Efficient Frontier

Sunday morning. Friend over chai. Says he is 100 pct equity. Asks why he would add gold to the book if gold compounds slower than equity.

Asked myself the same question the first time I read Markowitz. The honest answer is the one nobody believes until they see the math.

Adding a lower return asset can leave the portfolio better off.

Take the toy. Indian equity at 13 pct expected return and 19 pct annualised vol. Gold at 8 pct return and 15 pct vol. Correlation around 0.15 in calm regimes. Pure equity book sits at 13 / 19. Drop a 20 pct gold sleeve in. The portfolio now sits at 12.0 pct return and 15.9 pct vol.

One pp of return surrendered. Three pp of volatility absorbed. The brochure number went down. The portfolio went up.

The reason is in the variance formula. Portfolio variance is not a weighted average. It carries a cross term. Two times weight one, weight two, vol one, vol two, correlation. When correlation is low the cross term shrinks, and the whole variance shrinks with it. That is the line of math seventy years old that still trips most people on first reading.

Run the mix lower. Seventy thirty lands at 11.5 / 14.7. Sixty forty at 11.0 / 13.7. The minimum variance point in this toy sits at 36 / 64 equity gold, with portfolio vol of 12.6 pct. That number is below pure equity vol of 19, and below pure gold vol of 15. The mix is calmer than either ingredient. That is not a bug in arithmetic. That is the cross term doing its job.

The judgment that follows. Gold compounds slower than equity. So does cash. So do most hedges. If you reject every diversifier on its return number alone, you reject the frontier. The question is not what an asset earns alone. The question is what it does to the portfolio when the rest of the book is bleeding.

How correlation, variance and the frontier sit underneath every multi asset book