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# 金融代写|利率理论代写Portfolio Theory代考|FE630 OPT IMAL PORTFOL IOS WITH SHRINK AGE AND EMPIRIC AL BAYE S

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## 金融代写|利率理论代写Portfolio Theory代考|OPT IMAL PORTFOL IOS WITH SHRINK AGE AND EMPIRIC AL BAYE S

The optimization process tends to put higher (lower) weights on the assets with higher (lower) means. Due to parameter uncertainty, the extreme estimates in the mean vector for one period (estimation) are likely to be closer to the central estimates for the next period, which is the investment period. An optimizer that merely uses point estimates takes extreme positions and experiences poor performance during the investment period. The phenomenon is more serious for the more risk-tolerant investors who load up more on the extreme mean returns.
Frost and Savarino (1986) show that although optimization based on diffuse priors is an improvement over the classical substitution approach, the uncertainty in the mean is still too high to make the Markowitz framework more appealing than passive indexing strategies. The estimates and resulting portfolio weights still vary too much from period to period. This section discusses how portfolio performance can be improved with informative priors.

James and Stein (1961) introduce shrinkage estimators, which although they are biased are more efficient than the standard, maximum likelihood estimator (MLE) for estimating multivariate means. Their shrinkage estimator is:
$$\mu_{J S}=(1-\alpha) m+\alpha \mu_{0} i$$
where $\mathrm{m}$ is the $\mathrm{MLE}, \mu_{0}$ is a single central value toward which shrinkage occurs, and i is a vector of ones. The scalar coefficient, $\alpha$, is designed to optimally pull the estimate to a common value $\mu_{0}$. Shrinkage reduces the impact of parameter uncertainty in a vector of means by bringing extreme estimates closer to a central value. It replaces the sample estimates of the mean vector with a linear combination of this estimate and the central value, thereby reducing the cross-sectional dispersion of these means.

## 金融代写|利率理论代写Portfolio Theory代考|INCORPOR AT ING ECONOMIC INFORMAT ION IN THE PRIOR

The investor may want to incorporate her own subjective views and economic considerations into the prior density. This may also result in posterior mean estimates with a smaller cross-sectional dispersion than that of the sample means. For example, the prior views can center the mean returns on the CAPM. In the absence of additional information on specific returns, capitalization weights are good weights toward which to shrink an optimal portfolio. Indeed, an extreme of the passive investment framework involves simply replacing expected returns with betas, because the CAPM states that expected excess returns are proportional to betas. This reduces the uncertainty in the mean because betas are generally estimated more precisely than sample means.

An investor may have private information on some of the assets arising from proprietary analysis. She views the CAPM prediction for expected returns as prior information. She may have an econometric model to predict abnormal expected returns in excess of the CAPM for some but not all assets. In this spirit, Black and Litterman (1991) specifically account for the fact that active managers do not have private information on every asset in their investment universe. These authors notice that portfolio managers often modify only a few elements of the vector of means for which they have private information. They show that this practice has a large and undesirable impact on the entire vector of weights. Instead, they show how to incorporate both the private views and market equilibrium into the optimization.

# 利率理论代写

## 金融代写|利率理论代写Portfolio Theory代考|OPT IMAL PORTFOL IOS WITH SHRINK AGE AND EMPIRIC AL BAYE S

Frost 和 Savarino (1986) 表明，尽管基于扩散先验的优化是对经典替代方法的改进，但均值的不确定性㐷然太高，无法使
Markowitz 框架比被动紊引策略更具吸引力。不同时期的估计和由此产生的投资组合权重仍然相差很大。本节讨论如何利用信息

James 和 Stein (1961) 引入了收皕估计器，尽管它们有偏差，但在估计多元均值时比标准的最大似然估计器 (MLE) 更有效。他

$$\mu_{J S}=(1-\alpha) m+\alpha \mu_{0} i$$

## 金融代写|利率理论代写Portfolio Theory代考|INCORPOR AT ING ECONOMIC INFORMAT ION IN THE PRIOR

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。