The Diversification Effect

Everyone knows diversification is the one free lunch in economics: you can lower your risk without lowering returns by seeking a portfolio that diversifies idiosyncratic risk away. But did you know it can increase one's returns, too?

From Harvey and Erb:

...of the 36 individual commodity futures that Gorton and Rouwenhorst (2005) studied, 18 had geometric excess returns that were greater than zero and 18 had geometric excess returns that were less than zero.

The average annual standard deviation of the 36 commodity futures was 30%. Bottom line, the average return of the average commodity futures was not statistically different from zero. Alternatively, the average commodity futures had an average geometric “risk premium” of zero.

Interestingly, though, the return of an equally weighted, and monthly rebalanced, portfolio of commodity futures in Figure 2 had a statistically significant return of about 4.5%.

Basically, they present the following rule of thumb to estimate the portfolio diversification factor (ie, that subracted from the arithmetic return to get the geometric return): 0.5*(1-1/k)*(avg var)*(1-avg corr)

where k is the number of positions. Harvey and Erb call this 'turning water into wine', because in futures, a portfolio of futures with individual geometric returns of 0, can rise to 4.5%, merely by equal weighting the futures (or thereabouts). I guess this is pretty trivial given that a geometric return is the arithmetic return minus the variance divided by 2 (approximately), but then again, all mathematics is a riff on tautologies.