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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]

Portfolio optimization chooses asset allocations to optimize an objective under specified constraints. In a one-period mean-variance optimization model, the return vector $R$ has mean $\mu$ and covariance matrix $\Sigma$. Portfolio weights $w$ give expected return ${\mu }^{T}w$ and return variance $w^T\Sigma w$. A standard problem isThe constraints describe a fully invested portfolio, a target expected return and no short selling. Different assumptions produce different optimization problems. Harry Markowitz made the dependence on covariance and the tradeoff between mean and variance central to portfolio selection.^[ref]

For two uncorrelated assets with expected returns $4\%$ and $8\%$, each with standard deviation $20\%$, a weight $w$ in the first gives variance $0.04(w^2+(1-w{)}^2)$. Its unconstrained minimum is at $w=1/2$, yielding mean $6\%$ and standard deviation $\sqrt{0.02}\simeq 14.14\%$. A target of at least $7\%$ instead requires $w\le 1/4$, making the constrained minimum $w=1/4$. The same covariance matrix therefore gives a different solution when the objective's feasible set changes.

Estimated means and covariances are model inputs, rather than known future outcomes. Sampling uncertainty, transaction costs and additional allocation restrictions can materially change the resulting weights.