论文题目：Portfolio optimization selection technologies论文语种：英文您的研究方向：金融是否有数据处理要求：是您的国家：德国您的学校背景：德国一流大学要求字数：A4纸30页 小四字体 行距1.5倍(354×30=10620字)论文用途：硕士金融 Essay 指导补充要求和说明：实证分析，最好使用R软件进行分析，其他如EVIEW也可以。
Introduction of Markowitz’ Mean-Variance approach
Harry M. Markowitz published the preliminary portfolio model in his paper ‘Portfolio Selection’ in the 1952 Journal of Finance. In this paper, Markowitz revealed the basic problem of investing which is, given uncertain outcomes, how should an investor own risky securities and build an efficient frontier, and how should an investor select his optimal investing strategy given his own preference. He then introduced the famous two formulations to the development of the portfolio theory: the expected rate of return and the expected risk measure.2.1 The assumptions of the Mean-Variance approach(1) The Market is efficient. The securities’ prices can reflect their initial value. Every investor can get enough information of the market and be aware of the probability distribution of the possible returns.(2) Investor behavior assumptions. Investors will always seek “the second opinion.” When presented with spectrum of alternatives, investors will consider all expected rate of return over a specified period. Furthermore, investors’ decisions are based on two variables: the level of expected rate of return and the expected risk.Investors always seek for portfolios with higher expected returns and lower risk. In other words, the preference of the investors is risk averse.(3) Every asset’s rate of return is normally distributed.(4) All investors have the same expected single period investment horizon.
2.2 Markowitz portfolio rate of returnThe portfolio’s expected rate of return in Mean-Variance approach can be presented either in method one: where is the initial wealth, is the terminal wealth.Or in method two: where is the expected return of security i, is the proportion of the portfolio’s initial value invested in security i.
2.3 Portfolio riskThe measure of portfolio risk in Markowitz’s mean-variance approach can be presented as follow: where is the covariance( a measure of the relationship between two random variables) of returns between security i and security j. is the standard deviation which measures how much the return of a portfolio or stock moves around the average return.#p#分页标题#e#
2.4 Markowitz Mean- Variance ModelThe Markowitz model is all about maximizing return and minimizing risk simultaneously. We should reach a single portfolio of risky assets with the least possible risk that is preferred than other portfolios with same level of return. Our optimal portfolio will be on the ray connecting risk free investments to the risky portfolio and the ray becomes tangent to our set of risky portfolios that it has the highest slope.The point is shown as B in Figure2.1 Figure2.1 Combinations of risk less asset in a risky portfolio (Gruber et al. )Our task is to determine the portfolio with greatest ratio of excess return to standard deviation. Mathematically we should maximize: where is later co called Sharp ratio.This function is subjected to the constraint, The above problem can be solved by Lagrangian multipliers.
2.5 The efficient frontierThe efficient frontier (sometimes “Markowitz frontier”) curve is a graphic presentation of a set of portfolios that offer the maximum rate of return for any given level of risk or portfolios that give the minimum risk for given level of return. Mathematically the efficient frontier is “the intersection of the set of portfolio with minimum variance (MVS) and the set of portfolios with maximum return.”The efficient frontier is convex because the risk-return characteristics of a portfolio change in a non-linear way as its component weightings are changed. The shape of the efficient frontier is a hyperbola.The region above the frontier is unachievable by holding risky assets alone. Points below the frontier are not optimal. So a rational investor will hold a portfolio only on the efficient frontier.Figure 2.2 shows the optimal selection of the portfolio on the efficient frontier.In Figure 2.2 the CAL is so called the Capital Allocation Line which discusses the possibility of lending and borrowing at a risk free rate of interest.For every point of the individual assets, there will be at least one portfolio that can be constructed and has the risk and return corresponding to that point. Portfolio on the efficient frontier are optimal in both the sense that they offer maximal expected return for some given level of risk and minimal risk for some given level of expected return.What Markowitz started in the early 1960s was continued through the development of the capital market theory, whose final product is the capital asset pricing model (CAPM), which allowed a Markowitz efficient investor to estimate the rate of return for any risky security or asset.#p#分页标题#e#
2.6 Critique on the mean-variance approachAlthough the mean-variance approach is developed as early as 1952, it is still often applied in practice today. However, this approach is subjected to criticism as well.The most serious drawback of the mean-variance approach is that it is formed on one-period-model, which means we take consideration only a single transaction period. At the beginning of the period the investor complies his portfolio according to the chosen mean-variance criterion and holds this portfolio until the end of this period. During this period of time the investor will not change his portfolio no matter what happens on the stock market. Meantime, the stock prices are changing through the expected rate of return or the variance, as well as covariance with the return of other stocks. But there is no modeling of the stock price in the time lapse.In order to eliminate this disadvantage we observe the more-period models. In this case the investors can trade in either more finitely points in time or even continuously in a time lapse. This is accomplished by a more-period modeling of the stock prices. (For example the Black-Scholes-Model)Another drawback is using the variance as a measure of risk. The minimization of the variance basically punishes the desired positive deviations of the portfolio rate of return from its expected return. Therefore there are many works in literature replace the variance of portfolio rate of return by other risk measure, such as the Value-at-Risk. www.ukassignment.orgThe second major drawback of using variance is that by using variance as the measure of uncertainty the investor assumes that the distribution of returns can be described by its first two moments. However the variance does not capture all of the risk born from non-normal distributions.