Risk and return of P = wX + (1 − w)Y
Simulated joint returns
All in X
All in Y
Your portfolio
Minimum variance
Mixes at the current ρ
Mixes at ρ = −1, 0, +1
Put a fraction w of your money in X and the rest in Y. Move the weight and the correlation and watch the portfolio's mean and risk update, term by term. The left chart traces every mix from all-Y to all-X; the right chart shows the joint draws behind it.
← All labsBased on Chapter 1, Probability and Uncertainty, of Polson and Sokolov, Bayes, Artificial Intelligence, and Deep Learning (vsokolov.org/html/_book).
Risk and return of P = wX + (1 − w)Y
Simulated joint returns
Tortoise and Hare both sell 1.5 cars on average. Tortoise sells 1 or 2 (Var 0.25), Hare sells 0 or 3 (Var 2.25). Assume they are independent and splits the business 50/50. The variance falls to 0.625 with the same expected sales.
\(E(wX + (1 - w)Y)\) \(= w \cdot E(X) + (1 - w) \cdot E(Y)\). Mixing never changes the mean in a surprising way. The portfolio dot always sits between the two assets vertically.
\(\mathrm{Var}(aX + bY)\) \(= a^2\mathrm{Var}(X) + b^2\mathrm{Var}(Y)\) \({} + 2ab \cdot \mathrm{Cov}(X, Y)\). The last piece is the diversification term. With \(\mathrm{Cov} > 0\) it adds risk; with \(\mathrm{Cov} < 0\) it removes risk.
At \(\rho = +1\) the mixes lie on a straight line between \(X\) and \(Y\), so there is no benefit. At \(\rho = -1\) the curve touches \(\mathrm{sd} = 0\). Every \(\rho\) below +1 bends the curve to the left of that line.
Set the derivative of \(\mathrm{Var}(P)\) in \(w\) to zero. For Tortoise and Hare at \(\rho = 0\) this gives \(w^* = 2.25 / 2.5 = 0.9\), with Var 0.225. That is less risky than Tortoise alone, because a little Hare averages away some of Tortoise's own noise.
| Mix | ρ | Var(P) | sd(P) |
|---|---|---|---|
| All Tortoise | – | 0.250 | 0.500 |
| All Hare | – | 2.250 | 1.500 |
| 50/50 | 0 | 0.625 | 0.791 |
| w* = 0.9 | 0 | 0.225 | 0.474 |
| w* = 0.75 | −1 | 0.000 | 0.000 |
Takeaway for students. The benefit of mixing depends on the correlation, and correlations change with the regime. The 2022 lesson: stocks and bonds fell together when both reacted to the same Fed shock, and the diversification from bonds disappeared. Compare the two SPY / TLT presets. Growth and value move together (\(\rho \approx +0.85\)), so mixing them is not really diversification.
The coin that pays $2 or $0 has \(E(X) = \$1\) and \(\mathrm{Var}(X) = 1\). One play never pays $1, so the mean is not the whole story.
Covariance has awkward units (cars × cars, or %²). Dividing by both SDs gives a number between −1 and +1.
The Polymarket basket spreads bets over a World Cup, a tech market-cap race and a US primary. With \(\rho \approx 0\) between them, \(n\) equal bets cut the variance roughly by a factor of \(n\).