On Approximating Expected Log-Utility of. Mutual Fund Investments

Transcription

1 Reports on Economics and Finance, Vol.,, no., HIKARI Ltd, ttps://doi.org/./ref.. On Approximating Expected LogUtility of Mutual Fund Investments Xuejiang Jin and Cunui Yu Matematics Department, Farmingdale State College, SUNY Broadollow Road, Farmingdale, New York, USA Copyrigt Xuejiang Jin and Cunui Yu. Tis article is distributed under te Creative Commons Attribution License, wic permits unrestricted use, distribution, and reproduction in any medium, provided te original work is properly cited. Abstract In tis paper, considering te utility measured as a logaritmic function associated wit rate of return, by examining istorical data of a set of mutual funds, we study different approximating metods of te expected utility and compare teir related approximation errors. Keywords: Logutility, Expected Value, Approximation. Introduction Utility function is an important economics concept. It is a matematical way to represent satisfactions or preferences. One general restriction is tat te utility function, defined on real number and real value, be increasing, continues and concave. One commonly used utility measure is logutility U(x) = ln( + x).. Approximation Metods In tis section, we will approximate utility,eu, using a function of mean and variance. All tese tree metods ad been introduced in [] and []. Let e = E(r), v = Var(r), σ = v, were r is te rate of return. A) Midpoint approximation E[U(r)] U(e σ) + U(e + σ) () B) Taylor series approximation

2 Xuejiang Jin and Cunui Yu Markowitz [] introduced a metod to approximate EU based on Taylorseries around e: U(r) = U(e) + U (e)(r e) +.U (e)(r e) + Since e E r v E r e E[U(r)] U(e) +.U"(e)v () ( ), [( ) ], ten C) Tree points quadratic approximation Levy and Markowitz [] introduced an alternate way class of estimating functions tat were selected in wic te quadratic was fit to tree points: (e kσ, U(e kσ)), (e, U(e)), (e + kσ, U(e + kσ)) Write te quadratic functions as: Q k (r) = a k + b k (r e) + c k (r e) Ten, and E[Q k (r)] = a k + c k v a k = U(e) c k = b k = U(e + kσ) U(e kσ) kσ U(e + kσ) + U(e kσ) U(e) k σ From tat we got E[Q k (r)] = U(e) + U(e+kσ)+U(e kσ) U(e) k Clearly, if k =, ten E[Q (r)] = U(e+σ)+U(e σ), wic is same as te midpoint estimation in equation (). In fact, Levy and Markowitz [] also pointed out if k, ten E[Q k (r)] U(e) +.U"(e)v, same as Tylor series approximation in equation (). For readers convenience, we provide a proof of k result as follows. Let = kσ. Since σ is a finite constant,

3 On approximating expected logutility of mutual fund investments U(e + kσ) + U(e kσ) U(e) lim k k σ U(e + ) + U(e ) U(e) U(e + ) U(e) U (e) U (e ) = U (e) U(e) U(e ) Of course, people can also take te second derivative of te numerator and denominator on te first step wit respect to k and ten apply L Hospital s rule.. Empirical Results If we coose k =., ten E[Q. (r)] [U(e +.σ) + U(e.σ)] U(e). () For montly return of ten mutual funds from Jan. to Jan., Table provides results of tese tree estimations stated in equations (), () and (), for U(r) = ln( + r). Name E(r) U[E(r)] E[U(r)] Q (r) Q. (r) Q (r) ACMTX AAAAX AAXAX UTAYX ATHA MQIFX

4 Xuejiang Jin and Cunui Yu GRECX BIPSX UDPIX AFBIX Table : U(r) = ln( + r), r is te montly return rate. Table sows tat all tree metods (Q, Q., Q ) presented ere can be considered very good approximations of E(U(r)). In fact, te correlation coefficients (easily calculated by using CORREL function in Excel) between E[U(r)] and eac of Q, Q., Q are over.. It may also be noticed tat column values in U[E(r)] are greater tan values in E[U(r)], wic matces Jensen s inequality of E[U(r)] U[E(r)] for U(r) = ln( + r) is a concave function. For reader s convenience, steps of obtaining Table in Excel are also provided as follows:. Download and import montly data of tese mutual funds from Jan. to Jan.. A good data resource is yaoo finance webpage.. Evaluate montly returns r of eac mutual fund.. Evaluate average montly return, simple mean e = E(r) and simple variance VAR(r) of eac mutual fund.. Evaluate U[E(r)] by calculating LN( + e).. Evaluate E[U(r)] by calculating te average of LN(r).. Evaluate Q (r), Q. (r) and Q (r) by calculating Q (r) = LN(+e σ)+ln(+e+σ) ; Q. (r) = [LN( + e.σ) + LN( + e +.σ)] LN(e); Q (r) = LN( + e) (+LN(+e)) VAR(r) ; Here, e = E(r) and σ = SQRT(VAR(r)).. Furter Disscussions In tis paper, we used mutual funds data and logutility to work on te approximation. It can also be worked on oter types of financial data like price of stocks, options, or edge funds [] and oter types of utility functions like exponential utility U(x) = e ax for some a > or power utility U(x) = bx b for some b <. It is a good example to sow ow numerical analysis applied to economics and finance. For readers wit advanced numerical analysis skills, tey may want to test oter type of approximations. For example, Cebysev series estimation, wic is one kind of ortogonal polynomial approximations, states tat

5 On approximating expected logutility of mutual fund investments U(r) = a n T n (r) were te Cebysev polynomials of te first kind are defined by te recurrence relation: T (r) = T (r) = r T n+ (r) = rt n (r) T n (r) n= ln, n = If U(r) = ln( + r), ten a n = { π( ) n, n >. Interested readers may want to ceck on tis approximation metod. n References [] Harry Markowitz, Portfolio Selection: Efficient Diversification of Investment, Yale University,. []. H. Levy and H. M. Markowitz, Approximating Expected Utility by a Function of Mean and Variance, Te American Economic Review, (), no.,. [] W. Fung and D. Hsie, Is meanvariance analysis applicable to edge funds?, Economics Letters, (), no.,. ttps://doi.org/./s() Received: April, ; Publised: May,