> [!tldr] Tangent Portfolio > In the [[Efficient Portfolio Frontier]] setup, introduce a riskless asset of return rate $r$; then obviously the point $(r,0)$, i.e. all in the riskless asset is now in the feasible set; furthermore, any portfolio $\Pi$ can be linearly combined with $(r,0)$ to yield a mix of portfolios $\{ \lambda \Pi + (1-\lambda)B ~|~ \lambda \in \mathbb{R}\},$which forms a line on the $\mu$-$\sigma$ plane. The optimal of those lines is the line tangent to the frontier passing through $(r,0)$: > > ![[Tangent Portfolio.png#invert|center|w60]] > Those portfolios are the **tangent portfolios**. Since the slope of a straight line in this space is the common **sharpe ratio** of all portfolios on the line, *the tangent portfolios are hence the ones with the highest sharpe ratio among all feasible portfolios*. The above makes no assumption about investors (except that holding any amount of an asset is allowed, including non-integer and/or negative lots), and is purely a mathematical result of making portfolios. ### The Market Portfolio and CAPM Under the further assumption that all investors are rational (e.g. as in the [[Capital Market Pricing Model|CAPM]]), they would all opt for the tangent portfolio $T$ (at least a linear combination of $T$ and the risk-free bond $B$) -- this means that *$T$ has the exact same proportion of assets as the entire market*, as it would become the only product (along with $B$) on the market. As a result, it is also called the **market portfolio**. As a result, if an asset $A$ has a beta of $\beta_{A}$, then its "fair" expected return is $\mathbb{E}[r_{A}]=r+\beta_{A}(\mathbb{E}[r_{M}]-r),$where $r$ is the risk-free rate, and $r_{M}-r$ is the **market risk premium**, $r_{M}$ being the return of the tangent/market portfolio. It is "fair" in the sense that: - It can't be higher, else it would contradict the definition of a tangent portfolio: adding a infinitesimal amount of $A$ to $M$ (i.e. investing in $\epsilon A+(1-\epsilon)M$) will give a larger sharpe ratio, contradicting the optimality of $M$.