Markowitz Optimization
with Two Assets

Markowitz optimization is a mathematical framework, introduced by Harry Markowitz in 1952, for constructing a portfolio that maximises expected return for a given level of risk — or equivalently, minimises risk for a target return. With two assets it reduces to finding the optimal weight w invested in Asset 1 (and 1−w in Asset 2) that sits on the efficient frontier.

The portfolio expected return is a weighted average of the two assets' individual expected returns: E(Rₚ) = w · E(R₁) + (1 − w) · E(R₂) where w is the weight in Asset 1 and E(R₁), E(R₂) are their respective expected returns. The relationship is always linear — diversification does not affect expected return, only risk.

Portfolio variance depends on the individual variances and the covariance between the two assets: σ²ₚ = w²σ²₁ + (1−w)²σ²₂ + 2w(1−w)σ₁₂ where σ₁₂ = ρ₁₂ · σ₁ · σ₂ is the covariance and ρ₁₂ is the correlation coefficient. Unlike expected return, variance is a non-linear (quadratic) function of the weights — this is the source of diversification benefit.

Correlation ρ ∈ [−1, +1] is the key driver of diversification:

• ρ = +1 — perfect positive correlation; no diversification benefit. Portfolio risk is a simple weighted average of individual risks.
• ρ = 0 — uncorrelated assets; combining them reduces portfolio risk below the weighted average.
• ρ = −1 — perfect negative correlation; a specific weight w* can reduce portfolio risk to zero (complete hedge).

Lower correlation always produces a better (leftward) efficient frontier.

The MVP is the portfolio with the lowest possible variance across all weight combinations. For two assets, the closed-form solution is: w*₁ = (σ²₂ − σ₁₂) / (σ²₁ + σ²₂ − 2σ₁₂) The MVP sits at the leftmost point of the efficient frontier on a risk–return plot. Portfolios below it are inefficient because a higher return can be achieved at the same risk level by moving up the frontier.

The efficient frontier is the set of portfolios that offer the highest expected return for each level of risk. For two assets it is a curve (hyperbola) in risk–return space traced by varying w from 0 to 1. Only the upper half of the curve — from the MVP upward — is efficient. An investor selects their preferred point on this curve based on their risk tolerance.

Yes — when ρ < σ₁/σ₂ (roughly, when correlation is sufficiently below 1), the MVP has a standard deviation lower than both σ₁ and σ₂. This is the mathematical proof of diversification: combining imperfectly correlated risky assets can produce a portfolio that is safer than its least-risky component. It is one of the most important results in all of finance.

The Sharpe Ratio measures reward per unit of risk: S = (E(Rₚ) − Rᶠ) / σₚ where Rᶠ is the risk-free rate. When a risk-free asset exists, the optimal risky portfolio is the one with the highest Sharpe Ratio — called the Tangency Portfolio. It is the point where a line from Rᶠ is tangent to the efficient frontier. All rational investors hold a mix of this portfolio and the risk-free asset.

The core assumptions are:

• Mean–variance preferences — investors care only about expected return and variance (implying either normal returns or quadratic utility).
• Single period — the analysis is for one holding period; returns and covariances are stable.
• No transaction costs or taxes — weights can be adjusted freely.
• Divisible assets — any fraction of an asset can be held.
• Inputs are known — expected returns, variances, and the correlation are treated as given. In practice, estimating these reliably is the main empirical challenge.

With a single asset an investor can only choose how much to invest — risk and return are fixed. Adding a second asset introduces the covariance term into the portfolio variance formula, creating a continuous opportunity set (the frontier curve instead of just two points). The investor now has a genuine trade-off to optimise. The two-asset model is the pedagogical foundation for the N-asset case, where the covariance matrix Σ (N×N) replaces the single covariance σ₁₂, and the optimisation becomes a quadratic programming problem.