Effective Number of Bets

Counting the number of positions in a portfolio tells you almost nothing about how diversified it actually is. A hundred stocks all in the same crowded momentum trade behave, in a crash, like a single position; two positions with zero correlation to each other and to everything else can carry more real diversification than that hundred-stock book.

The effective number of bets (ENB) fixes this by working off the portfolio's risk contributions rather than its position count. One common construction decomposes total portfolio variance into contributions from uncorrelated ("principal") risk factors, computes each factor's share of total risk $w_i$, and then applies the same formula used for the Herfindahl concentration index inverted:

$$ENB = \frac{1}{\sum_i w_i^2} .$$

In plain English: if risk were spread perfectly evenly across $N$ independent factors, each $w_i = 1/N$ and $ENB = N$ exactly — the formula returns the true count. As risk concentrates into fewer factors, $\sum w_i^2$ rises and $ENB$ falls below the raw factor count.

Worked example: a portfolio whose risk decomposes into three uncorrelated factors with shares $w = (0.8, 0.15, 0.05)$ has $\sum w_i^2 = 0.64 + 0.0225 + 0.0025 = 0.665$, so $ENB = 1/0.665 \approx 1.5$ — despite having three named risk factors, the portfolio behaves, risk-wise, like it's carrying barely more than one and a half independent bets.