Each dot is an asset positioned by its risk σ (annualised volatility) and expected return R. The three bright anchor assets (A, B, C) feed a real two-fund Markowitz optimisation: their covariance matrix Σ is inverted to solve, for every target return, the portfolio weights that minimise variance — that curve is the efficient frontier. The Capital Market Line is the straight line from the risk-free rate tangent to that frontier; its slope is the market Sharpe ratio. Drag any dot to reshape σ/R live, or drag the gold marker at σ=0 to move Rf — the frontier, CML and Security Market Line all recompute every frame.
σp(R)² = (A·R² − 2B·R + C) / D [Markowitz two-fund theorem]
E(Ri) = Rf + βi·(E(Rm) − Rf) [CAPM]
βi = ρ(i,m)·σi / σm
- Risk-free rate Rf — the anchor of both the CML (in σ/R space) and the SML (in β/R space); raising it lifts both lines and rotates the CML's slope.
- Market return E(Rm) — the SML's slope pivot; a higher market return demands more compensation per unit of beta.
- Market volatility σm — every asset's beta is scaled by 1/σm, so a calmer market makes any fixed correlation translate into a larger beta.
- Drag an asset — changes its σ/R directly; anchor assets (A/B/C) also reshape the efficient frontier and tangency portfolio in real time.
Real-world relevance: fund managers use this exact machinery — Markowitz optimisation to size a diversified portfolio, then CAPM/SML to judge whether an individual stock's expected return compensates for the systematic (market) risk it adds.