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## avatest™帮您通过考试

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## 金融代写|投资组合代写Portfolio Theory代考|ESTIMATING MODELS WITH EXCESS RETURNS

When excess returns $\left(R^e\right)$ are used to estimate and test asset pricing models, the moment conditions (pricing equations) are
$$E\left(m R^e\right)=0_N \cdot$$
Let $m=\theta_0-\left(\theta_1 f_1+\cdots+\theta_K f_K\right)$. In this case, the mean of the SDF cannot be identified or, equivalently, the parameters $\theta_0$ and $\left(\theta_1, \ldots, \theta_K\right)$ cannot be identified separately. This requires a particular choice of normalization. One popular normalization is to set $\theta_0=1$, in which case $m=1-\left(\theta_1 f_1+\cdots+\theta_K f_K\right)$. An alternative (preferred) normalization is to set $\theta_0=1+\theta_1 E\left(f_1\right)+\cdots+\theta_K E\left(f_K\right)$, in which case $m=1-\theta_1\left[f_1-E\left(f_1\right)\right]-\cdots-\theta_K\left[f_K-E\left(f_K\right)\right]$ with $E(m)=1$. These two normalizations can give rise to very different results (see Kan and Robotti, 2008; Burnside, 2010).

Kan and Robotti (2008) argue that when the model is misspecified, the first (raw) and the second (de-meaned) normalizations of the SDF produce different GMM estimates that minimize the quadratic form of the pricing errors. Hence, the pricing errors and the $p$-values of the specification tests are not identical under these two normalizations. Moreover, the second (de-meaned) specification imposes the constraint $E(m)=1$ and, as a result, the pricing errors and the $\mathrm{HJ}$-distances are invariant to affine transformations of the factors. This is important because in the first normalization, the outcome of the model specification test can be easily manipulated by simple scaling of factors and changing the mean of the SDF. This problem is not only a characteristic of linear SDFs but also arises in nonlinear models. The analysis in Burnside (2010) further confirms these findings and links the properties of the different normalizations to possible model misspecification and identification problems discussed in the previous two subsections.

## 金融代写|投资组合代写Portfolio Theory代考|CONDITIONAL MODELS WITH HIGHLY PERSISTENT PREDICTORS

The usefulness of the conditional asset pricing models crucially depends on the existence of some predictive ability of the conditioning variables for future stock returns. While a large number of studies report statistically significant coefficients for various financial and macro variables in in-sample linear predictive regressions of stock returns, several papers raise the concern that some of these regressions may be spurious. For example, Ferson, Sarkissian, and Simin (2003) call into question the predictive power of some widely used predictors, such as the term spread, the book-to-market ratio, and the dividend yield. Spurious results arise when the predictors are strongly persistent (near unit root processes) and their innovations are highly correlated with the predictive regression errors. In this case, the estimated slope coefficients in the predictive regression are biased and have a nonstandard (nonnormal) asymptotic distribution (Elliott and Stock, 1994; Cavanagh, Elliott, and Stock, 1995; Stambaugh, 1999). As a result, $t$-tests for statistical significance of individual predictors based on standard normal critical values could reject the null hypothesis of no predictability too frequently and falsely signal that these predictors have predictive power for future stock returns. Campbell and Yogo (2006) and Torous, Valkanov, and Yan (2004) develop valid testing procedures when the predictors are highly persistent and revisit the evidence on the predictability of stock returns.

Spuriously significant results and nonstandard sampling distributions also tend to arise in long-horizon predictive regressions, where the regressors and/or the returns are accumulated over $r$ time periods so that two or more consecutive observations are overlapping. The time overlap increases the persistence of the variables and renders the sampling distribution theory of the slope coefficients, $t$-tests and $R^2$ coefficients, nonstandard. Campbell (2001) and Valkanov (2003) point out several problems that emerge in long-horizon regressions with highly persistent regressors. First, the $R^2$ coefficients and $t$-statistics tend to increase with the horizon, even under the null of no predictability, and the $R^2$ is an unreliable measure of goodness of fit in this situation. Furthermore, the $t$-statistics do not converge asymptotically to well-defined distributions and need to be rescaled to ensure valid inference. Finally, the estimates of the slope coefficients are biased and, in some cases, not consistently estimable. All these statistical problems provide a warning to applied researchers and indicate that the selection of conditioning variables for predicting stock returns should be performed with extreme caution.

## 金融代写|投资组合代写Portfolio Theory代考|ESTIMATING MODELS WITH EXCESS RETURNS

$$E\left(m R^e\right)=0_N \cdot$$

Kan和Robotti(2008)认为，当模型被错误指定时，SDF的第一次(原始)和第二次(去均值)归一化会产生不同的GMM估计，从而使定价误差的二次形式最小化。因此，在这两种归一化下，规范测试的定价误差和$p$ -值是不相同的。此外，第二个(去均值)规范施加了约束$E(m)=1$，因此，定价误差和$\mathrm{HJ}$ -距离对因子的仿射变换是不变的。这很重要，因为在第一个归一化中，模型规范测试的结果可以通过简单的因子缩放和改变SDF的平均值来轻松地操纵。这个问题不仅是线性sdf的一个特点，而且在非线性模型中也会出现。Burnside(2010)的分析进一步证实了这些发现，并将不同归一化的特性与前两小节中讨论的可能的模型错误规范和识别问题联系起来。

## 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。

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## 金融代写|投资组合代写Portfolio Theory代考|GMM Estimation and Evaluation of Asset Pricing Models in SDF Form

Using Equation 3.11, the pricing errors of the $N$ test assets can be expressed as
$$g(\theta)=E[m(1+R)]-1_N=E\left[(1+R) \tilde{f}^{\prime} \theta\right]-1_N=D \theta-1_N,$$
where $D=E\left[(1+R) \tilde{f}^{\prime}\right]$. Let $t=1,2, \ldots, T$ denote the number of time series observations on the test assets and the factors. The sample analog of the pricing errors is given by
$$g_T(\theta)=\frac{1}{T} \sum_{t=1}^T\left(1+R_t\right) \tilde{f}t^{\prime} \theta-1_N .$$ For a given weighting matrix $W_T$, the GMM estimator of $\theta$ minimizes the quadratic form $$g_T(\theta)^{\prime} W_T g_T(\theta)$$ and solves the first-order condition $$D_T^{\prime} W_T\left(D_T \theta-1_N\right)=0$$ where $D_T=\frac{\partial g_T(\theta)}{\partial \theta^{\prime}}=\frac{1}{T} \sum{t=1}^T\left(1+R_t\right) \tilde{f}_t^{\prime}$. Solving this system of linear equations for $\theta$ yields
$$\hat{\theta}=\left(D_T^{\prime} W_T D_T\right)^{-1}\left(D_T^{\prime} W_T 1_N\right)$$

## 金融代写|投资组合代写Portfolio Theory代考|Beta-Pricing Models and Two-Pass Cross-Sectional Regressions

From Equation 3.13, the expected-return errors of the $N$ assets are given by
$$e=E[R]-B \gamma .$$
A popular goodness-of-fit measure used in many empirical studies is the crosssectional $R^2$. Following Kandel and Stambaugh (1995), this is defined as
$$R^2=1-\frac{Q}{Q_0},$$
where $Q=e^{\prime} W e, Q_0=e_0^{\prime} W e_0, e_0=\left[I_N-1_N\left(1_N^{\prime} W 1_N\right)^{-1} 1_N^{\prime} W\right] E[R]$ represents the deviations of mean returns from their cross-sectional average and $W$ is a positive-definite weighting matrix. Popular choices of $W$ in the literature are $W=$ $I_N$ (ordinary least squares $[\mathrm{OLS}]$ ), $W=\operatorname{Var}[R]^{-1}$ (generalized least squares $[\mathrm{GLS}]$ ), and $W=\Sigma_d^{-1}$ (weighted least squares [WLS]), where $\Sigma_d$ is a diagonal matrix containing the diagonal elements of $\Sigma$, the variance-covariance matrix of the residuals from the first-pass time series regression. In order for $R^2$ to be well defined requires assuming that $E[R]$ is not proportional to $1_N$ (the expected returns are not all equal) so that $Q_0>0$. Note that $0<R^2<1$ and it is a decreasing function of the aggregate pricing-error measure $Q$. Thus, $R^2$ is a natural measure of goodness of fit.

As emphasized by Kan and Zhou (2004), $R^2$ is oriented toward expected returns whereas the HJ-distance evaluates a model’s ability to explain prices. With the zero-beta rate as a free parameter, the most common approach in the asset pricing literature, Kan and Zhou show that the two measures need not rank models the same way. Thus, both measures are of interest, with the choice depending on the economic context and perhaps the manner in which a researcher envisions applying the models.

## 金融代写|投资组合代写Portfolio Theory代考|GMM Estimation and Evaluation of Asset Pricing Models in SDF Form

$$g(\theta)=E[m(1+R)]-1_N=E\left[(1+R) \tilde{f}^{\prime} \theta\right]-1_N=D \theta-1_N,$$

$$g_T(\theta)=\frac{1}{T} \sum_{t=1}^T\left(1+R_t\right) \tilde{f}t^{\prime} \theta-1_N .$$对于给定的权重矩阵$W_T$, $\theta$的GMM估计量最小化了二次形式$$g_T(\theta)^{\prime} W_T g_T(\theta)$$并解决了一阶条件$$D_T^{\prime} W_T\left(D_T \theta-1_N\right)=0$$，其中$D_T=\frac{\partial g_T(\theta)}{\partial \theta^{\prime}}=\frac{1}{T} \sum{t=1}^T\left(1+R_t\right) \tilde{f}_t^{\prime}$。求解$\theta$产量的线性方程组
$$\hat{\theta}=\left(D_T^{\prime} W_T D_T\right)^{-1}\left(D_T^{\prime} W_T 1_N\right)$$

## 金融代写|投资组合代写Portfolio Theory代考|Beta-Pricing Models and Two-Pass Cross-Sectional Regressions

$$e=E[R]-B \gamma .$$

$$R^2=1-\frac{Q}{Q_0},$$

## 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。

Posted on Categories:Investment Portfolio, 投资组合, 金融代写

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## 金融代写|投资组合代写Portfolio Theory代考|THE EFFICIENT FRONTIER AND ASSET PRICING

The previous section detailed the various scenarios of portfolio optimization available to an investor. The chapter now adds the assumptions of homogeneous information about $\mu, V, R_f$, market efficiency, and frictionless and costless trading to show the implications for equilibrium of portfolio theory.

Recall the case with $N$ risky assets and a risk-free asset. This section now shows how the Sharpe-Lintner CAPM (Sharpe, 1964) follows directly. If all investors have the same information $\mu, V$, they all agree on the tangency portfolio, which is $T$ in Figure 2.3. In equilibrium, demand meets supply and this tangency portfolio must be the capitalization-weighted portfolio of all risky assets, also known as the market portfolio. Therefore, the cap-weighted market portfolio is the tangency portfolio on the efficient frontier. It is the mean-variance efficient portfolio because no other portfolio has a higher Sharpe ratio. This is the basis for indexing investment. The CAL defined by the market is called the capital market line (hereafter, the CML). It is the optimal CAL given the assumptions made at the beginning of the section.

Consider now a risk-free rate $R_f$ and a frontier of two risky assets, $i$ and $M$. Quadratic optimization shows that the portfolio with the maximum Sharpe ratio, the tangency portfolio, has a weight as shown in Equation 2.6:
$$w_i^=\frac{\left(\mu_i-R_f\right) \sigma_M^2-\left(\mu_M-R_f\right) \sigma_{i M}}{\left.\left(\mu_i-R_f\right) \sigma_M^2+\left(\mu_M-R_f\right) \sigma_i^2-\mu_i+\mu_M-2 R_f\right) \sigma_{i M}}$$ Let us apply this result to the context of equilibrium, with $M$ representing the market portfolio, and $i$ representing any security. In equilibrium, $M$ already contains $i$ in the optimal amount because it is already the mean-variance efficient, tangency portfolio of the frontier of all securities in the economy. Therefore, the weight $w_i^$ must be zero in equilibrium. Now set the numerator in Equation 2.6 to equal to zero. This immediately yields the well-known CAPM equation
$$\mu_i=R_f+\left(\mu_M-R_f\right) \beta_{i M}$$
where beta $\left(\beta_{i M}\right)$ is now considered with respect to the market portfolio. In Figure 2.4 , the solid lines show a two-asset frontier of the market and a security $P_1 \cdot M$ is the tangency portfolio of that frontier because the expected return of $P_1$ was set equal to the CAPM.

## 金融代写|投资组合代写Portfolio Theory代考|ACTIVE MANAGEMENT AND THE INFORMATION RATIO

This section discusses active management and portfolio performance evaluation. The best-known measure of performance, the Sharpe ratio, discussed in the previous sections, is only valid to rank mutually exclusive investments. The Sharpe ratio does not indicate how to optimally combine competing funds.

The previous section explains how in equilibrium, the capitalization-weighted market portfolio $\mathrm{M}$ achieves the best Sharpe ratio. In the active asset allocation framework, the manager identifies securities that may help improve upon the market portfolio’s Sharpe ratio. This section introduces the information ratio, widely used in quantitative active asset management, which indicates how a security contributes to the Sharpe ratio of a portfolio. The reasoning will parallel the Sharpe-Lintner CAPM proof seen above, incorporating the fact that the expected returns of some securities differ from the CAPM prediction and therefore will improve upon the Sharpe ratio of the market. Departures from the CAPM are modeled via Jensen’s (1968) apha, $a$, as shown in Equation 2.9:
$$\mathrm{E}\left(\mathrm{R}{\mathrm{i}}\right)=\alpha_1+\mathrm{R}_f+\beta_i \mathrm{E}\left(\mathrm{R}{\mathrm{M}}-\mathrm{R}{\mathrm{f}}\right)$$ Equation 2.9 nests the CAPM, in which case $a$ is 0 . To estimate alpha and beta, Jensen runs the time series regression shown in Equation 2.10: $$R{i t}-R_{f t}=\alpha_i+\beta_i\left(R_{M t}-R_{f t}\right)+\varepsilon_{i t},$$
where $R_{i t}$ is the return on the asset $\mathrm{i} ; R_{f t}$ is the risk-free rate; $R_{M t}$ is the market index return; and $\varepsilon$ is the random error of the regression, also known as the unsystematic or idiosyncratic return. The regression in Equation 2.10 also estimates the standard deviation of the idiosyncratic return $\sigma_{\varepsilon}$. In fact, it performs the variance decomposition for security $i$, shown in Equation 2.11:
$$\sigma_i^2=\beta_i^2 \sigma_M^2+\sigma_{\varepsilon, i}^2$$

## 金融代写|投资组合代写Portfolio Theory代考|THE EFFICIENT FRONTIER AND ASSET PRICING

$$w_i^=\frac{\left(\mu_i-R_f\right) \sigma_M^2-\left(\mu_M-R_f\right) \sigma_{i M}}{\left.\left(\mu_i-R_f\right) \sigma_M^2+\left(\mu_M-R_f\right) \sigma_i^2-\mu_i+\mu_M-2 R_f\right) \sigma_{i M}}$$让我们把这个结果应用到均衡的背景下，$M$代表市场投资组合，$i$代表任何证券。在均衡中，$M$已经包含了最优数量的$i$，因为它已经是经济中所有证券边界的均值-方差有效切线投资组合。因此，在平衡状态下，权重$w_i^$必须为零。现在将方程2.6中的分子设为0。这立即产生了著名的CAPM方程
$$\mu_i=R_f+\left(\mu_M-R_f\right) \beta_{i M}$$

## 金融代写|投资组合代写Portfolio Theory代考|ACTIVE MANAGEMENT AND THE INFORMATION RATIO

$$\mathrm{E}\left(\mathrm{R}{\mathrm{i}}\right)=\alpha_1+\mathrm{R}f+\beta_i \mathrm{E}\left(\mathrm{R}{\mathrm{M}}-\mathrm{R}{\mathrm{f}}\right)$$公式2.9嵌套CAPM，此时$a$为0。为了估计alpha和beta, Jensen运行了公式2.10所示的时间序列回归:$$R{i t}-R{f t}=\alpha_i+\beta_i\left(R_{M t}-R_{f t}\right)+\varepsilon_{i t},$$

$$\sigma_i^2=\beta_i^2 \sigma_M^2+\sigma_{\varepsilon, i}^2$$

## 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。

Posted on Categories:Investment Portfolio, 投资组合, 金融代写

## avatest™帮您通过考试

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## 金融代写|投资组合代写Portfolio Theory代考|INFINITE PAYOFFS, LIMITED COSTS-THE ST. PETERSBURG PARADOX

In a previous chapter, we briefly discussed Daniel Bernoulli’s critique of expected value theory. As a counterargument, Bernoulli proposed a gamble: An individual begins with an initial stake of $\$ 2$and we flip a coin. The initial$\$2$ bankroll is doubled every time heads comes up, and when tails comes up the game is over and the winnings are the value of the bankroll. What is interesting about this game is that it has an infinite expected value:
$$\sum_{n=1}^{\infty} \operatorname{Pr}i[\text { Heads }] \times V_i$$ where $\sum{n=1}^{\infty} \operatorname{Pr}i[$ Heads $]$ is the probability of getting heads on the $i^{\text {th }}$ coin flip, and $V_i$ is the payoff of that flip. The probability of winning the first flip is $1 / 2^1$ and the payoff is $2^1$. The probability of winning the second flip is $1 / 2^2$ and the payoff is $2^2$. We can generalize the expected value of the gamble as $$\sum{n=1}^{\infty} \frac{1}{2^i} \times 2^i$$
And it becomes clear that the sum is infinity:
$$\frac{1}{2^1} \times 2^1+\frac{1}{2^2} \times 2^2+\frac{1}{2^3} \times 2^3+\cdots=1+1+1+\cdots=\infty \text {. }$$

## 金融代写|投资组合代写Portfolio Theory代考|QUESTIONS OF PROBABILITY, QUESTIONS OF DOUBT

In 1953, as a critique of Von Neumann and Morgenstern’s axiom of independence, economist Maurice Allais ${ }^8$ conducted an experiment. He offered his subjects two sets of choices:
1A: $\$ 1,000,000$with$100 \%$probability 1B:$\$1,000,000$ with $89 \%$ probability $\$ 5,000,000$with$10 \%$probability$\$0$ with $1 \%$ probability
The second choice set:
2A: $\$ 1,000,000$with$11 \%$probability$\$0$ with $89 \%$ probability
$2 \mathrm{~B}: \quad \$ 5,000,000$with$10 \%$probability$\$0$ with $90 \%$ probability
Allais found that people tended to choose choice 1A over 1B and also chose $2 \mathrm{~B}$ over $2 \mathrm{~A}$. Though it is not immediately obvious, this is a contradiction!
From the first choice set we learn that $u(1 A)>u(1 B)$, or
$$v(\ 1,000,000)>0.89 v(\ 1,000,000)+0.10 v(\ 5,000,000),$$
which can be simplified to
$$0.11 v(\ 1,000,000)>0.10 v(\ 5,000,000) .$$
In other words, we learn that an $11 \%$ chance of gaining $\$ 1,000,000$carries more utility than a$10 \%$chance of gaining$\$5,000,000$. We learn from the second choice set that $u(2 A)<u(2 B)$, or
$$0.11 v(\ 1,000,000)<0.10 v(\ 5,000,000)$$
which directly contradicts the first choice set!
Allais concluded that the axiom of independence cannot be a valid one because it fails to predict “reasonable people choosing between reasonable alternatives.” Markowitz rebutted that people choosing the wrong alternative acted irrationally, but that people are irrational does not negate the axiom. So the behavioral-normative split was formed.

## 金融代写|投资组合代写Portfolio Theory代考|INFINITE PAYOFFS, LIMITED COSTS-THE ST. PETERSBURG PARADOX

$$\sum_{n=1}^{\infty} \operatorname{Pr}i[\text { Heads }] \times V_i$$其中$\sum{n=1}^{\infty} \operatorname{Pr}i[$头像$]$是$i^{\text {th }}$抛硬币得到头像的概率，$V_i$是抛硬币的收益。第一次抛掷获胜的概率是$1 / 2^1$，收益是$2^1$。第二次抛掷获胜的概率是$1 / 2^2$，收益是$2^2$。我们可以将赌博的期望值概括为$$\sum{n=1}^{\infty} \frac{1}{2^i} \times 2^i$$

$$\frac{1}{2^1} \times 2^1+\frac{1}{2^2} \times 2^2+\frac{1}{2^3} \times 2^3+\cdots=1+1+1+\cdots=\infty \text {. }$$

## 金融代写|投资组合代写Portfolio Theory代考|QUESTIONS OF PROBABILITY, QUESTIONS OF DOUBT

1953年，作为对冯·诺伊曼和摩根斯特恩的独立公理的批判，经济学家莫里斯·阿莱${ }^8$进行了一项实验。他给实验对象提供了两组选择:
1A: $\$ 1,000,000$与$100 \%$的概率 1B:$\$1,000,000$带$89 \%$概率$\$ 5,000,000$带$10 \%$概率$\$0$带$1 \%$概率

2A: $\$ 1,000,000$带$11 \%$概率$\$0$带$89 \%$概率
$2 \mathrm{~B}: \quad \$ 5,000,000$与$10 \%$概率$\$0$与$90 \%$概率

$$v(\ 1,000,000)>0.89 v(\ 1,000,000)+0.10 v(\ 5,000,000),$$

$$0.11 v(\ 1,000,000)>0.10 v(\ 5,000,000) .$$

$$0.11 v(\ 1,000,000)<0.10 v(\ 5,000,000)$$

## 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。

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## 金融代写|投资组合代写Portfolio Theory代考|Goals-Based Reporting

I remember, early in my career, I was excited to show a client what a good job we had done in the recovery post-2008. Through some effective stock picks, we had generated substantial alpha relative to the market as a whole. During the meeting, I pointed out the alpha figure and, much to my disappointment, my client responded, “What does that mean?” Unfortunately, rather than take the hint, I dove headlong into an explanation, equations and all, of how important alpha was as a risk-adjusted measure of returns. It did not help. My client did not care about his risk-adjusted performance relative to a benchmark. My client cared about achieving his goals! Alpha, to him, was a meaningless statistic.
So much of our current client reporting paradigm is consistent with the quip at the top of the chapter. We, as an industry, spend inordinate time and ink presenting metrics and data that we care about, and quite little on what it is the client cares about. Financial plans are a staple, of course, and that is good. But when it comes to monthly or quarterly performance reporting, the financial plan is drawered, and the meaningless metrics come back in force.
I want to explore how we might update our reporting to be consistent with a goals-based framework, to move away from the myriad meaningless metrics toward the metrics real people actually care about. I do not claim that this is the only way reporting should be done; by contrast, I am a bit of a neophyte when it comes to the challenges of client reporting. Like most practitioners, I rely on my technology services to generate and deliver reports without much second thought. This discussion is my attempt at a second thought on the matter, and it is my sincere hope that this discussion prompts others to add their wisdom and experience to the conversation. Hopefully, then, the industry can coalesce around a new and more meaningful norm for client reporting.
Rather than walk through the deficiencies of existing reports and how we might fix them, let’s begin this discussion with a blank slate. Given the goals-based framework, what information is relevant and meaningful to the investor? As we have discussed, goals-based investing sits at the intersection of the “big world” and the investor’s world. It stands to reason, then, that we need some reporting on both. First and foremost, an investor cares about her world-the world of her dreams, wants, wishes, and needs. Is the portfolio manager helping her achieve those objectives or not? But those objectives can only be achieved with the help from the big world of capital markets because the big world represents the opportunity set from which our investor can draw. If, for instance, the opportunity set only offered below-average returns, it is important our investor understands that the firm’s sub-par performance relative to her need is not necessarily due to insufficient skill on the part of the portfolio manager, but rather due to a poor opportunity set (and the inverse is also true, so this cuts both ways!). Good goals-based reporting should strike the right balance between the two (Figure 9.1 ).

## 金融代写|投资组合代写Portfolio Theory代考|Fragility Analysis of Goals-Based Inputs

I really had no way of effectively answering this question, so I sat on it for years-until I came upon Nassim Taleb’s theory of fragility. Most practitioners are familiar with Taleb’s iconoclastic style and more laymen-oriented work, such as The Black Swan. His book Antifragility is a must-read, but it was his technical work on the topic of system fragility that triggered a possible path to answer this latent question of mine. In 2012, with some coauthors, Taleb published a paper for the International Monetary Fund offering a simple heuristic for detecting fragility in any modellable system. ${ }^1$ These ideas were later expanded into more technical definitions, ${ }^2$ but the heuristic is sufficient for our needs.
In a nutshell, Taleb argues that linear increases in event significance often result in exponential increases in harm to the system. As an example: running a stop sign at 15 miles per hour is a fender-bender-likely a few thousand dollars in damage. Running a stop sign at 45 miles per hour is a hospital visit-likely tens of thousands of dollars in bodily damage, not to mention the possible loss of life. A $3 \mathrm{x}$ increase in event size (speed) resulted in a $10 x$ to $20 \mathrm{x}$ increase in harm (financial cost and bodily risk). In derivatives trading, this is known as gamma risk, or convexity.
Taleb uses this starting point to suggest a simple rule. Take a model of the system. Perturb the inputs, one at a time, equally to the upside and downside. Take the average of the output and subtract it from the baseline. If the result is negative, then negative convexity is present and the system is fragile with respect to that input. If the result is positive, then positive convexity is present and the system is antifragile (that is, it gains from extreme movements). If the result is 0 , then the system is robust (immune to movements). What is more exciting is that the accuracy of the system model is of secondary importance. As Taleb and Douady put it, “A wrong ruler will not measure the height of a child, but it will certainly tell us if he is growing.” It is not the absolute result with which we are concerned-that is, we are not concerned whether 8 comes out if 4.5 goes in. Rather, we are concerned with the drama associated with the changes-if 8 comes out when 4.5 goes in and 25 comes out when 5 goes in, we have some ground to say that getting this input right is very important.
More formally: Let $f(\cdot)$ be your model of a system and let $\alpha$ be an input to your model (since we are perturbing the model one variable at a time, $\alpha$ represents any input to the model). From here, we perturb $f(\cdot)$ by $\pm \Delta$, or some constant amount, and subtract the baseline from the average result:
$$\Xi=\frac{f(\alpha-\Delta)+f(\alpha+\Delta)}{2}-f(\alpha)$$

## 金融代写|投资组合代写Portfolio Theory代考|Fragility Analysis of Goals-Based Inputs

$$\Xi=\frac{f(\alpha-\Delta)+f(\alpha+\Delta)}{2}-f(\alpha)$$

## 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。

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## 金融代写|投资组合代写Portfolio Theory代考|Allocating Wealth Across Goals and Across Investments

Granularity is an obvious question at this stage-more granularity is better, but we can quickly run into unreasonable computation times and memory needs. If $\varrho$ is our level of resolution $(\varrho=100$ for $1 \%$ intervals, $\varrho=20$ for $5 \%$ intervals, $\varrho=10$ for $10 \%$ intervals, etc.) and $N$ is the number of goals, then we must calculate and hold in memory $\varrho \times N$ portfolios. Five goals run with a resolution of 100 yields $5 \times 100=500$ portfolio calculations that must be made and held in memory. In the end, the practitioner has to make this decision based on the application at hand and the computational resources available. It may well be that a resolution of 20 or 10 is sufficient for a particular application. I do not believe there is one right answer here.

Once we have generated optimal portfolios for each goal given each potential level of wealth, we use the results of these optimal investment allocations to inform the optimal across-goal allocation. Because of the discrete nature of our portfolio allocation results, I recommend using a Monte Carlo engine to simulate various across-goal allocations and their effects on total utility. Obviously, we are trying to find the across-goal allocation that yields the highest utility. For each simulation of across-goal allocation, we match the optimal portfolio for that level of across-goal allocation and return a probability of achievement. That probability is the input used in the utility function.

Finally, we match the optimal across-goal allocation with the optimal within-goal portfolio weights and return the optimal aggregate portfolio (or keep them separate, whichever the implementation strategy demands).

That is the procedure summary. Now, let’s tackle the first-stage optimization algorithm.
Define:
investment universe of $k$-number of potential investments.
necessary level of resolution, $\log$ this as $\varrho .^2$
$\pi(\omega)$ returns the parameters of our chosen cdf, given portfolio weights, $\omega .{ }^4$ parameters

## 金融代写|投资组合代写Portfolio Theory代考|CASE STUDY

Let’s consider a case study together to help illustrate the procedure. For interested readers, I have included the relevant $\mathrm{R}$ code script as part of the book supplement. Again, I want to stress that the actual procedure may not be optimal from a computational perspective-I would encourage other practitioners and researchers to develop their own approach-but it suffices for my purposes.

A client joins our firm. Our first step as advisors is to spend ample time in conversation. We need to fully understand her goals, her tax situation, her ethical constraints (more on that later), and so on. We must also ensure that she has reasonable expectations, both of herself and us. We should never take on a client with unreasonable expectations or a client mandate that is not within our ability. This conversation is, then, a two-way street, determining whether this client will fit within our process as well as for the client to get a handle on her financial goals and financial picture.

Another objective at this stage is to help the client dream a little. I have found that, very often, clients do not have a clear picture of their goals. One of an advisor’s jobs is to help the client crystalize her objectives. They are changeable, of course, and communicating that point is important, as well; clients will have a harder time committing to 30 -year objectives that they feel can never be updated. This involves plenty of listening as clients talk themselves through their needs, wants, wishes, and dreams. We need to spend time forecasting our client’s psychological state (“how would you feel if. ..”), as well as forecasting their financial state (“what is your pattern of raises…”). Client homework is not uncommon after the first meeting or two.

After ample conversation, we determine that our new client has the following goals in her goals-space:

• She wants to leave an estate to her children of $\$ 5,000,000$, planned in 30 years from now. • She needs to maintain her lifestyle expenses starting in 10 years, and we estimate that she will need$\$5,157,000$ to do that.
• Our client would like to purchase a vacation home in 4 years, and her estimated price is $\$ 713,500$. • If possible, our client would like to donate$\$8,812,000$ to her alma mater sometime around 18 years from now. This donation carries naming rights to a building on campus.

## 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。

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## 金融代写|投资组合代写Portfolio Theory代考|Portfolio Selection

This chapter is an introduction to the theory of portfolio selection, which together with capital asset pricing theory provides the foundation and the building blocks for the management of portfolios. The goal of portfolio selection is the construction of portfolios that maximize expected returns consistent with individually acceptable levels of risk. Using both historical data and investor expectations of future returns, portfolio selection uses modeling techniques to quantify “expected portfolio returns” and “acceptable levels of portfolio risk” and provides methods to select an optimal portfolio.

The theory of portfolio selection presented in this chapter, often referred to as mean-variance portfolio analysis or simply mean-variance analysis, is a normative theory. A normative theory is one that describes a standard or norm of behavior that investors should pursue in constructing a portfolio rather than a prediction concerning actual behavior.

Asset pricing theory goes on to formalize the relationship that should exist between asset returns and risk if investors behave in a hypothesized manner. In contrast to a normative theory, asset pricing theory is a positive theory-a theory that hypothesizes how investors behave rather than how investors should behave. Based on that hypothesized behavior of investors, a model that provides the expected return (a key input for constructing portfolios based on mean-variance analysis) is derived and is called an asset pricing model.

Together, portfolio selection theory and asset pricing theory provide a framework to specify and measure investment risk and to develop relationships between expected asset return and risk (and hence between risk and required return on an investment). However, it is critically important to understand that portfolio selection is a theory that is independent of any theories about asset pricing. The validity of portfolio selection theory does not rest on the validity of asset pricing theory.

It would not be an overstatement to say that modern portfolio theory has revolutionized the world of investment management. Allowing managers to quantify the investment risk and expected return of a portfolio has provided the scientific and objective complement to the subjective art of investment management. More importantly, whereas at one time the focus of portfolio management used to be the risk of individual assets, the theory of portfolio selection has shifted the focus to the risk of the entire portfolio. This theory shows that it is possible to combine risky assets and produce a portfolio whose expected return reflects its components, but with considerably lower risk. In other words, it is possible to construct a portfolio whose risk is smaller than the sum of all its individual parts!

## 金融代写|投资组合代写Portfolio Theory代考|Utility Function and Indifference Curves

There are many situations where entities (i.e., individuals and firms) face two or more choices. The economic “theory of choice” uses the concept of a utility function to describe the way entities make decisions when faced with a set of choices. A utility function assigns a (numeric) value to all possible choices faced by the entity. The higher the value of a particular choice, the greater the utility derived from that choice. The choice that is selected is the one that results in the maximum utility given a set of constraints faced by the entity.

In portfolio theory too, entities are faced with a set of choices. Different portfolios have different levels of expected return and risk. Typically, the higher the level of expected return, the larger the risk. Entities are faced with the decision of choosing a portfolio from the set of all possible risk-return combinations, where when they like return, they dislike risk. Therefore, entities obtain different levels of utility from different risk-return combinations. The utility obtained from any possible risk-return combination is expressed by the utility function. Put simply, the utility function expresses the preferences of entities over perceived risk and expected return combinations.
A utility function can be expressed in graphical form by a set of indifference curves. Exhibit $3.1$ shows indifference curves labeled $u_1, u_2$, and $u_3$. By convention, the horizontal axis measures risk and the vertical axis measures expected return. Each curve represents a set of portfolios with different combinations of risk and return. All the points on a given indifference curve indicate combinations of risk and expected return that will give the same level of utility to a given investor. For example, on utility curve $u_1$, there are two points $u$ and $u^{\prime}$, with $u$ having a higher expected return than $u^{\prime}$, but also having a higher risk. Because the two points lie on the same indifference curve, the investor has an equal preference for (or is indifferent to) the two points, or, for that matter, any point on the curve. The (positive) slope of an indifference curve reflects the fact that, to obtain the same level of utility, the investor requires a higher expected return in order to accept higher risk.

## 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。

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## 金融代写|投资组合代写Portfolio Theory代考|CAP-M and diversification

For any portfolio $P$
\begin{aligned} \sigma^2(P) & =w^{\prime} \Sigma w \ & =w^{\prime}\left(\underline{\beta} \underline{\beta^{\prime}} \sigma^2(R)+\Sigma_e\right) w \ & =\left(\sum_{i=1}^n w_i \beta_i\right)^2 \sigma^2(R)+\sum_{i=1}^n w_i^2 \sigma^2\left(e_i\right) \end{aligned}
Now let $w_i=\frac{1}{n}$. Then
$$\sigma^2(P)=\bar{\beta}^2 \sigma^2(R)+\frac{1}{n} \overline{\sigma^2(e .)}$$
and so
$$\sigma(P) \longrightarrow \bar{\beta} \sigma(R) .$$
This reaffirms that $\beta_i$ is a measure of the contribution of the $i^{\text {th }}$ security to the risk of the portfolio. $\beta_i \sigma(R)$ is called the market, or undiversifiable, risk of security $i . \sigma\left(e_i\right)$ is called the non-market risk, unsystematic risk, unique risk or residual risk of equity $i$. This risk is diversifiable.

## 金融代写|投资组合代写Portfolio Theory代考|The Sharpe-Lintner-Mossin CAP-M

The CAP-M is what is known as an equilibrium model. The market participants as a whole act to put the market into equilibrium.

A number of additional simplifying assumptions (over and above those of Markowitz) are made in the CAP-M which are thought to be not too far removed from reality, yet are useful in order to simplify (or even make possible) the derivation of the model. Of course, a set of such assumptions is necessary in any economic model. In this model, they are:

1. Short sales are allowed.
2. There is a risk free rate for lending and borrowing money. The rate is the same for lending and borrowing, and investors have any amount of credit.
3. There are no transaction costs in the buying and selling of capital assets.
4. Similarly, there are no income or capital gains taxes.
5. The market consists of all assets. (No assets are exclusively private property.)

## 金融代写|投资组合代写Portfolio Theory代考|CAP-M and diversification

$$\sigma^2(P)=\bar{\beta}^2 \sigma^2(R)+\frac{1}{n} \overline{\sigma^2(e .)}$$

$$\sigma(P) \rightarrow \bar{\beta} \sigma(R) .$$

## 金融代写|投资组合代写Portfolio Theory代考|The Sharpe-Lintner-Mossin CAPM

1. 允许卖空。
2. 资本咨的买卖没有交易成本。
3. 同样，也设有所得祝或洛本利得㙂。
4. 市场由所有咨产组成。 (没有资旁完全是私有财产）

## 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。

Posted on Categories:Investment Portfolio, 投资组合, 金融代写

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## 金融代写|投资组合代写Portfolio Theory代考|The expected return and risk of a portfolio of assets

Suppose we have a portfolio with $n$ assets, the $i^{t h}$ of which delivers a return $R_{t, i}$ at time $t$. This return has a mean $\mu_{t, i}$ and a variance $\sigma_{t, i}^2$. Suppose the proportion of the value of the portfolio that asset $i$ makes up is $w_i\left(\right.$ so $\sum_{i=1}^n w_i=1$ ).

What is the mean and standard deviation of the return $R$ of the portfolio? All known values are assumed to be known at time $t$, and the $t$ will be implicit in what follows. We can suppress the subscript $t$ as long as we understand that all of the parameters are dynamic and we need to refresh the estimates on a daily basis.
$$\mu:=\mathbb{E}[R]=\mathbb{E}\left[\sum_{i=1}^n w_i R_i\right]=\sum_{i=1}^n w_i \mathbb{E}\left[R_i\right]=\sum_{i=1}^n w_i \mu_i$$
and
ance matrix. So, the return on the portfolio has
\begin{aligned} & \mathbb{E}[R]=w^{\prime} \mu \ & \sigma(R)=\sqrt{w^{\prime} \Sigma w} \end{aligned}

## 金融代写|投资组合代写Portfolio Theory代考|The benefits of diversification

Let us consider some special cases. Suppose the assets are all independent, in particular, they are uncorrelated, so $\rho_{i j}=\delta_{i j}$. ( $\delta_{i j}$ is the indicator function.) Then $\sigma^2(R)=\sum_{i=1}^n w_i^2 \sigma_i^2$. Suppose further that the portfolio is equally weighted, so $w_i=\frac{1}{n}$ for every $i$. Then
$$\sigma^2(R)=\sum_{i=1}^n \frac{1}{n^2} \sigma_i^2=\frac{1}{n} \sum_{i=1}^n \frac{\sigma_i^2}{n} \longrightarrow 0$$
as $n \longrightarrow \infty$. If we accept that variance is a measure of risk, then the risk goes to 0 as we obtain more and more assets.

Suppose now that the portfolio is equally weighted, but that the assets are not necessarily uncorrelated. Then
\begin{aligned} \sigma^2(R) & =\sum_{i=1}^n \sum_{j=1}^n \frac{1}{n^2} \sigma_{i j} \ & =\frac{1}{n} \sum_{i=1}^n \frac{\sigma_i^2}{n}+\frac{n-1}{n} \sum_{i=1}^n \sum_{j=1, j \neq i}^n \frac{\sigma_{i j}}{n(n-1)} \ & =\frac{1}{n} \overline{\sigma_i^2}+\frac{n-1}{n} \overline{\sigma_{i j, i \neq j}} \ & \longrightarrow \frac{\sigma_{i j, i \neq j}}{\longrightarrow} \text { as } \longrightarrow \infty \end{aligned}
The limit is the average covariance, which is a measure of the undiversifiable market risk.

## 金融代写|投资组合代写Portfolio Theory代考|The expected return and risk of a portfolio of assets

$$\mu:=\mathbb{E}[R]=\mathbb{E}\left[\sum_{i=1}^n w_i R_i\right]=\sum_{i=1}^n w_i \mathbb{E}\left[R_i\right]=\sum_{i=1}^n w_i \mu_i$$

$$\mathbb{E}[R]=w^{\prime} \mu \quad \sigma(R)=\sqrt{w^{\prime} \Sigma w}$$

## 金融代写|投资组合代写Portfolio Theory代考|The benefits of diversification

$$\sigma^2(R)=\sum_{i=1}^n \frac{1}{n^2} \sigma_i^2=\frac{1}{n} \sum_{i=1}^n \frac{\sigma_i^2}{n} \longrightarrow 0$$

$$\sigma^2(R)=\sum_{i=1}^n \sum_{j=1}^n \frac{1}{n^2} \sigma_{i j} \quad=\frac{1}{n} \sum_{i=1}^n \frac{\sigma_i^2}{n}+\frac{n-1}{n} \sum_{i=1}^n \sum_{j=1, j \neq i}^n \frac{\sigma_{i j}}{n(n-1)}=\frac{1}{n} \overline{\sigma_i^2}+\frac{n-1}{n} \overline{\sigma_{i j, i \neq j}} \quad \longrightarrow \frac{\sigma_{i j, i \neq j}}{\longrightarrow} \text { as } \longrightarrow \infty$$

## 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。

Posted on Categories:Investment Portfolio, 投资组合, 金融代写

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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## 金融代写|投资组合代写Investment Portfolio代考|INVESTMENT COMPANIES

Investment companies (ICs) include open-end mutual funds, closed-end funds, and unit trusts. Shares in ICs are sold to the public and the proceeds invested in a diversified portfolio of securities. There is a wide range of ICs that invest in different asset classes and with different investment objectives. ICs can be actively managed or passively managed versus an index. Actively managed ICs offer investors to generate alpha returns as well as beta returns. Passively managed ICs offer only beta returns.
Types of Investment Companies
There are two primary types of ICs: open-end funds and closed-end funds.
Open-End Funds (Mutual Funds)
Open-end funds, commonly referred to simply as mutual funds (MFs), are portfolios of securities, mainly stocks, bonds, and money market instruments. There are several important aspects of MFs. First, investors in MFs own a pro rata share of the overall portfolio. Second, the investment manager of the MF manages the portfolio, that is, buys some securities and sells others.

Third, the value or price of each share of the portfolio, called the net asset value (NAV), equals the market value of the portfolio minus the liabilities of the MF divided by the number of shares owned by the MF investors. That is,
$$\text { NAV }=\frac{\text { Market value of portfolio }-\text { Liabilities }}{\text { Number of shares outstanding }}$$
For example, suppose that a MF with 20 million shares outstanding has a portfolio with a market value of $\$ 315$million and liabilities of$\$15$ million. The NAV is
$$\mathrm{NAV}=\frac{\ 315,000,000-\ 15,000,000}{20,000,000}=\ 15$$

## 金融代写|投资组合代写Investment Portfolio代考|Closed-End Funds

The shares of a closed-end fund $(\mathrm{CEF})$ are very similar to the shares of common stock of a corporation. The new shares of a CEF are initially issued by an underwriter for the fund. And after the new issue, the number of shares remains constant. This is the reason such a fund is called a “closed-end” fund. After the initial issue, there are no sales or purchases of fund shares by the fund company as there are for MFs. The shares are traded on a secondary market, either on an exchange or in the over-the-counter market.

The NAV of CEFs is calculated in the same way as for open-end funds. However, the price of a share in a CEF is determined by supply and demand, so the price can fall below or rise above the net asset value per share. Shares selling below NAV are said to be “trading at a discount,” while shares trading above NAV are “trading at a premium.” Newspapers list quotations of the prices of these shares under the heading “Closed-End Funds.” Some sources also list the NAV and the discount or premium of the shares.

Consequently, there are two important differences between MFs and CEFs. First, the number of shares of a MF varies because the fund sponsor will sell new shares to investors and buy existing shares from shareholders. Second, by doing so, the share price is always the NAV of the fund. In contrast, CEFs have a constant number of shares outstanding because the fund sponsor does not redeem shares and sell new shares to investors (except at the time of a new underwriting). Thus, the price of the fund shares will be determined by supply and demand in the market and may be above or below NAV, as discussed above.

Under the Investment Company Act of 1940, CEFs are capitalized only once. They make an initial public offering (IPO) and then their shares are traded on the secondary market, just like any common stock. The number of shares is fixed at the IPO; CEFs cannot issue more shares. Since CEFs are traded like stocks, the cost to any investor of buying or selling a $\mathrm{CEF}$ is the same as that of a stock. The obvious charge is the stock broker’s commission. The bid-offer spread of the market on which the stock is traded is also a cost.
Exhibit $2.4$ summarizes the differences between MFs and CEFs.

## 金融代写|投资组合代写Investment Portfolio代考|INVESTMENT COMPANIES

$$\text { NAV }=\frac{\text { Market value of portfolio }-\text { Liabilities }}{\text { Number of shares outstanding }}$$

$$\mathrm{NAV}=\frac{\ 315,000,000-\ 15,000,000}{20,000,000}=\ 15$$

## 金融代写|投资组合代写Investment Portfolio代考|Closed-End Funds

$\mathrm{CEF}$ 的资产净值计算方式与开放式其金相同。然而，CEF 的股票价格由供求关系决定，因此价格可能低于或高于每股净资产值。

## 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。