02 Portfolio Optim. for 2 Assets

tags: kkk

Portfolio optimization with 2 assets

Given two risky assets as follows:

$$\{\begin{array}{l}\text{Asset 1: }\mu ={\mu }_1,\sigma ={\sigma }_1\\ \text{Asset 2: }\mu ={\mu }_2,\sigma ={\sigma }_2\end{array}$$
We can use a weight vector

$[w_1,w_2{]}^T$, with

$w_1+w_2=1$ to allocate these two assets to have overall mean return

$\mu$ and variance

${\sigma }^2$:

$$\{\begin{array}{rcl}\mu & =& w_1{\mu }_1+w_2{\mu }_2\\ {\sigma }^2& =& w_1^2{\sigma }_1^2+w_2^2{\sigma }_2^2+2w_1{w}_2{\sigma }_{12}\end{array}$$
Instead of using two parameters in the above expression, we can use only a single parameter

$w$, with

$w_1=w$ and

$w_2=1-w$:

$$\{\begin{array}{rcl}\mu & =& w{\mu }_1+(1-w){\mu }_2\\ {\sigma }^2& =& w^2{\sigma }_1^2+(1-w{)}^2{\sigma }_2^2+2w(1-w){\sigma }_{12}\end{array}$$

$$\begin{array}{rcl}\frac{d{\sigma }^2}{dw}& =& 2w{\sigma }_1^2-2(1-w){\sigma }_2^2+2(1-2w){\sigma }_{12}\\ & =& 2w({\sigma }_1^2+{\sigma }_2^2-2{\sigma }_{12})-2{\sigma }_2^2+2{\sigma }_{12}\\ & =& 0\end{array}$$

$$\Rightarrow w=\frac{{\sigma }_2^2-{\sigma }_{12}}{{\sigma }_1^2+{\sigma }_2^2-2{\sigma }_{12}},1-w=\frac{{\sigma }_1^2-{\sigma }_{12}}{{\sigma }_1^2+{\sigma }_2^2-2{\sigma }_{12}}$$
Therefore when the minimum variance occurs at the above weights, we have

$$\mu =\frac{{\mu }_1({\sigma }_2^2-{\sigma }_{12})}{{\sigma }_1^2+{\sigma }_2^2-2{\sigma }_{12}}+\frac{{\mu }_2({\sigma }_1^2-{\sigma }_{12})}{{\sigma }_1^2+{\sigma }_2^2-2{\sigma }_{12}}=\frac{{\mu }_1{\sigma }_2^2+{\mu }_2{\sigma }_1^2-{\sigma }_{12}({\mu }_1+{\mu }_2)}{{\sigma }_1^2+{\sigma }_2^2-2{\sigma }_{12}}$$

$${\sigma }^2=\frac{({\sigma }_2^2-{\sigma }_{12}{)}^2{\sigma }_1^2}{({\sigma }_1^2+{\sigma }_2^2-2{\sigma }_{12}{)}^2}+\frac{({\sigma }_1^2-{\sigma }_{12}{)}^2{\sigma }_2^2}{({\sigma }_1^2+{\sigma }_2^2-2{\sigma }_{12}{)}^2}+\frac{2({\sigma }_2^2-{\sigma }_{12})({\sigma }_1^2-{\sigma }_{12}){\sigma }_{12}}{({\sigma }_1^2+{\sigma }_2^2-2{\sigma }_{12}{)}^2}=\frac{{\sigma }_1^2{\sigma }_2^2-{\sigma }_{12}^2}{{\sigma }_1^2+{\sigma }_2^2-2{\sigma }_{12}}$$ (You need to verify this by yourself!)

We can go one step further to find the equation defining the relationship betweeen

$\mu$ and

$\sigma$. First of all, we have

$$w=\frac{{\mu }_2-\mu }{{\mu }_2-{\mu }_1},1-w=\frac{\mu -{\mu }_1}{{\mu }_2-{\mu }_1}$$
Therefore

$$\begin{array}{rcl}{\sigma }^2& =& {\left(\frac{{\mu }_2-\mu }{{\mu }_2-{\mu }_1}\right)}^2{\sigma }_1^2+{\left(\frac{\mu -{\mu }_1}{{\mu }_2-{\mu }_1}\right)}^2{\sigma }_2^2+2\left(\frac{{\mu }_2-\mu }{{\mu }_2-{\mu }_1}\right)\left(\frac{\mu -{\mu }_1}{{\mu }_2-{\mu }_1}\right){\sigma }_{12}\\ & =& \frac{1}{({\mu }_2-{\mu }_1{)}^2}[({\mu }^2-2{\mu }_2\mu +{\mu }_2^2){\sigma }_1^2+({\mu }^2-2{\mu }_1\mu +{\mu }_1^2){\sigma }_2^2-2({\mu }^2-({\mu }_1+{\mu }_2)\mu +{\mu }_1{\mu }_2){\sigma }_{12}]\\ & =& \frac{1}{({\mu }_2-{\mu }_1{)}^2}[({\sigma }_1^2+{\sigma }_2^2-2{\sigma }_{12}){\mu }^2-2({\mu }_2{\sigma }_1^2+{\mu }_1{\sigma }_2^2-({\mu }_1+{\mu }_2){\sigma }_{12})\mu +{\mu }_2^2{\sigma }_1^2+{\mu }_1^2{\sigma }_2^2-2{\mu }_1{\mu }_2{\sigma }_{12}]\end{array}$$
Since

${\sigma }_1^2+{\sigma }_2^2-2{\sigma }_{12}\ge 2({\sigma }_1{\sigma }_2-{\sigma }_{12})\ge 0$, the above equation is a hyperbola on the

$\sigma -\mu$ plane. It can reduce to a parabola if

${\sigma }_1^2+{\sigma }_2^2-2{\sigma }_{12}=0$.

Note that we can express

${\sigma }_{12}$ as follows:

$${\sigma }_{12}={\sigma }_1{\sigma }_2{\rho }_{12},$$
where

${\rho }_{12}$ is the correlation coefficient for the return rates of assets 1 and 2, with

$-1\le {\rho }_{12}\le 1$. This leads to the following expressions of the overall

$\mu$ and

$\sigma$:

$$\{\begin{array}{rcl}\mu & =& w_1{\mu }_1+w_2{\mu }_2\\ {\sigma }^2& =& w_1^2{\sigma }_1^2+w_2^2{\sigma }_2^2+2w_1{w}_2{\sigma }_1{\sigma }_2{\rho }_{12}\end{array}$$

Let's discuss 3 cases when

${\rho }_{12}$ is equal to 1, 0, and -1, respectively.

• When
  ${\rho }_{12}=1$, we have
  
  $${\sigma }^2=w_1^2{\sigma }_1^2+w_2^2{\sigma }_2^2+2w_1{w}_2{\sigma }_1{\sigma }_2=(w_1{\sigma }_1+w_2{\sigma }_2{)}^2\Rightarrow \{\begin{array}{rcl}\mu & =& w_1{\mu }_1+w_2{\mu }_2\\ \sigma & =& |w_1{\sigma }_1+w_2{\sigma }_2|\end{array}$$ As
  $w_1$ is changing from 0 to 1, the above equations represent a line connecting
  $({\sigma }_1,{\mu }_1)$ (when
  $w_1=1$ and
  $w_2=0$) and
  $({\sigma }_2,{\mu }_2)$ (when
  $w_1=0$ and
  $w_2=1$). So the minimum variance is
  $min({\sigma }_1^2,{\sigma }_2^2)$.
• When
  ${\rho }_{12}=0$, we have
  
  $${\sigma }^2=w_1^2{\sigma }_1^2+w_2^2{\sigma }_2^2$$ By using Cauchy-Schwartz inequality, we have
  
  $$(w_1^2{\sigma }_1^2+w_2^2{\sigma }_2^2)({\sigma }_1^{-2}+{\sigma }_2^{-2})\ge (w_1+w_2{)}^2=1$$ Therefore the minimum variance can be derived as follows:
  
  $${\sigma }^2=w_1^2{\sigma }_1^2+w_2^2{\sigma }_2^2\ge ({\sigma }_1^{-2}+{\sigma }_2^{-2}{)}^{-1}=\frac{{\sigma }_1^2{\sigma }_2^2}{{\sigma }_1^2+{\sigma }_2^2}$$ The equality holds when
  
  $$w_1^2{\sigma }_1^2/{\sigma }_1^{-2}=w_2^2{\sigma }_2^2/{\sigma }_2^{-2}\Rightarrow w_1=\frac{{\sigma }_2^2}{{\sigma }_1^2+{\sigma }_2^2},w_2=\frac{{\sigma }_1^2}{{\sigma }_1^2+{\sigma }_2^2}$$ And the corresponding overall
  $\mu$ can be expressed as
  
  $$\mu =\frac{{\mu }_1{\sigma }_2^2+{\mu }_2{\sigma }_1^2}{{\sigma }_1^2+{\sigma }_2^2}$$
• When
  ${\rho }_{12}=-1$, we have
  
  $${\sigma }^2=w_1^2{\sigma }_1^2+w_2^2{\sigma }_2^2-2w_1{w}_2{\sigma }_1{\sigma }_2=(w_1{\sigma }_1-w_2{\sigma }_2{)}^2$$
  In this case, we can achieve zero risk by setting
  
  $$w_1{\sigma }_1=w_2{\sigma }_2\Rightarrow w_1=\frac{{\sigma }_2}{{\sigma }_1+{\sigma }_2},w_2=\frac{{\sigma }_1}{{\sigma }_1+{\sigma }_2}.$$ Therefore the minimum variace is 0, and the corresponding return is
  $\frac{{\sigma }_2{\mu }_1+{\sigma }_1{\mu }_2}{{\sigma }_1+{\sigma }_2}$.

The plot of efficient frontiers with various values of

${\rho }_{12}$ when

$({\sigma }_1,{\mu }_1)=(0.15,0.2)$ and

$({\sigma }_2,{\mu }_2)=(0.25,0.3)$ is shown next. In particular, if we restrict

$w$ to be within the interval

$[0,1]$, then the feasible area for the efficient frontiers of varying correlation coefficients is a triangle with tips at

$({\sigma }_1,{\mu }_1)$,

$({\sigma }_2,{\mu }_2)$, and

$(0,\frac{{\sigma }_1{\mu }_2+{\sigma }_2{\mu }_1}{{\sigma }_1+{\sigma }_2})$.

And here is another plot for

$({\sigma }_1,{\mu }_1)=(0.2,0.1)$ and

$({\sigma }_2,{\mu }_2)=(0.1,0.3)$.

Exercises

1. In PO for 2 assets, when will the efficient frontier reduce to a straight line?
2. In PO for 2 assets, when will the efficient frontier reduce to a parabola?
3. In PO for 2 assets, when will the overall risk go to zero? What are the weights when this happens?
4. In PO for 2 assets, can you derive the general formula for minimum-variance portfolio?
   • What is the minimum variance?
   • What is the corresponding return and weights?
5. Given two risky assets as follows:
   
   $$\{\begin{array}{l}\text{Asset 1: }\mu =0.2,\sigma =0.1\\ \text{Asset 2: }\mu =0.3,\sigma =0.2\end{array}$$ Under the following conditions, what are the corresponding minimum variances when we achieve minimum-variance portfolio?
   • ${\rho }_{12}=1$
   • ${\rho }_{12}=0$
   • ${\rho }_{12}=-1$
6. Given two risky assets as follows:
   
   $$\{\begin{array}{l}\text{Asset 1: }\mu =0.2,\sigma =0.1\\ \text{Asset 2: }\mu =0.3,\sigma =0.2\end{array}$$ And the correlation coefficient of these two assets is
   ${\rho }_{12}=0$. We want to perform portfolio optimization with investment weigthing of
   $w_1$ and
   $w_2$ for assets 1 and 2, respectively.
   • What are the overall
     $\mu$ (return) and
     $\sigma$ (volatility) when
     $w_1=0.4$ and
     $w_2=0.6$?
   • What are the overall
     $\mu$, overall
     $\sigma$, and
     $w_1$ for achieving the minimum-variance portfolio?
7. Given two risky assets as follows:
   
   $$\{\begin{array}{l}\text{Asset 1: }\mu =0.2,\sigma =0.1\\ \text{Asset 2: }\mu =0.3,\sigma =0.2\end{array}$$ And the correlation coefficient of these two assets is
   ${\rho }_{12}=0.4$. We want to perform portfolio optimization with investment weigthing of
   $w_1$ and
   $w_2$ for assets 1 and 2, respectively.
   • What are the overall
     $\mu$ (return) and
     $\sigma$ (volatility) when
     $w_1=0.4$ and
     $w_2=0.6$?
   • What are the overall
     $\mu$, overall
     $\sigma$, and
     $w_1$ for achieving the minimum-variance portfolio?