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1. Market Portfolio Weights

Initially, the weight vector $\mathbf{w}$ required to construct simple portfolio was determined by minimising the variance $\sigma_p$ subject to being feasible (weights sum to 1) and such that it attained a given desired mean return $\mu_p$. We then briefly saw how to construct a feasible portolio from any two frontier portfolios.

Now we create a portfolio $p$ comprised of a feasible portfolio $q$ and a risk free asset $r_o$. Notably, the risk free asset has the following properties

\begin{align}
\mathbb{E} [r_o] &= r_o \qquad \text{it is a certainty}\\
\text{variance} (r_o) &= 0 \qquad \text{(risk-free)}
\end{align}
In this setting the minimum variance portfolio $q$ holds some risk and so always has a better return $\mu_q$ than the risk-free rate $r_o$. The portfolio $p$ comprised of a feasible portfolio $q$ and a risk free asset $r_o$.

\begin{align}
r_p = \underbrace{\alpha r_q}_{\mu_q(g+h)} + \underbrace{(1 – \alpha)r_o}_{(1 – \mu)g}
\end{align}
which confirms that we can create a feasible portfolio of $q$ and $r_o$. Assuming weight vector $\mathbf{w}_q$ for portfolio $q$,
$$\mu_q = \mathbf{w}_q^T\mathbf{e} \qquad\text{mean return on }q$$
the expected portfolio return and
$$\sigma_q^2=\mathbf{w}_q^T {V} \mathbf{w}_q\qquad\text{variance of }q$$
$$\mathbf{w}_q^T \mathbf{1}= 1\qquad\text{feasible}$$

The line portfolio $q$ dwells on has slope$$\dfrac{\mu_q \; – \; r_o}{\sigma_q}$$and $\mu$ intercept$$r_o$$The investment line of the portfolio $q$ is therefore
$$\mu = r_o + \dfrac{\mu_q \; – \; r_o}{\sigma_q} \sigma$$To get the most out of the line we should choose $q$ so that it maximises the slope

We choose the efficient portfolio on the upper limb of the hyperbola whose tangent cuts through $r_o$. This portfolio is called the market portfolio. The resulting line is the capital market line.

1. Market Portfolio Weights

The weights of the market portfolio are the weights which maximise the slope while remaining feasible. We seek to maximize
$$f(\mathbf{w}) = \dfrac{\mu_q – r_o}{\sigma} =\dfrac{\mathbf{w}^T\mathbf{e} \; – \;r_o}{\sqrt{\mathbf{w}^T V \mathbf{w}}}$$ subject to
$$g_1(\mathbf{w}): \quad \mathbf{w}^T \mathbf{1} -1 = 0$$
With one constraint, introduce a single new parameter $\lambda$, to an objective function, the Lagrangian function which is unconstrained
$$\mathcal{L} (\mathbf{w},\lambda) = \dfrac{\mathbf{w}^T\mathbf{e} \; – \;r_o}{\sqrt{\mathbf{w}^T V \mathbf{w}}}\; -\; \lambda (\mathbf{w}^T \mathbf{1} -1).$$
First Order Conditions. Let $\mathbf{w}_m$ be the solution vector and $\lambda^*$ the solution multiplier:
\begin{align}
\dfrac{\partial \mathcal{L}}{\partial{\lambda}} &= -(\mathbf{w}_m^T \mathbf{1} -1)= \mathbf{0} \tag{1.1}
\end{align}
\begin{align}\nabla \mathcal{L}(\mathbf{w}_m, \lambda^*) &= \mathbf{0} \tag{1.2} \end{align}Recall the quotient rule;
\begin{align}
\Bigl( \dfrac{u}{v} \Bigr)’ = \dfrac{vu’ \; – \; uv’}{v^2}
\end{align}
\begin{align}
\Rightarrow \; \nabla_{\mathbf{w}_m}\mathcal{L}(\mathbf{w}_m, \lambda^*) &= \dfrac{(\mathbf{w}_m^T V \mathbf{w}_m)^{1/2} \mathbf{e} \; – \;(\mathbf{w}_m^T\mathbf{e} \; – \;r_o)\frac{1}{2}(\mathbf{w}_m^T V \mathbf{w}_m)^{-1/2}2V\mathbf{w}_m}{\mathbf{w}_m^T V \mathbf{w}_m} \; – \lambda^* \mathbf{1} = \mathbf{0}\end{align}
recall that
\begin{align}
\mu_m &= \mathbf{w}_m^T \mathbf{e} \qquad \qquad \text{mean of market portfolio}\\
\sigma_m^2 &=\mathbf{w}_m^T V \mathbf{w}_m \qquad \text{variance of market portfolio}\end{align}
hence
\begin{align}
\nabla_{\mathbf{w}_m}\mathcal{L}(\mathbf{w}_m, \lambda^*) &= \dfrac{\sigma_m \mathbf{e} \; – \;\dfrac{\mu_m \; – \;r_o}{\sigma_m}V\mathbf{w}_m}{\sigma_m^2} \; – \lambda^* \mathbf{1} = \mathbf{0}
\end{align}
divide top and bottom by $\sigma_m$
\begin{align}
& \dfrac{ \mathbf{e} \; – \;\dfrac{\mu_m \; – \;r_o}{\sigma_m^2}V\mathbf{w}_m}{\sigma_m} \; – \lambda^* \mathbf{1} = \mathbf{0}
\end{align}
re-arranging
\begin{align}
\mathbf{e} \; – \; \dfrac{\mu_m \; – \;r_o}{\sigma_m^2} V\mathbf{w}_m \; = \lambda^* \sigma_m \mathbf{1}\\ \\
\underbrace{\dfrac{\mu_m \; – \;r_o}{\sigma_m^2}}_{\text{Sharpe Ratio}} V\mathbf{w}_m \; = \mathbf{e} \; – \; \lambda^* \sigma_m \mathbf{1}\tag{1.3}
\end{align}
multiply both sides by $\mathbf{w}_m^T$
\begin{align}
\dfrac{\mu_m \; – \;r_o}{\sigma_m^2}\; \mathbf{w}_m^T V\mathbf{w}_m \; &= \mathbf{w}_m^T \mathbf{e} \; – \; \lambda^* \sigma_m \mathbf{w}_m^T \mathbf{1}\\ \\
\dfrac{\mu_m \; – \;r_o}{\sigma_m^2} \; \sigma_m^2 \; &= \mu_m \; – \; \lambda^* \sigma_m \cdot 1 \\ \\
\mu_m \; – \; r_o &= \mu_m \; – \; \lambda^* \sigma_m\\ \\
\Rightarrow \quad \lambda^*&= \dfrac{r_o}{\sigma_m}
\end{align}
substitute into $1.3$
\begin{align}
\dfrac{\mu_m \; – \;r_o}{\sigma_m^2}\; V\mathbf{w}_m \; &= \mathbf{e} \; – \; r_o \mathbf{1}\\ \\
\dfrac{\mu_m \; – \;r_o}{\sigma_m^2}\; \mathbf{w}_m \; &= V^{-1}(\mathbf{e} \; – \; r_o \mathbf{1})\tag{1.4}\end{align}
multiply by $\mathbf{1}^T$ and observe from $1.1$
\begin{align}
\dfrac{\mu_m \; – \;r_o}{\sigma_m^2} \; \underbrace{\mathbf{1}^T \mathbf{w}_m}_{=1} &= \mathbf{1}^T V^{-1}(\mathbf{e} \; – \; r_o \mathbf{1})\\ \\
\Rightarrow \quad \dfrac{\mu_m \; – \;r_o}{\sigma_m^2}\; &= \mathbf{1}^T V^{-1}(\mathbf{e} \; – \; r_o \mathbf{1})
\end{align}
substituting this in $1.4$ we have the answer to the big question: what is the weight vector of the market porfolio
\begin{align}
\mathbf{w}_m \; &= \dfrac{V^{-1}(\mathbf{e} \; – \; r_o \mathbf{1})}{\mathbf{1}^T V^{-1}(\mathbf{e} \; – \; r_o \mathbf{1})} \end{align}