Mean Variance Portfolio and CAPM

Let $R_i$ be the return on asset $i\in 1,\dots ,N$

Then the expected return on asset $i$ is given by:

$$E[R_i]={\mu }_i\to {\mu }_{N\times 1}$$

which can be used to make a vector of returns for each asset.

Now we can construct the variance-covariance matrix of returns, which is given by:

$$V[R_i],\mathrm{Cov}[R_i,R_j]\to V_{N\times N}$$

The variance-covariance matrix captures the volatility of each asset and the correlation between different assets. Now we can express the return of the portfolio:

$$R_p=x_1{R}_1+\cdots +x_N{R}_N$$

Where $\sum x_i=1$ and $x_i$ is the weight of asset $i$ in the portfolio. The expected return of the portfolio can be expressed as:

$$E[R_p]=x_1{\mu }_1+\cdots +x_N{\mu }_N=x^T\mu =u_p$$

As well the variance of the portfolio is

$$V[R_p]=V_p^2=\sum _{i=1}^N\sum _{j=1}^N{x}_i{x}_j\mathrm{Cov}[R_i,R_j]=x^{T}Vx$$

Minimum Variance Problem

Problem 1

$$\{\begin{array}{ll}{\min }_{(x_1,\dots ,x_N)}& \frac{V_p^2}{2}\\ s.t.& {\sum }_{i=1}^N{x}_i=1\end{array}$$

Proposition 16

The optimal weights that solve the minimum variance problem are given by

$$\hat{x}=\frac{V^{-1}\cdot l}{l^T\cdot V^{-1}\cdot l}$$

As well, the minimum variance is given by

$${\beta }_0=\frac{1}{l^T\cdot V^{-1}\cdot l}$$

The expected value or “mean” is then

$${\alpha }_0=\frac{u^T\cdot V^{-1}l}{l^T\cdot V^{-1}\cdot l}$$

The proof for the above uses the Lagrange Multiplier

If $R_i$ are normal then

$$\frac{W(T)-W}{W}=R_p\sim N({\alpha }_0,{\beta }_0)$$ $$W(T)=W(1+R_p)\sim N(W(1+{\alpha }_0),W^2{\beta }_0))$$

De Miguel et. al. R.F.S. (2009), tells us that we should just do

$$X_i=\frac{1}{N}$$

Problem 2

$$\{\begin{array}{ll}\min & \frac{1}{2}{x}^{T}Vx-t{\mu }^{T}x\\ s.t.& l^{T}x=1\end{array}$$

The parameter $t$ basically describes how much weight we put on the expected return of the portfolio. If $t=0$ then we are just doing the minimum variance problem, and if $t\to \infty$ then we are just trying to maximize the expected return of the portfolio.

This is not a practical problem, because how do you pick $t$? It is very arbitrary.

Because it is impractical, there are two variations to this problem.

Problem 3 - Fixed Mean Problem

$$\{\begin{array}{ll}\min & \frac{1}{2}{x}^{T}Vx\\ s.t.& l^{T}x=1\\ & {\mu }^{T}x=u_p\end{array}$$

Problem 4 - Fixed Variance Problem

$$\{\begin{array}{ll}\min & {\mu }^{T}x\\ s.t.& l^{T}x=1\\ & \frac{1}{2}{x}^{T}Vx=\frac{1}{2}{V}_p^2\end{array}$$

**does this

stay bold**