Two-Asset Portfolio Calculator

The expected return is the easy half: weight each asset’s expected return by its share of the portfolio and add, E(Rp) = wA·RA + wB·RB. Risk needs the full Markowitz formula: the portfolio variance is wA²σA² + wB²σB² + 2wAwBρσAσB, and the portfolio standard deviation is its square root. The third term carries the correlation ρ between the assets — it is what makes portfolio risk depend on how the pieces move together, not just on how risky each piece is alone.

Because the assets don’t move in perfect lockstep. The weighted average wAσA + wBσB is what risk would be if ρ = 1 — every wiggle in one asset matched by the other. Whenever ρ < 1, some of one asset’s bad days land on the other’s good days and partially cancel, so the combined standard deviation comes out below the average. That gap is the diversification benefit, and it costs nothing in expected return — the reason diversification is often called the only free lunch in investing.

It is the mix with the lowest possible risk given the two assets’ volatilities and correlation: wA* = (σB² − ρσAσB) ÷ (σA² + σB² − 2ρσAσB). Counterintuitively, it usually is not 100% of the safer asset — with the defaults here, holding about 7% stocks alongside bonds is less risky than holding bonds alone, because low-correlation stock swings offset some bond swings. When the formula lands outside 0–100%, the long-only answer clamps to the nearer endpoint.

Estimate it from history — the correlation of monthly or annual returns over a period you consider representative — or take a published figure for your asset pair. US stocks and Treasury bonds have historically ranged roughly from −0.3 to +0.4 depending on the decade and inflation regime, so a value near 0 to 0.3 is a common baseline. Be humble about it: correlations drift, and they famously rise toward 1 in crises, exactly when you want diversification most. Test a pessimistic (higher) ρ to see how much benefit survives.

Yes — the same mathematics generalizes to any number of assets: expected return stays the weighted sum, and variance becomes wᵀΣw, the weight vector times the covariance matrix, so an n-asset portfolio sums n² covariance terms. Nothing conceptual changes: pairwise co-movement still drives total risk, and as n grows, the covariances — not the individual variances — dominate. The two-asset case is the standard classroom version because every effect is visible by hand, but real optimizers solve exactly this problem in matrix form.