# Is Equity Cheap?

This is the question which is dominating financial media at the moment. Case-Schiller point out that their cyclically adjusted PE ration (CAPE10), is above historically adjusted norms. My thesis in this post is that the CAPE10 actually indicates that the stock market is cheaply-fairly valued depending on the expected future path of interest rates. In fact, I am going to use Shiller’s own data to explain why I think he is wrong.

Equity Valuations

The most theoretically sound way to value equities is the present discounted cash flow. That is to say, the current intrinsic value of a stock is the sum of all future profits discounted by the risk free rate. Of course, we can never be sure about the future, so is born the concept of a risk premium, that equities should be valued slightly lower than their fair value because we have a high degree of uncertainty about the future. My preferred way of saying this in maths is:

$\frac{\text{Price}}{\text{current earnings}} = \sum^{\infty}_{t=0} \exp((-R+G-\rho)t) \approx \frac{1}{R-G+\rho}$

where we have defined that R is the risk free interest rate, G is the growth of the company, and $\latex \rho$ is the equity risk premium, which is at heart, the uncertainty about growth and interest rates. The approximation follows from the formula for a geometric sum if one assumes that $R-G+\rho$ is small. Obviously, such a valuation builds in some fairly crucial assumptions. Firstly, that growth is smaller than the interest rate and the risk premium. A share whose earnings growth continually out earns interest rate and uncertainty premiums is worth infinity. Of course, this occurs because we assumed growth was a fixed parameter, which it is not, but such simplifications are sufficient for our current purpose.

Using the Case Shiller data, and using the above analysis, I have plotted the PE ratio for ten year US treasuries along side the CAPE10 ratio for the S&P500, since 1950.

Notice the incredible correlation between bond returns and equity returns over the proceeding decades as tail risks of WW2, the cold war, and thermonuclear war abated.

So the question, “are stocks cheap”, is now seen in its proper context. Stocks are cheap if you expect interest rates to stay at their current levels for an extended period. If you expect that interest rates will normalise around 5% in the not to distant future, then stocks are fairly valued. I see think that interest rates are still some years away from 5%, and see rather evidence of continued tail risk from the Eurozone’s self inflicted misery. Notice quite what an aberration the tech bubble was. Even at the end of the dot com crash stocks were not especially cheap compared to bonds. Notice that stocks were incontrovertibly cheap post financial crisis in 2009. Anyway, I am cautiously optimistic in the long run/medium run for stocks, but still worried about the Eurozone. The ongoing crises in Europe seems temporarily in the background of investor’s minds, either because they believe that the ECB will bow to pressure and follow the other central banks, or because they believe that liquidity from other parts of the world will alleviate Europe’s problems despite ECB intransigence. I would not bet on either.

Disclosure, I am 30% in cash, 70% long equities, with 0% European exposure.