An interactive version of a two-asset mean-variance model I built — combine any two assets in every proportion, trace out the resulting risk/return frontier, and find the exact combination that maximizes the Sharpe ratio.
Combining two risky assets almost never just averages their risk — because they don't move in perfect lockstep, the combination can be less risky than either asset alone. That's diversification, and mean-variance analysis is the tool for finding exactly how much of it you can get.
A portfolio's expected return is just the weighted average of its assets' expected returns — no surprises there. Its risk is not: because of that cross term, as long as the two assets aren't perfectly correlated (ρ < 1), the portfolio's standard deviation is less than the weighted average of the two individual standard deviations. Lower correlation means more diversification benefit — the two assets are more likely to cushion each other's bad days.
The Sharpe ratio measures compensation per unit of risk taken — excess return (over the risk-free rate) earned for every unit of standard deviation. Among every possible mix of the two assets, the one with the highest Sharpe ratio is the single best risky combination to hold, regardless of how risk-averse you personally are — that's the two-fund separation result underlying the tangency portfolio below.
The Capital Allocation Line (CAL) is what you get by mixing the risk-free asset with the best risky combination — the tangency portfolio T, the point on the two-asset frontier with the highest Sharpe ratio. Its slope is that maximum Sharpe ratio. Every point along the CAL — lending at the risk-free rate to move left of T, or borrowing at it to move right — dominates every other combination of the two risky assets on a risk-adjusted basis. That's why the highest-Sharpe portfolio isn't just "a good one" — it's the pivot point the entire efficient set of overall portfolios is built from.
Enter two assets' expected returns, standard deviations, and their correlation, plus a risk-free rate. The chart traces every long-only combination of the two — 0% to 100% in one — and marks the minimum-variance and maximum-Sharpe points.
Here, the curve traced by every long-only combination of the two assets. More generally (with many assets), it's the set of portfolios offering the highest expected return for each level of risk.
The single combination of the two assets with the lowest possible standard deviation — the leftmost point on the frontier.
The combination where a line from the risk-free rate is tangent to the frontier — equivalently, the single highest-Sharpe-ratio combination of the two risky assets.
Every portfolio you can build by mixing the risk-free asset with the tangency portfolio — moving along it by lending (left of T) or borrowing (right of T) at the risk-free rate.
Lower (or more negative) correlation bows the frontier further to the left — more diversification benefit, and a lower-risk minimum-variance portfolio — without changing either asset's own risk or return.
Every investor, regardless of risk tolerance, should hold the same tangency portfolio of risky assets — they differ only in how much of the risk-free asset they mix in alongside it.