The Equity Risk Premium

Nobody holds stocks as a favour. A government bill pays you a known amount on a known date; a share pays you whatever is left over after everyone else has been paid, on no schedule, and can halve in a year. If both were expected to return the same, no sane person would own the share. So stocks have to be priced to offer more, and that extra expected return is the equity risk premium, or ERP.

It is compensation, not a gift. You are being paid to accept that your wealth will occasionally fall by a third at the exact moment you most need it.

Definition

Write $r_e$ for the expected return on the broad equity market and $r_f$ for the risk-free rate. Then

$$\text{ERP} = r_e - r_f .$$

In words: how much more, per year, you expect to earn by owning the market than by owning a safe bill. Everything downstream depends on it. The Capital Asset Pricing Model (CAPM) turns the ERP into a required return for any single stock by scaling it with Beta (β). Every discounted-cash-flow valuation uses it in the denominator. Every "is my backtest actually good?" question compares a strategy against it.

Two decisions have to be made before the number means anything: which risk-free asset (a 3-month bill or a 10-year bond?), and which kind of average (arithmetic or geometric?). Change either and the answer moves by two full percentage points.

Route one: look at history

The obvious approach is to measure what equities did earn over bills, and assume the future rhymes. Over roughly a century of US data:

	Stocks	T-bills	Premium
Arithmetic mean	~11.7%	~3.3%	~8.4%
Geometric mean	~9.8%	~3.3%	~6.5%

Worked example: how much does a century actually tell us? Take the arithmetic premium of 8.4% and ask how precisely it is measured. Annual equity excess returns have a standard deviation of roughly 20%. With $n = 96$ years, the standard error of the mean is

$$\text{SE} = \frac{20\%}{\sqrt{96}} = \frac{20\%}{9.8} \approx 2.0\% .$$

Plain English: even with a full century of data, our estimate of the average wobbles by about two percentage points. A 95% confidence interval is roughly 8.4% ± 4%, i.e. 4.4% to 12.4%. That is not a precise number. It is barely a number at all. To halve that error bar you would need four centuries of data — and the economy would not sit still while you collected it.

Route two: back it out of today's prices

Instead of asking what equities earned, ask what they must return given what people are paying today. Using the Gordon growth relation, a market with cash yield $y$ (dividends plus buybacks, as a fraction of price) growing at rate $g$ forever has expected return $r_e = y + g$, so

$$\text{ERP} = y + g - r_f .$$

In words: your return comes from the cash you get paid now, plus the rate at which that cash grows, and the premium is whatever is left after subtracting the safe rate.

Worked example. Suppose the index pays a 1.5% dividend yield and buys back another 1.5% of its shares, so $y = 3.0\%$. Long-run nominal cash-flow growth is assumed at $g = 4.0\%$. The 10-year Treasury yields 4.2%. Then

$$\text{ERP} = 3.0\% + 4.0\% - 4.2\% = 2.8\% .$$

Less than half the historical figure. That is not a contradiction — it is the point. High past returns came partly from valuations rising, and a market that has already re-rated upward offers less going forward. Historical premiums are backward-looking; implied premiums move with prices, falling as markets rally and spiking in crashes.

The equity premium puzzle

A realised premium of 6 to 8 percent is, embarrassingly, far larger than standard economic models can justify. To make an investor demand that much for a risk as modest as equity volatility, you need implausibly extreme risk aversion. This is the equity premium puzzle (Mehra and Prescott, 1985), and forty years of proposed fixes — rare disasters, habit formation, long-run risk, survivorship in the US sample — have not fully closed it. Part of the honest answer is that the realised premium was probably larger than the expected one: the US was the century's winner, and we are looking at its chart.

What it means in practice

• Sizing conviction. If a century of data gives ±4%, your three-year backtest tells you close to nothing about expected return. Estimate risk from data; be far more humble about estimating return.
• Discount rates. A DCF is enormously sensitive to the ERP. Moving it from 4% to 6% can cut a long-duration valuation by a third.
• A hurdle. A long-only equity strategy has to beat the premium it is passively collecting, not zero, before it counts as skill.

Key terms

• Risk-free rate — the return on a default-free government security, usually a short bill.
• Arithmetic vs geometric mean — simple average of yearly returns vs the compounded growth rate; geometric is always lower.
• Implied ERP — the premium backed out of current prices and expected cash flows.
• Equity premium puzzle — the observed premium is too large for standard models to explain.