porfolio optimization with beta distributed returns and exponential utility ron davis college of...
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![Page 1: Porfolio Optimization with beta distributed returns and exponential utility Ron Davis College of Business San Jose State University Presented June 8, 2002](https://reader030.vdocuments.mx/reader030/viewer/2022032801/56649d535503460f94a30146/html5/thumbnails/1.jpg)
Porfolio Optimization with beta distributed returns and exponential utility
Ron DavisCollege of BusinessSan Jose State University
Presented June 8, 2002
Hawaii Conference on Statistics
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PRESENTATION OUTLINE
Generalized Beta on [A, B] Estimating beta parameters from return
data Formulating the beta-portfolio model Evaluating beta distributed gambles Creating the Maximal Value Frontier Example results
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Generalized beta formulas
)(
))(()(
)()(
11
BxAfor
AB
xB
AB
Ax
ABxf
)( ABA
)1(
AB
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Moment Constrained Least Square Fit
Order Data points x(1)<x(2)<…<x(n)
)()1(
.).()1(
)()(
:
)),,,,/)1(2/1()((1
2
nxBxA
devstsamplesAB
dataofmeanxBAA
tosubject
BAnknbetainvkxMinn
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USTB CDF COMPARISON
0
0.2
0.4
0.6
0.8
1
1.2
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008
USTB return
Cum
ulat
ive
Prob
abili
ty
beta
data
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GBIT CDF COMPARISON
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05
GBIT return
Cum
ulat
ive
Prob
abili
ty beta
data
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CBLT CDF COMPARISON
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08
CBLT return
Cum
ulat
ive
Prob
abili
ty
beta
data
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GBLT CDF COMPARISON
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1
GBLT return
Cum
ulat
ive
Prob
abili
ty
beta
data
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SLCO CDF COMPARISON
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15
SLCO return
Cum
ulat
ive
Prob
abili
ty
beta
data
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SSCO CDF COMPARISON
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3
SSCO RETURN
Cum
ulat
ive
Prob
abili
ty beta
data
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Beta-Portfolio model Maximize Subject to P_A = AX P_B = BX P_µ = RX P_var = XtCX P_apb = (P_µ-P_A)(P_B-P_µ)/P_var – 1 P_a = P_apb*(P_µ-P_A)/(P_B-P_A) P_b = P_apb – P_a
)_,_,_,_( BPAPbPaPCE
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Exponential Utility
Functional form: Risk Tolerance parameter Value Additivity property If G1 and G2 are independently
distributed gambles, then
/1)( xexU
)()()&( 2121 GCEGCEGGCE
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CE-value of beta[a,b,A,B]
Solve
.
)(certainty
)( //
gamblebetathefor
valuecashequivalentx
dxexfeB
A
xx
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CE-value of beta[a,b,A,B]
Solution for CE-value
Use power series for exp term Integrate term by term Sum until remainder sufficiently small
B
A
x dxexfx /)(ln
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MAXIMAL VALUE FRONTIER
Let RiskTolerance vary from eps to inf Solutions obtained constitute the
“Maximal Value Frontier” This is the theoretically “correct”
generalization of the “mean-variance efficient frontier” of Markowitz-Sharpe theory to the asymmetric case
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Ibbotson Associates times series
Treasury Bills Intermediate-Term Government Bonds Long-Term Corporate Bonds Long-Term Government Bonds Large Company Stocks Small Company Stocks 12 yrs of monthly data, Jan 1990-Dec 2001
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beta-model ALLOCATIONRiskTolerance from 50 to 155
Note: USTB between 98600 and 100000 throughout
-200
0
200
400
600
800
1000
1200
1400
1600
0 20 40 60 80 100 120 140 160 180
Risk Tolerance
$ In
vest
men
t
GBIT
CBLT
SSCO
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beta-model ALLOCATIONRisk Tolerance from 155 to 5960
-20000
0
20000
40000
60000
80000
100000
120000
0 1000 2000 3000 4000 5000 6000 7000
Risk Tolerance
USTB
GBIT
SLCO
SSCO
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beta-model ALLOCATIONSRisk Tolerance from 6000 to 20000
0
10000
20000
30000
40000
50000
60000
70000
80000
90000
6000 8000 10000 12000 14000 16000 18000 20000
Risk Tolerance
GBIT
CBLT
GBLT
SLCO
SSCO
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beta-model ALLOCATIONSRisk Tolerance from 20000 to 233200
0
10000
20000
30000
40000
50000
60000
70000
80000
90000
100000
110000
20000 70000 120000 170000 220000
Risk Tolerance
$ In
vest
men
t GBLT
SLCO
SSCO
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CONCLUSIONS
Generalized beta fits return data rather well
CE-value of generalized beta is computable using VBA custom function
Beta-portfolio model takes into account min and max as well as mean and var
Maximal Value Frontier is not the same as mean-variance Efficient Frontier
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Summing up
If you are doing portfolio optimization with asymmetric distributions
Compute the Maximal Value Frontier by varying Risk Tolerance
Rather than mean-variance Efficient Frontier analysis
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For More Information
Vol 6, Advances in Mathematical Programming and Financial Planning
Published by Elsevier Science, 2001 Web site
http://www.mathproservices.com Email: [email protected]