Portfolio Optimization
with Conditional Value-at-Risk
and Chance Constraints
David L. Olson
University of Nebraska
Desheng Wu
University of Toronto; University of Reykjavik
Risk & Business
• Taking risk is fundamental to doing business
– Insurance
• Lloyd’s of London
– Hedging
• Risk exchange swaps
• Derivatives/options
• Catastrophe equity puts (cat-e-puts)
– ERM seeks to rationally manage these risks
• Be a Risk Shaper
Financial Risk Management
• Evaluate chance of loss
– PLAN
• Hubbard [2009]: identification, assessment,
prioritization of risks followed by coordinated
and economical application of resources to
minimize, monitor, and control the probability
and/or impact of unfortunate events
– WATCH, DO SOMETHING
Our Paper
• PLAN
– Markowitz [1952] risk = variance
• Control by diversifying
• Take advantage of correlation to get build-in hedging
– Generate portfolios on efficient frontier
• Chance constrained programming
• Value-at-risk
• Conditional value-at-risk
Value-at-Risk
• One of most widely used models in financial
risk management (Gordon [2009])
• Maximum expected loss over given time
horizon at given confidence level
– Typically how much would you expect to lose 99%
of the time over the next day (typical trading
horizon)
• Implication – will do worse (1-0.99) proportion of the
time
VaR = 0.64
expect to exceed 99% of time in 1 year
Here loss = 10 – 0.64 = 9.36
Finland 2010
Use
• Basel Capital Accord
– Banks encouraged to use internal models to measure
VaR
– Use to ensure capital adequacy (liquidity)
– Compute daily at 99th percentile
• Can use others
– Minimum price shock equivalent to 10 trading days
(holding period)
– Historical observation period ≥1 year
– Capital charge ≥ 3 x average daily VaR of last 60
business days
Finland 2010
VaR Calculation Approaches
• Historical simulation
– Good – data available
– Bad – past may not represent future
– Bad – lots of data if many instruments (correlated)
• Variance-covariance
– Assume distribution, use theoretical to calculate
– Bad – assumes normal, stable correlation
• Monte Carlo simulation
– Good – flexible (can use any distribution in theory)
– Bad – depends on model calibration
Finland 2010
Limits
• At 99% level, will exceed 3-4 times per year
• Distributions have fat tails
• Only considers probability of loss – not
magnitude
• Conditional Value-At-Risk
– Weighted average between VaR & losses
exceeding VaR
– Aim to reduce probability a portfolio will incur
large losses
Finland 2010
Optimization
Maximize f(X)
Subject to: Ax ≤ b
x≥0
Finland 2010
Minimize Variance
Markowitz extreme
Min Var [Y]
Subject to: Pr{Ax ≤ b} ≥ α
∑ x = limit
= to avoid null solution
x≥0
Finland 2010
Chance Constrained Model
• Maximize the expected value of a probabilistic
function
Maximize E[Y] (where Y = f(X))
Subject to: ∑ x = limit
Pr{Ax ≤ b} ≥ α
x≥0
Finland 2010
Maximize Probability
Max Pr{Y ≥ target}
Subject to: ∑ x = limit
Pr{Ax ≤ b} ≥ α
x≥0
Finland 2010
Minimize VaR
Min Loss
Subject to: ∑ x = limit
-Loss = initial value - z1-α √[var-covar] + E[return]
where z1-α is in the lower tail, α= 0.99
x≥0
• Equivalent to the worst you could experience at the given
level
Finland 2010
Demonstration Data
• 5 stock indexes
– Morgan Stanley World Index (MSCI)
– New York Stock Exchange Composite Index (NYSE)
– Standard & Poors 500 (S&P)
– Shenzhen Composite (China)
– Eurostoxx 50 (Euro)
Data
Daily – 1992 through June 2009 (4,292 observations)
MSCI
NYSE
0.00018 0.00027
Mean
Covariance(MSCI) 9.69E-05 9.91E-05
Covariance(NYSE)
0.000135
Covariance(S&P)
Covariance)China)
Covariance(Euro)
S&P
China
Euro
0.000252
0.000783
0.0003
0.0001
7.21E-06
0.000101
0.000137
7.89E-07
9.13E-05
0.000147
-2.2E-06
8.9E-05
0.000543
3.32E-06
0.000204
Correlation
China uncorrelated
Eurostoxx low correlation with first 3
MSCI
MSCI
NYSE
S&P
China
Eurostoxx
NYSE
S&P
China
1
0.867571
1
0.841261 0.975329
1
0.031434 0.002916 -0.00794
1
0.722692 0.551672 0.51456 0.009975
Eurostoxx
1
Distributions
• Used Crystal Ball software
– Chi-squared, Kolmogorov-Smirnov, AndersonDarling for goodness of fit
• Results stable across methods
• Student-t best fit
– Logistic 2nd, Normal & Lognormal 3rd or 4th
– IMPLICATION:
• Fat tails exist
• Symmetric
Impact of Distribution on VaR
Fat tails matter
120
100
80
Return
VaR(t)
60
VaR(logistic)
VaR(normal)
40
20
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Models
• Maximize expected return s.t. budget ≤ 1000
• Minimize Variance s.t. investment = 1000
• Maximize probability{return>specified level} for
levels [1000, 950, 900, and 800].
• Maximize expected return s.t. probability{return
≥ specified level} ≥ α for α [0.9, 0.8, 0.7, and 0.6].
• Minimize Value at risk for an α = 0.99
• Minimize CVaR constrained to attain given return
Optimization Solutions
• Excel SOLVER
– Maximize return linear
– Others nonlinear
• Generalized Reduced Gradient
• Some instability in solutions across runs
Simulated Solutions to evaluate
• Monte Carlo Simulation
– Crystal Ball
– 10,000 runs of one year each (long-term view)
• Correlation: Daily (short-term)
– Crystal Ball allows use of correlation matrix
• Correlation: Annual data (245 days)
– Couldn’t reasonably enter that many within
software
– Used Cholesky decomposition
Optimization Solutions
Objective
Max E[return]
Min Variance
Max Pr{E[Ret]>1000}
Max Pr{E[Ret]>950}
Max Pr{E[Ret]>900}
Max Pr{E[Ret]>800}
CC {Pr>.9[Ret>800]}
CC{Pr>.8[Ret>800]}
CC{Pr>.8[Ret>900]}
CC{Pr>.7[Ret>900]}
Min VaR at 0.99 level
MSCI
0
123.4
0
0
0
0
0
0
0
0
113.1
NYSE
0
876.6
671.6
877.3
938.3
964.4
547.3
0
408.6
0
886.9
S&P China Euro
0 1000.0
0
0
0
0
0
166.6 161.8
0
91.5
31.2
0
61.7
0
0
35.6
0
0
208.3 244.4
0
571.5 428.5
0
255.8 335.6
0
885.7 114.3
0
0
0
Solution Expected Performances
Objective
Return
Variance
Pr{>1000}
Pr{>950}
Pr{>900}
Pr{>800}
Max E[return]
1275.9*
546422
0.6429
0.6672
0.6908
0.7353
Min Variance
1067.4
28818*
0.6516
0.7501
0.8302
0.9320
Max Pr{E[Ret]>1000}
1106.6
46131
0.6865*
0.7614
0.8243
0.9130
Max Pr{E[Ret]>950}
1088.9
34200
0.6813
0.7680*
0.8384
0.9305
Max Pr{E[Ret]>900}
1082.3
31396
0.6755
0.7666
0.8400*
0.9340
Max Pr{E[Ret]>800}
1076.9
29890
0.6686
0.7629
0.8388
0.9350*
CC {Pr>.9[Ret>800]}
1116.5
55736
0.6856
0.7543
0.8132
0.9000
CC{Pr>.8[Ret>800]}
1194.4
207316
0.6623
0.7004
0.7362
0.8000
CC{Pr>.8[Ret>900]}
1127.8
69152
0.6830
0.7454
0.8000
0.8841
CC{Pr>.7[Ret>900]}
1254.2
437035
0.6470
0.6740
0.7000
0.7487
Min VaR at 0.99 level
1067.6
28819
6519
7505
8304
9321
Simulation – Max Return
Simulation – Max Return
•
•
•
•
•
•
•
•
•
Trials
Mean
Median
Standard Deviation
Variance
Skewness
Kurtosis
Minimum
Maximum
10,000
1,217.16
996.42
883.36
780,316.87
2.51
16.69
0.00
12,984.16
Simulation – Min Variance
Simulation – Min Variance
•
•
•
•
•
•
•
•
•
Trials
Mean
Median
Standard Deviation
Variance
Skewness
Kurtosis
Minimum
Maximum
10,000
1,054.31
1,034.20
208.39
43,424.47
0.5960
3.72
354.32
2,207.00
Comparison
Model
Max return
Model
Model Model CVaR
Return Variance VaR
1275.9 546422
1649 1649
Sim
Sim
Sim
return Variance VaR
1217.2 780317 849
Min variance
1067.4
28818
374
1063
1054.3
43424 353
Max Prob{Ret>1000}
1106.6
46372
452
1102
1082.4
47101 320
Max Prob{Ret>950}
1088.9
34200
392
1085
1070.0
40820 327
Max Prob{Ret>900}
1082.3
31396
379
1078
1065.3
40397 377
Max Prob{Ret>800}
1076.9
29890
373
1073
1061.3
40629 332
Max Ret st Pr{Ret>800}>0.9
1116.5
55736
498
1111
1088.8
54031 325
Max Ret st Pr{Ret>800}>0.8
1194.4
207316
604
1184
1146.1
250461 532
Max Ret st Pr{Ret>900}>0.8
1127.8
69152
557
1122
1096.4
67862 383
Max Ret st Pr{Ret>900}>0.7
1254.2
437035
1446
1446
1194.8
579255 762
Min VaR at the 0.99 level
1076.1
29724
226
1072
1061.3
41039 343
CVaR Models
ratio f(α)/(1-α)
Ratio
1.0
1.08
1.10
1.13
1.15
1.18
1.20
1.22
1.222
1.2225
MSCI
0
0
127.9
0
0
0
0
0
0
0.3
NYSE
0
0
230.6
414.9
0
0
0
0
55.7
44.7
S&P
China
1000
931.7
292.1
286.7
598.7
456.0
360.9
265.8
132.6
133.6
0
68.3
148.0
298.4
401.3
544.0
639.1
734.2
735.4
737.2
Euro
0
0
201.4
0
0
0
0
0
76.3
84.2
Model Results
Return
constraint
1.0
1.08
1.10
1.13
1.15
1.18
1.20
1.22
1.222
1.2225
CVaR
Return Variance Min
Max
VaR
1052
1056.7
33906
433
2094
309
1062
1066.5
32765
516
2434
290
1074
1078.7
42570
528
4281
298
1097
1104.6
91784
424
7685
361
1113
1122.3
144582
390
10025
443
1130
1141.5
225217
314
5367
501
1143
1156.3
305746
248
6071
571
1156
1171.1
400289
183
6774
635
1156
1171.3
400923
184
6742
702
1157
1171.6
402773
181
6752
634
Correlation Makes a Difference
Daily Models t-distribution
0.80
0.70
0.60
0.50
Return(correlated)
0.40
Return(uncorrelated)
0.30
0.20
0.10
0.00
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Correlation impact on Variance
Daily Models t-distribution
3 outliers – China mixed with others
1600.00
1400.00
1200.00
1000.00
Return(correlated)
800.00
Variance(correlated)
Variance(uncorrelated)
600.00
400.00
200.00
0.00
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Correlation impact on Value-at-Risk
Daily Models t-distribution
Directly proportional to Variance
120.00
100.00
80.00
Return(correlated)
60.00
VaR(correlated)
VaR(uncorrelated)
40.00
20.00
0.00
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Conclusions
• Can use a variety of models to plan portfolio
• Expect results to be jittery
– Near-optimal may turn out better
– Sensitive to distribution assumed
• Trade-off – risk & return