# 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. $f(\mathbb{E}[x])$?

• By market-weighting the P/B ratios you will get a bias towards companies with relative high P/Bs as market value is part of the equation. I'd assign a book value weight based on the total book value. Or more simple, divide the market value of the sector by its total book value. Further it would be wise to have a closer look at each company to make sure they are comparable, in other words make some correction for excess cash etc. – Tim Jan 16 '17 at 17:01
• @Umang Gupta. Is there any further information you need before you accept one of the answers below? – David Addison Apr 25 '17 at 18:49

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.

1. 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.

1. 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.

The answer to your question would depend on what is the intended use case for the number. If you follow convention and use it for similar use cases the second choice would be the right number to look at. http://www.indexologyblog.com/2014/02/07/inside-the-sp-500-pe-and-earnings-per-share/

If the book values and market values of each company are $B_i$ and $M_i$ respectively, then the book-to-market ratio for the sector is

\begin{align} B/M & = \frac{\sum_i B_i}{\sum_i M_i} \\ & = \frac{\sum_i M_i (B_i/M_i)}{\sum_i M_i} \\ & = \sum_i \frac{M_i}{\sum_j M_j} \frac{B_i}{M_i} \\ & = \sum_i W_i \frac{B_i}{M_i} \end{align}

where $W_i=M_i/\sum_jM_j$, i.e. the book-to-market ratio for the sector is the market-weighted average of the book-to-market ratios for the individual stocks.

The price-to-book ratio for the sector is the inverse of this, i.e.

\begin{align} M/B & = \frac{1}{\sum_i W_i \frac{B_i}{M_i}} \\ & = \left( \sum_i W_i \left(\frac{M_i}{B_i}\right)^{-1}\right)^{-1} \end{align}

so the price-to-book ratio is the weighted harmonic mean of the price-to-book ratios for the individual stocks, where the weights are the market values.

Alternatively, you can use the equivalent definition

\begin{align} M/B & = \frac{\sum_i M_i}{\sum_i B_i} \\ & = \frac{\sum_i B_i (M_i/B_i)}{\sum_i B_i} \\ & = \sum_i \frac{B_i}{\sum_j B_j} \frac{M_i}{B_i} \\ & = \sum_i W^B_i \frac{M_i}{B_i} \end{align}

where $W^B_i = B_i/\sum_j B_j$, so the price-to-book ratio for the sector can also be expressed as the book-weighted average of the price-to-book ratios of the individual stocks.