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# 金融代写|利率理论代写Portfolio Theory代考|FNCE463 Brief Overview of the Bayesian Process for Stock Returns

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## 金融代写|利率理论代写Portfolio Theory代考|Brief Overview of the Bayesian Process for Stock Returns

An investor may have prior views on parameters even before conducting an empirical analysis. The views can be, for example, that betas are likely to be close to 1 , that returns on some risky assets are likely to be equal to the CAPM for lack of better information, or that variances or correlations could be equal. These views are summarized in a prior density, $\mathrm{p}(\mu, \mathrm{V})$ for the means and covariance matrix, and could be quite vague. The investor may not have any views, in which case the prior distribution is made very vague so as to have no impact on the analysis. This is referred to as a diffuse prior.

In a standard analysis, one typically estimates parameters by maximizing the likelihood function, which is the density of the data-here the returns $\mathrm{R}$ – given a value of the parameters $\mathrm{p}(\mathrm{R} \mid \mu, \mathrm{V})$. This process yields the classic maximum likelihood (ML) estimator. In the Bayesian setup, the likelihood and the priors are combined and result in the so-called posterior density of the parameters $\mathrm{p}(\mu, \mathrm{V} \mid \mathrm{R})$. This density represents the investor’s knowledge after observing the data. Quantitatively, this combination is done in an optimal way with the use of Bayes theorem: One can show that the posterior $\mathrm{p}(\mu, \mathrm{V} \mid \mathrm{R})$ is proportional to the product $\mathrm{p}(\mu, \mathrm{V}) \mathrm{p}(\mathrm{R} \mid \mu, \mathrm{V})$. The posterior density is found by simply multiplying the likelihood by the prior density. Estimates of the parameters typically reported can include the mean and the standard deviation of the posterior distribution. Now, the investor wants to represent the density of future returns, summarizing her knowledge. To do so, she could simply use the distribution of the returns, such as normal or lognormal, substituting her best ML estimate of the parameters, $\mathrm{p}\left(\mathrm{R}{\mathrm{T}+1} \mid \mu{\mathrm{MLE}}, \mathrm{V}{\mathrm{MLE}}\right)$. Decision theory shows that this is suboptimal. Instead, she must rely on the predictive density of the future returns, which averages out the uncertainty in the parameters. Formally, the predictive density of the asset returns for time $T+1$ is shown in Equation 2.15: $$P\left(R{T+1} \mid R\right)=\int p\left(R_{T+1} \mid R, \mu, V\right) p(\mu, V \mid R) d \mu d V,$$
where the integration is done on the range of the mean and variance parameters. The first term in the integral is the density of the future return, given the mean and variance. This is what the substitution approach uses, simply replacing the parameter with an estimate. However, it does not incorporate the fact that these estimates are uncertain. Therefore, it overstates the investor’s precision about the future returns. The second term is the posterior distribution of the parameters. It represents the knowledge on the parameters after observing the data.

## 金融代写|利率理论代写Portfolio Theory代考|Bayesian Portfolio Optimization with Diff use Priors

Klein and Bawa (1976) show that computing and then optimizing expected utility around the predictive density is the optimal strategy. The chief reason is that the mere substitution of point estimates of the parameters in the variance of a portfolio, in its CE or its Sharpe ratio, clearly omits the uncertainty about these estimates, which must be accounted for, especially by risk-averse investors. Bawa, Brown, and Klein (1979) incorporate parameter uncertainty into the optimal portfolio problem. They mostly use diffuse priors to compute the predictive density of the parameters and maximize expected utility for that predictive density.
For the case of $N$ assets, the main result is that the predictive density of returns has a larger variance than the sample estimate of $\mu$ and $\mathrm{V}$ suggests. In fact, it is larger by a factor $(1+1 / T)(T+1)(T-N-2)$. This factor modifies the optimal allocation, especially when $\mathrm{N}$ is sizable relative to $\mathrm{T}$. Relative to portfolios based on point estimates, Bayesian optimal portfolios take smaller positions on the assets with a higher risk. The term $(1+1 / T)$ is the correction due to the uncertainty in the mean. Consider, for example, the risky versus risk-free asset allocation. With a diffuse prior, the predictive density of the (single) future return is normal with mean $\mathrm{m}$ (the estimate) and variance $\mathrm{s}^{2}(1+1 / \mathrm{T})$, where $\mathrm{s}$ is the sample estimate. Intuitively, the future variance faced by the investor is the sum of the return’s variance given the mean $\mathrm{s}^{2}$ and the variance of the estimate $s^{2} / \mathrm{T}$. Computing the Merton allocation with respect to this predictive density of returns lowers the allocation on the tangency portfolio in Equation $2.3$ by the factor $1+1 / \mathrm{T}$.

# 利率理论代写

## 金融代写|利率理论代写Portfolio Theory代考|Brief Overview of the Bayesian Process for Stock Returns

$$P(R T+1 \mid R)=\int p\left(R_{T+1} \mid R, \mu, V\right) p(\mu, V \mid R) d \mu d V$$

## 金融代写|利率理论代写Portfolio Theory代考|Bayesian Portfolio Optimization with Diff use Priors

Klein 和 Bawa (1976) 表明，计算然后围绕预测密度优化预期效用是最佳策略。主要原因是仅仅用投诏组合方差㣍数的点估计青

Brown 和Klein (1979) 将参数不确定性纳入最优投资组合问题。他们主要使用扩散先验来计算参数的预则密度，并最大化该预测

$(1+1 / T)(T+1)(T-N-2)$. 这个因洯会修改最优分配，尤其是当 $\mathrm{N}$ 相对于 $\mathrm{T}$. 相对于基于点估计的投诏组合，贝叶斯最优 投资组合在风险较高的资产上持有较小的头寸。期限 $(1+1 / T)$ 是由于均值的不确定性而导致的修正。例如，考虑风险诏产配置

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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。