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).


1 Answer 1


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).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.