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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|风险和利率理论代写Market Risk, Measures and Portfolio Theory代考|Return

From the above example we see that the risk can be reduced by diversification. In this section we discuss how to minimise risk when investing in two stocks.

Suppose that we buy $x_1$ shares of stock $S_1$ and $x_2$ shares of stock $S_2$. The initial value of this portfolio is
$$V_{\left(x_1, x_2\right)}(0)=x_1 S_1(0)+x_2 S_2(0) .$$
When we design a portfolio, usually its initial value is the starting point of our considerations and it is given. The decision on the number of shares in each asset will follow from the decision on the division of our wealth, which is our primary concern and is expressed by means of the weights defined by
$$w_1=\frac{x_1 S_1(0)}{V_{\left(x_1, x_2\right)}(0)}, \quad w_2=\frac{x_2 S_2(0)}{V_{\left(x_1, x_2\right)}(0)} .$$
If the initial wealth $V(0)$ and the weights $w_1, w_2, w_1+w_2=1$, are given, then the funds allocated to a particular stock are $w_1 V(0), w_2 V(0)$, respectively, and the numbers of shares we buy are
$$x_1=\frac{w_1 V(0)}{S_1(0)}, \quad x_2=\frac{w_2 V(0)}{S_2(0)} .$$
At the end of the period the securities prices change, which gives the final value of the portfolio as a random variable
$$V_{\left(x_1, x_2\right)}(1)=x_1 S_1(1)+x_2 S_2(1) .$$
To express the return on a portfolio we employ the weights rather than the numbers of shares since this is more convenient.

The return on the investment in two assets depends on the method of allocation of the funds (the weights) and the corresponding returns. The vector of weights will be denoted by $\mathbf{w}=\left(w_1, w_2\right)$, or in matrix notation
$$\mathbf{w}=\left[\begin{array}{c} w_1 \ w_2 \end{array}\right],$$
and the return of the corresponding portfolio by $K_{\mathrm{w}}$.

## 金融代写|风险和利率理论代写Market Risk, Measures and Portfolio Theory代考|Attainable set

Finding the risk of a portfolio requires, apart from the risks of the components and the weights, some knowledge about their statistical relationship.
Recall from [PF] the notion of covariance of two random variables, $X, Y$ :
$$\operatorname{Cov}(X, Y)=\mathbb{E}[(X-\mathbb{E}(X))(Y-\mathbb{E}(Y)]=\mathbb{E}(X Y)-\mathbb{E}(X) \mathbb{E}(Y),$$
with $\operatorname{Cov}(X, X)=\operatorname{Var}(X)=\sigma_X^2$ in particular. Applying the Schwarz inequality ([PF, Lemma 3.49]) to $X-\mathbb{E}(X)$ and $Y-\mathbb{E}(Y)$ we obtain
$$|\operatorname{Cov}(X, Y)| \leq \sigma_X \sigma_Y .$$
This leads immediately to an inequality, that we leave as an exercise.
Exercise 2.2 Suppose that random variables $X, Y$ have finite variances. Show that $\sigma_{X+Y} \leq \sigma_X+\sigma_Y$.

Let us introduce the following notation for the covariance of the returns on the stocks $S_1, S_2$ :

$$\sigma_{i j}=\operatorname{Cov}\left(K_i, K_j\right),$$
for $i, j=1,2$. In particular,
\begin{aligned} &\sigma_{11}=\operatorname{Cov}\left(K_1, K_1\right)=\operatorname{Var}\left(K_1\right)=\sigma_1^2, \ &\sigma_{22}=\operatorname{Cov}\left(K_2, K_2\right)=\operatorname{Var}\left(K_2\right)=\sigma_2^2 . \end{aligned}
From (2.3) we see that
$$\sigma_{12}=\sigma_{21} .$$
If the returns are independent, then we have $\sigma_{12}=0$.

# 风险和利率理论代写

## 金融代写|风险和利率理论代写市场风险、措施和投资组合理论代考|回报

.

$$V_{\left(x_1, x_2\right)}(0)=x_1 S_1(0)+x_2 S_2(0) .$$

$$w_1=\frac{x_1 S_1(0)}{V_{\left(x_1, x_2\right)}(0)}, \quad w_2=\frac{x_2 S_2(0)}{V_{\left(x_1, x_2\right)}(0)} .$$

$$x_1=\frac{w_1 V(0)}{S_1(0)}, \quad x_2=\frac{w_2 V(0)}{S_2(0)} .$$

$$V_{\left(x_1, x_2\right)}(1)=x_1 S_1(1)+x_2 S_2(1) .$$

$$\mathbf{w}=\left[\begin{array}{c} w_1 \ w_2 \end{array}\right],$$
，对应投资组合的回报用$K_{\mathrm{w}}$表示

## 金融代写|风险和利率理论代写市场风险、度量和投资组合理论代考|可实现集

. .

$$\operatorname{Cov}(X, Y)=\mathbb{E}[(X-\mathbb{E}(X))(Y-\mathbb{E}(Y)]=\mathbb{E}(X Y)-\mathbb{E}(X) \mathbb{E}(Y),$$
，特别是$\operatorname{Cov}(X, X)=\operatorname{Var}(X)=\sigma_X^2$。将Schwarz不等式([PF，引理3.49])应用到$X-\mathbb{E}(X)$和$Y-\mathbb{E}(Y)$，我们得到
$$|\operatorname{Cov}(X, Y)| \leq \sigma_X \sigma_Y .$$

$$\sigma_{i j}=\operatorname{Cov}\left(K_i, K_j\right),$$
for $i, j=1,2$。特别是
\begin{aligned} &\sigma_{11}=\operatorname{Cov}\left(K_1, K_1\right)=\operatorname{Var}\left(K_1\right)=\sigma_1^2, \ &\sigma_{22}=\operatorname{Cov}\left(K_2, K_2\right)=\operatorname{Var}\left(K_2\right)=\sigma_2^2 . \end{aligned}

$$\sigma_{12}=\sigma_{21} .$$

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assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师