Porfolio Optimization with beta distributed returns and exponential utility Ron Davis College of...
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Transcript of Porfolio Optimization with beta distributed returns and exponential utility Ron Davis College of...
Porfolio Optimization with beta distributed returns and exponential utility
Ron DavisCollege of BusinessSan Jose State University
Presented June 8, 2002
Hawaii Conference on Statistics
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
Generalized beta formulas
)(
))(()(
)()(
11
BxAfor
AB
xB
AB
Ax
ABxf
)( ABA
)1(
AB
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
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
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
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
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
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
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
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
Exponential Utility
Functional form: Risk Tolerance parameter Value Additivity property If G1 and G2 are independently
distributed gambles, then
/1)( xexU
)()()&( 2121 GCEGCEGGCE
CE-value of beta[a,b,A,B]
Solve
.
)(certainty
)( //
gamblebetathefor
valuecashequivalentx
dxexfeB
A
xx
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
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
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
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
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
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
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
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
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
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]