1	APPLICATIONS OF THE CE-VALUE FUNCTION PAK Ron Davis Mathematical Programming Services http://www.mathproservices.com/cevalues San Jose State University http://www.cob.sjsu.edu/facstaff/davis_r
2	PRESENTATION OUTLINE Background on Exponential Utility Status of the CE-VALUE FUNCTION PAK The CE-VALUE MAXIMAL FRONTIER Application to Portfolio Gamma Model Application to Portfolio Scenario Optimization Application to DPL Decision Analysis
3	Exponential Utility U(x) = 1 - exp(-x/T) T = Risk Tolerance What is the maximum wager you would make on a triple-or-nothing bet with even odds? Multiply that by 2 to get T.
4	Certain Equivalent (CE) Value Discrete Distributions Continuous Distributions
5	VALUE ADDITIVITY Let G1 and G2 be two independent gambles G1&G2 is the joint receipt of G1 & G2 CE(G1&G2) = CE(G1) + CE(G2) Holds for all risk tolerances
6	INFORMATION VALUE Concept Extensions Certain Equivalent Value of Perfect Info CE\|PI - Max{CE(Ai)\|Prior Info} Certain Equivalent Value of Sample Info CE\|SI - Max{CE(Ai)\|Prior Info}
7	CE-VALUE FUNCTION PAK Add-In for EXCEL DLL for Visual Basic DLL for C++ library of value functions computes CE from T and distribution parameters
8	CE-Value Functions: Discrete Binomial; Negative Binomial Geometric; HyperGeometric Pascal Poisson Uniform Finite Discrete - {x_i,p_i}
9	CE-Value Functions: Continuous Uniform; Histogram Normal; logNormal Beta; Gamma; Inverse Gamma Exponential; Erlang; Chi-Square Triangular Laplace; Rayleigh; Logistic Weibull
10	Example CE-Value Functions Normal Finite Discrete Gamma
11	Maximizing the CE-Value Maximize CE(T,distribution parameters) By changing decision variables X Where distribution parameters = g(X) Repeat for range of risk tolerance values Converts stochastic programming problem to a parameterized family of nonlinear programs
12	The CE-VALUE MAXIMAL FRONTIER The set of decision variable vectors which optimize the CE value for SOME risk tolerance constitute the CE-Value Maximal Frontier Other solutions are CE-Value dominated and hence sub-optimal
13	Concept Extension The mean-variance efficient frontier is the CE-Value Maximal Frontier for the portfolio problem under normal assumptions. The CE-Value Maximal Frontier is the extension to the asymmetric or discrete distribution case
14	EXAMPLE APPLICATIONS Portfolio Gamma Model Portfolio model with Scenarios Decision Analysis models in DPL (or Precision Tree?)
15	Portfolio Formulation Enhancement Minimize Variance subject to expected return >= r0 Maximize CE-Value subject to budget constraint mean-variance efficiency is the same as CE-Value maximality in this case.
16	Portfolio Gamma Formulation Maximize CE-Value subject to constraints computing portfolio min portfolio mean portfolio variance
17	Backtest Results with Portfolio Gamma Model
18	Portfolio Optimization with Multiple Scenarios Minimize Variance subject to constraint on expected return Maximize CE-Value of expected returns where r_i is the expected return under the ith scenario
19	Min Variance Allocations: Multiple Scenarios
20	MAX CE-Value Allocations Multiple Scenarios
21	Performance Comparisons: Min Var vs. Max CE-Value
22	Conclusions Exponential Utility is preferred because it least to Value Additivity CE-Value functions can be developed in closed form, or series expansion numerical integration Parametric optimization wrt Tolerance yields the CE-Value Maximal Frontier CE-Value Maximization is the proper generalization of mean-var efficiency for the asymmetric case.