## Monday, May 29, 2017

### Price Earning Ratios for Portfolios When Some Firms Have Negative Earnings

The Price Earnings Ratio of a Portfolio Where Some Firms Have Negative Earnings

Issue:  How does one create an accurate measure of price in relation to earnings for a portfolio of stocks when some firms in the portfolio have negative earnings?

Question:    A person owns three stocks Apple, Microsoft, and Twitter.   The stock price and earnings per share for the three stocks are listed below.

 Earnings Per Share and Price Per Share of Three Stocks AAPL MSFT TWTR E 8.52 2.27 -0.62 P 153.61 69.96 18.23

A person invests 40 percent of her funds in Apple, 40 percent in Microsoft and 20 percent in Twitter.   What is the PE ratio of this portfolio?

Background:   In a previous post I explain that negative PE ratios have no economic meaning.   Firms with negative earnings generally have a lot in common with high PE startups that have small current earnings and high expectations for future earnings.   Moreover, a firm with negative earnings that is close to zero will have a lower PE ratio than a firm with huge losses.

Some analysts appear to average all firms (both firms with negative and positive earnings) to get a portfolio PE.   This approach is nonsense.

Other analysts simply omit information on PE ratios for firms with negative earnings.   This approach ignores important information by understating the deviation between perceived value of stocks measured by price and earnings for the current year.

How does one create an accurate measure of price in relation to earnings for a portfolio of stocks when some firms in the portfolio have negative earnings?

My Solution:   My solution is based on the statistic (P-E)/P.  The advantage of the (P-E)/P statistic compared to the P/E statistic is that it has the same economic meaning for firms with positive and negative earnings.   This statistic compares the amount the value of the price of stock exceeds earnings for one year.   This difference represents future expected earnings, liquidation value or some combination of the two.

When earnings are negative as in the case of Twitter (1-P)/P will exceed 1.0.

Using algebra, it is easy to show that there is a one to one relationship between (P-E)/P and P/E.   Denote (P-E)/P as f.  Use algebra (shown in appendix below) to show

P/E=1/(1-f)

My procedure for finding the P/E of the portfolio involves three steps:

·      Find (P-E)/P for all three stocks
·      Take weighted average of (P-E)/P for all three stocks
·      Plug weighted average of (P-E)/P into formula for P/E shown above to get portfolio P/E.

Step One:

The value of (P-E)/P for the three stocks are presented below.

 Earnings Per Share and Price Per Share of Three Stocks AAPL MSFT TWTR E 8.52 2.27 -0.62 P 153.61 69.96 18.23 (P-E)/P 0.94 0.97 1.03 1/(1-(P-E)/P) 18.03 30.82 -29.40 P/E 18.03 30.82 -29.40

Note the correspondence between (P-E)/P and P/E is correct.

Step Two:

Find weighted average of (P-E)/P where weights are 0.40 of Apple, 0.40 for Microsoft, and 0.20 for Twitter.   This can be done with the SUMPRODUCT function in Excel.

The SUMPRODUCT of the (P-E)/P vector with the portfolio share vector (0.40,0.40,0.20) is 0.971637.

Step Three:   Transform the weighted average of (P-E)/P to a P/E ratio.

1/(1-0.971637)  or 35.25.

Concluding Thoughts:    Many analysts citing PE ratios for Tech stock ETFs claim that Tech stocks are not overvalued.   The analysis presented here suggests this claim is incorrect.  I suspect these analysts are overvaluing the Tech sector by failing to accurately incorporate information about firms with negative earnings in their calculation of Tech Sector PE ratios.

My finance post discusses the literature on PE ratios and looks at the frequency distribution of the PE ratios in Vanguard Tech fund VGT.

I am concerned that we have entered a bubble that will not end nicely.

A Note:   When firms are near bankruptcy (P-E)/P could be quite large because earnings are a very large negative number.   The PE of this firm might not have a large impact on the P/E ratio because the low share price of a firm near bankruptcy will lead to a portfolio share near zero.   However, if portfolio share estimates are not up to date the inclusion of firms near bankruptcy in the PE calculation might overstate financial risk because the investor cannot lose more than the value of the outstanding shares.

The Tech firms in VGT have negative earnings with high stock prices that have risen in future months.    These high-flying firms need to be included when assessing portfolio risk.

Appendix:    Show that P/E = 1/(1-f) where f is (P-E)/P using P and E data for Microsoft.     The algebra is shown below.

0.94453 = (P-E)/P

Rearrange this equation to solve for P/E.

(1-0.94453)*P =E

or

P/E= 1/(1-0.94453)

The more direct way to get P/E is simply divide P by E.   Confirm that both statistics equals to 18.029 for Microsoft.