This appendix collects the mathematical derivations behind statements made in the asset-pricing chapters. It is prose and math only — no code.

Optimal mean–variance portfolio weights

We seek the portfolio weights \(\omega\) that minimize portfolio variance subject to (i) the weights summing to one and (ii) achieving a target expected return \(\bar\mu\). With \(\Sigma\) the covariance matrix, \(\mu\) the expected returns, and \(\iota\) a vector of ones, the Lagrangian is

$$ \mathcal{L}(\omega,\lambda,\delta) = \omega'\Sigma\omega - \lambda(\omega'\mu - \bar\mu) - \delta(\omega'\iota - 1), $$

with multipliers \(\lambda\) and \(\delta\) for the two constraints. The first-order condition with respect to \(\omega\) is

$$ 2\Sigma\omega - \lambda\mu - \delta\iota = 0 \quad\Longrightarrow\quad \omega = \tfrac{1}{2}\Sigma^{-1}(\lambda\mu + \delta\iota). $$

Substituting back into the two constraints gives two linear equations in \(\lambda\) and \(\delta\), whose solution yields the optimal weights as a combination of \(\Sigma^{-1}\iota\) and \(\Sigma^{-1}\mu\). Setting the target-return constraint aside (dropping the \(\mu\) term) recovers the minimum-variance portfolio,

$$ \omega_\text{mvp} = \frac{\Sigma^{-1}\iota}{\iota'\Sigma^{-1}\iota}. $$

The tangency portfolio maximizes the Sharpe ratio

With a risk-free rate \(r_f\) and excess returns \(\tilde\mu = \mu - r_f\iota\), consider maximizing the Sharpe ratio

$$ \max_\omega \frac{\omega'\tilde\mu}{\sqrt{\omega'\Sigma\omega}} \quad\text{s.t.}\quad \omega'\iota = 1. $$

The objective is scale-invariant in \(\omega\), so the first-order conditions imply the optimal direction \(\omega \propto \Sigma^{-1}\tilde\mu\). Renormalizing so the weights sum to one gives the tangency portfolio

$$ \omega_\text{tan} = \frac{\Sigma^{-1}\tilde\mu}{\iota'\Sigma^{-1}\tilde\mu}, $$

which is exactly the risky-asset portfolio every investor holds in the CAPM, independent of risk aversion. Its slope in mean–standard-deviation space is the maximum attainable Sharpe ratio, the capital market line.

The CAPM relation

Writing the efficient-portfolio first-order condition as \(\tilde\mu = \big(\iota'\Sigma^{-1}\tilde\mu\big)\,\Sigma\,\omega_\text{tan}\), the \(i\)-th row reads \(\tilde\mu_i = \big(\iota'\Sigma^{-1}\tilde\mu\big)\,\mathrm{Cov}(r_i, r_\text{tan})\). Dividing by the same expression evaluated for the tangency portfolio's own variance gives

$$ \tilde\mu_i = \beta_i\,\tilde\mu_\text{tan}, \qquad \beta_i = \frac{\mathrm{Cov}(r_i, r_\text{tan})}{\mathrm{Var}(r_\text{tan})}, $$

the CAPM: an asset's expected excess return is proportional to its beta with the (market) tangency portfolio, with the tangency portfolio's excess return as the price of risk.

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Study notes following the Tidy Finance curriculum by Scheuch, Voigt, Weiss, and Frey. Prose is my own, licensed CC BY-NC-SA 4.0.