Include covariance when combining two assets

MPT evaluates expected returns together with variances and covariances. Consider a fully invested, unlevered, long-only portfolio with weights w and 1−w both positive. If both standard deviations are positive and correlation ρ < 1, portfolio standard deviation is below wσA+(1−w)σB. Zero weights or zero volatility do not guarantee a strict reduction. The same statement cannot be transferred unchanged to short positions.

Illustrative annual inputs: A expected return 8%, volatility 20%; B expected return 4%, volatility 10%; equal 50% weights and correlation 0.2. Expected return is 6%. Variance is 0.5²×0.2² + 0.5²×0.1² + 2×0.5×0.5×0.2×0.1×0.2 = 0.0145. Standard deviation is√0.0145 ≈ 12.0416%, below the weighted average 15% but above B's 10%. Diversification does not mean lower volatility than every constituent.

With the same inputs but correlation 1, volatility is 15%. Historical correlation need not persist, and computing this 50:50 mix does not establish that it is an efficient or optimal portfolio. CFA Part I explains portfolio volatility and the role of correlation.

What the efficient frontier means

For specified assets, weight constraints, expected returns and covariance inputs, the efficient frontier is the boundary of efficient feasible combinations offering the highest expected return at a given volatility. It can also be explored by finding minimum variance for a target expected return, excluding inefficient combinations when another offers higher expected return with lower risk. Changing inputs or constraints changes the boundary.

Integration of Risk-Free Assets and the Capital Allocation Line

A risk-free asset in the model must match the investment horizon and currency. A capital allocation line (CAL) can combine any modeled risky portfolio with that risk-free asset; the line tangent to the efficient frontier identifies the tangency portfolio. Selling an actual government bond early or holding it in another currency can depart from the risk-free assumption. The precise point of tangency designates the Tangency Portfolio, which maximizes the Sharpe ratio among the modeled risky-asset combinations. It is equivalent to the market portfolio only under additional CAPM assumptions, including homogeneous expectations and unrestricted borrowing and lending at the risk-free rate. Under Tobin's separation theorem, the model lets investors adjust risk by combining the tangency portfolio with a risk-free asset. Real-world results depend on estimates, costs, borrowing access, and investment constraints.

Check what the inputs and model are for

CFA Part II distinguishes CAL, CML and additional CAPM assumptions. Use returns with matched periods and currencies and label expected returns and covariance as estimates. Costs, weight restrictions and borrowing conditions change the feasible set and frontier. Standard deviation is the model's risk measure, not maximum loss or a guarantee of liquidity when cash is needed.