Transcript The Efficient Conditional Value-at

THE ACADEMY OF ECONOMIC STUDIES BUCHAREST
DOCTORAL SCHOOL OF FINANCE AND BANKING
The Efficient Conditional Valueat-Risk/Expected Return Frontier
Student: Stan Anca Mihaela
Supervisor:Professor Moisa Altar
Contents
• Objectives
• VaR, CVaR, ER-properties and
optimization algorithms
• Methodology
• Empirical Application
• Concluding remarks
Objectives
• Construct the efficient CVaR/Expected
Return frontier
• Analyze CVaR’s performance as a proxy
variable for VaR
• Use CVaR as a risk tool in order to
efficiently restructure portfolios
• Analyze the impact of transaction costs
VaR-alternative definitions
1.
VaRk   k x  min   : x,   k
  x,   
p y dy

 
f x, y 
τ
~
r α dF ~
r
2. LPM α τ ,r    τ  ~

 E max0 ,τ  ~
r α



Pr ~
rp  rp*  1  c



 VaR  rp*
LPM0 rp* , ~
rp  1  c  rp*  LPM01 1  c
CVaR
• The expected losses exceeding VaR calculated with a precise
confidence level:
ESk  CVaRk   k x   1  k 1
 f x, y  p y dy
f  x, y  k  x 
ESk  CVaRk  VaRk  E f x, y   VaRk / f x, y  VaRk 
• In terms of lower partial moments, CVaR can be defined as a lower
partial moment of order one with   rp*

  
*
~
LPM1 r , rp  E max 0, rp  ~
rp
*
p

Expected Regret
• The mean value of the loss residuals, the
differences between the losses exceeding a
fixed threshold and the threshold itself.

G x     f x, y     p y dy
y m
u
 max0, u
VaR/CVaR Comparison
Translation equivariant
VaR
+
CVaR
+
Positively homogeneous
+
+
Convex
Comonotone additive
+
+
-
Monotonic w.r.t. SD(1)
+
+
Monotonic w.r.t. SD(2)
-
+
Monotonic w.r.t. MD(2)
-
+
Coherent
-
+
VaR
•
•
•
•
•
•
VaR/CVaR Comparison
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
Easily applied to backtesting
Established as a standard
risk measure
Consistent with first order
stochastic dominance
CVaR
•
•
•
•
•
Simple convenient
representation of risks (one
number)
Measures downside risk
Applicable to nonlinear
instruments, such as
options, with non-symmetric
(non-normal) loss
distributions
Not easily applied to efficient
backtesting methods
Consistent with second
order stochastic dominance
VaR/CVaR Comparison
•
•
•
•
•
does not measure losses
exceeding VaR
reduction of VaR may lead to
stretch of tail exceeding VaR
non-sub-additive and nonconvex:
– 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
•
•
•
accounts for risks beyond VaR
(more conservative than VaR)
convex with respect to portfolio
positions
coherent in the sense of Artzner,
Delbaen, Eber and Heath:
–
•
•
•
(translation invariant, subadditive, positively
homogeneous, monotonic w.r.t.
Stochastic Dominance1)
continuous with respect to
confidence level   consistent at
different confidence levels
compared to VaR
consistency with mean-variance
approach: for normal loss
distributions optimal variance
and CVaR portfolios coincide
easy to control/optimize for nonnormal distributions, by using
linear programming techniques
CVaR optimization

•
•
•
•
. Notations:
x = (x1,...xn) = decision vector (e.g., portfolio positions)
y = (y1,...yn) = random vector
yj = scenario of random vector y , ( j=1,...J )
f(x,y) = loss functions
  x =CVaR at  confidence level
  x =VaR at  confidence level
CVaR Optimization
• Rockafellar and Uryasev (1999) have shown that both  x and  x
can be characterized in terms of the function F defined on
X   by:
F  x,      1   
1





f
x
,
y


p y dy

y m
  x   min F x,  
 , x
  x  left endpointof A x
A  x   arg min F  x,  
 
By solving the optimization problem we find an optimal portfolio x* ,
corresponding VaR, which equals to the lowest optimal , and minimal CVaR,
which equals to the optimal value of the linear performance function.
CVaR Optimization
• When the function F is approximated using
scenarios, the problem is reduced to LP
with the help of a dummy variable:
min  1   
1
 ,x
m
min    z j
 ,x
 p j y j   
m
j 1
j 1
z j  0, z j  y j   , j  1....J
  (1   J ) 1

ER Optimization
F x,      1    G x 
1
If the function G is approximated using scenarios, the problem can be reduced to
a linear programming problem, having the same constraints as the CVaR
optimization problem and with the objective function
min  p j y j   
J
x
j 1

Optimization problem
m
min    z j
 ,x
j 1
z j  0, z j  y j   , j  1....J
  (1   J ) 1
The constraint on return takes the form:
n
 qi xi ri     0
i 1
The balance constraint that maintains the total value of the portfolio less
transaction costs:
n

i 1
qi xi0
n

i 1
ci qi xi0
n
 xi   q i xi
i 1
Optimization problem
n
q x
i
i 1
xi0
0
i


 c q 

n
i 1
i
i
i


n
  i   q i xi
i 1

  i   i  xi
We impose bounds on the position changes:
i

 max
0  i  i

 max
0  i  i
We also consider that the positions themselves can be bounded:
ximin  xi  ximax
We do not allow for an instrument i to constitute more than a given percent
of the total portfolio value:
n
q i xi   i  xi q i
i 1
Optimization Problem
• Size of LP
• For n instruments and J scenarios, the
formulation of the LP problem presented
above has n+2 equalities, 3n+J+1 variables
and n+J inequalities.
Empirical Application-Data
• Portfolio consisting of 5 Romanian equities traded on Bucharest
Stock Exchange – ATB, AZO, OLT, PCL, TER, selected by taking
into account the most actively trading securities in the analyzed
period.
• 450 daily closing prices between 03/05/2001 to 12/18/02
70
Series: PORTFOLIO
Sample 1 450
Observations 450
60
50
40
30
20
10
0
-0.05
0.00
0.05
0.10
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.001071
0.000611
0.101912
-0.068113
0.018174
0.237032
5.559054
Jarque-Bera
Probability
127.0030
0.000000
Data
140
200
Series: ATB
Sample 1 450
Observations 450
120
100
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
80
60
40
-0.000192
0.000000
0.152908
-0.162167
0.046259
-0.589530
7.225231
Series: PCL
Sample 1 450
Observations 450
160
120
80
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.001590
0.000000
0.141651
-0.125163
0.031772
0.783280
7.761276
Jarque-Bera
Probability
471.0724
0.000000
40
20
Jarque-Bera
Probability
0
-0.15
-0.10
-0.05
0.00
0.05
0.10
360.8016
0.000000
0
0.15
-0.10
240
Series: AZO
Sample 1 450
Observations 450
200
-0.05
0.00
0.05
0.10
0.15
160
Series: TER
Sample 1 450
Observations 450
140
120
160
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
120
80
40
0
-0.250
Jarque-Bera
Probability
0.002024
0.000000
0.392855
-0.239779
0.037359
1.961602
33.88792
18177.28
0.000000
100
80
60
40
20
0
-0.125
0.000
0.125
0.250
0.375
-0.10
-0.05
0.00
0.05
0.10
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.001965
0.000000
0.134975
-0.124454
0.023316
0.283138
9.690548
Jarque-Bera
Probability
845.3268
0.000000
Data
140
Series: OLT
Sample 1 450
Observations 450
120
100
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
80
60
40
20
0
-0.10
Jarque-Bera
Probability
-0.05
0.00
0.05
0.10
-0.000115
0.000000
0.129678
-0.099372
0.032195
0.173737
4.533149
46.33654
0.000000
Substitution error
VaRx  - VaR of port foliox
x VaR  port foliowit h t heminimalvalue of VaR, amongt hoseresult edfromCVaR opt imizaton
i
VaR min  VaR xvaR   minimalvalue of VaR
xCVaR - port foliowit h t heminimalvalue of CVaR
Rx  - ret urnof port foliox;
R min  minimalvalue of port folioret urn
R max - maximalvalue of port folioret urn
EVaR 
ER 
VaRxCVaR   VaRmin
100%
VaRmax  VaRmin
RxCVaR   RxVaR 
100%
Rmax  Rmin
Substitution error
200 scenarios
Portfolio
xi  100
300 scenarios
Portfolio
EVaR
37.58%
ER
-8.08%
10.69%
xi  200
1.00%
-1.01%
0%
0%
17.90%
4.04%
xi  500
4.93%
7.07%
6.06%
-1.01%
0%
0%
4.93%
7.07%
0%
0%
xi  100
xi  300
xi  400
ER
EVaR
-7.07
xi  200
xi  300
xi  400
xi  500
Substitution error
EVaR
Without restriction/without transaction costs
With restriction/without transaction costs
With restriction/with transaction costs
0%
0.57%
0.26%
ER
0%
8.08%
3.03%
CVaR Efficient Frontier Without
Transaction Costs
-3
2.5
-3
CVaR Efficient Frontier
x 10
2.5
ER for  =0.9VaR0
Empiric VaR
Function VaR
Function CVaR
2
Return
Return
ER for  =0.6VaR0
0.5
0
0
2
2.5
3
3.5
4
Risk
4.5
5
5.5
-0.5
1.5
6
2.5
3
3.5
4
x
2.510
Empiric VaR
Function ER
-3
x 10
VaR ER Comparison
VaR ER
for
VaR ER
for ER
VaR
for ER
VaR
for ER
VaR
for
1.5
1.5
R
et
ur
n
0.5
5
4
2
1
4.5
Risk
x 10
2
1
=0.9Va0
=0.8Va0
R
R
=0.7Va
0
R
=0.6Va
0
R
=0.5Va
0
R
0.5
0
0
-0.5
0.5
2
4
ER Efficient Frontier
x 10
ER for  =0.5VaR0
1
0.5
-3
Return
ER for  =0.7VaR0
1.5
1
2.5
ER for  =0.8VaR0
2
1.5
-0.5
ER Comparison
x 10
1
1.5
2
2.5
Risk
3
3.5
4
4.5
4
x 10
0.5 2
2.5
3
3.5
Risk
4
4.5
5
x
10
4
CVaR Efficient Frontier Without
Transaction Costs
-3
2.5
-3
CVaR ER Comparison
x 10
2.5
Empiric VaR from CVaR
Empiric VaR from ER
2
Return
Return
1.5
1
0.5
0
0
2
2.5
3
3.5
4
-3
2.5
3
3.5
4
4.5
Risk
x 10
-3
2.5
Empiric VaR
Markowitz
2
4
x 10
CVaR Markowitz Comparison. Sigma
x 10
Empiric Sigma from CVaR
Empiric Sigma from Markowitz
2
1.5
Return
1.5
1
1
0.5
0.5
0
0
-0.5
1.5
2
4
Mark owitz Efficient Frontier
x 10
-0.5
4.5
Risk
Return
1
0.5
2.5
Empiric VaR from CVaR
Empiric VaR from Markowitz
2
1.5
-0.5
CVaR Markowitz Comparison. VaR
x 10
2
2.5
3
Risk
3.5
4
4.5
4
x 10
-0.5
1.6
1.8
2
2.2
2.4
Risk
2.6
2.8
3
4
x 10
Restructuring the initial portfolio
Rest CVaR
VaR
25132.3811147459
x1
x2
44.02
11.44
CVaR
33233.9881474066
x3
381.82
x4
ER
12394.4801945106
Markowitz
17831.1009011683
x5
67.12
65.30
--------------------------------------------------------------------------------
-------------------------------------------------------------------------------Rest ER
VaR
23605.9960542892
x1
x2
65.33
22.43
CVaR
34836.8025753390
x3
x4
358.76
ER
10122.7638791596
Markowitz
17147.3982854023
x5
36.33
Expected
Return
0.001071
Alpha
7,500
VaR
27,429.48
CVaR
37,302.43
ER
12,428.36
Markowitz
18,729.29
89.05
--------------------------------------------------------------------------------
Rest Markowitz
VaR
22812.7095783657
CVaR
35215.9043438216
x1
x2
x3
55.15
26.54
385.60
x4
38.86
ER
12065.2682437272
Markowitz
17094.4715337914
x5
83.95
--------------------------------------------------------------------------------
Restructuring the initial portfolio
• However, the restructured portfolios are not efficient with respect to
their return level, they lie on the “inefficient”, lower section of the
boundary. For a CVaR of 33,239.28 we can find, for instance, on the
CVaR efficient frontier a portfolio (x1=16.68, x2=43.49, x3=183.63,
x4=68.82, x5=92.58) that has an expected return of 0.001492 (instead
of 0.001071) – this suggests that the” efficient” portfolio, offering
maximum return for a given minimal risk level can be achieved by
lowering the position in the first asset (ATB) that is the most risky
one and has a negative expected return and by investing more in the
second (AZO) and fifth asset (TER) that have the highest expected
return.
76 0.001492 23432.3311 17759.0986 23432.3311 33239.2890 16.68 43.49 183.63 68.82 92.58
The impact of transaction costs
Transaction Costs
-3
2.5
-3
CVaR Efficient Frontier
x 10
2.5
VaR ER for  =0.9VaR0
Empiric VaR
Function VaR
Function CVaR
2
VaR ER for  =0.8VaR0
2
VaR ER for  =0.7VaR0
VaR ER for  =0.6VaR0
1.5
Return
1.5
Return
VaR ER Comparison
x 10
1
VaR ER for  =0.5VaR0
1
0.5
0.5
0
0
-0.5
1.5
-0.5
2
2.5
3
3.5
4
4.5
5
Risk
-3
2.5
4
-3
x 10
2.5
3
Risk
3.5
4
4.5
4
x 10
CVaR ER Comparison
x 10
Empiric VaR from CVaR
Empiric VaR from ER
2
Empiric VaR
Function ER
2
2.5
5.5
ER Efficient Frontier
x 10
2
1.5
Return
Return
1.5
1
0.5
0.5
0
0
-0.5
0.5
1
1
1.5
2
2.5
Risk
3
3.5
4
4.5
4
x 10
-0.5
1.5
2
2.5
3
Risk
3.5
4
4.5
4
x 10
Transaction Costs
-3
2.5
-3
Markowitz Efficient Frontier
x 10
2.5
ER for  =0.9VaR0
Empiric VaR
Markowitz
2
Return
Return
ER for  =0.6VaR0
0.5
0
0
2
-3
2.5
3
Risk
3.5
4
4.5
-0.5
2
2.5
3
3.5
Risk
4
x 10
CVaR Markowitz Comparison. VaR
x 10
ER for  =0.5VaR0
1
0.5
-0.5
1.5
x
2.510
Empiric VaR from CVaR
Empiric VaR from Markowitz
2
-3
4
4.5
5
4
x 10
CVaR Markowitz
Comparison. Sigma Empiric Sigma from CVaR
Empiric Sigma from Markowitz
2
1.5
Return
ER for  =0.7VaR0
1.5
1
1.5
R
e 1
t
u 0.5
r
n 0
1
0.5
0
-0.5
ER for  =0.8VaR0
2
1.5
2.5
ER Comparison
x 10
2
2.5
3
3.5
Risk
4
4.5
4
x 10
0.51.5
2
2.5
Risk
3
x
10
4
The restructured portfolios
-------------------------------------------------------------------------------Rest CVaR
VaR
22149.5403559007
x1
43.66
x2
CVaR
29754.7450725926
x3
12.56
x4
331.11
ER
Markowitz
11675.1857348015
15883.2579071336
x5
58.95
58.27
--------------------------------------------------------------------------------
-------------------------------------------------------------------------------Rest ER
VaR
21049.0041262429
x1
59.38
x2
CVaR
31138.2995033972
x3
16.91
317.30
x4
ER
Markowitz
9655.3558249694
15368.7518388624
x5
33.36
80.83
--------------------------------------------------------------------------------
Rest Markowitz
VaR
23585.5755
x1
65.27
CVaR
33415.627
ER
12619.3345
x2
x3
x4
22.41
358.46
36.34
--------------------------------------------------------------------------------
Markowitz
10094.4715337914
x5
89.13
The restructured portfolios
-------------------------------------------------------------------------------Rest ER
-10,000
VaR
CVaR
20357.5049116300
x1
56.45
76 0.001492
ER
31114.7744204140
x2
18.55
x3
320.05
Markowitz
11653.6911834453
x4
38.02
15339.6377734087
x5
75.23
Rest CVaR
x1
x2
x3
x4
x5
With transaction costs
8.97%
2.78%
27.79%
33.22%
27.25%
Without transaction costs
9.938%
3.403%
26.916%
32.586%
27.157%
21215.6582
15930.8490
21215.6582
29789.3824
18.63
40.28
153.64
61.98
81.97
The restructured portfolio
Market Value VaR CVaR ER
ER-with transaction costs
922,653 2.29% 3.38% 0.98%
CVaR-with transaction costs
922,625 2.44% 3.23% 1.20%
ER-without transaction costs
1,030,543 2.28% 3.37% 1.05%
CVaR-without transaction costs
1,030,543 2.40% 3.23% 1.27%
Concluding Remarks
• CVaR is a conceptually superior risk measure to VaR
• It can be used to efficiently manage and restructure a
portfolio (other applications include the hedging of a
portfolio of options, credit risk management (bond
portfolio optimization) and portfolio replication).
• Direction for further development:
– Conditional Drawdown-at-Risk
– Risk measures consistent with third or higher order stochastic
dominance criteria
References
•
•
•
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