From two assets to an efficient allocation

Learning goals

• Compute portfolio return, risk and the minimum-variance weight.
• Connect risk aversion, the allocation line and a market portfolio.

Draft · approximately 60 minutes, excluding further revision.

Weights apply to both return and covariance

For fully invested weights w and 1−w, expected portfolio return is wμA+(1−w)μB. Variance is w²σA²+(1−w)²σB²+2w(1−w)Cov(A,B). Standard deviations cannot generally be averaged. Diversification depends on co-movement; a portfolio with highly correlated assets can remain concentrated in one economic risk.

When unconstrained and the denominator is positive, the minimum-variance weight in A is [σB²−Cov(A,B)]/[σA²+σB²−2Cov(A,B)]. With no shorting, check whether that weight lies in [0,1] and compare feasible boundaries. An efficient portfolio offers the highest expected return at its risk level; the lower branch below the minimum-variance portfolio is dominated when a higher-return portfolio has the same risk.

Choose risk after choosing the risky portfolio

Mixing a risk-free asset with risky portfolio P at weight y gives expected return rf+y(μP−rf) and volatility |y|σP. The capital allocation line has slope (μP−rf)/σP for y≥0. Under quadratic mean–variance utility U=E[R]−Aσ²/2, using decimal units and A>0, unconstrained y*=(μP−rf)/(AσP²). Borrowing limits, unequal borrowing/lending rates and nonnormal losses can invalidate this simple allocation.

The capital market line is the special allocation line using the market portfolio under equilibrium assumptions. It relates expected return to total risk for efficient combinations. CAPM instead relates an individual asset’s expected return to beta, its covariance with the market divided by market variance. Do not place every individual stock on the capital market line.

A two-asset covariance calculation with units

A portfolio invests 60% in asset A with 20% volatility and 40% in asset B with 10% volatility; correlation is 0.25. Covariance is 0.25 × 0.20 × 0.10=0.005. Portfolio variance=0.6²× 0.2²+0.4²× 0.1²+2 × 0.6 × 0.4 × 0.005=0.0184, giving volatility≈13.5647%. The weighted average individual volatility 16% is not the portfolio volatility because it omits covariance.

If expected returns are 8% and 4%, portfolio expected return is 6.4% regardless of correlation, but its risk changes when correlation changes. Diversification reduces some risk without guaranteeing protection in every stress event. Estimated correlations can change precisely when liquidity is scarce; supplement a historical matrix with scenario analysis rather than treating an optimized weight as certain.

Formula and assumptions

σp² = w²σA²+(1−w)²σB²+2w(1−w)Cov(A,B); wA,min = (σB²−Cov)/(σA²+σB²−2Cov); y* = (μP−rf)/(AσP²)

Diversification and a risk-aversion choice

1. A and B each have 20% volatility and zero correlation. A 50/50 portfolio has variance 0.25×0.04 + 0.25×0.04 = 0.02, so volatility = 14.1421%.
2. For μP=8%, rf=2%, σP=20% and A=3, y*=0.06/(3×0.04)=0.5. Expected return = 5%, volatility = 10%.
3. The resulting allocation reflects the stated utility model and estimates; changing risk aversion or expected returns changes the answer. It is an educational calculation, not an allocation recommendation.

Calculation lab

Errors to catch

• Averaging volatility or forgetting the covariance term.
• Treating an unconstrained optimum as feasible when borrowing or shorting is prohibited.

Original knowledge check

Answer every question before checking. Review explanations after submitting; retry starts a new attempt.

Sources and coverage status

These original lessons are drafts. Public outlines are used to check scope; the official learning-outcome text and proprietary question banks are not reproduced here. Follow the source for exam-year requirements and updates.

• Reference · www.cfainstitute.org