Equity Risk Premium

The question the ERP is the answer to

Imagine you have a dollar and a one-year horizon. Hand it to the US Treasury and you will receive, with near-certainty, that dollar back plus a few cents of interest — the risk-free rate r_f. Hand it instead to the S&P 500 index and the outcome is a distribution: you might earn 30%, you might lose 30%, the long-run mean has been somewhere around 10% nominal. Why would anyone take the second deal?

The standard answer is that investors are risk-averse and need to be compensated for bearing the year-to-year variability. That compensation, expressed as an annual return premium, is the equity risk premium:

ERP = E[r_market] − r_f

It is conceptually simple and operationally fundamental. Almost every quantitative valuation in modern finance is built on top of it — the cost of equity in CAPM, the equity weight of WACC, project hurdle rates, pension-fund strategic asset allocation, life-cycle target-date funds. A 1-percentage-point error in the ERP propagates into a 15-25% error in the present value of a long-duration cash-flow stream. So what is the right number?

Three measurements, three numbers

The ERP cannot be observed directly — only its components can. As a result three distinct families of estimate circulate, each with its own loyalists.

Measurement	Method	US 2024 value	Best used for
Historical realised (arithmetic)	Mean of annual excess returns since 1928	~8.0%	One-period expected return
Historical realised (geometric)	Compound growth of $1 invested in stocks vs bills	~6.5%	Multi-year compounding, long horizons
Implied (Damodaran-style DCF)	IRR that equates index level to forecast cash flows	~4.5%	Forward-looking cost of equity
Forward proxy (E/P − 10Y)	S&P forward earnings yield minus 10-year Treasury	~0%	Quick valuation signal, market timing
Survey (Duke CFO, Graham–Harvey)	Asked: what 10-year ERP do you assume?	~4–5%	Cross-check on practitioner consensus
Cross-country (Dimson–Marsh–Staunton)	1900–present, 23 markets, equal-weighted	~5% world, ~6% US	International discount rates

The spread is wide. A skeptic could pick a number anywhere from 0 to 8 and find published support for it. Which is why the practitioner literature has converged on a defensive convention: cite both the historical and the implied figure, justify which you used, and stress-test the valuation across the range.

Arithmetic vs geometric — Jensen's wedge

The arithmetic mean of annual excess returns and the geometric (compound) mean differ by approximately half the variance — Jensen's inequality applied to the log function:

r_geom ≈ r_arith − σ² / 2

For US stocks σ ≈ 0.20 annualised, so σ²/2 ≈ 0.02 — the two-percentage-point gap exactly matches the 8% arithmetic vs 6.5% geometric numbers. Which is "right"? Both, for different jobs.

• For a one-period discount rate — what return should I expect next year? — the arithmetic mean is unbiased.
• For a multi-period compound projection — how much will $1 grow to over 30 years? — the geometric mean is the unbiased terminal-wealth estimator.
• For a cash-flow valuation with declining horizons (year 1, year 2, …, year N), the right weighted average lies somewhere between. Ibbotson's classic textbook recommendation is the arithmetic; Damodaran prefers the geometric; many practitioners split the difference at 6%.

The disagreement is not academic. Discounting a 30-year cash flow at 9% vs 7% (1.5 percentage-point ERP difference at β = 1) changes its present value by a factor of about 1.8.

The implied ERP — Damodaran's monthly update

The historical premium is backward-looking and survivorship-biased: it weights the US (the best-performing market of the 20th century) too heavily, and it incorporates an enormous amount of luck. The implied ERP avoids both problems by asking what discount rate would make today's market price equal to a forecast of future cash flows. Aswath Damodaran of NYU Stern publishes the standard calculation monthly.

The set-up is a multi-stage free-cash-flow model. Take the S&P 500 index level on day t. Use bottom-up analyst forecasts of next year's earnings, dividend yield, and buybacks. Assume a steady-state growth rate at year 5 or 10 — Damodaran uses the 10-year Treasury yield as a long-run nominal growth proxy. Solve for the internal rate of return that makes the present value of those cash flows equal the index level:

P_index = Σ CF_t / (1 + r)^t  + TV_N / (1 + r)^N

ERP_implied = r − r_f

As of early 2024 the resulting implied US ERP is about 4.5%. That is roughly 2 points below the historical geometric figure and tells you something specific: investors today price the S&P 500 as if it will outperform Treasuries by less than the past century's average. Whether you interpret that as "the world is genuinely safer now" or "the market is rich and the prospective premium has been compressed" is a forecasting judgement, not a measurement.

The Mehra–Prescott equity premium puzzle

In their 1985 paper "The Equity Premium: A Puzzle," Rajnish Mehra and Edward Prescott took a standard consumption-based asset-pricing model and asked what equity premium it predicted. The model uses constant-relative-risk-aversion (CRRA) utility,

u(c) = c^(1−γ) / (1 − γ)

and the Euler equation that relates asset returns to consumption growth:

E[r_market − r_f] ≈ γ × Cov(r_market, Δc/c)

Plugging in measured US data — mean consumption growth μ_c ≈ 1.8%, σ_c ≈ 3.6%, correlation with stock returns ρ ≈ 0.4 — the predicted ERP is about 0.35 × γ percent. To match the observed ERP of about 6% requires γ on the order of 30. But at γ = 30 a representative agent is so risk-averse that she would pay $49,000 to avoid a 50/50 gamble between $50,000 and $100,000. Nobody behaves that way. So either the data, the preferences, or the model is wrong.

Forty years of macrofinance has been the answer to that question.

Four resolutions

The literature has produced four broad families of fix, each empirically and intellectually distinct. Most modern macrofinance models combine two or three.

Resolution	Key reference	Core idea	Risk aversion needed
Habit formation	Campbell–Cochrane 1999	Utility depends on c − X, where X is slow-moving habit. Effective risk aversion rises in recessions.	γ effective varies; static γ ≈ 2
Long-run risk	Bansal–Yaron 2004	Tiny, persistent component in expected consumption growth + stochastic volatility, with Epstein–Zin preferences.	γ ≈ 10, ψ > 1
Rare disasters	Rietz 1988; Barro 2006	Add a small-probability catastrophic drop in consumption. Calibrated from international 20th-century data.	γ ≈ 3–4
Epstein–Zin preferences	Epstein–Zin 1989	Recursive utility separates risk aversion from elasticity of intertemporal substitution.	Allows γ ≠ 1/ψ
Behavioural / loss aversion	Benartzi–Thaler 1995	Investors evaluate stocks myopically over short windows; losses hurt 2.25× more than gains help.	Loss aversion, not γ
Limited participation	Mankiw–Zeldes 1991	Stockholders' consumption is more volatile than aggregate; the right covariance is bigger.	γ ≈ 5–7

Habit and long-run risk dominate the modern macrofinance toolkit because they explain a wider set of moments simultaneously — not just the equity premium level but the volatility of returns, the predictability of returns by the price-dividend ratio, and the term structure of equity risk. Rare disasters provide the cleanest theoretical fix and have a natural empirical anchor in international depression and war histories. Epstein–Zin preferences are not really a substantive resolution on their own but an enabler — they allow the long-run-risk and habit mechanisms to generate large premia without forcing implausible auxiliary assumptions.

From ERP to a single stock's cost of equity

The Capital Asset Pricing Model (Sharpe 1964, Lintner 1965, Mossin 1966) takes the ERP and applies it to a specific security via the security's beta:

r_e = r_f + β × ERP

β = Cov(r_i, r_market) / Var(r_market)

β > 1 means the stock amplifies market moves and demands a return above the market; β < 1 means the stock dampens market moves and demands less. β is estimated by regressing the stock's excess returns on the market's, typically over 60 months. A few representative numbers as of 2024:

Sector	Typical β	r_f	ERP	r_e
Utilities	0.5	4.5%	5%	7.0%
Consumer staples	0.7	4.5%	5%	8.0%
Broad market	1.0	4.5%	5%	9.5%
Industrials	1.2	4.5%	5%	10.5%
Technology	1.3	4.5%	5%	11.0%
Small-cap growth	1.5	4.5%	5%	12.0%

That r_e then becomes the cost-of-equity input to the WACC:

WACC = (E / V) × r_e + (D / V) × r_d × (1 − t_c)

Discount free cash flows at the WACC, sum to a present value, divide by share count — that is essentially every equity research valuation in the world. The ERP is the load-bearing input.

Worked example — how much does the ERP move a valuation?

Consider a steady-state company expected to generate $100m of free cash flow next year, growing at 3% in perpetuity. Capital structure is 80% equity, 20% debt; β = 1.1; r_f = 4.5%; r_d = 6%; t_c = 21%. Let us value this firm under two ERP assumptions, 5% (implied) and 7% (historical arithmetic).

ERP = 5%:
  r_e = 4.5 + 1.1 × 5 = 10.0%
  WACC = 0.8 × 10.0 + 0.2 × 6 × (1 − 0.21) = 8.95%
  V = 100 / (8.95% − 3%) = $1,681m

ERP = 7%:
  r_e = 4.5 + 1.1 × 7 = 12.2%
  WACC = 0.8 × 12.2 + 0.2 × 6 × (1 − 0.21) = 10.71%
  V = 100 / (10.71% − 3%) = $1,297m

A 2-point change in the assumed ERP shrinks the firm's value by 23%. Now imagine a research analyst arguing for a price target: the choice of historical vs implied ERP is not a footnote, it is the conclusion. This is why the ERP is — quietly — the most important and least-discussed number in corporate finance.

International evidence — Dimson, Marsh, Staunton

The Credit Suisse Global Investment Returns Yearbook (originally Dimson–Marsh–Staunton 2002, updated annually) extends the analysis to 23 markets from 1900. The headline findings:

• The unweighted-average real equity premium over bills across 23 countries is 4.6% (geometric), 6.0% (arithmetic).
• The US figure of 5.4% geometric is above the world average, partly reflecting survivorship — the US dataset has not been interrupted by world war, hyperinflation, or political destruction.
• The cross-country premium is correlated with realised volatility: more volatile markets (Italy, Japan, Germany) earned higher premia.
• Emerging markets, where data are shorter and noisier, show realised premia of 7–9% in some long samples (Brazil, India) but with much wider error bars.
• The premium has declined modestly in the second half of the 20th century: 4.4% (1900–1949) vs 4.0% (1950–2020).

For an international project valuation, the practitioner add-on is a country-risk premium scaled either by sovereign CDS spreads or by the volatility ratio σ_equity / σ_bond — both methods popularised by Damodaran. A Brazilian discount rate might be the US ERP plus 300 basis points of country risk, multiplied through CAPM with a local-currency β.

Forward ERP — what 2024 is telling us

The most digestible forward indicator is the spread between the S&P 500 forward earnings yield (1/forward-P/E) and the 10-year Treasury yield. It is a crude approximation to the implied ERP — it ignores buyback yield and treats earnings as if they were cash dividends — but its movements track Damodaran's full calculation closely.

Period	Forward E/P	10Y Treasury	Forward ERP	Interpretation
1982 trough	~12%	~13%	~−1%	Stocks cheap vs rates
2000 dot-com peak	~3.5%	~6%	~−2.5%	Historic peak overvaluation
2009 GFC trough	~9%	~3%	~6%	Generational entry point
2020 COVID trough	~6%	~0.7%	~5.3%	Attractive vs rates
2024 (late)	~4.6%	~4.5%	~0.1%	Historic compression

A near-zero forward ERP has historically been a poor entry signal: in every previous instance (1968, 1987 pre-crash, 1999–2000) it preceded a multi-year period of below-trend equity returns. The 2024 reading does not guarantee that history will rhyme, but it does say that the market is pricing equities and bonds for the same expected return — an unusual configuration.

Where the ERP shows up in practice

• Corporate valuation. Every DCF model uses ERP via the WACC. Equity research, M&A, IPO pricing, fairness opinions — all rest on it.
• Capital budgeting. Companies set internal hurdle rates (the minimum project IRR required to allocate capital) using the WACC. A 5% vs 7% ERP shifts the hurdle by 1.5–2 percentage points and changes which projects get funded.
• Strategic asset allocation. Pension funds and endowments solve a mean-variance problem in which the ERP is the equity return assumption. A 0.5-point change can swing the optimal equity weight by 5–10 percentage points.
• Regulated industries. Utility regulators set allowed returns using r_e from CAPM. The ERP assumption is litigated explicitly in every rate case.
• Lifecycle investing / target-date funds. The age-based glide path from equities to bonds is calibrated against an assumed ERP.
• Insurance liabilities. Some insurance reserves are discounted at expected portfolio returns, which depend on the ERP.
• Litigation damages. Lost-profits valuations in commercial litigation routinely depend on which ERP the expert defends in cross-examination.

Common pitfalls

• Confusing the historical ERP with an expectation. The realised premium is one path of a stochastic process. Using 6.5% for the next 30 years assumes the next 30 will look like the last 96. They may not.
• Mismatching the risk-free rate. The ERP is defined relative to either bills or bonds — never both at once. Using a 30-year Treasury yield as r_f and then adding a bills-based ERP double-counts the term premium.
• Currency mismatch. An international valuation should discount cash flows in the currency they are denominated in. Mixing a USD ERP with EUR cash flows is a category error.
• Forgetting Jensen's wedge. Quoting an arithmetic 8% as if it were a compound growth rate inflates terminal values by σ²/2 per year — about 2 points compounded over 30 years is a 79% overstatement.
• Treating implied ERP as a forecast. The Damodaran implied number is what the market is pricing today, not what the market will realise. Mean-reverting forecasters add or subtract from it; pure-implied users should not.
• Ignoring beta-ERP correlation. Betas are not invariant — a stock's beta typically rises during market stress, which is exactly when the ERP also rises. The product β × ERP is more volatile than its components suggest.
• Survivorship bias. US-only historical ERPs overstate the global figure because the US did not suffer the catastrophic 20th-century interruptions that shrank other markets' returns.