Is the Equity Premium Getting Smaller?

The Mehra-Prescott argument

To really understand what Mehra and Prescott did, you would have to spend an inordinate amount of time poring over their 1985 paper – unless you are an academic economist who is already well-versed in the jargon and the form of mathematics that is practiced among economists. But I will attempt to relate the gist of it.

Like almost all of academic economics, which is obsessed with “marginal” effects (that is, infinitesimal changes in one variable that occur as a result of infinitesimal changes in another variable), Mehra and Prescott focus on a marginal rate, the rate of substitution of consumption in one short time period for consumption in the next short time period.

The problem with this marginal, short-term approach is that to apply it to long-term effects, you have to string together a very long series of short terms (an infinite series, if the marginal calculations are over infinitesimal time). But economics has never mastered that stringing-together process. Applications of the theory to the long term are often extrapolated in a wholly unsophisticated manner as an afterthought, with a marked reduction in rigor as compared to the short-term theorizing.

In the Mehra and Prescott case, let’s consider the marginal rate of substitution of consumption in one short time period – say a year – for consumption in the previous year. To help understand this better, let’s switch temporarily from comparing preferences in consumption from one year to the next to comparing preferences for bread and cheese.

If you would be willing to give up 1% of your cheese to have 2% more bread, your marginal rate of substitution of bread for cheese is 2. You therefore have a preference for cheese, because you are stingier with it – you’ll only give up 1% of it even to get 2% more bread.

Now this doesn’t tell you quite enough though, because to get 4% more bread you might not give up 2% of your cheese – because, perhaps, then you wouldn’t have enough cheese to go with all that bread, so what would be the point of more bread? What you really want to know is not the amounts of bread and cheese but their utilities – how useful are they to you. You really want to know the marginal utility of substitution of bread for cheese.

Now let’s go back to consumption from year to year. If the marginal rate of substitution of future consumption (next year) for current consumption (this year) were 2, you’d need a 100% return on investment to make it desirable to make the substitution. Why is that? Because to be able to consume 2% more next year, you would only be willing to sacrifice 1% of your consumption this year. To get that extra 2% consumption next year, you’d have to take the 1% of consumption that you’d be willing to give up this year, invest it and reap 2% extra next year – a 100% required return on investment (if consumption would otherwise be constant).

Hence, the marginal rate of substitution of future consumption (next year) for current consumption (this year) must only be about 1.01 or 1.02 – which would imply you only need about a 1-2% return on your investment to make the substitution.

But that’s if you know for sure you’re going to get that 1% or 2% return – in other words, if the investment is risk-free.

What if it’s a risky investment like equities? You might get a whole range of returns, from -30% to +50%. And furthermore, 50% more consumption might not be twice as desirable as 25% more consumption – that is, it might not have twice as much utility. So what you need is the utility of consumption. You also need to know if the expected value of the utility of consumption next year is great enough to warrant substitution of 1% of this year’s consumption for an uncertain amount of next year’s consumption, given your marginal rate of substitution.

The key to solving this is to know the utility function – how much utility there is in being able to consume 10% more, 25% more, 50% more, etc., and how much (dis)utility there is in consuming 20% or 30% less.

Mehra and Prescott make an assumption about the mathematical form of this utility function – one that is supposedly in accord with assumptions about the “rationality” of the investor and market equilibrium precepts.

When they make this assumption, they discover that the expected return on the uncertain investment (the equity investment) can’t be more than 0.35% greater than the certain return on the riskless investment.

As you can see, a large number of assumptions need to be made in this process of reasoning. Some of those assumptions test credulity to the breaking point – try reading the Mehra-Prescott paper to see.

And then when you’re done – and if you believe all of it – all you know is that if you invest a percent of what you would otherwise consume this year in equities and then cash it in next year in order to consume it next year, you’ll need an ERP of no more than 0.35%.

But what if you won’t consume it next year? What if you won’t consume your investment for 30, 40 or 50 years? It’s not easy – if it’s even possible – to extend the Mehra-Prescott analysis to answer that question. In fact, I doubt that any of the 5,000 other papers and books that cited the Mehra-Prescott work tried to answer it either.

A way of looking at it that starts with the long term

Few individual and institutional investors need to invest for only one year. For individuals and households saving to receive income in retirement, and for endowment, foundation and pension funds, the horizon is more like 30, 40, or 50 years.

Let us assume that for each private corporation, the investor has a choice of investing in its 30-year bond or in its equity. Assume furthermore that the investor diversifies among many high-quality companies, so that the impact of defaults on the aggregate 30-year bond return is very small, and the variability of the equity investments in aggregate is equal to the market’s variability. (To eliminate reinvestment risk, one could even assume the bonds are purchased in the form of bond strips.)

How much of an expected ERP would an investor require (and hence, what price would the investor be willing to pay) to invest in the company’s stock rather than in its bond?

A standard model of investment returns that is widely recognized is the lognormal model: periodic returns, when continuously compounded, follow a normal distribution that is independent from period to period. (I will deal with “fat tails” later.)

What many people in the investment field don’t realize is how skewed this distribution becomes over long time periods – that is, how few potential results reside on a long, high-return tail, and how many are clustered around the low-return mode. Figure 2 below, showing the distribution of final wealth resulting from a dollar growing for 30 years, illustrates this result.

Figure 2. Probability distribution of 30-year ending wealth resulting from investment of an initial $1

The assumptions used to create this graph were that the expected ERP was 3.5% in excess of a 30-year bond return of 4.5%, and the equity standard deviation was 16%. With these assumptions, the ending wealth for the bond would be $3.75, while the mode of the distribution of equity returns – that is, the most likely result, the one represented by the peak of the distribution – is an ending wealth of $3.80, only 5 cents higher than the virtually certain ending wealth with the bond. Notice how much of the distribution is clustered around that $3.80 mode.

The expected ending wealth is much greater, $8.98, but most of that figure is due to the low-probability high-wealth outcomes on the right hand tail of the distribution. In fact, if we invoke the concept of utility and down-weight the utility of those high outcomes (on the assumption that, for example, ending wealth of $40 is not twice as useful as ending wealth of $20), then we might find that the expected utility of ending wealth with the equity investment was not very much higher than the utility of ending wealth with the bond. And there is a significant chance that ending wealth for the equity investment will be less than for the bond.

Table 1 below shows the mean, median, and mode for several different expected ERP assumptions.

Table 1. Ending wealth probability distribution parameters with various equity premia and a 16% standard deviation

The numbers in this table are subject to interpretation, but I believe most people would agree with some conclusions. First, an ERP of 3% would be insufficient given that the most likely result ($3.30) would be ending wealth less than with the nearly riskless bond. Similarly, an ERP of 3.5% seems not quite sufficient either. On the other hand, ERPs greater than 5% are overly rich and would be bid down by investors willing to invest for a lower expected ERP.

This leaves us with the conclusion that an ERP somewhere between 4% and 5% might be sufficient. This conclusion roughly agrees with the conclusions of Elroy Dimson, Paul Marsh and Mike Staunton of the London Business School, based on their compilation of the international history of realized returns from 1900 to 2007 for developed country equities.¹ However, the number is lower than the 6% realized U.S. ERP from 1889 to 1978 of Mehra and Prescott’s 1985 paper.

Why should that be? Note that the volatility of equity returns was higher in the U.S. prior to about 1945 than it has been since, with a standard deviation of returns of about 20% in the earlier period as opposed to the more recent 16%. This may have given rise to an assumption on the part of investors that investment in equities entailed a high level of risk.

Let us redo the figures in Table 1 but assuming a standard deviation of 20%. The results are shown in Table 2.

Table 2. Ending wealth probability distribution parameters with various equity premia and a 20% standard deviation

Notice how much this assumption of higher volatility exaggerates the skew of the wealth distribution. (An assumption of fat tails – that is, probabilities of extreme results greater than in a normal distribution – would produce a similarly skewed distribution.) Now an ERP less than 6% may not be sufficient.

These tables suggest that if a conservative assumption is desired in, say, a Monte Carlo simulation, it may be reasonable to assume an ERP of 4-4.5% in combination with a standard deviation of equity returns of 16%. If, on the other hand, the modeler believes that equities are becoming more volatile in perception and in reality, a combination of a 6% ERP with a standard deviation of 20% may be warranted.

While this analysis relies on uncertain assumptions and an imprecise and unscientific interpretation of investor preferences, it has the virtue of avoiding false precision. It may be better as a guide to the future than the Mehra-Prescott analysis and most of the follow-on articles it has spawned. Furthermore, as a theoretical construct, it has the attractive characteristic of saying that what works in practice also works in theory.

Michael Edesess, a mathematician and economist, is a visiting fellow with the Centre for Systems Informatics Engineering at City University of Hong Kong, a partner and chief investment officer of Denver-based Fair Advisors and a research associate at EDHEC-Risk Institute. In 2007, he authored a book about the investment services industry titled The Big Investment Lie, published by Berrett-Koehler. His new book, The Three Simple Rules of Investing, co-authored with Kwok L. Tsui, Carol Fabbri and George Peacock, has just been published by Berrett-Koehler.

Read more articles by Michael Edesess