Shiller Asset Pricing

Every Tom, Dick, and Harry imagines himself to be a savvy investor. People don’t realize how badly they are outmatched by the big fish. Investment management is a capital-intensive industrial enterprise. In Advances in Financial Machine Learning, Prado gives the analogy of what happened to gold mining:

In less than a hundred years, the Spanish treasure fleet quadrupled the amount of precious metals in circulation throughout Europe. Those times are long gone, and today prospectors must deploy complex industrial methods to extract microscopic bullion particles out of tons of earth. … The discovery of investment strategies has undergone a similar evolution. … Individuals searching nowadays for macroscopic alpha, regardless of their experience or knowledge, are fighting overwhelming odds. The only true alpha left is microscopic, and finding it requires capital-intensive industrial methods.

People are easily fooled by anecdotal evidence. Everyone knows someone who made great returns by calling something or the other. They may themselves have had the pleasant experience of making a great deal of money. The problem is that it is, almost without exception, pure luck — you just can’t repeat it. Taleb calls it getting ‘fooled by randomness’. Of course, if you know a bunch of people picking stocks, some of them are going to have a great run simply due to randomness. The truth is that, like the lottery, stock picking is a tax on regular folk.

In the technical literature, there is a word for retail investors. They are called noise traders. Systematically extracting money from them are informed traders — the big fish. These are not necessarily people trading on insider information. But they have access to resources you cannot possibly imagine. Stock investing functions as a gigantic machine for extraction of profits from retail investors to wholesale investors.

Since the market cataclysm began people have been asking me for investment advice. I told my Facebook friends (if you are a regular reader of the Policy Tensor connect with me on Facebook) that they shouldn’t bother trying to pick winners. Here’s what I wrote:

1. Don’t try to pick stocks. You may have good reason to believe that a company will do well with certainty. But you don’t have any way of knowing whether its stock is overpriced or underpriced at any given time. Even if you have a valuation model that tell you what a fair price for a stock would be, the market can defy you essentially forever. In any case, the big fish have armies of stock pickers who do this for a living. And even their track record is atrocious. There is no reliable way to do it. Just pick a broad diversified index with minimal admin fees.

2. If you haven’t already sold your stocks, stay invested and ride out the storm. Selling now will guarantee that you book a devastating loss. More importantly, you will likely miss the rally when it comes. Forget about the stock market; go read a book. The mayhem will eventually end. You can pat yourself on the back later for having the stomach to ride it out.

3. If you are sitting on cash, you need to decide when to buy back in. You will not be able to time the market bottom with any precision. We know that stock markets rebound well before recessions end. This is particularly true of deep V-shaped recessions like this one. So by the time fundamentals start looking good, it’s too late. What should you do?

4. The best strategy is to have a firm, set-in-advance, historically-determined cutoff for a basic valuation tool like the PE ratio or the earnings yield. When the value drops below that level, buy back in and sit tight. The most reliable valuation tool out there is Robert Shiller’s cyclically-adjusted price-earnings ratio, the CAPE.

5. The current value of the CAPE for the SP500 is 21.45, which is the 30th percentile of its value since the fall of the Berlin Wall. I would say when it hits 18.5 (the 10th percentile over the same period), dive back in. That’ll probably happen soon enough. Or, if you strongly believe that there is ways to fall, wait till the CAPE hits 13.32 — the level it hit at the bottom of the Great Recession. In any case, make the decision beforehand, tune out the noise, and stick to it. This is the best that a retail investor can hope to do. There is no known way to beat this strategy. Unless, of course, you want to invest in my vol strategy.

Since then, the CAPE has dropped below 21 — the market moves fast these days. The next figure displays the long-term behavior of Shiller’s CAPE, along with a more recent alternative he has come up with, the Total Return CAPE, which adjusts for changes in corporate buybacks and payout policies.

CAPE.png

After writing that note on Facebook I became curious about whether the CAPE is priced into the cross-section of stock returns. In English: whether stocks that are more sensitive to the CAPE exhibit higher returns than stocks that are less so. Since a high value of CAPE predicts poor returns, stocks that go up more when the CAPE does are more exposed to overvaluation that stocks that are less sensitive to the same. Since investors must be compensated for this risk, they must sport higher returns than their less sensitive counterparts.

In order to test this theory, I extracted monthly returns from July 1926 to September 2018 for 25 portfolios sorted by market cap and book-to-market ratio from Kenneth French’s website. Then I computed betas for each of them by regressing their excess returns (in excess of the risk-free rate) on the risk factors in the time series, using OLS. Then I regressed mean excess returns in the cross-section, again using OLS, to obtain the prices of risk. This is called the standard 2-pass regression for cross-sectional asset pricing.

I used the monthly returns on the Total Return CAPE as a risk factor. I compared it to the Fama-French benchmark risk factors: Market, Size, and Value. Since the portfolios are sorted precisely on Size and Value, it would be incredible if our risk factor outperformed them. The question is how close it can get to the very factors from which the portfolios are constructed in explaining the cross-section of expected excess returns. The answer is pretty close indeed. The next figure shows the cross-sectional performance of CAPE. This is quite striking.

CAPE_priced.png

How well does the TR-CAPE do compared to the Fama-French benchmarks? Table 1 reports our estimates. A good pricing model is one for which the price of risk (the slope in the cross-sectional regression) is large and significant, the zero-beta rate (the intercept in the cross-sectional regression) vanishes; whose R^2 is high and mean squared error is low. The best metric is the mean absolute pricing error (MAPE) that sums to the absolute value of the zero-beta rate and the mean of the absolute value of the errors. We can see that the TR-CAPE performs splendidly. The price of this risk is high and significant. It explains nearly 60 percent of the cross-sectional variation in expected excess returns — more than Market and more than the Size factor! Moreover, its mean absolute pricing error is small — much smaller than the Market model and nearly as small as Size. It also sports the smallest mean squared error.

Table 1. Cross-sectional Regressions. 
CAPE MKT SMB HML
Zero-beta rate 1.377 3.278 -0.762 -0.853
std error 0.207 0.404 0.729 0.484
Price of risk 1.211 1.592 3.223 2.491
std error 0.166 0.345 0.524 0.357
R^2 0.585 0.516 0.522 0.670
Adj. R^2 0.567 0.495 0.502 0.656
MSE 0.121 0.275 1.099 0.353
MAPE 1.623 3.676 1.550 1.296
Source: Robert Shiller, Kenneth French, author’s computations. Std errors are Newey-West’s heteroskedasticity robust standard errors. MAPE=Mean absolute pricing error. Estimates in bold are significant at the 5 percent level. Estimates for single factor models only. The test universe is 25 portfolios sorted on SMB and HML obtained from Kenneth French’s website. The data is for monthly returns from July 1926 to Sep 2018.

The bottom line is that Shiller’s TR-CAPE is strongly priced into the cross-section of expected stock excess returns. This makes a lot of sense. Much more sense, in fact, than Size and Value. For there is no theoretical reason why stocks of firms with smaller market size or higher book-to-market ratios should sport higher returns. The empirical performance of these factors is rather an anomaly to be explained. On the other hand, it makes ample sense that stocks that tend to get overpriced during asset price booms (as captured by the CAPE) pose a systematic risk to your marginal investor and should therefore compensate her for that risk.

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