What Return can we Expect from Stocks?

Part 2: The answer

There are two broad categories of methods for coming up with a long-term expected return for the stock market: one is to look to history and to assume that the future will be like the past, and the other is to conduct an economic analysis. Under the latter heading, there are several methods and variations on those methods.

Looking backward: Projecting the past onto the future

If we begin by looking at the history of stock market returns, we’ll grow accustomed to the numbers and develop a feel for the possible and the credible.

Most treatments of the stock market’s returns begin in January of 1926, for no better reason than our best database of all stock prices and dividends arbitrarily contains nothing earlier.⁴ From the beginning of 1926 to the end of 2010, the annual rate of return on the stock market was 9.9%. If we look back only fifty years, to the beginning of 1961, the annual rate of return was 9.8%, which is very similar. But we do, indeed, have records—not as good, but the result of continually improving data research and management—going back into the nineteenth century, and if we start in 1900, then, over the last 111 years, the annual rate of return was 9.4%.  In each case, we’re taking 31 December 2010 as the end point.  I can remember, ten years ago, looking back to the beginning of 1926 and seeing an annual rate of return of 11.3%. This demonstrates that history, and the long-term record, is not what it used to be.

The most widely quoted expert on historical stock market returns is Jeremy Siegel, of the Wharton School of Business, who latterly has made a career of stock market forecasts based on history.  He observes that extended periods of below-average market returns have been followed by extended periods of above-average returns (and vice versa), and, given that we’re emerging from a more-than-ten-year period of below-average returns, he argues that we have good reason to expect that the next five to ten years will see above-average returns.  That may be right, but it is an intermediate-term forecast, and our perspective is long term.  Professor Siegel’s average return, that is, the average that he thinks may be exceeded in the intermediate term, is still the historical long-term rate of return.⁵ (This prediction of a return reversal, based on patterns, presumes that the stock market as a whole is not priced efficiently, in the sense we considered in an earlier essay.)

Given the numbers I just presented,  it is fair to say that an annual rate of return between 9% and 10% is a pretty good summary of the historical long term, and, in the absence of any further information, this would be a good estimate for the long-term future. But we do have further information.

Looking forward: Economic analysis

In an earlier essay, I explained that the value of an investment should equal the sum of all future cash flows from that investment, converted into the common currency of the present day.⁶  The conversion rates, or, as I called them, the “discount factors,” are, in effect, the rates of return. Just as the rate of return converts the present value into the future value, so, too, can the rate of return work in reverse and convert the future value back into the present value. This is true for the stock market as a whole, just as much as it is of any of the market’s constituent company stocks.

Valuing the market’s future cashflows

Consequently, one way of estimating the future stock market return is to estimate the future cash flows to all the stocks in the market, and to work out what constant rate of return would equate those cash flows to the present value of the market, namely, its price. What is the price of the market?  Well, we have indices of the market, like the S&P 500.  If we can describe future cash flows not in total dollars, but in dollars scaled to the prices of those indices, we can work out an estimate of the rate of return.  Fortunately, there are various commercial services, including Standard & Poor’s (S&P), that provide these scaled values of past (and for one or two years in the future) cash flows.

If you have owned a stock, you probably never thought of receiving a “cashflow.”  You thought you were receiving dividends.  So, for a first pass at an estimate of the rate of return, let’s try to equate future dividends to the current price of the market. (I’m using the value of the S&P 500 on December 31, 2010, which was 1257.64.) The return that I find is roughly 6.2%.  That’s very low.

But future dividends don’t represent all the cashflows that an investor might receive from a company or from the market.  Sometimes, companies return value to shareholders by buying up their stock in the market.  Sometimes, companies hoard cash for a long time, before finally relenting and returning it to the shareholders as regular dividends, a one-time dividend, or a share repurchase.  Warren Buffett’s Berkshire Hathaway famously pays no dividend at all. It’s really a matter of what a company could pay out in dividends, not what it actually pays out.  So, at the upper limit of what the future cashflows might be, we can redo our calculation on the assumption that the value of the market equates the value of all future earnings, of which only some portion may be paid out in dividends, to the current price.  The return that I find when I do this is roughly 10.8%.  That’s in line with the long-term historical return of the market, though a bit high.⁷

But we can be quite sure that not all of a company’s (or the market’s) earnings will ever be returned to the shareholders, who have claim to only a portion of the earnings, so our 10.8% estimate of the market’s return is an upper limit.  And the 6.2% return estimated from dividends is a lower limit.  Let’s split the difference and shade it down, and say that (according to this particular methodology), our estimate of the future return on the U.S. stock market is about 8%.

That range of 4.6 percentage points (between 6.2% and 10.8%) is very wide and could result in a stupendous difference in amounts of wealth at some point in the future.  It shows up the high degree of uncertainty in our forecast.  Furthermore, I spared you the details of my calculations, but you can be sure that my estimates of future dividends and future earnings (or the rate of growth of dividends and earnings from current levels) were very rough indeed. Still, there is a lesson here beyond the uncertainty: The range of estimates is mostly below the long-term historical rate of return on the market.

But this isn’t the only way of estimating the future return on the stock market.

Introducing the equity risk premium

One alternative way of thinking about the stock market’s rate of return is that it is a layer cake of returns, one stacked atop another.  In its simplest form, this idea suggests that the market’s rate of return has two layers: the rate you would earn from a riskless investment, and the rate the market pays you for taking on the risk that things might not turn out well.  This second component is called the equity risk premium.  (Remember that when I say that the market is paying a return, I don’t mean that the market has a man behind a curtain setting prices; rather, the net result of institutions and individuals, like you and me, judging prices and trading among themselves in the market is that stock prices have to compensate these participants for taking risks.)

There is, indeed, a riskless investment, which has a more-or-less knowable rate of return.  U.S. government Treasury bills, which pay you back after three months, are as close to absolutely safe as it is possible to find in this corruptible world. (Forget the silly political persiflage you may have heard of the U.S. government’s being bankrupt.) So the rate of return on Treasury bills is what financial economists usually regard as the riskless rate of return. But because Treasury bills are good for only three months (after which the money is returned to you and, presumably, you reinvest it in new Treasury bills), there’s a mismatch to our time scale for long-term market returns. Instead, for our purpose of estimating the long-term stock market return, it may be more fitting to take as our riskless foundation the U.S. government twenty-year bond (which matures in twenty years), which is equally free of the risk of default. And it is generally the case, during most short periods of time, and always the case over long periods, that the longer a bond takes to mature, the higher the interest rate on it.⁸

So, we can use historical data to estimate the equity risk premium as the difference in rate of return between the stock market and the U.S. government twenty-year bond. And if we take the period from 1925 to the end of 2010, we find that this annual difference averages 4.0%.

One especially useful feature of bonds, which is not shared by stocks, is that (assuming that the issuer of the bond doesn’t default), you know what rate of return you will have realized when the bond reaches its maturity date.⁹ So, I can look online and see that right now, the twenty-year U.S. Treasury bond has an interest rate of about 4.8%.  I add to this the equity risk premium, and—Presto!—I have a forecast of 8.8% for the stock market’s rate of return over the next twenty years.

But projecting the long-run historical equity risk premium into the future is nearly as crude and uninformative as projecting the historical return.  The equity risk premium has varied considerably over time. That’s the whole point of an equity risk premium.  If it were certain, there would be no risk, and its value would be 0. So our original question could, with scarcely any change of meaning, be reworded as: “What equity risk premium can we expect over the long term?”

Our little calculation has served mainly as an introduction to the concept of an equity risk premium.

Forecasting the equity risk premium

So the best way to forecast the market’s rate of return is to apply economic arguments to estimate the equity risk premium, and then to add this to the risk-free rate of return.  And the best version of this method that I have seen depends explicitly on the limit to the economy’s growth.

It stands to reason that the companies whose stocks are traded in the stock market cannot forever grow faster than the economy, because, if they could, all the companies that constitute the market would be bigger than the economy that contains them. If you pay attention to the nightly business reports on the radio and television, you receive frequent updates on what the growth rate of the U.S. economy has been, and what the growth rates of other economies, especially China’s, have been.  If you pay more attention to the glamorous Chinese economy than to the U.S., you may forget what is a realistic long-term growth rate for a mature economy, like ours.  It’s probably in the vicinity of 3% (net of inflation) a year, not the 8% to 10% we’ve been seeing in China. (The White House and the Congressional Budget Office are currently forecasting slightly less that 3%.)  Using the assumption of 3% per year for the U.S., Bradford Cornell, an economist at CalTech, has worked out that the long-term equity risk premium should be no more than 4% to 5% per year, after allowing for inflation.¹⁰ Let’s add this to the twenty-year risk-free rate of return, which as we saw is currently 4.8%. Then Cornell’s estimate is in the range of an upper limit of 8.5% to 9.8% per year.¹¹ He specifically states that this is an estimate of the upper limit, not the expected value. Notice that Cornell’s value appears to be in line with the historical value. But that was an actual value, and Cornell’s is an upper limit.

Cornell’s paper was published in early 2010, but his calculations were as of the end of 2008, when the stock market was much lower than it was at the end of 2010, which is my reference point.

If the market were efficient, meaning that it always priced stocks correctly, the market could not, by definition, be priced too low with respect to its true value.  But as I explained in an earlier essay on market efficiency,¹² there is reason to suspect that, while the market is very good at pricing stocks relative to each other, it may sometimes get their value in aggregate wrong.  Cornell believed that the market was underpriced at its recent depth, close to the time he was writing, and that it would therefore bounce back.  This future bounce is reckoned into his calculation.  But since the market bounced a cumulative 46% over 2009 and 2010 (and more in the last few months), some of his equity risk premium has already been realized. If we were to repeat Cornell’s calculations today, our upper limit on the forecast rate of return for the market would be lower.

So, our economic analyses have given us estimates of the stock market’s long-term future return of between 6.2% and 10.8% (and probably less than the middle of that range, around 8%); of about 8.8% (by a method that is mostly a projection of history); and with an upper limit of 8.5% to 9.8%, based on the most rigorous analysis, but with outdated information, which, if updated, would almost certainly result in lower numbers.

In light of these values, I feel most comfortable with a forecast return of about 7% for the long-run return on the stock market, though it may reflect a soupçon of pessimism.¹³

So much for return; what about risk?

The vagueness of our estimate of expected return doesn’t fully comprehend the risk that things might turn out worse.  There’s a subtle and potentially confusing distinction here.  We need estimates of two quantities: the expected rate of return, which is our best estimate of the future rate of return, and the measure of expected risk, representing what might be wrong with our estimate, and how bad the stock market return might turn out to be.

Recall our initial definition of the expected rate of return, which related it to the myriad possible ways the annual returns might be realized.  We’ve just arrived at our expected rate of return without having to calculate all the infinite possible sequences of annual returns to the future, along with their probabilities.  Conceptually, however, it still fits our definition of expected return: Implicitly, if the multitudinous possible market trajectories were to be weighed in the balance, our rate of 7% per year represents the return around which the greater possible rates of return are in equipoise with the lower possible rates of return.

When we look at the expected return this way, we see it situated in the context of investment risk, the possibility that the return (and our investments) will turn out badly.

If you were ever a student in a first-semester statistics class, you may recall that your teacher or your textbook would bestow on you, for problem sets, the expected value and the amount of variability as two precise numbers, as if they were knowable realities.  But we’re in the dark cave of investing, and we see only the shadows of these values on the cave wall.¹⁴ The vagueness of our estimate of the expected rate of return reflects its unknowability, which is not the full measure of the investment risk. The quantity of risk itself is also, ultimately, unknowable.

I know of no better way of estimating long-term future risk than to look backward at the variability of past returns, though there are complementary ways of interpreting the historical numbers. There are no economic methodologies for estimating long-term future risk. (There are economic and statistical methods for estimating risk over the short-term future, but our time scale is vastly greater than is appropriate for these methods.) We know that we can lose a lot of money in the stock market in one year; in 2008 alone, the stock market lost -37%.  But year-to-year fluctuations tend to cancel each other. (Not entirely though; it’s a common error to think so, and I’ll have more to say on this in a moment.)

Let’s consider a holding period of ten years. That is, we assume that we invested some money in the stock market and held onto that money for ten years.  We’ll start by looking backward from the end of December 1935, because we have values of actual returns going back to the beginning of 1926. We find that over those ten years, we earned an annual rate of return that was 3.8% better than if we had just kept our money in cash (or, more precisely, Treasury bills, which, as distinct from loose change, is what an economist means by “cash”). We’re once again looking at the actual equity risk premium. Now we move forward to the end of January 1936 and look back ten years again.  We keep stepping forward a month at a time, and looking back ten years.  This way, we have 901 rolling ten-year holding periods. Of these, 150 had stock market returns that were worse than just staying in cash.  We can repeat this exercise for rolling twenty-year holding periods. There were 781 such periods from 1926 through 2010, and during these, the stock market always came out better than being in cash.  But sometimes not by much: For all that stock market variability, there were 27 twenty-year periods where the annual rate of return on the stock market beat cash by less than a percentage point. Less than a percentage point per year can add up to a significantly different amount of money over twenty years, but still, this isn’t a spectacular outcome.

You should not infer from this history, however, that if your time horizon is twenty years or more, you can be certain of coming out ahead if you put your money in the stock market.  The market hasn’t signed a notarized contract to deliver to you a positive return over any time horizon, and the only natural constraint on its bad behavior is that can’t lose more than 100%.

An alternative way of getting at the long-term risk of the stock market, still using history, is to infer the chances of specific bad outcomes from the frequency with which various returns occurred in the past. That is, given the market’s past behavioral patterns, we can infer the likelihood of its misbehavior in the future.

If the pattern of past monthly or annual returns looks like a bell curve (to take one very convenient pattern), then well-established mathematical properties of the bell curve will allow us to calculate the probability of an annual rate of return, for any chosen time horizon, that is worse than any arbitrary cut-off we might establish.¹⁵

What, for example, is the probability that the annual rate of return will be less than zero over a twenty-year time horizon? To answer this (on the assumption of a bell curve), I need the two numbers my statistics teacher would give me: the expected annual rate of return above the return on cash, and a value representing the volatility. I shan’t go much into the latter,¹⁶ which is too technical for this essay, but for the expected rate, let’s use our estimate of 7%. Subtracting the historical average rate of return on cash, we get 3.3%. Then, using the standard mathematics of the bell curve, I calculate that there is a probability of 21.5%, which is to say, about a 1-in-5 chance, that over the next twenty years, an investment in stocks will turn out to be worse than just keeping money in cash. Because the risk changes with time horizon, the corresponding figures for ten years are 28.8%, or a little worse than a 1-in-4 chance, and for thirty years are 14.4%, or about a 1-in-6 chance. So, even though we’ve never seen a twenty-year span, let alone a thirty-year span, in which the stock market’s rate of return was worse than keeping money in cash, we can calculate the probability of such a risk in the future.

If you have been paying close attention, you’ll have noticed that the 1-in-4 chance for a ten year interval is much worse that what we saw in the past, 150 cases out of 901 (which is only about 1-in-6). Is all our complex mathematics overestimating the risks? Yes and no.

In an earlier essay, I explained that the familiar bell curve is not a good model for the behavior of stock returns.¹⁷ In particular, it leads to overestimates of the small risks, but underestimates of big risks.  A rate of return that matches that of cash may not be so terrible.  So, it is, indeed, possible that our 1-in-6 estimate of the chance of a stock market rate of return worse than cash for a thirty-year span is an exaggeration of the risk.  But if some possibilities are less probable, then other possibilities have to be more probable, and it is correspondingly likely that the chance of a really bad outcome, like having a result worse than cash by at least -5% per year (which means, relative to cash, a cumulative decline of -79% over 30 years!), is greater than I’d calculate it to be.

Table 1 presents the risks of bad outcomes relative to keeping money safe in cash, for different time horizons, and different standards of what a bad outcome might be:

Table 1

When contemplating this table, bear in mind that these values depend on our estimate of 7% for the expected rate of return on the stock market (and, for that matter, on the historical rate of return of 3.6% on cash), and on historical statistics for volatility. A higher expected return on the stock market would result in less risk, a lower expected return in higher risk. And because they assume that the pattern of possible future returns resembles a bell curve, which we know is incorrect, the values in the lower right of the table are probably too low; that is, the risks are greater than they suggest. Also, remember that the returns are rates; -3% per year over ten years corresponds to a cumulative decline of -26.3%, and -5% per year over ten years corresponds to a cumulative decline of -40%.

Monte Carlo simulation

There’s yet another statistical method of estimating the risks of the stock market. Although I won’t squeeze a result out of it now, it’s worth knowing what the method is.  It simulates the stock market’s possible returns over time, and starting about ten or twelve years ago, it has been heavily used by financial planners.  It is the computerized equivalent of the following: write each historical annual difference between the stock market’s return and the return on cash on a slip of paper,¹⁸ put all the slips into an upturned hat, and stir them around.  Take out slip, write down the number as the return for year 1, put the slip back into the hat, and stir. Take out a slip, write down the number as the return for year 2, put the slip back, stir.  Repeat this for each year through year 30. You now have a possible sequence of returns for the next thirty years. Calculate the rate of return that corresponds to this sequence.  Repeat this exercise a few thousand times, so that you have a few thousand possible sequences of returns for the coming thirty years, and a few thousand possible rates of return. This exercise is called a Monte Carlo simulation. If you sort through the resulting rates of return, they’ll form a pattern, something like the bell curve, but not exactly, because the pattern will reflect the way the actual historical data deviate from the bell curve.¹⁹

With this curve, you will be able to count and see what percentage of all the rates of return were below 0%, below -3%, and so forth, and the resulting risk estimates might be fairer than those of our purely analytical computations.

Expected return and risk behave differently over tme

Notice that long-term expected risk behaves differently from the long-term expected rate of return. The expected rate of return over the short term should be the same as the expected rate of return in the long term, at least in the absence of information to the contrary. But also in the absence of additional information, risk, if defined as the likelihood of a rate of return below a given benchmark, decreases over time.  The short-term outlook for risk (by our definition) is greater than the long-term outlook.

Notice also that all our methods of estimating risk, even the count of historical twenty-year periods, treat risk as purely a game of numbers, without regard to causes.  But, as we assumed when we calculated the expected rate of return on the stock market, returns are economic phenomena, created by expectations of cashflows and suitable discount factors.  If the market behaves badly, it will likely be the reflection of a broad outlook among traders for dire problems with the economy at large. Seen this way, the statistical analysis of risk takes into account only symptoms, not underlying causes, and if you firmly believe that the economy has next to no chance of facing a disaster, then our estimate of a 1-in-170 risk of a cumulative decline of -79% with respect to cash over the next thirty years will, in your judgment, be much too high.

All the estimates in this essay are based on either economic or statistical reasoning. If you are in thrall to the occult and believe that the stock market is driven by forces not based in economics, like astrological harmonies or mysterious patterns of long waves, you may come up with very different estimates of return and risk, but I daresay that ultimately, you’ll have to agree that the growth of the market will be limited by actual economic forces that constrain the size of the companies it represents.  And, of course, on the downside, returns are constrained by a cumulative value of -100%. Unless you choose to go into debt, you can’t lose more than everything with which you began.

But wait! There’s less!

The stock market returns that we have been estimating are returns before taxes are paid on them.  If you invest in the stock market though a taxable account, your returns will be reduced by income taxes.  There are all sorts of clever and legal ways of avoiding or reducing taxes on investments, such as balancing realized capital losses against realized capital gains, but our interest here has been in ideal theoretical outcomes for the market, not the results of clever investment and tax strategies.  A straightforward investment in the stock market through a taxable account will certainly have its returns reduced by, at the very least, the income tax on qualified dividends, currently 15% at the Federal level. Moreover, income taxes can add insult to injury.  In 2008, when the stock market produced a return of -37%, it nonetheless paid dividends. Dividends always represent a positive return, so, if we ignore those, the price return in 2008 was worse, -38.5%.  But the dividends, which slightly counteracted the market decline, were taxed.  After applying a tax rate of 15% to the dividends received that year, the after-tax total return on the market in 2008 was actually -37.3%. If our expectation for the long term is of a rate of return of 7% on the stock market, and of that return, 2% comes from dividends, then the after-tax return on the stock market will be about 6.7%. That reflects just Federal tax; your state may impose an additional tax on the dividends. And it doesn’t matter if you, as an individual (as distinct from my readers who represent non-profit, and therefore tax-exempt, organizations) have invested your money through a tax-advantaged account, like an IRA.  When you withdraw your retirement money, you’ll be paying tax on it.²⁰

The results are diminished further when you recall that all the values of the market’s rate of return that we are considering include inflation.  Take out inflation to reflect the actual purchasing power of the returns, and you find that the market’s expected rate of real return is actually much smaller. So if inflation is, say, 2.5% per year, then the real expected after-tax return on the market is down to 4.2%.

The typical return

The expected annual rate of return isn’t what you might call the “typical” annual return.  The typical return that you might expect is the average of the returns that you’ll see over the coming years.  As I noted near the beginning of this essay, the average is greater than the rate (unless there’s no variation from year to year).  A simple mathematical method allows us to estimate the average, or typical, return of the market, given the rate of return and the volatility of returns.  Using this method, we can convert an annual rate of return of 7% into a typical annual return of about 9%.²¹ That’s nearly 3 percentage points less than the historical average annual return of 11.9% since 1926. In 2010, the market returned even more, 15.1%.  Of course, we may seldom observe the precise typical return, but, given the nature of the bell curve, actual one-year returns that are close to it should occur with some frequency.

Summary

To recapitulate the forecasts in one place: We find that the expected annual total rate of return on the market is probably around 7%, though our uncertainty of this estimate is such that we’ve seen a range from 6.2% to 10.8%, neither of which is very credible by itself. The expected long-term rate of return is less than the historical long-term rate of return, which, depending upon our choice of historical period, has been between 9% and 10% per year. Furthermore, Federal taxes, to say nothing of state taxes, will lower the expected rate for stock market investments held in taxable accounts; if the forecast is 7%, then the after-tax forecast is about 6.7% or less.  If we take out an assumed inflation rate of 2.5%, then the pre-tax rate is about 4.5% and the after-tax rate is about 4.2%. The risks may be represented very crudely by Table 1, though the numbers there probably underrepresent the true risks of disastrous outcomes. And the typical annual return that we can expect to see in the future is about 9%.

For the investor, stock market return is not destiny. On the one hand, there are investments, like bonds, that we normally expect to have lower returns (thanks to lower risk), but there are also investments that may well have higher returns. The rate of return on the stocks of smaller companies, in aggregate, has exceeded the rate of return on the stock market as a whole, and there is reason to believe that this will continue, though the returns are at the expense of greater risk. It’s entirely possible that the returns to the stock markets of emerging economies will also be larger than the returns to the stock markets of developed economies, at least until those economies emerge. And, before they do, there are the so-called “frontier markets,” like Pakistan’s and Tanzania’s, whose returns are even riskier than those of the emerging markets.

Ingenious stock selection, if someone can succeed at it, or clever strategies that actually work in practice and not merely on paper, or even just a combination of risk-taking and good luck might produce much higher returns than the market.  For example, if you had had perfect one-month foresight, so that you sold all your stocks immediately before months when the stock market went down, and then bought back into the market immediately before months when it went up, your stock market investment from 1926 through 2010 would have earned an annual rate of return of 32.7%.

Regardless, however, of whether you think you can get greater returns through straightforward proven methods, or through unproven and improbable ingenuity, the long-term rate of return on the stock market is a benchmark against which we will measure our expectations, and just maybe, around which we will plan our economic futures.

Adam Jared Apt, CFA, is a financial advisor and the owner of Peabody River Asset Management, based in Cambridge, MA.

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