## How Leveraged Should Your Stock Market Investments Have Been?

**J. Bradford DeLong and Siyuen Chen**

Here http://www.econ.yale.edu/~shiller/data.htm we have Robert Shiller's working, updated data series for the U.S. stock market over the long run: starting in 1871—back when the stock market was overwhelmingly railroads—with the Cowles Commission Index then spliced to the S&P Composite http://delong.typepad.com/2017-06-30-shillerdata.csv.

If you had taken 1 in real value, invested it in the stock market in January 1871, reinvested the dividends, and paid no taxes, you would have 16,000 today. Such are the returns to patience, risk-bearing, diversification, and capital ownership in the extraordinary economic boom that was the extended American century.

By contrast, if you had taken your money and invested and reinvested it in 10-year U.S. government bonds—again, without taxes—today you would have only 37 in real value. "Only". That 1-to-37 in real value is itself powerful testimony to how scarce financial capital has been in the economy over the past century and a half even when it has to pay the enormous safety and liquidity penalty that the market has exacted.

Take a look at the stock and bond portfolios as they compound here:

The cumulative real return from investing in the S&P Composite—or in its earlier Cowles Commission extension—and reinvesting your dividends, without taxes, is the jagged olive line. The cumulative real return from investing and reinvesting in the Treasury long bond market is the green line.

The 1-to-37 is itself testimony to the remarkable value of thrift and patience. But it is the extreme 450-fold gap between 16,000 and 37 that must flabbergast. With only a moderate time horizon—ten years, say—you would have to be a real idiot to put any of your money in bonds relative to a diversified index of stocks. The outperformance is remarkable.

But suppose you do not think that simply putting all of your money in stocks is aggressive enough, given the extraordinary equity return premium that the past century and a half has shown. What would have happened if you had leveraged up: borrowed money, thus put more than 100 percent of your net wealth in stocks, and rebalanced every month to keep a constant leverage ratio?

Now this is a thought experiment. It assumes that you can go short the Treasury bond market without transactions costs (you can't). It assumes you can rebalance your portfolio every month (you can't). It supposes you can reinvest tax free. The figure below shows five leverage ratios: the bond portfolio (that gives you a β of zero), the all-stocks portfolio (that gives you the risk of the stock market—a β of one), and then three portfolios in which you borrow your wealth, twice your wealth, and three times your wealth, respectively, and invest your wealth and your borrowings in the stock market, giving you portfolios with β values relative to the Cowels-S&P composite of 0, 1, 2, 3, and 4.

How well would these reinvested portfolios have done since 1871? The figure below shows:

The β = 4 portfolio flames out spectacularly in the Great Crash of 1929: just one month that sees a 25% collapse in the stock market bankrupts you.

The β = 3 portfolio escapes bankruptcy in the Great Depression by the narrowest of margins, falling from 2500 in September 1929 to 2.4 in May 1932. Even as late as August 1942, an investor would still have been richer investing in bonds alone than in the β = 3 portfolio, borrowing twice their wealth and putting it all on stocks. (And such an all-bond investor would have slept more comfortably at night as well, if they were not worried about inflation risk.) Thereafter, in the 75 years that separates us from the nadir of Allied and the zenith of Nazi fortunes in World War II, the β = 3 portfolio compounds from 11 dollars in real value to today's 107,000.

The β = 2 portfolio loses only 98% of its value in the Great Depression, and is the champion: standing now at 285,000 in real value for each 1 dollar of real value invested in 1871.

The β = 1 portfolio—all stocks and no extra leverage—produces the roughly 16,000 dollars that we saw before: thirty-five times its β = 0 all-bonds brother, and one-seventeenth fourth of its β = 2 sister.

And the β = 0 all-bonds portfolio turns one dollar of real value in 1871 into 37 today.

We can narrow things down a little further: if we limit ourselves to portfolios with constant leverage and thus a constant β, the portfolio that does the best since 1871 has β = 2.28. That portfolio escapes bankruptcy in 1932 by 1 percent of its peak-1929 value, and today yields 325,000 dollars in real value for every dollar invested in 1871 and reinvested since:

Being all in bonds at β = 0 turns 1 of real value into 37 over the past century and a half. Moving from bonds to stocks—from β = 0 to β = 1—multiplies your portfolio value today by an additional 450-fold. Borrowing your wealth and adding leverage—β = 2—gets you from 16,000 to 285,000: an additional nearly 17-fold wealth amplification. And then going from β = 2 to β = 2.28 gets you an additional 40% to 325,000.

Beyond that, however, your portfolio would have faced gambler's ruin: the β = 3 portfolio would today be worth only ("only") 107,000; the β = 4 portfolio would today be worth zero, having crashed in the Great Crash of 1929.

You may say: Wait a minute. You may say: Stocks have a higher expected and average return than bonds, so the β = 3 portfolio's expected and average return should be three times that of stocks minus two times that of bonds, while the β = 2 portfolio's return—at two times that of stocks minus one times that of bonds—is less. 145 years is long enough for the central limit theorem to have considerable bite. So why does the β = 3 portfolio appear less lucrative than the β = 2 one?

You are indeed correct to think that the expected value as of 1871 of what the β = 3 portfolio would be today is higher than that of the β = 2 portfolio. But in this case expected values are not typical. Because of reinvestment and compounding, the distribution of cumulative returns approaches not a normal distribution—where the mean outcome, the median outcome, and the modal outcome are all the same—but instead approaches a log-normal distribution—or, in the case of β = 4, and almost in the cases of β = 3 and β = 2, worse—where the mean is vastly in excess of the median and modal outcome. The cumulative returns on the reinvested portfolios are best thought of as single draws from these near log-normal distributions and thus as estimates of the median rather than of the expected value. The β = 3 portfolio had a long upper tail—which we did not see. And the β = 4 distribution had an even longer and more extravagant tail—but the result we see is the goose-egg.

This analysis, however, is too simple. The implicit assumption that monthly returns are drawn from an unchanging process—that β should not be varied—is itself very wrong. There is a huge amount of mean reversion in stock prices: that has been the point of Shiller's career as an economist. Better portfolios—ex post better, at least—would reduce leverage when market prices rose above Shiller fundamentals (and raise leverage when market prices fell below them). And even allowing for mean reversion, there is no reason to believe that the stochastic process generating returns has the same return distribution today as it had in the 1890s is surely false.

There is lots more work to be done!

But do check my work here!: http://delong.typepad.com/2017-07-06-shiller-data-stock-market-portfolios-optimal-%CE%B2-equals-2.28.ipynb