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Harry Markowitz developed modern portfolio theory, making the tradeoff between expected return and variability a mathematical allocation problem. His 1952 Portfolio Selection by Harry Markowitz emphasizes that the variability of a combination depends on relationships between its components, not just on each component considered alone.^[ref]

For an original toy example, let two return random variables $X,Y$ each have variance one. Equal weighting givesWith zero covariance the result is $1/2$; with covariance one it is one. The calculation explains why counting assets alone does not determine diversification. Expected values, covariance estimates and allocation constraints supply distinct inputs to the optimization.

The 1990 economics prize recognized his theory of portfolio choice. The awarding academy's account identifies both the 1952 paper and his expanded 1959 treatment.^[ref]

Robert Litterman developed the Black-Litterman model with Fischer Black. Their 1992 paper makes equilibrium expected returns a neutral starting point for portfolio modeling, then incorporates views that can differ in their confidence. The contribution addresses how uncertain inputs enter modern portfolio theory, rather than treating every proposed expected return as equally reliable.^[ref]

An elementary Bayesian inference analogy illustrates the role of uncertainty. For an unknown quantity $\mu$ with normal distribution prior $N(m_0,s_0^2)$ and an independent observation error giving $y\mid \mu \sim N(\mu ,s_y^2)$, Gaussian conditional expectation gives the posterior meanThe more uncertain input receives less weight. This scalar example explains a modeling principle; the full Black-Litterman model treats related expected returns jointly, with a covariance matrix describing their uncertainty.