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In mean-variance optimization with unrestricted borrowing/lending, the upper efficient ray through the risk-free point has slope $\theta =\sqrt{(\mu -r1{)}^T{\Sigma }^{-1}(\mu -r1)}$. If $1^T{\Sigma }^{-1}(\mu -r1)>0$, it touches the risky efficient frontier at the positive-premium normalized market portfolio. The Sharpe ratio there equals the slope. If this normalization is negative, the normalized tangency portfolio instead has negative excess return; the upper ray uses a short position in that portfolio and is not tangent to the upper risky branch.

The capital asset pricing model givesThus it has the same affine relation between expected returns and a single systematic exposure. When the factor is the market portfolio's excess return, the loadings are the beta of an asset, ${\lambda }_0=r_f$, and ${\lambda }_1=\mathbb{E}{r}_M-r_f$. More generally, for a nondegenerate exact one-factor model and a market portfolio with loading $b_M\ne 0$, ${\beta }_i=b_i/b_M$, and the pricing relation can be rewritten using that exposure.

The interpretations and assumptions differ. The CAPM is an equilibrium relation obtained with a mean-variance optimization framework, common beliefs, and suitable borrowing/lending and market-clearing assumptions. Arbitrage pricing theory uses absence of arbitrage, factor structure, and in realistic models diversification; it does not identify a generic factor with the market portfolio. Moreover the CAPM permits asset-specific risk, whereas the exact model in part (a) has no residual risk. The two formulas coincide for a suitable market factor, but no-arbitrage factor pricing by itself does not establish the CAPM's equilibrium assumptions.

Let $\mu$ be the vector of expected returns and $\Sigma$ the positive definite covariance matrix of risky returns. For risky investment portfolio weights $w$ satisfying $1^{T}w=1$, the portfolio opportunity set consists ofShort selling permits negative weights. For a fixed expected return, the least possible variance is attained on the left boundary of this set. Its upper branch is the mean-variance efficient frontier: every other attainable point has either a larger standard deviation for the same expected return or a smaller expected return for the same risk. The lower boundary is therefore inefficient.

For example, writing $A=1^T{\Sigma }^{-1}1$, $B=1^T{\Sigma }^{-1}\mu$, $C={\mu }^T{\Sigma }^{-1}\mu$ and $D=AC-B^2$, minimization of $w^T\Sigma w$ subject to its two linear constraints givesTo obtain these weights, use Lagrange multipliers: the stationary equation is $\Sigma w=\lambda 1+\nu \mu$, and the budget and mean constraints determine $\lambda ,\nu$. Positive definiteness makes this the unique constrained minimum. Thus the boundary is a hyperbola in risk/mean coordinates.

Adding the risk-free asset allows borrowing and lending at $r$. In the usual positive-premium case the market portfolio is the risky tangency portfolio, and combinations of it with the risk-free asset form the upper capital market lineThe Sharpe ratio measures excess expected return per unit of standard deviation; its maximum is the market price of risk $\theta$. Points between the risk-free point and the tangency point lend some wealth; points beyond it borrow to leverage the market portfolio. The first panel illustrates the actual data here. The second anticipates the change in part (g), where the normalization of the tangency direction reverses its orientation.

Write $d=\mu -r1$ and let $x$ denote the risky holdings as fractions of total wealth. The remaining fraction $1-1^{T}x$ is in the risk-free asset. Consequently their excess expected return is $d^{T}x$ and their variance is $x^T\Sigma x$. The Cauchy-Schwarz inequality in the inner product defined by $\Sigma$ givesEquality with positive excess return holds precisely for $x=cz$, $c>0$, where $z={\Sigma }^{-1}d$. Thus solving $\Sigma z=d$ determines both the efficient risky direction and the upper capital market line slope. Provided $s=1^{T}z>0$, the normalized market portfolio and its Sharpe ratio areHere the excess means are $(7,14,21{)}^T$, so the required equations areThe leading principal minors of $\Sigma$ are $4$, $35$ and $539$, so Sylvester's criterion verifies positive definiteness and the solution is unique.

The risky weights of the normalized market portfolio must sum to one. Since $1^{T}z=38/11>0$, divide each component by this sum:Every weight is positive in this case, although the optimization permitted short selling.

The expected return of the market portfolio is the weighted mean of the asset returns:For its variance, use $\Sigma z=d$ rather than expanding nine terms:Hence its return standard deviation isMeans and standard deviations here are both measured in percentage points, consistently with the given covariance matrix.

All three individual risky assets now have negative excess expected returns. The covariance matrix and the risky portfolio opportunity set remain unchanged, but the risk-free point moves above all three individual means. Solving the new tangency equations givesThe sign of the normalized tangency portfolio now matters. Normalizing to a unit risky budget givesand its Sharpe ratio is $-\sqrt{4555/77}$. This point touches the lower risky frontier, rather than being an efficient positive-premium market portfolio.

The positive-premium efficient direction is still $x=cz$, $c>0$. It shorts the first two assets, holds a hedging long position in the third, and has negative total risky investment. Its risk-free investment exceeds its initial wealth: the net proceeds of short selling are lent. Equivalently it takes a negative position in the normalized tangency portfolio. ThusNegative excess means therefore do not eliminate efficient risky investments when unrestricted short selling is allowed. The second diagram shows both orientations; it would be incorrect to interpret the negative normalized ratio as a negative slope for the upper efficient ray.

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.