#### CAPE and the Growth Problem

The way Robert Shiller calculates CAPE (from his book Irrational Exuberance as illustrated by his data) is by taking the current real price and dividing that by the average real earnings for the last 10 years.)

$$\text{CAPE} = \frac{\text{Real Price}}{\text{Average 10 Year Real Earnings}}$$

The goal is to "normalize" earnings because it's going to vary depending on things like the business cycle. The problem with this calculation is when there's a substantial amount of real growth.

Imagine two hypothetical stocks. Both stocks earn a return on capital that is equal to their cost of capital. As a result, if the company decides to retain earnings for growth, this adds no value. We are exchanging \$1 in dividends today for \$1 worth of future dividends. As a result, the stock should be valued the same if they choose to pay out dividends or retain earnings for growth.

So here are company A and B. Company A pays out all of its earnings while company B retains a sufficient amount to achieve 4% real growth. I'll assume all of the figures are inflation adjusted figures and that inflation affects prices in the same way.

One simple way this could be achieved is if the company performed share buybacks instead of paying dividends. This is equivalent to receiving a dividend and reinvesting it (ignoring taxes). As a result, both companies ought to trade at the same forward PE since the growth neither adds nor subtracts from the value of the company (if return on equity were higher than cost of equity, that would not be the case.)

Here's what it looks like:

One consideration here is whether or not we ought to use the forward earnings in our average or trailing earnings. I calculated both for illustrative purposes. But whichever the case, it's clear that even though these companies should be worth the same, Shiller's CAPE says that company B is more expensive (forward CAPE is 17.78) and company A is a better value (forward CAPE of 15). This is clearly a deficiency of Shiller's CAPE.

#### Alternative CAPE

In the previous piece, I gave an alternative means of calculating CAPE that still serves the function of normalizing earnings while also addressing the growth problem. Instead of averaging the last 10 years of earnings we normalize earnings by averaging ROE and multiplying it by current book value (one could do the same with ROIC and current invested capital.) The result is an alternative CAPE calculation:

$$\text{Normalized Earnings} = \text{Average}(ROE) \times BV$$

$$\text{CAPE}_\text{ALT} = \frac{\text{Price}}{\text{Average}(ROE) \times BV}$$

So that's the calculation.

#### Regarding the Data

Previously I only had about 14 years of data. I now have about 35 years of data with a few caveats. Here's my data source.

- The data is only annual instead of quarterly.
- S&P changed the way they calculate book value (by including intangible assets). They adjusted this back to 1992. The authors (of the above chart) used individual stock data to calculate book value back to 1979.
- I didn't actually have the data so I estimated from the chart.

#### S&P 500 Return on Equity

So here's the historical ROE for the S&P 500 (both the annual figures and the 10 year rolling average).

While year over year ROE can be quite volatile, the "normalized" ROE is on average about 13% and stayed within 11.4% to 14.2% range.

#### S&P 500 Shiller versus Alt CAPE

So we'll look at this in two different ways. First, here's the relevant CAPE ratios:

As of January 2014, Shiller's CAPE (calculated using annual instead of monthly figures) was at 24.09 versus my alternative CAPE coming in at 19.76.

The red and blue lines represent the (harmonic) average over this period. One thing that's interesting to note is that Shiller's CAPE is actually slightly above its average but my alternative CAPE is right at the average.

Another way to look at this, and personally I find this more insightful, is to invert it to get a

*cyclically adjusted earnings yield*(CAEY):

This chart just inverts the above chart. Shiller's yield comes in at 4.15% versus the alternative yield of 5.06%.

So where should they be at? I don't know. Given our low interest rate environment, 5% doesn't sound too far off.

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