Calculating an Equal-Weighted Index P/E with the Harmonic Mean
Summary
The document explains how to calculate the price-to-earnings ratio of an equal-weighted equity index from the constituent stocks' P/E ratios. Its answer gives the harmonic mean: take the number of stocks and divide by the sum of the reciprocal constituent P/E ratios. This differs from a simple arithmetic average because an index-level P/E is total portfolio market value divided by the earnings attributable to the portfolio.
A holding-company analogy derives the result. Equal weighting means the portfolio owns different fractions of each company, with smaller market-cap companies requiring a larger ownership fraction to receive the same portfolio allocation. Summing the earnings associated with those holdings and dividing portfolio value by that earnings total produces the harmonic-mean expression. The derivation assumes the constituent P/Es and earnings are defined consistently. The document does not discuss how to handle firms with zero or negative earnings, which can make ordinary P/E ratios or their reciprocals unsuitable.
Key ideas
- An equal-weighted index P/E is the harmonic mean of its constituents' P/E ratios.
- The harmonic mean follows from dividing total portfolio market value by portfolio earnings.
- Equal portfolio allocations imply different ownership fractions across companies with different market capitalizations.
- The calculation depends on meaningful, consistently defined constituent P/E ratios.
- Zero or negative earnings require special treatment that the document does not specify.
Tags
Full text
# Calculate PE ratio of equal-weighted index
# Calculate PE ratio of equal-weighted index
I need to calculate `Price-to-Earnings Ratio (PE Ratio)` of an Equal-weighted index.
```
Sum of market caps
P/E for Market-cap Weighted index = -------------------
Sum of Earnings
Sum of market caps Shares1*Price1 + Shares2*Price2 + ...
P/E for Equal Weighted index = ?? ---------------------------- OR --------------------------------------
(Sum of Earnings/# of stocks) Shares1*EPS1 + Shares2*EPS2 + ...
```
where Shares1, Shares2, etc are number of shares in the portfolio (replicating the index).
## Answer by Alper (score 1)
https://quant.stackexchange.com/a/74563
You just need to take the (simple) harmonic mean of the P/E's of the stocks in the index given that it's an equal-weighted equity index.
Suppose you have $N$ stocks in the equal-weighted index. Then, using the harmonic mean formula, the index's P/E would be calculated as:
$$\text{P/E}_{\text{index}} = \dfrac{N}{\mathop {\sum}\limits_{i=1}^{N}\frac{1}{\text{P/E}_i}} \tag{1}$$
where $\text{P/E}_i$ is the P/E of stock $i$ and calculated as $\frac{M_i}{E_i}$ with $M_i$ being the stock $i$'s market cap and $E_i$ its earnings.
The logic behind the use of the harmonic mean can be explained by using the holding company analogy which I also used in my answer to another question about index P/E's.
Suppose the assets of a holding company $H$, whose market cap is $R_H$, comprise of $N$ stocks with equal weights. The holding company's earnings, $E_H$, would then be the sum of the earnings of the stocks, $E_i$, weighted by the holding company's respective ownership ratios, $x_i$, in each. Then, the holding company's P/E would be:
$$\begin{align} \text{P/E}_{H} &= \dfrac{R_H}{E_H} \\ &= \dfrac{R_H}{\mathop {\sum}\limits_{i=1}^{N}x_iE_i}. \tag{2} \end{align}$$
Assuming $M_i$ again denotes the market cap of the stock $i$, we can figure out $x_i$ as follows:
$$\begin{align} x_iM_i &= \dfrac{R_H}{N} \\ x_i &= \dfrac{R_H}{NM_i}. \tag{3} \end{align}$$
Substituting (3) into (2), we find:
$$\begin{align} \text{P/E}_{H} &= \dfrac{R_h}{\mathop {\sum}\limits_{i=1}^{N}\left(\dfrac{R_H}{NM_i}\right)E_i} \\ &= \dfrac{R_H}{\dfrac{R_H}{N} \mathop {\sum}\limits_{i=1}^{N}\left(\dfrac{1}{M_i}\right)E_i} \\ &= \dfrac{N}{\mathop {\sum}\limits_{i=1}^{N}\dfrac{E_i}{M_i}} \\ &= \dfrac{N}{\mathop {\sum}\limits_{i=1}^{N}\dfrac{1}{\frac{M_i}{E_i}}} \\ &= \dfrac{N}{\mathop {\sum}\limits_{i=1}^{N}\frac{1}{\text{P/E}_i}} \tag{4}\end{align}$$
which is the same as the equation (1).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.