Lecture Notes on the Equity Premium: Part I

Lecture Notes on the Equity Premium: Part I
J. Bradford DeLong
March 2, 2006

There is, somewhere, a marginal investor: somebody just about indifferent between stocks and bonds. If the expected return to stocks were a little higher, he or she would move more money to stocks. If the extra risk associated with holding stocks were a little higher, he or she would move more money to stocks. In either case, that increase in demand for stocks would push their prices up, and so push the relative returns on stocks--which are the dividends that will be paid out on stocks in the future divided by the stocks' current price--down. But this marginal investor fears the risk as much as he or she values the return, and so does not move more money into stocks, and that is why the current price of stocks and the current value of the equity risk premium are what they are.

Now let's look at patterns of returns over the twentieth century, and try to figure out what this marginal investor has been seeing and thinking to make the equity risk premium what it has been, and then try to figure out what the current equity risk premium is.

Our first candidate marginal investor is someone who has some wealth that they want to invest for one year, and is considering whether to invest it in (relatively safe) one-year government bills (a "bill" is a short term bond) or in a (riskier) diversified portfolio of stocks. Figure 1 below plots the difference in returns for these two strategies for each year from 1900 to 2004--with the return differentials ranked from lowest to highest. 1931, with its (geometric) return differential of -60% (yes, 1931 was a bad year for the stock market) is at the far left; 1933 (the rich may not have voted for Roosevelt, but they certainly voted with their dollars that his "New Deal" was going to add value to American corporations) is at the far right.

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Given this distribution of one-year returns over the twentieth century, the marginal one-year investor looks at it and reasons as follows:

Stocks certainly pay a healthier and higher return on average: over the twentieth century the inflation-adjusted real stock return has averaged 6.6% per year, while the real one-year Treasury bill return has averaged only 1.7% per year for an average gap of 4.9% per year. But look at all that extra risk: there's a 35% chance that you will do worse with stocks than bonds, a 10% chance that you will--relatively--lose more than 1/5 of your wealth if you put it in stocks, and in 1931 the (geometric) relative stock return was -60%! Considering the portfolio positions I already hold, the extra expected return to moving a little more of the wealth I'll need to spend next year out of bills and into stocks is simply not worth the risk.

Thus we understand why a--rational, reasonable, well-informed--marginal investor putting money away to spend in a year would not decide that stocks are underpriced, that the equity risk premium is appropriate. And so such a marginal investor would not put downward pressure on the equity risk premium.

The problem--the reason that the large value of the equity risk premium is called a "puzzle" is that the marginal one-year investor is not the only possible marginal investor. Consider the marginal twenty-year investor: somebody 40 with ten-year-old children who is putting money away to spend on his or her children's college, or somebody 50 saving for expenditures at 70 after they have retired. This marginal investor has to be satisfied with the configuration of asset returns as well. And what does the distribution of twenty-year-returns--either buy and hold a distributed portfolio of stocks (reinvesting the dividends) or buying and rolling over short-term Treasury bonds--look like? The answer is shown in Figure 2, which plots the twenty-year return differential over the twentieth century. The average return differential is (of course) the same: 4.9% per year. Over twenty years that cumulates to a lot: e^{20 x .049} = 2.67. But much more important is the lower tail: only 4% of the time do stocks do worse than bills over a twenty-year horizon. And the worst observation is the twenty years starting in 1965, when investing in stocks yields -0.84% per year less than investing in bills--a relative wealth loss of 17%.

What kind of investor would turn up a 96% chance of gain, associated with an expected more-than-doubling of relative wealth, that carries with it only a 4% chance of any relative loss and a maximum loss of 17% of relative wealth? Where is this twenty-year marginal investor, and what is he or she possibly thinking? All such twenty-year marginal investors should be furiously pulling much more money out of short-term bills and investing it into diversified portfolios of stocks, thus making the equity risk premium lower and stocks less of an overwhelmingly attractive long-run investment. That is the equity premium puzzle.

One possible answer that initially looked promising was that the marginal twenty-year investors had already moved all the money they could out of the short-term money market: that it was hard to short-sell Treasury bills (a large part of the value of them, after all, is certainty of immediate liquidity, which nobody other than the government can promise), and that the holders of Treasury bills are overwhelmingly institutions that needed them for specific institutional liquidity or transaction-cost-minimization reasons. The problem with this possible answer is that the same equity premium puzzle emerges when we look at twenty-year differential returns between diversified stocks and investment-grade corporate or longer-term bonds, which are extraordinarily widely held, and which do not have the special certain-liquid-value properties of short-term Treasury bills. Figure 3 shows the distribution analogous to that of Figure 2, where this time the marginal twenty-year investor is considering the relative returns to investing in a diversified stock portfolio (and reinvesting the dividends) as opposed to investing in a bond portfolio (and rolling maturing bonds over into new issues).

This time the lower tail is even smaller: in only 1% of the years in the twentieth century would investing in bonds for twenty years outperformed investing in stocks; and in that year--1929--the twenty-year returns to bonds would be only 8% ahead of the twenty-year returns to stocks. The equity premium puzzle is not due to the specific liquidity or collateral or other institutional properties of Treasury bills that make them especially valuable.

We have climbed into the box of the equity premium puzzle, which has now been locked and sealed with us on the inside. How are we going to get out of it? There are four possible roads, four possible arguments. They are:

1. The twenty-year-horizon marginal investors are very, very risk averse--so averse to risk that they should be too scared to dare get into the bathtup (Daniel Altman cleaned this up a bit).
2. There are no twenty-year-horizon marginal investors. Institutional features of Wall Street force everybody able to mobilize significant money to have a very, very short time horizon.
3. The distribution of actual returns over the twentieth century does not match the true ex ante distribution of returns: we've been very lucky. If we hadn't been so lucky, relative stock returns would not look nearly so impressive and there would be no equity premium puzzle.
4. Investors over the course of the past century have misjudged the return distribution: the true ex ante long-run return distribution matches the ex post distribution we see in Figures 2 and 3, but for one of a number of possible reasons investors have greatly overestimated the risk associated with lon-run diversified stock market positions.

To be continued...