Two assets, one formula: the variance of a pair

The two-asset case contains the entire mechanism, and it is small enough to hold in one line. Everything later in this track is this formula with more terms.

Put weights of 0.55 and 0.45 on holdings with volatilities of 31 and 22 percent. The first two terms are 0.55 squared times 0.31 squared = 0.02907025, and 0.45 squared times 0.22 squared = 0.00980100. They sum to 0.03887125 and they do not move for any value of the correlation. Only the third term moves. At a correlation of 0.15 it is 0.00506385, giving a variance of 0.04393510 and a volatility of 20.96 percent. At 0.65 it is 0.02194335, giving 24.66 percent. At minus 0.40 it is negative 0.01350360, giving 15.93 percent.

The boundary case is the one to memorise. At a correlation of exactly 1 the variance is 0.07263025 and the volatility is 26.95 percent, which is also 0.55 times 31 plus 0.45 times 22. A perfectly correlated pair's volatility is the plain weighted average of its parts. Every diversification benefit that exists anywhere is the gap between that weighted average and the actual answer, and the gap exists only because the correlation is below one.

That identity has a direct collateral consequence. A blended advance rate is a weighted average of per-asset advance rates, which is arithmetic that behaves as though every correlation were 1. Track 1 lesson 3 introduced lending value as market value times an advance rate; the blending step is linear, so it cannot represent diversification at all. Whatever credit a schedule gives for holding different things has to come from somewhere else, and lesson 24 shows where.