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The expected excess return obeys $\mathbb{E}(R_i-r)={\beta }_i\mathbb{E}(R_M-r)$ in the mean-variance market model. The beta of an asset is a covariance-to-market-variance ratio.

Let initial wealth be $w$ and let ${\theta }_i$ be the number of units held of risky asset $i$; the bank investment is $w-{\theta }^T{S}_0$. Put $b=\mu -(1+r)S_0$. Terminal wealth is normal, with mean $(1+r)w+{\theta }^{T}b$ and variance ${\theta }^{T}V\theta$. The expectation of the exponential utility isFor $\gamma >0$ and positive definite covariance matrix $V$, maximizing this is equivalent to maximizing the certainty equivalent ${\theta }^{T}b-\gamma {\theta }^{T}V\theta /2$. Thus the optimal investment isAll investors have proportional risky holdings. A market portfolio can therefore be represented by ${\theta }_M=V^{-1}b$, with any positive scalar multiple giving the same return. Suppose its initial value $P={\theta }_M^T{S}_0$ is positive and $b\ne 0$. Set $B=b^T{V}^{-1}b$. For returns $R_i=S_1^i/S_0^i-1$ and $R_M={\theta }_M^T{S}_1/P-1$,The beta of an asset is its covariance with the market return divided by the market variance. ConsequentlyThis is the capital asset pricing model. Since ${\beta }_i={\rho }_i{\sigma }_i/{\sigma }_M$, dividing the excess-return identity by ${\sigma }_i$ gives the Sharpe ratio identityThese ratio statements require nonzero variances and nonzero initial portfolio value. If $V$ is singular and $b$ is in its range, replace $V^{-1}$ by its pseudoinverse; any kernel holding can be added to an optimizer. If $b$ has a component in the kernel, it gives a deterministic excess gain that can be scaled without bound, so there is no finite optimal portfolio. If $b=0$, there is no distinguished risky market portfolio and its beta ratios are undefined.