# 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).

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).