Suppose that You have the two different opportunities to invest, You can invest in Risk Free asset and the typical Risky asset.

The Green line represents the Efficient Frontier of risky assets.

The Blue lines represents the Capital Market Line, a trade off between the invest on risk, no-risk assets.

### The Risk Free asset:

\begin{equation}
Var( r_f ) = \sigma_f^2
\end{equation}
\begin{equation}
Cov(r_i,r_j) = 0; \forall i \neq j

\end{equation}

**Sharpe Ratio:** offers a ratio to compare how much are the premium in the mean per every unit of risk (variance).

\begin{equation} Sharpe Ratio = \frac{ ( E[r_j]-r_f ) }{\sigma_j} \end{equation}

Investor separate have to decide between risk free or risky asset in a \( \alpha \ in \ [0,1] \) so:

**Expected return:**
\begin{equation}
E[r_{portfolio}] = \alpha\ E[r_{RiskyPortfolio}] + (1-\alpha)r_{RiskFree}
\end{equation}
**Variance:**
\begin{equation}
Var[ r_{Portfolio} ] = \sigma^{2}_{Portfolio} = \alpha^2\sigma^2_{RiskyPortfolio}

\end{equation}

\begin{equation} \sigma_{Portfolio} = \alpha\sigma_{RiskyPortfolio} \end{equation}

### Capital Asset Pricing Model

**Systematic Risk:** also called market rick, represent the non-diversifiable risk. In the CAPM is represented by \( \beta \).

**Model:**
\begin{equation}
E[r_{Risk Premium}] = \beta E[r_{Market Risk}]

\end{equation}
\begin{equation}
r_{Stock} = r_{Risk Free Asset} + \beta( r_{Market} - r_{Risk Free Asset} )
\end{equation}
\begin{equation}
Var( r_{Risky Asset} ) = \beta^{2} Var( r_{Market} ) + Var( r_{Asset Specific Risk} )

\end{equation}
**Security Market Line:** it is the graphs of the CAPM, Xaxis = \(\beta\) Yaxis = \( E[r]\).

**The Market Portfolio:** is the sum of all investor's portfolios, have \( r_{market}\) return. For CAPM is an efficient portfolio.

**The Three-Factor Model:** uses the CAPM model adding some macroeconomics factors.