index - What is the better representative of a P-B ratio for a sector?

What is a better representative of a P-B ratio for a sector, for using it as a factor to predict future returns on that sector? The market weighted average of P-B for all names in that index, or market weighted average of all prices/ market weighted average of all book values, basically $\mathbb{E}[f(x)]$ vs. $f(\mathbb{E}[x])$?

Answer

This is a question of how to aggregate ratios.

I see your two options, and raise you one more.

1. Method 1: The mean (or median) value of Price-to-Book values for individual securities

$$f(E[x]) = \frac{\sum_{i}^{n} \frac{P_i}{B_i}}{n} $$

Pros: Relatively simple to calculate and gives an idea of what the typical company's Price-to-Book is.

Cons: Individual company values can tend towards extremes. Because it is a ratio, you will get asymtpotically small and large numbers in your data-set which skew the results. For example, a relatively large cap company can have an extremely high or even negative book value. Likewise, relatively small cap companies, through any number of accounting anomalies (anyone remember Chinese reverse take-overs and variable interest entities?), can have very large book values.

2. Method 2: The aggregate value

$$E[f(x)] = \frac {\sum_{i}^{n} {P_i}}{\sum_i^n B_i} $$

Pros: Also relatively simple to calculate. Resilient to outliers and small-cap anomalies.

Cons: Tends to be skewed towards larger cap companies which may not give a good idea of the typical company's price-to-book.

3. Method 3: The harmonic mean defined by: $$\frac{1}{f(E[x^{-1}])} = \frac{n}{(\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}+...\frac{1}{x_n})} $$

Pros: useful for taking the average of certain ratios. In this case, where $x_i$ represents a security's price-to-book, it is essentially taking the mean of the inverse of price-to-book (i.e., book-to-market). Taking the mean of the book-to-market will be more resilient to outliers since market caps are on the denominator and cannot be zero or less than zero. You will thereby eliminate the problem of asymptotic values.

Cons: It may be inappropriate in certain situations.

Personally, I prefer the sector aggregate method because it gives an unbiased estimate of the actual mean. In certain cases, when I am trying to get an idea of the typical company's multiple, I will take the median value. In situations where it seems to appropriate to take the harmonic average, I can't help but to think that I should've just started with the inverse as my baseline.