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: ron@mathproservices.com