
Weapon 3 2 0 0 1 0
Weapon 4 0 1 1 0 1
Weapon 5 1 1 0 1 0
Optimal solution of WTA with uncertain probabilities:
Optimal solution of WTA with known probabilities:
0.4
0.5
0.6
0.7
0.8
0.9
1
0.5 0.6 0.7 0.8 0.9
Average probability of hitting a target
Co
st o
f fi
rin
g
Statistics Applications Regression
a
x
y
tanb
.
.
.
..
. .
.
jj
jjj yy ˆDiscrete case:
nP j 1)(
jj xbay ˆ
Linear estimation:
yy ˆDeviation is a random value with density
xbay ˆEstimationLinear
)(p
),,,(ˆEstimationNonlinear 1 maaxy
maa ,,1 are unknown parameters
Statistics Applications (cont'd)Distribution of random deviation
Deviation,
Fr
eq
ue
nc
y
Statistics Applications (cont'd)
Quadratic metrics:
N
jj
aa m 1
2
,,1
min
Discrete case:
Continuous case:
dpmaa
)(min 2
,,1
Regression in general case:
Parameters can be found using different metrics
)var(
),cov(],[
)var(
),cov(][
x
yxbxE
x
yxyEa
Solution in the case of linear estimation :
- metrics, which can be expressed as function of deviation, .
)(min)ˆ,(min,,,, 11
gEyyEmm aaaa
)ˆ,( yy
22ˆ)ˆ,( yyyy
xbay ˆ
maa ,,1
Statistics Applications (cont'd)Absolute deviation:
N
jj
aa m 1,,
||min1
Discrete case:
Continuous case:
dpmaa
)(||min,,1
||ˆ)ˆ,( yyyy
In the case of the discrete distribution and linear regression function the problemcan be reduced to LP
.
,..
min1
,,,, 1
jjj
jjj
N
jj
uuba
yxbau
xbayuts
uN
Statistics Applications (cont'd)
Probability 10% Probability 10%
Mean of the right tail Mean of the left tail
1 2min CVaR ( ) CVaR ( )
9.021
Statistics Applications (cont'd)Reduction to LP (discrete case):
1 1
2 2
1 11 1 1
1
1 12
1
(1 ) [ ]
(1 ) [ ]
N
jj
N
jj
N
N
.,1,0,0
,,
,0,..
)1()1(min
21
2211
1
12
121
11
112
,,,
,
21
21
Njzz
uzuz
xbayts
zzNuu
jj
jjjj
N
jjjjj
N
jjj
zzuu
ba
jj
FACTOR MODELS:PERCENTILE and CVaR REGRESSION
1 , ..., qX X
Y factors from various sources of information
failure load
0 1 1 , ..., ,q qY c c X c X where is an error term
= direct estimator of percentile with confidence
0 1 1 , ..., q qc c X c X
10% points below line:
= 10%
X
Y
Percentile regression (Koenker and Basset (1978))
CVaR regression (Rockafellar, Uryasev, Zabarankin (2003))
PERCENTILE ERROR FUNCTION and CVaR DEVIATION
Statistical methods based on asymmetric percentile error
functions: [(1 )( ) ]E
= positive part of error
= negative part of error
CVaRFailure
Success
Percentile
Mean
CVaR deviation
Accident lost
Payment
Accident lost
Payment
Premium
Deductible
Payment
Payment
Premium
Deductible
Payment
Payment
Premium
Deductible
Payment
CDF
1
Payment
CDF
Premium
Deductible
0p
1
0p- probability of driving without accident
Example 1: Nikkei Portfolio
Distribution of losses for the NIKKEI portfolio with best normal approximation(1,000 scenarios)
Ex. 1: Nikkei Portfolio (cont'd)
NIKKEI Portfolio
Ex. 1: Nikkei Portfolio (cont'd)
Ex. 1: Hedging – Minimal CVaR Approach
Ex. 1: One Instrument Hedging
Ex. 1: One Instrument Hedging (cont'd)
Ex. 1: Multiple Instrument Hedging CVaR approach
Ex. 1: Multiple-Instrument Hedging
Ex. 1: Multiple-Instrument Hedging: Model Description
Ex. 1: Multiple-Instrument Hedging: Optimization Problem
n
dpT
RyXx
yyxzym )(]),()[()1(min,
J
j
Tj
T Jdpn 1
1 ]),()[()(]),()[( xzymyyxzymRy
Formulation:
Approximation for discrete case:
J
j
TjJ
1
1
,]),()[()1(min
xzym
Xx
Optimization problem:
Example 2: Portfolio Replication Using CVaR
• Problem Statement: Replicate an index using instruments. Consider impact 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 of j-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 Nonlinear Transaction Costs and Minimal Transaction Unit Constraints,Tokyo Institute of Technology, CRAFT Working paper 00-07,(2000).
1, ,t T tIj tp 1, ,j n 1, ,t T T
TIT
jx
t1
n
j t jj
p x
1
| ( ) /( ) |n
t j t j tj
I p x I
tI
1, ,j n
t1
( , ) ( ) /( )n
t t j t j tj
f x p I p x I
Ex. 2: Portfolio Replication Using CVaR (Cont’d)
Index and optimal portfolio values in in-sample region, CVaR constraint 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
rtfo
lio
val
ue
(US
D)
portfolio
index
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 7 13
19
25
31
37
43
49
55
61
67
73
79
85
91
97
Day number in out-of-sample region
Po
rtfo
lio
val
ue
(US
D)
portfolio
index
Ex. 2: Portfolio Replication Using CVaR (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
Po
rtfo
lio
val
ue
(US
D)
portfolioindex
Ex. 2: Portfolio Replication Using CVaR (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 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97
Day number in out-of-sample region
Po
rtfo
lio
va
lue
(U
SD
)
portfolioindex
Ex. 2: Portfolio Replication Using CVaR (Cont’d)
Relative underperformance in in-sample region, CVaR constraint 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
Dis
crep
ancy
(%
)
active
inactive
Ex. 2: Portfolio Replication Using CVaR (Cont’d)
Relative underperformance in out-of-sample region, CVaR constraint 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
Dis
cre
pa
nc
y (
%)
active
inactive
Ex. 2: Portfolio Replication Using CVaR (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
0.02 0.01 0.005 0.003 0.001
omega
Va
lue
(%
)
in-sample objectivefunction
out-of-sample objectivefunction
out-of-sample CVAR
Ex. 2: Portfolio Replication Using CVaR (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 w 0.01, CVaR constraint is active. • Decreasing of CVaR causes an increase of objective function (mean absolute deviation) in the in-sample
region (Column 2). • Decreasing of CVaR causes a decrease of objective function in the out-of-sample region (Column 3).
However, this reduction is data specific, it was not observed for some other datasets.
CVaR in-sample (600 days) out-of-sample (100 days) out-of-sample CVaR
level w objective function, in % objective function, in % in %
0.02 0.71778 2.73131 4.88654
0.01 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
Ex. 2: Portfolio Replication Using CVaR (Cont’d)
In-sample-calculations: 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. Probability of 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
= (Ψ(VaR) -- [546/600 - 0.9]/[1 - 0.9] = 0.1
• Since “splits” VaR probability atom, i.e., Ψ(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 optimal solution. VaR equals the smallest optimal *.
Ex. 2: Portfolio Replication Using CVaR (Cont’d)
Example 3: Asset Liability Management (ALM)
Active members
Non-Active Members
Risk Management
Pension Fund
Payments? Future liabilities?
Ex. 3: ALM – Data
• ORTEC Consultants BV in Holland provided a dataset for a pension fund
• 10000 scenarios: liabilities of a fund and returns for 13 assets (indices)
• To simplify interpretation of results, mostly, we considered only four assets:
1. cash, 2. Dutch bonds index, 3. European equity index, 4. Dutch real estate index.
• However, the approach can easily handle 100,000 assets.
Ex. 3: ALM – Data Statistics
Ex. 3: ALM – Notations • Parameters = total initial value of all assets
= total amount of wages
= total amount of liabilities due at the start of the planning period
= lower bound of funding ratio
• Random data = random rate of return in asset class i, ( i = 0,…,N )
= liability that needs to be met or exceeded
• Decision variables = contribution rate
= total invested amount in asset class i, ( i = 0,…,N )
0
0A
ir
y
0W
ix
L
Ex. 3: ALM – Minimization of Contributions
minimize
subject to
with high certainty
free
• constraint on funding ratio, can be defined using CVaR risk measure
• loss function:
• with high certainty => CVaR w
0
1N
ψ i if ( x ,r ,L ) L ( r )x ,
0
11
N
i i( r )x ,L
0 , 0,...,ix i N
,y
y
0
11
N
i i( r )x ,L
0 0 0
N
ix A W y
0
11
N
i i( r )x ,L
Ex. 3: ALM – Linear Programming Formulation
minimize (1)
subject to
free
0 , 0,...,ix i N
0 0 0
N
ix A W y
J
jj
z w
0
1 1N
j ij i ji
L ( r )x z , j ,...,J
0 , 1,...,jz j j
y ,
y
Ex. 3: ALM – Properties of Solutions • Let w = 0 and y *, x *, * is an optimal solution of problem (1). Consider a
new funding ratio coefficient ' = t . New solution vector equals
(t (y * + A0/W0) - A0/W0 ,t x *, t * * ) .
• Optimal relative portfolio allocations DO NOT DEPEND upon funding ratio coefficient (which is a risk parameter)!!!
• Linear efficient frontier: contribution rate linearly depends upon the funding ratio parameter !!!
• VaR linearly depends upon the funding ratio parameter !!!
• If L is deterministic, then optimal relative portfolio allocations DO NOT DEPEND upon CVaR risk parameter w !!! Also, VaR and contribution rate of optimal solution linearly depend upon w !!!
Ex. 3: ALM – Linear Dependence of Contribution on
E fficient frontier
-2
-1.5
-1
-0.5
0
0.5
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
Psi
Contribution rate
Ex. 3: ALM – Linear Dependence of Portfolio on
Optimal portfolio allocation
0.0000
0.2000
0.4000
0.6000
0.8000
1.0000
1.2000
1 2 3 4 5 6 7 8 9 10 11
Psi
re NL
eq EM
bond cp NL
Ex. 3: ALM – Return Maximization
maximize
subject to
free
• Contribution to the fund, , is fixed
0 , 0,...,ix i N
0 0 N
ix A B0
J
jj
z w
0
1 1N
j ij i ji
L ( r )x z , j ,...,J
0 , 1,...,jz j j
,
i
J
j
N
iji xr
J
1 0
)1(1
0B
Return maximization and Contribution minimization (random L)
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
-50.00% -30.00% -10.00% 10.00% 30.00% 50.00% 70.00% 90.00% 110.00%
CVaR constraint parameter w
Po
rtfo
lio r
etu
rn
Return max" Contrib min"
Ex. 3: ALM – Results
Optimal portfolio allocations (stochastic L)
0%10%
20%30%
40%50%60%
70%80%
90%100%
CVaR constraint parameter w
Po
rtfo
lio
allo
cati
on
s priv eq, m=11.5, std=30
eq UK, m=8.5, std=18
eq J P, m=10, std=28.6
eq US, m=10.5, std=15
bond UK, m=5, std=1.9
bond US, m=6.4, std=0.9
Ex. 3: ALM – Results (cont'd)
Optimal portfolio allocations (deterministic L)
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
CVaR constraint parameter w
Po
rtfo
lio
allo
cati
on
s
eq EM, m=16.Std=37.3
eq US, m=10.5,std=15
bond US, m=6.4,std=0.9
Ex. 3: ALM – Results (cont'd)