金融代写|风险和利率理论代写Market Risk, Measures and Portfolio Theory代考|EC4443

金融代写|风险和利率理论代写Market Risk, Measures and Portfolio Theory代考|Derivation of CAPM

In this section we derive the Capital Asset Pricing Model formula for the expected return of a risky security. Before doing so we need the following definition.
Definition $5.2$
We call
$$\beta_i=\frac{\operatorname{Cov}\left(K_i, K_{\mathrm{m}}\right)}{\sigma_{\mathrm{m}}^2}$$
the beta factor of the $i$-th security.
It will turn out that the beta factor is directly related to the systematic risk of a security. We discuss this later on. First we state the famous CAPM formula.
Theorem $5.3$ (CAPM)
Suppose that the risk-free return $R$ is lower than the expected return of the minimal variance portfolio (so that the market portfolio $\mathbf{m}$ exists). Then, for each $i \leq n$, the expected return $\mu_i$ of the $i$-th asset in the portfolio is given by the formula
$$\mu_i=R+\beta_i\left(\mu_{\mathrm{m}}-R\right) .$$
Proof As we know, the capital market line is tangent to the minimum variance line at the market portfolio point $\left(\sigma_{\mathbf{m}}, \mu_{\mathbf{m}}\right)$ (see Figure 4.8). Consider all portfolios built by means of the market portfolio and the $i$-th security. They form a hyperbola which we claim to be tangent to the capital market line at $\left(\sigma_{\mathrm{m}}, \mu_{\mathrm{m}}\right)$. Suppose that, on the contrary, this hyperbola intersects the CML. This clearly contradicts the fact that the slope of CML is maximal, see Figure $5.1$

We compute the slope of the tangent line to the hyperbola at $\left(\sigma_{\mathbf{m}}, \mu_{\mathbf{m}}\right)$ and then we will use the fact that the slope of CML is the same. Denote the proportion of wealth invested in security $i$ by $x$ and that invested in the market portfolio by $1-x$. We use $\mathbf{x}$ to denote the portfolio $\mathbf{x}=(x, 1-x)$.

金融代写|风险和利率理论代写Market Risk, Measures and Portfolio Theory代考|Security market line

We start by presenting an alternative proof of Theorem 5.3. We do this in a slightly more general context, formulating the result for a portfolio instead of a single security.
Theorem $5.4$
Suppose that the risk-free return $R$ is lower than the expected return of the minimal variance portfolio (so that the market portfolio $\mathbf{m}$ exists). Then, for any portfolio w
$$\mu_{\mathrm{w}}=R+\beta_{\mathrm{w}}\left(\mu_{\mathrm{m}}-R\right) .$$
Proof From Theorem $4.10$ we know that
$$\mathbf{m}=\frac{1}{\gamma} C^{-1}(\mu-R \mathbf{1}),$$
for $\gamma=\mathbf{1}^{\mathrm{T}} \boldsymbol{C}^{-1}(\boldsymbol{\mu}-\boldsymbol{R} \mathbf{1})$. Applying Proposition 4.2,
$$\beta_{\mathrm{w}}=\frac{\operatorname{Cov}\left(K_{\mathrm{w}}, K_{\mathrm{m}}\right)}{\sigma_{\mathrm{m}}^2}=\frac{\mathbf{w}^{\mathrm{T}} C \mathbf{m}}{\mathbf{m}^{\mathrm{T}} C \mathbf{m}}=\frac{\frac{1}{\gamma} \mathbf{w}^{\mathrm{T}}(\boldsymbol{\mu}-R \mathbf{1})}{\frac{1}{\gamma} \mathbf{m}^{\mathrm{T}}(\mu-R \mathbf{1})} .$$
Since $\mathbf{w}^{\mathrm{T}} \boldsymbol{\mu}=\mu_{\mathrm{w}}, \mathbf{m}^{\mathrm{T}} \boldsymbol{\mu}=\mu_{\mathrm{m}}$ and $\mathbf{w}^{\mathrm{T}} \mathbf{1}=\mathbf{m}^{\mathrm{T}} \mathbf{1}=1$, this gives
$$\beta_{\mathrm{w}}=\frac{\mu_{\mathrm{w}}-R}{\mu_{\mathrm{m}}-R} .$$
Rearranging we obtain (5.2).
The above proof is shorter than our first proof of Theorem 5.3. The first proof, however, is more intuitive, showing that the beta factor arises from purely geometric considerations.

风险和利率理论代考

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| CAPM的推导

$$\beta_i=\frac{\operatorname{Cov}\left(K_i, K_{\mathrm{m}}\right)}{\sigma_{\mathrm{m}}^2}$$

$$\mu_i=R+\beta_i\left(\mu_{\mathrm{m}}-R\right) .$$

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|证券市场线

$$\mu_{\mathrm{w}}=R+\beta_{\mathrm{w}}\left(\mu_{\mathrm{m}}-R\right) .$$

$$\mathbf{m}=\frac{1}{\gamma} C^{-1}(\mu-R \mathbf{1}),$$

$$\beta_{\mathrm{w}}=\frac{\operatorname{Cov}\left(K_{\mathrm{w}}, K_{\mathrm{m}}\right)}{\sigma_{\mathrm{m}}^2}=\frac{\mathbf{w}^{\mathrm{T}} C \mathbf{m}}{\mathbf{m}^{\mathrm{T}} C \mathbf{m}}=\frac{\frac{1}{\gamma} \mathbf{w}^{\mathrm{T}}(\boldsymbol{\mu}-R \mathbf{1})}{\frac{1}{\gamma} \mathbf{m}^{\mathrm{T}}(\mu-R \mathbf{1})} .$$

$$\beta_{\mathrm{w}}=\frac{\mu_{\mathrm{w}}-R}{\mu_{\mathrm{m}}-R} .$$

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