Risk against return scatter plot

Four markers sit where a rough teaching version of each asset class falls: cash low and flat, bonds a little higher, a broad stock index higher again, and one company's stock highest on both axes. Up is more expected return. Right is more risk, measured as the standard deviation of yearly returns.

The solid green curve is every possible mix of two of them. The dashed straight line is where those same mixes would sit if the two assets always rose and fell together. At any height on the chart both lines carry the same expected return, so the horizontal gap between them is risk that mixing removes with nothing given up in exchange. That gap is what the panel calls risk saved by mixing, and it exists only because the two assets do not move in step.

When the curve bends far enough left, some mixes carry less risk than the safer asset held on its own. That point is the minimum-variance mix, and it is the reason a small holding of something volatile can lower the risk of the whole rather than raise it.

Expected return is a plain weighted average. With weight $w_1$ in the first asset and $w_2$ in the second, the two adding to 1:

$$E(r_p) = w_1 r_1 + w_2 r_2$$

Risk is not an average, and that difference is the whole story:

$$\sigma_p = \sqrt{w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \rho \sigma_1 \sigma_2}$$

Correlation $\rho$ runs from -1 to 1 and enters only through the last term. At $\rho = 1$ the square root collapses to $w_1 \sigma_1 + w_2 \sigma_2$, the weighted average, and the curve becomes the dashed straight line. Below 1 that last term shrinks, so the mix carries less risk than the average of its parts.

The returns plotted here are yearly rates before inflation. Turning one into what it buys is a separate step, done by the real return calculator.

Read this plainly: the four marker positions are round teaching figures picked to show the shape of the trade-off. They are not measured history, not a projection of what any asset class will return, and not advice about what to hold. Real standard deviations and real correlations move around, and correlations in particular tend to rise in exactly the falling markets where a low one would have helped most.

What does carry over is the shape. Expected return is linear in the mix and risk is not, so the two never move together point for point. That relationship holds whatever numbers you put in the corners.

This page is educational material about how the arithmetic works, not financial advice.