The Equivalence Property Of Mean-Variance Models Of Modern Portfolio Theory In Mathematical Finance

The Equivalence Property ofMean-Variance Models of Modern Portfolio Theory in MathematicalFinance Author: LAI MinMajor: Theory of Probability and Mathematical StatisticsTutor: SHI Ning-zhong ProfessorIt is known that the Modern Portfolio Theory was first proposed by Markowitz in 1950's [43] [45], and he was awarded the Nobel Prize in Economic Sciences in 1990. One important theme in the model is said to be risk-return model or variance-expectation model that finds the solution to minimum risk. In practice, there is another strategy to be chosen for an investor. We call it return-risk model or expectation-variance model, which finds the solution to maximum return in a given control of the risk. This paper shows that the above two models are equivalent and gives a method to obtain the solution for the latter model.In this paper, we consider the problem in the case of investment portfolios which no riskfree assets, contain riskfree assets under " short-selling " or without " short-selling " conditions. Suppose that there are riskfree assets and n risky assets to be considered by an investor whose returns are expressed by constant R0 and random vector x =(xx,x2,...,xn)T respectively. Then we denote its mean vector and covariance matrix, respectively, byEx - (Ex1,Ex2,... , Exn)T = n = (u1,u2,...,Un)T. Var(x) = E(x - u)(x -u)T = (Cov(xi,Xj)) =Suppose riskfrec assets and n risky assets' investment ratio are denoted by w0,w1,w2,... ,wn respectively, w = (w1,w2,..., wn)T, we say that w or (w0,wT)T is a portfolio if wT 1 =1 or w0+wT1 = 1, where 1 is the n- dimensional vectorwhose coordinates are all 1. For a portfolio, its expected return is wiui=wTuand its risk is Var(wTx) = wT w.Usually there are two strategies for an investor chooses portfolios are as follows:1. Given the lower bound for expected return by a, find an optimal portfolio w to minimize the risk;2. Given the upper bound for risk by 0, find an optimal portfolio w to maximize the expected return.(i). Under short-selling conditions, the portfolio models without riskfree assets, which contain n risky securities, are as follows:Let .Then if u's coordinates ui are not completely equal, ( > 0), we have:Theorem 2.1 (1). As far as model (I) is concerned, suppose w* is the optimalportfolio, when a >B/A,(2).When Theorem 2.2 As far as model (II) is concerned, suppose w is the optimal portfolio, when Synthesize Theorem 2.1 and Theorem 2.2, within these functions' relations, we know that the optimal solutions w* and w of the models (/) and (//) are the same. That is, the two optimal portfolios are equal.(ii).Under short-selling conditions, the portfolio models with riskfree assets, which contain n risky securities, are as follows:If u's coordinates ui are not completely equal, (K > 0), we have: Theorem 3.1 As far as model (I)' is concerned, suppose (w0*, (w*)T)T is the optimal portfolio, when a >R0,Theorem 3.2 As far as model (II)' is concerned, suppose (w0,wT)T is the optimal portfolio, thenSynthesize Theorem 3.1 and Theorem 3.2, within these functions' relations, we know that the optimal solutions (w0*,(w*)T)T and (w0,wT)T of the model (I)' and (II)' are the same. That is, model (I)' and (II)' are equivalent.(iii).Under without short-selling conditions, the portfolio models without riskfree assets, which contain n risky securities, are as follows:Where 1 = 1n is the n-dimensional vector whose coordinates are all 1. Clearly, a Without loss of generality, denote the covariance matrix of the first k assets returns , A11 is a k x k matrix.Denote A1 = (A11, A12) and A2 = (A21, A22). Let A = denote a (n - k)- dimensional vector of zeros. For simplicity, let 0/0 = 0. Then we have the following theorem:Theorem 4.1 As far as model (I)" is concerned, w* = (w1*,w2, ...,wn*)T, where w*i > 0 for i = 1, 2, ..., k and wj*= 0 for j = k + 1, ... , n is the optimal solution of model (I)" if and only ifWhere . The optimal solution isand the corresponding minimum risk isWhen = 0, which mean...