Portfolio Theory

Assume the Efficient Market Hypothesis.

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How to maximize returns while minimizing risk [=volatility =std. dev.]? → Since risk/return is proportional to each other, you choose one optimization goal.

• Single Stock (See Dividend Discount Model)
• Price of stock $i$ at time $t$: $S_i(t)$
• Return of stock $i$ at time $t$: $R_i(t)=\frac{S_i(t)-S_i(t_0)}{S_i(t_0)}+\frac{D_i}{S_i(t_0)}$
  • $S_i,D_i,R_i$ are all Random Variables
• Expected Return of stock $i$: ${\mu }_i=E[R_i]$
• Volatility (=Risk) of Stock $i$: ${\sigma }_i=\sqrt{Var[R_i]}$

Two-stock Portfolios §

• Covariance of stocks $i,j$: ${\sigma }_{i,j}=Cov[R_i,R_j]$
• Correlation of stock $i,j$: ${\rho }_{i,j}=\frac{{\sigma }_{i,j}}{{\sigma }_i,\cdot {\sigma }_j}$ such that $-1\le \rho \le 1$
• Weights $w,1-w$ for $(R_1,{\mu }_1,{\sigma }_1),(R_2,{\mu }_2,{\sigma }_2)$
• Portfolio Expected Return: ${\mu }_p=w{\mu }_1+(1-w){\mu }_2$
• Portfolio Variance:

$$\begin{array}{rlr}{\sigma }_p^2& =w^2{\sigma }_1^2+(1-w{)}^2{\sigma }_2^2+2w(1-w)\rho \cdot {\sigma }_1{\sigma }_2& \\ & =({\sigma }_1^2-2\rho {\sigma }_1{\sigma }_2+{\sigma }_2^2)w^2+2(\rho {\sigma }_1{\sigma }_2-{\sigma }_2^2)w+{\sigma }_2^2& \text{function of }w\end{array}$$

• Minimum Variance Portfolio:

$$\begin{array}{rl}W& =\frac{{\sigma }_2^2-\rho {\sigma }_1{\sigma }_2}{{\sigma }_1^2-2\rho {\sigma }_1{\sigma }_2+{\sigma }_2^2}\\ 1-W& =\frac{{\sigma }_1^2-\rho {\sigma }_1{\sigma }_2}{{\sigma }_1^2-2\rho {\sigma }_1{\sigma }_2+{\sigma }_2^2}\end{array}$$

N-Stock Portfolio §

$$\vec{R}=\left[\begin{array}{c}{R}_1\\ ⋮\\ R_N\end{array}\right]\vec{\mu }=\left[\begin{array}{c}{\mu }_1\\ ⋮\\ {\mu }_N\end{array}\right]\vec{w}=\left[\begin{array}{c}{w}_1\\ ⋮\\ w_N\end{array}\right]V=\left[\begin{array}{ccc}{\sigma }_1& \dots & {\sigma }_{1,N}\\ ⋮& \ddots & ⋮\\ {\sigma }_{N,1}& \dots & {\sigma }_{N,N}\end{array}\right]$$

• Known information: ${\sigma }_i,{\sigma }_{ij},{\rho }_{ij},{\mu }_i$ are all known
• Multiple Stocks (=Portfolio)
  • Position of portfolio $p$ at time $t$ (=amount invested): $V_p(t)={\sum }_{i=1}^N{n}_i{S}_i(t)$
  • Portfolio Return at time $t$ $R_p(t)={\vec{w}}^T\cdot \vec{R}$
  • Portfolio Expected Return at time $t$: ${\mu }_p(t)=\vec{w}\cdot \vec{\mu }$
  • Portfolio Variance: ${\sigma }_p^2={\sum }_{i=1}^N{w}_i^2{\sigma }_i^2+2{\sum }_{1\le i<j\le N}{w}_i{w}_j\rho {\sigma }_i{\sigma }_j={\vec{w}}^{T}V\vec{w}$

def. Feasible Portfolios. Set of tuples $(\sigma =\text{risk},\mu =\text{expected return})$ given we have many assets weighted $\vec{w}$.

• $F_p=\{({\sigma }_p,{\mu }_p)|w\in \mathbb{R}\}$ ← Short-selling is allowed
• Lower the correlation [= the more negatively correlated] $\propto$ the better diversification is.
  • When two stocks have perfectly negative correlation $\rho =-1$, the efficient frontier touches the $y$-axis (=${\sigma }_p=0$)

Efficient Frontier §

$${\text{min}}_{\vec{w}}{\sigma }_p=\sqrt{{\vec{w}}^{T}V\vec{w}}\text{such that}{\vec{w}}^T\vec{1}=1,{\vec{w}}^T\vec{\mu }={\mu }_p$$

• “Minimize the variance $\sigma$ such that the sum of weights are $1$ and the portfolio return is $\mu$ (a constant)”
• Key Assumption: neither the returns ${\mu }_i$ or risks ${\sigma }_i$ are identical, and there is no perfect correlation ${\rho }_{i,j}=\pm 1$
• Minimum for given return level ${\mu }_p$ at:

$$\vec{w}=\frac{C-B{\mu }_p}{AC-B^2}{V}^{-1}\vec{e}+\frac{{\mu }_{p}A-B}{AC-B^2}{V}^{-1}\vec{\mu }$$

• Efficient Frontier:

$$\{({\sigma }_p,{\mu }_p)|{\sigma }_p^2=\frac{A{\mu }_p^2-2B\mu \_p+C}{AC-B^2}\}$$

• …where $A,B,C$ are scalars:
  • $A=\vec{e}{V}^{-1}\vec{e}$
  • $B={\vec{\mu }}^T{V}^{-1}\vec{e}={\vec{e}}^T{V}^{-1}\vec{\mu }$
  • $C={\vec{\mu }}^T{V}^{-1}\vec{\mu }$