Stochastic Optimization ESI 6912

1
Stochastic OptimizationESI 6912
NOTES 7
Optimization using CV_at_R
Algorithms and Applications

• Instructor Prof. S. Uryasev

2
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

3
Value-at-Risk Definition

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
?)
4
Value-at-Risk Definition (contd)

Mathematical Definition
? - random variable
Remarks

• 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)

5
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.

6
Value-at-Risk Features (contd)

• 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.

7
Value-at-Risk Example

? - normally distributed random variable with
mean ? and standard deviation ?
8
Value-at-Risk Example (cont'd)

9
Conditional Value-at-Risk Definition

• Notations
• ? cumulative distribution of random variable
  ? ,
• ?? ?-tail distribution, which equals to zero
  for ? below VaR,
• and equals to (?- ?)/(1- ?) for ?
  exceeding or equal to VaR
• Definition CVaR is mean of ?-tail
  distribution ??

Cumulative Distribution of? , ?
10
Conditional Value-at-Risk Definition (contd)

• 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)
• ? (VaR) probability that ? does not
  exceed VaR or equal to VaR
• Property CVaR is weighted average of VaR
  and CVaR

11
Conditional Value-at-Risk Definition (contd)

y
c
n
e
u
q
Maximal value
e
VaR
r
F
Probability
1 - ?
CVaR
Random variable, ?
12
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.),

13
Conditional Value-at-Risk Features (cont'd)

CVaR
Risk
CVaR
CVaR-
VaR
x
CVaR is convex, but VaR, CVaR- ,CVaR may be
non-convex, inequalities are valid
VaR ? CVaR- ? CVaR ? CVaR
14
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

15
Conditional Value-at-Risk Features (cont'd)

• 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

16
CVaR Continuous Distribution, Example 1

? - normally distributed random variable with
mean ? and standard deviation ?
2?
17
CVaR Continuous Distribution, Example 1

? - normally distributed random variable with
mean ? and st. dev. ?
18
CVaR Discrete Distribution, Example 2

• ? does not split atoms VaR lt CVaR- lt CVaR
  CVaR,
• ? (?- ?)/(1- ?) 0

19
CVaR Discrete Distribution, Example 3

• ? splits the atom VaR lt CVaR- lt CVaR lt CVaR,
  
• ? (?- ?)/(1- ?) gt 0

20
CVaR Discrete Distribution, Example 4

• ? splits the last atom VaR CVaR- CVaR,
• CVaR is not defined, ? (? - ?)/(1- ?) gt 0
  

21
Formulation of Optimization Problem

Notations
? - random variable, x - vector of control
variables
Stochastic functions
loss or reward
Optimization problem
where for some of j
22
Examples

loss function
reward function,
1.
2.
3.
23
Examples (cont'd)

4.
5.
6.
24
Optimization Techniques

Notations
Theorem 1
a) ? -VaR is a minimizer of F with respect to
?
b) ? - CVaR equals minimal value (w.r.t. ? ) of
function F
Remark. This equality can be used as a definition
of CVaR ( Pflug ).
25
Optimization Techniques (cont'd)

26
Optimization Techniques (cont'd)

• 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

27
Reduction to LP

Discrete distribution (continuous distribution
can be approximated using scenarios).
In the case of discrete distribution
Reduction to LP by expanding the problem
dimension
28
Reduction to LP (cont'd)

• CVaR minimization
• min x?X 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

29
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

30
Financial Engineering Applications

Portfolio optimization

• 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 ?, ( j1,...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, (x1m1x2m2), and an uncertain value of
  the portfolio at the next day (x1?1x2?2), i.e.,
•  
• - f(x,?) (x1m1x2m2) (x1?1x2?2) 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 (? j1,?j2), j1,...J , are sample
  daily prices (e.g., historical data for J trading
  days).

31
Financial Engineering Applications (cont'd)

CVaR and Mean Variance normal returns
Mean return and variance
Return constraint
No-shorts and budget constraints

• 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

32
Financial Engineering Applications (cont'd)

Optimization problem formulations
CVaR minimization (assumption risk constraint is
active)
VaR minimization (assumption risk constraint is
active)
Variance minimization
33
Financial Engineering Applications (cont'd)

Optimization problem formulations (cont'd)
34
Example

Data
Portfolio Mean return
Portfolio Covariance Matrix
35
Example (cont'd)

Results
Optimal Portfolio with the Minimum Variance
Approach
Optimal VaR and CVaR with the Minimum Variance
Approach
36
Example (cont'd)

The Portfolio, VaR and CVaR with the Minimum CVaR
Approach
37
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
38
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)

39
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!)
40
WTA WITH UNCERTAIN PROBABILITIES

• Introduce uncertainty in the model by making
  probabilities pik dependent on the random
  parameter x pik pik(x)
• Assume a scenario model for stochastic
  parameters
• WTA with uncertain probabilities

CVaR constraint on risk of failure to destroy a
target

• Loss function Lk(x,x) quantifies the risk of not
  destroying target k

41
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(x) pi(xs)
• any aircraft can attack any target (ti
  5)
• destroy targets with 95 confidence (dk 0.95)
• 90 confidence level in CVaR constraints (a
  0.90)

Optimal solution of WTA with uncertain
probabilities
Optimal solution of WTA with known probabilities
42
Statistics Applications

Regression
Discrete case
Linear estimation
43
Statistics Applications (cont'd)

Distribution of random deviation
y
c
n
e
u
q
e
r
F
Deviation, ?
44
Statistics Applications (cont'd)

Regression in general case
- metrics, which can be expressed as function of
deviation, ? .
Parameters can be found
using different metrics
Quadratic metrics
Discrete case
Continuous case
Solution in the case of linear estimation

45
Statistics Applications (cont'd)

Absolute deviation
In the case of the discrete distribution and
linear regression function the problem can be
reduced to LP
46
Statistics Applications (cont'd)

47
Statistics Applications (cont'd)

Reduction to LP (discrete case)
48
FACTOR MODELS PERCENTILE and CVaR REGRESSION
factors from various
sources of information failure load
Percentile regression (Koenker and Basset
(1978)) CVaR regression (Rockafellar, Uryasev,
Zabarankin (2003))
49
PERCENTILE ERROR FUNCTION and CVaR DEVIATION
Success
Mean
Percentile
CVaR deviation
CVaR
Failure
50
Payment
Payment
Deductible
Premium
Accident lost
Accident lost
PDF
PDF
?
?
?
Deductible
Premium
Payment
Payment
51
PDF
PDF
?
?
?
Deductible
Premium
Payment
Payment
CDF
CDF
1
1
Deductible
- probability of driving without accident
Premium
Payment
Payment
52
Example 1 Nikkei Portfolio

Distribution of losses for the NIKKEI portfolio
with best normal approximation (1,000 scenarios)
53
Ex. 1 Nikkei Portfolio (cont'd)

NIKKEI Portfolio
54
Ex. 1 Nikkei Portfolio (cont'd)

55
Ex. 1 Hedging Minimal CVaR Approach

56
Ex. 1 One Instrument Hedging

57
Ex. 1 One Instrument Hedging (cont'd)

58
Ex. 1 Multiple Instrument Hedging CVaR approach

59
Ex. 1 Multiple-Instrument Hedging

60
Ex. 1 Multiple-Instrument Hedging Model
Description

61
Ex. 1 Multiple-Instrument Hedging Optimization
Problem

Formulation
Approximation for discrete case
Optimization problem
62
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

63
Ex. 2 Portfolio Replication Using CVaR (Contd)
12000
10000
8000
Portfolio value (USD)
portfolio
6000
index
4000
2000
0
1
51
101
151
201
251
301
351
401
451
501
551
Day number in-sample region
64
Ex. 2 Portfolio Replication Using CVaR (Contd)
12000
11500
11000
10500
Portfolio value (USD)
10000
portfolio
index
9500
9000
8500
1
7
13
19
25
31
37
43
49
55
61
67
73
79
85
91
97
Day number in out-of-sample region
65
Ex. 2 Portfolio Replication Using CVaR (Contd)
12000
10000
8000
Portfolio value (USD)
portfolio
6000
index
4000
2000
0
1
51
101
151
201
251
301
351
401
451
501
551
Day number in-sample region
66
Ex. 2 Portfolio Replication Using CVaR (Contd)
12000
11500
11000
10500
Portfolio value (USD)
portfolio
index
10000
9500
9000
8500
1
7
13
19
25
31
37
43
49
55
61
67
73
79
85
91
97
Day number in out-of-sample region
67
Ex. 2 Portfolio Replication Using CVaR (Contd)
68
Ex. 2 Portfolio Replication Using CVaR (Contd)
6
5
4
3
Discrepancy ()
active
2
inactive
1
0
-1
-2
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
69
Ex. 2 Portfolio Replication Using CVaR (Contd)
70

• Calculation results
• 
  
• CVaR constraint reduced underperformance of the
  portfolio versus the index both in

Ex. 2 Portfolio Replication Using CVaR (Contd)
71

• 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 lt 1- a , and
• ? (?(VaR) - a) /(1 - a) 546/600 - 0.9/1
  - 0.9 0.1
• Since a splits VaR probability atom, i.e.,
  ?(VaR) - a gt0, CVaR is bigger than CVaR- (lower
  CVaR) and smaller than CVaR ( upper CVaR,
  also called expected shortfall)
• CVaR- 0.004592779726 lt CVaR 0.005 lt
  CVaR0.005384596925
• CVaR is the weighted average of VaR and CVaR
• CVaR ? VaR (1- ?) CVaR 0.1
  0.001538627671 0.9 0.005384596925 0.005

Ex. 2 Portfolio Replication Using CVaR (Contd)
72
Example 3 Asset Liability Management (ALM)

Pension Fund
Non-Active Members
Active members
Payments? Future liabilities?
?
Risk Management
73
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.

74
Ex. 3 ALM Data Statistics
75
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 )

76
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
  gt CVaR ? w

77
Ex. 3 ALM Linear Programming Formulation

• minimize
  (1)
• subject to
• 
  
• 
  free
  

78
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
  !!!

79
Ex. 3 ALM Linear Dependence of Contribution on
?
80
Ex. 3 ALM Linear Dependence of Portfolio on ?

81
Ex. 3 ALM Return Maximization

• maximize
  
• subject to
• 
  
• 
  free
• 
  
• Contribution to the fund, , is fixed

82
Ex. 3 ALM Results
83
Ex. 3 ALM Results (cont'd)
84
Ex. 3 ALM Results (cont'd)