2 Returns, Risk, and Dependence
2.1 Returns have more than one definition
The simple return from time \(t-1\) to \(t\) is
\[R_t = \frac{P_t-P_{t-1}+D_t}{P_{t-1}},\]
where \(P_t\) is the ending price and \(D_t\) is a distribution received during the period. A log return without distributions is \(r_t=\log(P_t/P_{t-1})\); it adds across time, whereas simple returns compound multiplicatively.
2.2 Volatility and covariance
Variance describes dispersion about an estimated mean. Covariance measures how two return series vary together. If weights sum to one, portfolio return and variance are
\[E[R_p]=\mathbf w^\mathsf T\boldsymbol\mu, \qquad \operatorname{Var}(R_p)=\mathbf w^\mathsf T\boldsymbol\Sigma\mathbf w.\]
Correlation rescales covariance into the interval from \(-1\) to \(1\). Low correlation can reduce portfolio risk without eliminating risk in every possible scenario.
2.3 Estimation matters
Two analysts can use the same formula and obtain different portfolios because of sampling periods, dividend adjustments, missing returns, or assumptions about future dependence. A correlation estimate is a historical summary, not a promise.
Try it: Calculate monthly returns for two assets, then compare their individual volatilities with that of a 50/50 rebalanced portfolio. Decide whether the variance reduction survives in a stressed period.
Living chapter. This is an initial working treatment. Derivations, examples, and references will be expanded in future editions.