> [!tldr] Efficient Portfolio Frontier > Given a series of risky products $\{ X^{(1)},\dots,X^{(n)} \}$, we can construct all linear combinations of them (assuming that unlimited short selling is allowed) as portfolios. > > In the $\mathbb{R}^{2}$ plane of expected return $\mu$ plotted against portfolio risk (standard deviation) $\sigma$, the set of admissible$^{\ast}$ portfolios form a right-opening hyperbola$^{\ast\ast}$: > ![[Efficient Portfolio Frontier.png#invert|center]] > The upper half (which has the same $\sigma$ but larger $\mu$ compared with the lower half) is the **efficient portfolio frontier**. $^{\ast}$In this case **admissibility** is not exactly its sense in [[Decision Theory]], but has similar vides: the admissible portfolio maximizes the expected return while holding the risk constant; the lower frontier minimized the expected return instead. $^{\ast\ast}$It is a hyperbola because if $\beta$ are the coefficients of $X^{(1)},\dots,X^{(n)}$ in the portfolio, then $\mu$ is linear in $\beta$, while $\sigma$ is quadratic in $\beta$.