The puzzle begins after the reason for a premium has been granted. A share makes a residual claim on uncertain proceeds; a safer promise can pay when the shareholder receives little or nothing. Someone who dislikes exposure may therefore require a higher expected return to own the share. The difficulty is to account for the size of the difference. When economists put measured consumption risk and a specified account of preference into a model, the compensation the model requires can be much smaller than the historical reward. The explanation has then failed at a quantity. Saying once more that shares are risky cannot repair it. Either the danger has been measured inadequately, the account of how people bear it is incomplete, or the history against which the explanation is tested has been unusually fortunate.
The famous comparison is particular enough to be stated exactly. In their 1985 paper, Rajnish Mehra and Edward Prescott reported average annual real equity returns of 6.98 percent against 0.80 percent on their relatively riskless series for the United States over 1889–1978: a difference of 6.18 percentage points. These are arithmetic averages in a historical sample, not compound growth rates or a promise about the next century. The safer series joins different instruments across the period, including prime commercial paper in the early years and Treasury instruments later. Its construction matters, as does the period in which the stocks were held. The finding nevertheless gave the question its force. Could a model reproduce so large a spread while also reproducing a low return on the safer asset?
A return is partly the consequence of the price at which the claim was bought. Hold expected future payments fixed and a lower purchase price raises the expected return; a greater willingness to pay lowers it. A high premium can therefore express reluctance to own a claim whose payments disappear at a bad time. But a high realized return can also result from payments that turned out better than expected, or from a rise in the price at which the claim could later be sold. Historical averages supply evidence about the reward that occurred. The inference from that reward to the compensation investors expected requires an account of their beliefs and of how representative the observed years were. The purchaser standing before the uncertainty and the historian standing after it possess different information.
The relevant risk concerns the value of resources when payment arrives. Consider an invented two-state economy in which the states are equally likely, but a dollar delivered in the poorer state costs sixty cents today and a dollar delivered in the richer state costs forty. A claim paying eighty dollars in the poor state and a hundred and twenty in the rich costs ninety-six dollars. Its expected payment is a hundred, giving an expected return of about 4.17 percent. Reverse the payments, and the claim costs a hundred and four: its expected return is about minus 3.85 percent. Both claims have the same mean payment and the same spread of payments. The second is more valuable because it pays more precisely where an additional dollar is more valuable. A guaranteed hundred costs a hundred in this example. The premium belongs to the relation between the claim and the circumstances of consumption.
Mehra and Prescott’s benchmark gave this thought a demanding test. A representative household consumed the aggregate endowment, with constant relative risk aversion and time-additive preferences; a two-state process described consumption growth. Within their stated parameter range, and restricting the average risk-free return to between zero and four percent, the largest equity premium the model generated was 0.35 percentage points. The observed 6.18 was far away. This result belongs to the joint structure: the preference specification, the consumption process, the representative household, the parameter restrictions and the two returns being matched. It establishes a severe failure of that benchmark. It does not turn every larger observed premium into a violation of rational choice, or show that the model’s description exhausts the danger households could face.
Simply increasing aversion to risk does not leave the rest of the economy untouched. In the benchmark’s preference specification, the same parameter also affects the willingness to substitute consumption across time. If future consumption is expected to be more abundant, a very strong reluctance to shift consumption between dates can require a high return to induce saving; precautionary motives pull in another direction. A proposal that enlarges the equity premium must consequently answer for the safer rate as well. The bond is part of the puzzle, even when the drama is in the stock market. Mehra’s later exposition makes this joint discipline explicit: making one number look plausible at the expense of the other has explained neither price as part of a common economy.
One response enlarges the map of possible suffering. A sample may contain serious recessions and still fail to reveal the full probability and scale of a rare collapse. In Robert Barro’s rare-disaster model, a large fall in aggregate consumption makes the marginal value of consumption very high in precisely the state in which an equity claim pays poorly. A small probability can then carry substantial weight in price. The important multiplication is between probability, the loss and the value of resources at the moment of loss. It is insufficient to attach a frightening name to a remote event. The event must enter a coherent model with an exposure large enough, and a probability credible enough, to account for the prices being explained.
Such a model also clarifies two statements that can otherwise seem incompatible. A large historical equity return may reflect the fortunate absence of a disaster that investors reasonably feared. A seller of protection may meanwhile collect too little for the catastrophe the contract actually exposes it to. The first statement concerns the compensation embodied in an asset price and the outcomes a sample happened to contain; the second concerns whether a different promise has priced and funded its losses adequately. A rare event does not make every observed premium too large, or every insurance premium too small. Even “disaster” needs disaggregation. An output collapse and a default that strikes several kinds of claim together have different effects on their relative prices. The distribution of the damage among claims is part of the explanation.
A second response changes the way experience is assessed. Shlomo Benartzi and Richard Thaler combined loss aversion with the frequency at which investors evaluate an account. Someone who feels losses more strongly than equivalent gains and repeatedly closes the mental account can find volatile equity unattractive, even with a long planning horizon. In their simulations, evaluation at intervals of about a year was consistent with the historical premium under their chosen preference specification and return data. The evaluation period was doing work that the retirement date could not do by itself. A long period of ownership does not require a long period between judgments about whether one has won or lost.
A small invented calculation shows how the grouping can matter. Let a gamble pay a gain of a hundred and sixty or a loss of a hundred, with equal probabilities, and let a loss count twice as heavily as a gain of the same size. Assessed by itself, the gamble has an average subjective value of minus twenty. Assess two independent repetitions separately, and the values sum to minus forty. Assess their combined money outcome once, and the possibilities are a gain of three hundred and twenty, a gain of sixty, or a loss of two hundred, with probabilities one quarter, one half and one quarter. The average subjective value is now plus ten. Expected money is sixty under either method of accounting. The treatment of interim gains and losses has changed the appraisal of the same pair of opportunities.
These explanations direct attention to different evidence. A rare-disaster account needs a defensible account of states that may be poorly represented in the return sample. An evaluation-frequency account needs a defensible account of preferences and of how gains and losses are grouped. Neither is established merely because it can make one observed number fit. The fit must survive the other implications the explanation carries. Nor does the ability to change an evaluation period abolish a household’s need for cash. A person exposed to unemployment, a forced sale or a claim due tomorrow can have reasons to care about short-run losses that a mental-accounting illustration has deliberately left out. To call all reluctance excessive fear would be to assume away the circumstances whose price was being investigated.
The size of the premium therefore supplies a test of explanations rather than a direct measure of merit. Some compensation may be required to place a claim with someone able to bear its losses. The historical proceeds can also contain unexpected gains, institutional advantages and the effects of who had wealth to invest when prices were low. A model that accounts for the spread would explain a price under stated conditions; it would still leave the distribution of access and the justice of the proceeds to be considered. Conversely, finding the benchmark inadequate does not entitle us to erase the danger or declare the return unearned. The puzzle is productive because it prevents an attractive verbal justification from standing in for the missing quantitative account.
The premium is a price with an explanation still owing.
Whether fear is overpriced or danger underpriced depends on whose expectations, which payments, what consumption states and what purchase price are at issue. The safer asset must be explained alongside the risky one, and the realized sample must be distinguished from the distribution investors faced. The Hedge has identified the exposure a portfolio can remove; The Tail follows the catastrophe that ordinary observation may understate. Between them, the equity premium asks what the remaining exposure is worth to those who must carry it. The question has become sharper than the claim that owners are paid for taking risks. The argument remains open.