Probabilistic Models
• Value-at-Risk (VaR)
• Chance constrained programming
– Min variance
– Max return s.t. Prob{function≥target}≥α
– Max Prob{function≥target}
– Max VaR
Finland 2010
Value at Risk
Maximum expected loss given
time horizon,
confidence interval
Finland 2010
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
Average return
Variance
Covariance
with S
Covariance
with B
Stock S
0.148
0.014697
Bond B
0.060
0.000155
0.000468
SCIP G
0.152
0.160791
-0.002222
-0.000227
Finland 2010
Maximize Expected Value of
Probabilistic Function
• The objective is to maximize return:
Expected return = 0.148 S + 0.060 B + 0.152 G
• subject to staying within budget:
Budget = 1 S + 1 B + 1 G ≤ 1000
Pr{Expected return ≥ 0} ≥ α
S, B, G ≥ 0
Finland 2010
Solutions
Probability
{return≥0}
0.50
0.80
0.90
0.95
0.99
α
Stock
Bond
0
0.253
0.842
1.282
2.054
379.91
556.75
622.18
668.92
-
Finland 2010
Gamble Expected
return
1000.00
152.00
620.09
150.48
443.25
149.77
377.82
149.51
331.08
149.32
Minimize Variance
Min 0.014697S2 + 0.000936SB - 0.004444SG +
0.000155B2 - 0.000454BG + 0.160791G2
st
S + B + G  1000 budget constraint
0.148 S + 0.06 B + 0.152 G ≥ 50
• S, B, G ≥ 0
Finland 2010
Solutions
Specified
Gain
≥50
≥100
≥150
≥152
Variance
Stock
Bond
Gamble
106.00
2,928.51
42,761
160,791
406.31
500.00
-
825.30
547.55
-
3.17
46.14
500.00
1,000.00
Finland 2010
Max Probability
α
Stock
Bond
Gamble
3
4
4.5
4.8
4.9 and up
157.84
73.21
406.31
500.00
-
821.59
914.93
547.55
-
20.57
11.86
46.14
500.00
-
Finland 2010
Expected
return
75.78
67.53
64.17
61.48
0
Real Stock Data – Student-t fit
Finland 2010
Logistic fit
Finland 2010
Daily Data: Gains
Ford
IBM
Pfizer
SAP
WalMart
XOM
S&P
Mean
1.00084
1.00033
0.99935
0.99993
1.00021
1.00012
0.99952
Std. Dev
0.03246
0.02257
0.02326
0.03137
0.02102
0.02034
0.01391
Min
0.62822
0.49101
0.34294
0.81797
0.53203
0.51134
0.90965
Max
1.29518
1.13160
1.10172
1.33720
1.11073
1.17191
1.11580
Cov(Ford)
0.00105
0.00019
0.00014
0.00020
0.00016
0.00015
0.00022
0.00051
0.00009
0.00016
0.00013
0.00012
0.00018
0.00054
0.00011
0.00014
0.00014
0.00014
0.00098
0.00010
0.00016
0.00016
0.00044
0.00011
0.00014
0.00041
0.00015
Cov(IBM)
Cov(Pfizer)
Cov(SAP)
Cov(WM)
Cov(XOM)
Cov(S&P)
0.00019
Finland 2010
Results
Model
Max Return
Min Variance
Normal
Pr{>970}>.95
t
Pr{>970}>.95
t
Pr{>970}>.95
Pr{>980}>.9
t
Pr{>970}>.95
Pr{>980}>.9
Pr{>990}>.8
Max
Pr{>1000}
Ford
1000.00
0
-
IBM
Pfizer
SAP
WM
XOM
S&P
Return
Stdev
-
-
-
-
-
-
1000.84
32.404
45.987
90.869
30.811
127.508
116.004
588.821
999.76
13.156
398.381 283.785
-
-
222.557
95.277
-
1000.49
18.534
607.162 296.818
-
-
96.020
-
-
1000.63
23.035
581.627 301.528
-
-
116.845
-
-
1000.61
22.475
438.405 279.287
-
-
220.254
62.054
-
1000.51
19.320
16.275 109.867
105.586
38.748
174.570
172.244
382.711
999.91
13.310
Finland 2010