## 金融代写|风险和利率理论代写Market Risk, Measures and Portfolio Theory代考|FNCE463

2022年10月14日

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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|Risk and return

A portfolio constructed from $n$ different securities can be described by IIleans of the vector of weights
$$\mathbf{w}=\left(w_1, \ldots, w_n\right),$$
with the constraint $\sum_{j=1}^n w_j=1$. Denoting by $\mathbf{1}$ the $n$-dimensional vector
$$\mathbf{1}=(1, \ldots, 1),$$
the constraint can conveniently be written as
$$\mathbf{w}^{\mathrm{T}} \mathbf{1}=1 .$$

The attainable set is the set of all weight vectors $w$ that satisfy this constraint.

If short-selling is not possible, the condition $w_j \geq 0$ is added to the constraint, so in that case the attainable set becomes
$$\left{\mathbf{w}: \mathbf{w}^{\mathrm{T}} \mathbf{1}=1, w_j \geq 0 \text { for all } j \leq n\right} .$$
Unless stated otherwise, we shall assume availability of short sales.
Alternatively a portfolio is described by the vector of positions taken in particular components (numbers of units of assets)
$$\mathbf{x}=\left(x_1, \ldots, x_n\right) \text {. }$$
We have the following relations between the weights, prices and the numbers of shares:
$$w_j=\frac{x_j S_j(0)}{V(0)}, \quad j=1, \ldots, n$$
where $x_j$ is the number of shares of security $j$ in the portfolio, $S_j(0)$ is the initial price of security $j$, and $V(0)$ is the total money invested.

Denote the random returns on the securities by $K_1, \ldots, K_n$, and the vector of expected returns by
$$\boldsymbol{\mu}=\left(\mu_1, \ldots, \mu_n\right)$$
with
$$\mu_j=\mathbb{E}\left(K_j\right), \quad \text { for } j=1, \ldots, n$$

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

The purpose of this section is to provide geometric intuition as to the shape of the attainable set.

In the case when we have three risky assets, the third weight of a portfolio can be computed from the first two weights
$$w_3=1-w_2-w_1,$$
meaning that the attainable set is parameterised by $w_1$ and $w_2$. We can write the formulae for $\mu_{\mathrm{w}}$ and $\sigma_{\mathrm{w}}$ with respect to these two parameters as
\begin{aligned} \mu_{\mathrm{w}} &=w_1 \mu_1+w_2 \mu_2+w_3 \mu_3 \ &=w_1 \mu_1+w_2 \mu_2+\left(1-w_1-w_2\right) \mu_3, \end{aligned}
and
\begin{aligned} \sigma_w^2=& w_1^2 \sigma_1^2+w_2^2 \sigma_2^2+w_3^2 \sigma_3^2+2 w_1 w_2 \sigma_{12}+2 w_1 w_3 \sigma_{13}+2 w_2 w_3 \sigma_{23} \ =& w_1^2 \sigma_1^2+w_2^2 \sigma_2^2+\left(1-w_2-w_1\right)^2 \sigma_3^2+2 w_1 w_2 \sigma_{12} \ &+2 w_1\left(1-w_2-w_1\right) \sigma_{13}+2 w_2\left(1-w_2-w_1\right) \sigma_{23} . \end{aligned}
The plots of $\mu_{\mathbf{w}}$ and $\sigma_{\mathbf{w}}$ are given in Figure 4.1. The lines on the graphs represent the level sets $\left{\mu_{\mathrm{w}}=m\right}$ and $\left{\sigma_{\mathrm{w}}=c\right}$ for several values of $m$ and $c$

Since the third weight can be computed from the first two, the attainable set is represented as the $\left(w_1, w_2\right)$-plane in Figure 4.2. The vertices of the grey triangle represent investments in single assets. The point $(1,0)$ represents the first asset, $(0,1)$ the second asset, and since $w_3=1-w_1-w_2$, the point $(0,0)$ represents the third asset. The grey triangle consists of the points
$$\left{\left(w_1, w_2\right) \mid w_1, w_2 \geq 0, w_1+w_2 \leq 1\right},$$ and contains portfolios attainable without short-selling.
The level sets $\left{\mu_w=m\right}$ and $\left{\sigma_w=c\right}$ from Figure $4.1$ can be projected onto the $\left(w_1, w_2\right)$-plane in Figure 4.2. These are the straight lines and ellipses in Figure 4.2, respectively. The middle point of the ellipses is the minimum variance portfolio. In this particular figure, since the point lies outside of the triangle, we see that the minimum variance portfolio requires short selling. In Figure $4.2$ we also see that if short-selling is not allowed, then the smallest attainable $\sigma_w$ lies on the ellipse which is tangent to the grey triangle. The minimum variance portfolio without short-selling is the tangency point.

We now discuss the shape that the set of attainable portfolios takes in the $(\sigma, \mu)$-plane. We start with Figure $4.3$, where we see the plane corresponding to portfolios with $\mu_{\mathrm{w}}=m$, together with the plot of $\sigma_{\mathrm{w}}$. We see that there is a single point that has smallest attainable variance under the constraint $\mu_{\mathrm{w}}=m$. This is the point at the bottom of the intersection of the plane with the hyperbola. From the plot we also see that for $\mu_{\mathrm{w}}=m$ we can have portfolios with arbitrarily large $\sigma$. This leads to the conclusion that in the $(\sigma, \mu)$-plane, the set of portfolios with $\mu_{\mathrm{w}}=m$ is a horizontal half line, which is depicted in Figure 4.4. Intuitively one can think of Figure $4.4$ as the leftmost graph from Figure 4.3, rotated clockwise by ninety degrees, and projected onto the plane. Since the plot of $\sigma_w$ is a hyperbola, one is led to believe that the boundary of the attainable set on the $(\sigma, \mu)$-plane should also be a hyperbola. This is just a geometric intuition, and is by no means meant as a proof. We shall prove this fact later on.

# 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|风险和回报

$$\mathbf{w}=\left(w_1, \ldots, w_n\right),$$

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