# Stock prices and fundamentals

How low can stock prices go, and how worried should you be?

Since 1946, in an average year the stocks in the S&P500 offered a 3.5% dividend and went up in price 2% faster than inflation, for a combined real yield of 5.5%. At that rate, if you reinvested your dividends, you might expect to double the real value of your portfolio in 13 years.

The recent volatility in stock prices has led me to think some more about an old paper by Robert Shiller titled Do stock prices move too much to be justified by subsequent changes in dividends? In that paper, Shiller introduced the concept of a “perfect foresight” stock price, denoted P*(t), which is defined as the present value of the actual subsequent dividends over future periods, D(t+j):

P*(t) = D(t) + (1+r)-1 D(t+1) + (1+r)-2 D(t+2) + (1+r)-3 D(t+3) + …Here r denotes the rate of return, which from the numbers above we might take to be r = 0.055 at an annual rate.

Let’s consider what this formula would imply for stock prices under various scenarios for what dividends might do over the future. As a starting point, we can calculate the perfect foresight price P*(t) under a scenario in which real dividends simply grow at a constant 2% annual rate from their present values. The graph below plots 100 times the natural log of the actual dividends on the S&P500 through 2008:M12 and their future path from here on under this first scenario. Measuring the variable in this way (100 times the log) allows one to read approximate percentage changes off of the vertical axis; for example, a vertical drop of 10 units on this scale corresponds approximately to a 10% decrease in the inflation-adjusted value of total dividends paid on the index.

The black line in the figure below plots the observed historical value for the S&P500, while the blue line is the perfect foresight price P*(t) calculated from the formula given above under the specified scenario. When the actual price P(t) is above P*(t), it means that an investor who bought stocks at that date and held onto them forever would have earned a flow of subsequent dividends whose present value was less than the price paid, or, to put it another way, would achieve a real rate of return less than 5.5%. If you bought at a time when P(t) was less than P*(t), you would have enjoyed a better than 5.5% return. The figure reveals that, under the hypothesized scenario, we have just come out of a 20-year episode during which stocks were “overvalued”, but got back to a situation where they were again undervalued when the S&P500 fell below 830.

I don’t want to claim too much for these specific numbers. Different but equally defensible values for the discount rate and growth rate would give you a quite different value for the “fundamentals-justified” value for the S&P. The calculations I’m offering here are for illustrative purposes only. As they say, your mileage may vary.

Now let’s ask how much the stock price should change if we switch to an alternative, very different scenario for what’s about to happen to dividends. As an example, I decided to ask what would be the consequences if we’re just about to repeat what happened during the Great Depression. For this scenario, I simply took the historical trajectory of real dividends between 1931:M1 and 1936:M12 and pasted it onto the series beginning in 2009:M1, and supposed that afterward (from 2015 onward) real dividends go back to growing at 2% annually, with the Great Depression II keeping us stuck permanently with a level of dividends well below that implied by scenario 1. Thus scenario 2 looks like this.

The green line in Figure 4 below plots the behavior of P*(t) under this alternative scenario 2. If we’re about the enter Great Depression II, stocks are still overvalued, and the S&P500 would have to fall another 19% [all percent calculations reported here are logarithmic changes] to get down to 608, the value at which investors could again anticipate a 5.5% real rate of return given the horrible news ahead on dividends. This alternative path for P*(t) doesn’t get back to the current value for the actual S&P (which closed at 735 on Friday) until 2016.

But that doesn’t mean that if you buy stocks today you’d have to wait 7 years before you’re even, because in the mean time you’ll still collect dividends from your stocks. The typical stock will pay a significantly lower dividend in 2009 and 2010 than it did in 2008, if we’re about to repeat the depression, but you’ll still get something, and under the depression scenario, you get more shares per dividend as you reinvest the dividends at cheaper stock prices. Plus, the value of P*(t) isn’t going to stay put at 608. By construction, the value of P*(t) necessarily grows over time, even in a period when dividends are falling, because the date t value of P*(t) is by definition sufficiently low to ensure the 5.5% total return across a period of falling dividends. Between the reinvested dividends and expected increase (from 608) in P*(t), if you buy at 735 now you’ll only be down 13.5% by the end of the year. The table below reports your average annual return if you buy at 735, watch the market drop instantly to its depression-justified value, reinvest the dwindling dividends for the specified number of years, and then sell out assuming that stocks are valued at P*(t+j) at the date you sell. You’d break even after 4 years, and if you held tight until 2019, your average annual return would be 3.6%.

Of course, under this scenario you would do better waiting for the market to recognize the depression and wait to buy at 608 rather than now at 735. Moreover, given the historical tendency for over exuberance in upswings and excessive pessimism in downturns, you might expect the actual price to fall well below 600 in another depression, at which point there will be returns to be had well in excess of 5.5% if you time your moves just so.

But good luck with carrying out that particular scheme. After all, scenario 2 assumed we’re about to start another Great Depression, and hopefully it goes without saying that this need not necessarily happen. If it doesn’t, you may find yourself waiting for the S&P to fall below 600 until you’re both retired and dead. If the downside to investing now, even under the depression scenario, is better than a 3% average real rate of return over the next decade, I can live with that.