# 金融代写|风险和利率理论代写Market Risk, Measures and Portfolio Theory代考|MATH0094

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• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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## 金融代写|风险和利率理论代写Market Risk, Measures and Portfolio Theory代考|Proofs

If $\mu_1 \neq \mu_2$ and $\rho_{12} \in(-1,1)$, then the attainable set is a hyperbola with its centre on the vertical axis.

Proof For a more familiar notation we introduce the letters $x, y$ for the coordinates so that we have the following description of the attainable set:
\begin{aligned} y &=w \mu_1+(1-w) \mu_2, \ x^2 &=w^2 \sigma_1^2+(1-w)^2 \sigma_2^2+2 w(1-w) \sigma_{12} . \end{aligned}
The goal of further computations is to convert the above system of equations to the form
$$\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1,$$
from which we will be able to read off the properties of the hyperbola (see Figure 2.12).
Solving (2.19) for $w$
$$w=\frac{y-\mu_2}{\mu_1-\mu_2}$$
(note the relevance of the assumption $\mu_1 \neq \mu_2$ ) and nsering into (2.2U), we get
$$x^2=\frac{1}{A}\left[\left(y-\mu_2\right)^2 \sigma_1^2+\left(\mu_1-y\right)^2 \sigma_2^2+2\left(y-\mu_2\right)\left(\mu_1-y\right) \sigma_{12}\right],$$

where $A=\left(\mu_1-\mu_2\right)^2>0$. Simple computation gives
$$x^2=\frac{1}{A}\left[B y^2-2 C y+D\right],$$
where
\begin{aligned} &B=\sigma_1^2+\sigma_2^2-2 \sigma_{12}, \ &C=\sigma_1^2 \mu_2+\sigma_2^2 \mu_1-\sigma_{12}\left(\mu_1+\mu_2\right), \ &D=\sigma_1^2 \mu_2^2+\sigma_2^2 \mu_1^2-2 \sigma_{12} \mu_1 \mu_2 . \end{aligned}

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

The mean-variance analysis of asset portfolios carried out in the previous chapter was greatly simplified by considering portfolios of only two assets. This meant that the portfolio weights involved only a single variable, making basic calculus techniques available for finding the portfolio of minimum variance. For portfolios of more than two assets this no longer applies. We will need a method that will allows us to find minima of functions of many variables under constraints. (In portfolio theory the first natural constraint is that all weights need to add up to one.)

In this chapter we digress a little from portfolio theory. We present a general method that locates potential extreme points of functions under constraints, and, in a special case that suffices for our intended applications, enables us to classify them as maxima or minima. It turns out that the minimisation problem provides a system of equations whose solution provides a candidate for the minimum. The ‘method of Lagrange multipliers’ is a standard tool in advanced calculus, but the proofs we provide are frequently only sketched in standard textbooks.

# 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|证明

\begin{aligned} y &=w \mu_1+(1-w) \mu_2, \ x^2 &=w^2 \sigma_1^2+(1-w)^2 \sigma_2^2+2 w(1-w) \sigma_{12} . \end{aligned}

$$\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1,$$
，从中我们将能够读出双曲线的性质(见图2.12)。求解(2.19)$w$
$$w=\frac{y-\mu_2}{\mu_1-\mu_2}$$
(注意假设$\mu_1 \neq \mu_2$的相关性)并代入(2.2U)，我们得到
$$x^2=\frac{1}{A}\left[\left(y-\mu_2\right)^2 \sigma_1^2+\left(\mu_1-y\right)^2 \sigma_2^2+2\left(y-\mu_2\right)\left(\mu_1-y\right) \sigma_{12}\right],$$

where $A=\left(\mu_1-\mu_2\right)^2>0$。简单计算得到
$$x^2=\frac{1}{A}\left[B y^2-2 C y+D\right],$$

\begin{aligned} &B=\sigma_1^2+\sigma_2^2-2 \sigma_{12}, \ &C=\sigma_1^2 \mu_2+\sigma_2^2 \mu_1-\sigma_{12}\left(\mu_1+\mu_2\right), \ &D=\sigma_1^2 \mu_2^2+\sigma_2^2 \mu_1^2-2 \sigma_{12} \mu_1 \mu_2 . \end{aligned}

. .

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