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• W302819365 abstract "Introduction At the core of asset acquisition, revision, and rebalancing of asset holdings is the principle of diversification, and that is well-recorded in the existing literature. Since the classic works of Markowitz (1952), and Roy (1952), portfolio theory has become a fascinating area for further examination from both academics and practitioners in view of risk, uncertainty, and expectation. The research alluded to refers to the theoretical arguments for risk minimization or trade-off between return and risk of any portfolio. Markowitz's mean-variance frontier emphasizes that trade-off structure. In yet another classic piece, Tobin (1958) derives the mean-variance locus with the additional insight on the choice of a risk-loving or risk-averting investor. The analysis of diversification highlighting the principle of diversification and safety first initially was applied to domestic assets alone until Grubel (1968), Levy and Sarnat (1970, 1979), Solnik (1974), Losq (1979), Vaubel (1979), and Friend and Losq (1979), among others, brought portfolio structure into the setting of international markets. Since then it has been recognized that asset holdings in international capital markets certainly extend the efficiency envelope to the further benefits of investors. In all of the cited works and beyond, it is empirically established that international diversification reduces risk for a given return or increases return for a given risk. In this work, I emphasize the basic structure of Markowitz, followed by Roy, and extend their analyses with further comments and clarifications on those classic theoretical models. Next, I go beyond those paradigms of the Markowitz-Roy foundation and analyze the theoretical revision of optimal portfolio within a comparative static framework. The effects of portfolio revision are analyzed under two scenarios: (i) a portfolio manager who rebalances the investor's assets holding knows the investor's utility function, and (ii) utility function of the investor is unknown. It is ascertained that under both scenarios, the end result is identical. Theoretical Structure of Diversification Markowitz Framework Consider a rational investor who has $W and who decides to invest in 11 assets with the expected returns on these assets being [r.sub.1], [r.sub.2], [r.sub.3], ..., [r.sub.n], and the variance of returns on these assets being [[sigma].sup.2.sub.1], [[sigma].sup.2.sub.2], [[sigma].sup.2.sub.3], ..., [[sigma].sup.2.sub.n], respectively. The investor's expected portfolio return ([R.sub.p]) is given as: [R.sub.p] = [n.summation over (i=1)] [w.sub.i][r.sub.i] (1) where [w.sub.i] is the proportion of investible funds put in asset i (alternatively called weight for i = 1, 2, 3, ..., n), and [n.summation over (i=1)] [w.sub.i] = 1 (2) His or her portfolio risk, measured by variance ([[sigma].sup.2.sub.p]), is: [[sigma].sup.2.sub.1] = [n.summation over (i=1)] [n.summation over (i=1)] [w.sub.i] [w.sub.j] [[sigma].sub.i] [[sigma].sub.j] [[rho].sub.ij] = [n.summation over (i=1)] [n.summation over (i=1)] [w.sub.i] [w.sub.j] [[sigma].sub.ij] (3) Here [[rho].sub.ij] is the correlation coefficient, and [[sigma].sub.ij] is the covariance between the returns of i-th asset and j-th asset. In this n-asset portfolio, there are n terms involving variances of n assets, each multiplied by the squared value of its weight plus nC2 (n(n-1)/2) terms involving covariance terms (or correlation coefficient terms). In other words, expression (3) is as follows: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3A) or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3B) Note that the terms within the square brackets ([]) in the first part on the right side of equation (3B) is the non-removable component of the portfolio risk. If many of the [[rho].sub.ij] s in the second part of (3B) in the second brackets ({}) are negative, and the negative terms are added to the first component of portfolio risk, however, then the total portfolio risk becomes smaller in value. …" @default.
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• W302819365 date "2010-01-01" @default.
• W302819365 modified "2023-09-26" @default.
• W302819365 title "Asset Acquisition, Diversification, and Revision-Theoretic Exercises in Portfolio Theory" @default.
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