INVESMEN MODELS Ulrih Rieder University of Ulm, Germany Keywords: meanvariane ortfolio seletion, Markowitz model, minimum variane ortfolio, twofund searation, HARAutility, BlakSholes model, stohasti dynami rogramming, stohasti ontrol, martingale method Contents 1. Introdution. MeanVariane Portfolio Seletion.1 Markowitz Model. MeanVariane Portfolio with a Riskless Asset 3. Portfolio Seletion in Disrete ime 3.1 Stohasti Dynami Programming Aroah 3. HARAUtilities 4. Portfolio Seletion in Continuous ime 4.1 Stohasti Control Aroah 4. Martingale Method 5. Further Models Glossary Bibliograhy Biograhial Sketh Summary In this aer we exosit investment models in disrete and ontinuous time with a seial emhasis on solution methods. In a first art, we desribe the early work of Markowitz and obin. We define and solve the meanvariane ortfolio seletion roblem and formulate the twofund searation theorem. Multieriod disretetime models are investigated by stohasti dynami rogramming. For ontinuoustime ortfolio roblems, we exlain two solution methods: stohasti ontrol tehniques and the martingale aroah. In artiular, we derive otimal ortfolios for seial HARAutilities. he last setion identifies some extensions and further models. 1. Introdution he heart of investment models is the seletion of an otimal set of finanial assets. An investor endowed with a given amount of inome has to deide how many shares of whih asset he should hold at whih time instant in order to maximize his wealth at the time horizon and/or his exeted utility of onsumtion during the time interval [0, ]. he main objetive of investment analysis is to rovide otimal deisions for suh ortfolio seletion roblems. he earliest aroah for solving the ortfolio seletion roblem is the soalled meanvariane formulation. It was ioneered by Markowitz (195, 1959) and obin
(1958) and is only suited for oneeriod roblems. It still has great imortane in reallife aliations and is widely alied in risk management deartments of banks. he roblem of multieriod ortfolio seletion was roosed in the late sixties and early seventies by (among others) Mossin (1968), Samuelson (1969) and Fama (1970) in a disretetime setting. he work of Merton (1969, 1990) must be regarded as the real starting oint of ontinuoustime ortfolio theory. In what follows we will distinguish investment models in disrete and ontinuous time. In setion, we desribe the meanvariane aroah. Setion 3 is devoted to disretetime multieriod ortfolio seletion, setion 4 deals with solution methods for ontinuoustime models.. MeanVariane Portfolio Seletion In the formulation of the meanvariane ortfolio, we use the following notation: x is a vetor whose omonents denote the weight or roortion of the investor s wealth alloated to the n assets in the ortfolio. Obviously, the sum of these weights is equal to 1. e is a vetor of ones. r is the vetor of exeted returns of the n assets, where it is assumed that not all elements of r are equal, and Q is the n n ovariane matrix. We assume that Q is nonsingular. his essentially requires that none of the asset returns be erfetly orrelated with the returns of any ortfolio made u of the remaining assets; and that none of the assets or ortfolios of the assets be riskless. Note that Q is symmetri and ositive definite being a ovariane matrix. Finally, r is the investor s target return..3 Markowitz Model Following Markowitz (195) and obin (1958) the ortfolio seletion roblem an be stated as: minimize subjet to x Qx x r = r x e = 1 In roblem (1), we minimize the ortfolio variane subjet to two onstraints: first, the ortfolio mean must be equal to the target return r, and seond, the ortfolio weights must sum to unity, whih means that all wealth is invested. ehnially, we minimize a quadrati funtion subjet to linear onstraints. Sine x Qx is stritly onvex, the roblem (1) has a unique solution and we only need to onsider the firstorder onditions (see Nonlinear Programming). (1) Consider the Lagrange funtion of (1) λ1 λ L = x Qx ( x r r ) ( x e 1). () he firstorder onditions are
L = Qx λ1r λe = 0 x L = r x r = 0 λ 1 (3) (4) L λ = 1 x e = 0 (5) Solving for x we obtain n 1 1 x = Q r + e = Q r e = Q r e A r Where ( λ1 λ ) [ ][ λ1 λ] [ ] [ 1] a b r Q r r Q e A = = b r Q e e Q e hen x is alled a minimum variane ortfolio. Note that x is linear in its exeted return r. he result of this analysis an be stated as: heorem.1 Let Q be the ositive definite ovariane matrix, and r the vetor of exeted returns of the n assets where it is assumed that not all elements of r are equal. hen the minimum variane ortfolio with given target return r is unique and its weights are given by (6). Substituting (6) into the definition of the ortfolio variane yields σ : = x Qx = [ r 1] A [ r1] a br + r = a b. (6) (7) (8) In (8) the relation between σ and any given exeted return r is exressed as a arabola and is alled the minimum variane ortfolio frontier. he set of ortfolios having the highest return for a given variane is alled set of meanvariane effiient ortfolios. he global minimum variane ortfolio is the ortfolio with the smallest ossible variane for any exeted return. Its exeted return r g ist given by r g = b and its variane σ g is equal to 1. he weights of the global minimum variane ortfolio are
x g Q e =. (9) Equation (6) shows a twofund searation theorem, that the minimum variane ortfolio frontier an be generated by any two distint frontier ortfolios. heorem. (wofund searation) Let x a and x b be two minimum variane ortfolios with exeted return r a and r b resetively, suh that r a r b. a. hen every minimum variane ortfolio is a linear ombination of x a and x b. b. Conversely, every ortfolio, whih is a linear ombination of x a and x b, is a minimum variane ortfolio.. In artiular, if x a and x b are minimum variane effiient ortfolios, then αx a + (1 α)x b is also a minimum variane effiient ortfolio for α [0, 1]. Only two ortfolios are suffiient to desribe the entire effiient set. It is of ratial interest to selet two ortfolios whose means and varianes are easy to omute. One suh ortfolio is the global minimum variane ortfolio x g. he other effiient ortfolio ould be x m Q r = (10) b a. b with exeted return r m = a and variane σ b m = Bibliograhy O ACCESS ALL HE 13 PAGES OF HIS CHAPER, Visit: htt://www.eolss.net/eolsssamleallchater.asx Ingersoll J.E. (1987). heory of Finanial Deision Making, 474. otowa, NJ: Rowman and Littlefield. [extbook on multieriod investment models.] Korn R. (1997). Otimal Portfolios: Stohasti Models for Otimal Investment and Risk Management In Continuous ime,338. Singaore: World Sientifi. [Reent book on ontinuoustime ortfolio theory.] Luenberger D.G. (1998). Investment Siene, 494. New York: Oxford University Press. [Exellent textbook on investment models.] Markowitz H. (1959). Portfolio Seletion: Effiient Diversifiation of Investments, 344. New York: Wiley. [Early textbook on ortfolio seletion.]
Merton R.C. (1990). Continuousime Finane, 700. Cambridge, MA: B. Blakwell. [First book on ontinuoustime ortfolio theory.] Sethi S.P. (1997). Otimal Consumtion and Investment with Bankruty, 48. Boston: Kluwer Aademi Publishers. [Reent survey on onsumtion and investment models.] Biograhial Sketh Ulrih Rieder, born in 1945, reeived the Ph.D. degree in mathematis in 197 (University of Hamburg) and the Habilitation in 1979 (University of Karlsruhe). Sine 1980, he has been Full Professor of Mathematis and head of the Deartment of Otimization and Oerations Researh at the University of Ulm. His researh interests inlude the analysis and ontrol of stohasti roesses with aliations in teleommuniation, logistis, alied robability, finane and insurane. Dr. Rieder is EditorinChief of the journal Mathematial Methods of Oerations Researh.