The Equity Premium Puzzle

A simple guide from basics to theory

The Numbers

Over about 100 years in the U.S., stocks returned ~7% per year. Treasury billsi returned ~1%. The gap is the equity premiumi.

What we see in data

6.18%

What the model can do (max)

0.35%

The model can only explain a small premium. The real premium is about 18 times larger. That is the puzzle.

The puzzle in one picture

First: stocks return ~7%, bonds ~1%. The difference (6%) is the equity premium.

Stocks (S&P 500)

~7%

Bonds (T-bills)

~1%

The gap between the two bars = equity premium (~6%).

The model can only produce 0.35% premium. Same scale:

Observed (data)

6.18%

Model (max)

0.35%

0% ———————————————————————— 7%

Observed in data Model maximum

What does "model max" mean?

"Model max" means the highest equity premium the model can produce when you vary its parameters (α and β) over the ranges we think are plausible.

You try many combinations: α from 0 to 10, β from 0 to 1. For each pair, you compute the premium the model predicts. The best you can get—the maximum—is 0.35%. No plausible choice of α and β gives more than that.

The model

Preferences. One agent maximizes:

E Σ β^t (c_t^1−α − 1) / (1 − α)

β = discount factor. α = risk aversion.

Economy. One "tree" produces fruit (consumption) each period. Consumption growth follows a Markov process calibrated to U.S. data. Stocks = claim on the tree. Bonds = risk-free claim.

Pricing. In equilibrium, the agent is indifferent between holding stocks and bonds. The Euler equation pins down prices. From it we get:

ln(1 + premium) = α × σ²

where σ² is the variance of log consumption growth. So the premium rises with α and with consumption risk. With U.S. data, σ² ≈ 0.00125. With α ≤ 10, the premium stays small.

Why can the model only produce 0.35%?

1. Consumption is smooth. The growth rate of consumption does not vary much from year to year. In the data, its variance is small (~0.00125). So stocks are not that risky from the perspective of "when will I be able to consume?" They pay off more when consumption is already high, but consumption does not swing wildly.

2. The premium is proportional to risk aversion × consumption risk. The formula is: premium = exp(α × σ²) − 1. With small σ², you need large α to get a large premium. With α = 10 (the upper bound of the "plausible" range), the model gives about 1–2% premium in the simple formula. Mehra and Prescott used a discrete Markov chain; with their calibration and α ≤ 10, the maximum they could get was 0.35%.

3. We restrict α to be plausible. Most studies say α is between 1 and 5. Mehra and Prescott used 0 ≤ α ≤ 10 as a generous upper bound. If we allowed α = 50, the model could match 6%. But then people would be so risk-averse that they would pay huge sums to avoid small gambles. That does not match how people behave. So we stick to plausible α—and with that, the premium stays small.

What 6% really means

Six percent per year sounds small. But compounded over 90 years (1889-1978), $1 becomes very different amounts. Click to see.

Stocks

T-Bills

1889 — Click "Go" to start

Returns by decade

Average real annual returns. The premium exists in nearly every decade.

1889-99

S: 8.7% B: 4.5%

1900-09

S: 8.4% B: 2.0%

1910-19

S: 0.9% B:-3.5%

1920-29

S:15.0% B: 5.4%

1930-39

S: 0.5% B: 2.6%

1940-49

S: 6.3% B:-4.0%

1950-59

S:14.8% B: 1.0%

1960-69

S: 5.7% B: 1.3%

1970-78

S: 0.8% B:-0.3%

Stocks T-Bills

Stocks beat bonds in 7 of 9 decades. Even the two exceptions (1930s, 1970s) show tiny stock returns, not large bond returns. The premium is persistent, not a fluke of one era.

Find your risk aversion

You have $10,000. A fair coin is flipped. Would you take each of these bets?

Question 1 of 6

Heads: you win $6,000. Tails: you lose $1,000.

Risk aversion and the formula

Risk aversioni is how much you dislike risk. In the model it is measured by αi (the coefficient of relative risk aversion). Higher α means you dislike consumption going up and down more.

The equity premium in the model comes from this formula:

ln(1 + premium) = α × σ²

Here σ² is the variance of log consumption growth (~0.00125 in U.S. data). So: premium = exp(α × 0.00125) − 1.

When consumption growth varies, stocks are risky because they pay off more when consumption is already high (and you value that less). The premium is the reward for bearing that risk. It grows with α: more risk-averse people demand a bigger premium to hold stocks.

In the data, the variance of consumption growth is small (~0.00125). So to get a 6% premium, α must be large. Move the slider to see how the formula changes.

α = 10

With α = 10:

Premium = exp(α × 0.00125) − 1 = 1.26%

What this risk aversion means in practice

What world would fix the puzzle?

The puzzle exists because consumption is too smooth. Change the data parameters below and see what happens. What would the economy need to look like for the model to work with reasonable risk aversion (α ≤ 10)?

Consumption volatility (σ) 3.6%

Actual U.S. data: 3.6%. This is how much consumption growth varies year to year.

Stock-consumption correlation 0.40

Actual: ~0.40. How closely stock returns move with consumption.

Stock return volatility (σ_s) 16.7%

Actual: ~16.7%. How volatile stock returns are.

α needed to match 6.18%

plausible: ≤ 10

Cov(stocks, consumption)

0.0024

actual: ~0.0024

The Story in Three Papers

What they asked

Can standard theory explain why stocks return so much more than safe bonds?

Main equations

Objective:

max E₀ Σ_t=0^∞ β^t (c_t^1−α − 1) / (1 − α)

E₀ = expectation at time 0

β = discount factor (how much you value future vs. today)

c_t = consumption at time t

α = coefficient of relative risk aversion

Euler equation (pricing condition):

E_t[β (c_t+1/c_t)^−α R_t+1] = 1

E_t = expectation given information at time t

R_t+1 = gross return on an asset (stocks or bonds)

Equity premium (with lognormal consumption growth):

ln(1 + premium) = α × σ²

premium = equity premium (extra return on stocks over bonds)

σ² = variance of log consumption growth

Assumptions

One representative agenti with power utilityi
α (risk aversion) between 0 and 10; β (discount factor) between 0 and 1
Complete marketsi
Frictionlessi trading
Consumption growth follows a two-state Markov chain that matches U.S. data (mean, variance, serial correlation)
Dividend growth is perfectly correlated with consumption growth

The setup

Pure exchange economy (Lucas tree)
Stocks = claim on the tree; bonds = risk-free claim

Where does the formula come from?

Click through each step to see how Mehra and Prescott derive the equity premium formula.

Step 1. The investor wants to maximize lifetime utility:

max E₀ Σ β^t U(c_t)

They choose how much to consume now vs. save/invest for tomorrow.

Step 2. At the optimum, the investor is indifferent about buying one more unit of any asset. This gives the Euler equation:

U'(c_t) = β E_t[U'(c_t+1) R_t+1]

The cost of giving up consumption today (left) must equal the expected benefit of investing and consuming tomorrow (right).

Step 3. Plug in power utility U(c) = c^1-α/(1-α), so U'(c) = c^-α:

c_t^-α = β E_t[c_t+1^-α R_t+1]

Divide both sides by c_t^-α:

1 = β E_t[(c_t+1/c_t)^-α R_t+1]

Step 4. Write this for both stocks (R^s) and bonds (R^b), subtract:

E_t[(c_t+1/c_t)^-α (R^s_t+1 - R^b_t+1)] = 0

The equity premium must satisfy this condition.

Step 5. If consumption growth is lognormal, take logs and use the approximation:

E[R^s] - R^b ≈ α × Cov(R^s, Δ ln c)

With i.i.d. growth this simplifies to:

ln(1 + premium) = α × σ²

And that is the equity premium formula. With σ² ≈ 0.00125 and α ≤ 10, the right side is tiny. The puzzle.

The result

With plausible risk aversioni (α ≤ 10), the model gives at most 0.35% premium. Observed: 6.18%.

Consumptioni does not move enough. To justify 6%, people would need to be very risk-averse (α ≈ 50).

What Weil tried

Can we separate risk aversioni from intertemporal substitutioni? He used Kreps-Porteusi (or Epstein-Zini) preferences.

Main equations

Recursive utility (Kreps-Porteus / Epstein-Zin):

U_t = [(1−β)c_t^1−ρ + β(E_t U_t+1^1−γ)^{(1−ρ)/(1−γ)}]^1/(1−ρ)

U_t = utility at time t

β = discount factor

c_t = consumption at time t

γ = coefficient of relative risk aversion (separate from ρ)

ρ = parameter for intertemporal substitution (1/ρ = IES)

E_t = expectation given information at time t

With i.i.d. consumption growth, the equity premium depends only on γ:

ln(1 + premium) = γ × σ²

premium = equity premium

γ = risk aversion (ρ does not appear)

σ² = variance of log consumption growth

Assumptions

Same economy as Mehra-Prescott (Lucas tree, Markov dividend process)
But: Kreps-Porteusi preferences instead of time-additive expected utility
Two separate parameters: γ (risk aversion) and ρ (intertemporal substitution)
Complete marketsi, frictionlessi trading
Same calibration as Mehra-Prescott (two-state Markov, U.S. data)

The result

With i.i.d.i growth, the equity premiumi depends only on risk aversion (γ), not on intertemporal substitutioni. So the fix does not work.

He also added the risk-free rate puzzlei: if people dislike consumption swings, why is the risk-free ratei so low?

Try it: Weil's two independent knobs

Risk aversion (γ) 2.0

IES (1/ρ) 1.00

0.12%

Observed: 6.18%

0% Predicted equity premium 8%

Predicted EP

0.12%

needed: ~6.18%

Predicted risk-free rate

7.0%

observed: ~0.8%

Notice: moving the IES slider barely changes the equity premium -- it just makes the risk-free rate better or worse. The extra knob does not help.

What Kocherlakota did

He wrote a survey. He showed the puzzles are robust and come from only three assumptions. Any fix must relax at least one.

Main equations

Kocherlakota uses the same model as Mehra-Prescott. The key conditions are:

Equity premium condition:

E[(C_t+1/C_t)^−α (R_t+1^s − R_t+1^b)] = 0

C_t = per capita consumption at time t

α = coefficient of relative risk aversion

R_t+1^s = gross return on stocks

R_t+1^b = gross return on bonds

E = unconditional expectation

Risk-free rate condition:

β E[(C_t+1/C_t)^−α R_t+1^b] = 1

β = discount factor

If these hold with per capita consumption, the puzzles emerge. Any model that relaxes power utility, complete markets, or frictionless trading can avoid them.

Assumptions

Kocherlakota showed the puzzles come from only three things. Any fix must change at least one:

Power utilityi with plausible αi and βi
Complete marketsi
Frictionlessi trading

He showed the puzzles do not depend on the exact consumption process, sampling error, or the assumption that dividends equal consumption. Only these three matter.

Survey of fixes

In the decade after Mehra and Prescott, economists tried every route. Kocherlakota surveyed them all. Click each approach to see what it tried and why it fell short.

Badges show whether the approach can explain the equity premium (EP) and the risk-free rate (RF).

Epstein-Zin / Generalized Expected Utility

EP: No RF: Yes

▼

The idea: Separate risk aversion from intertemporal substitution, giving two independent knobs instead of one.

On the risk-free rate: It works. By setting IES independently, the model can match the low observed risk-free rate.

Why it fails on the EP: The equity premium is governed almost exclusively by risk aversion, regardless of IES. You still need implausibly high risk aversion to match the data.

Habit Formation

EP: No RF: Yes

▼

The idea: Your happiness depends on what you consumed yesterday. If you drove a BMW last year, going back to a cheaper car feels painful. This amplifies risk aversion.

On the risk-free rate: Helps a lot. Since people are "hooked" on their current level, they save more, pushing the rate down.

Why it fails on the EP: When consumption growth is hard to predict (which it is), the extra risk aversion from habits does not transfer into a higher equity premium in the formal model.

"Keeping Up with the Joneses"

EP: Partial RF: Partial

▼

The idea: People care about how they compare to everyone else. If everyone's consumption falls, you feel extra pain beyond just your own loss.

Why it falls short: Can match the EP, but requires an unrealistically extreme concern about relative standing. Also needs habit-like effects to explain the low risk-free rate simultaneously.

Incomplete Markets (Uninsurable Income Risk)

EP: No RF: No

▼

The idea: Real people cannot insure against everything -- job loss, health shocks. Individual consumption is more volatile than the aggregate.

Why it fails: In a long-lived economy, people smooth income shocks by saving during good years and spending during bad years (dynamic self-insurance). This works well enough that the incomplete-markets equilibrium looks very similar to the complete-markets one.

Borrowing Constraints

EP: No RF: Yes

▼

The idea: If people cannot borrow against future income, they hold extra safe assets as a buffer. This pushes the risk-free rate down.

Why it fails on the EP: Constrained investors are constrained in both markets. Both returns get pushed down together, leaving the spread roughly unchanged.

Transaction Costs

EP: Only if... RF: N/A

▼

The idea: Stocks are more expensive to trade than bonds. Part of the premium compensates for these costs, not for risk.

Why it falls short: For a buy-and-hold investor, the annualized cost shrinks the longer they hold. Only dramatically higher stock trading costs would explain the gap, and there is little evidence for that.

Rare Disasters

EP: Possible RF: Wrong

▼

The idea: Investors fear a small probability of catastrophe -- a Great Depression, a war. Even if these did not happen in the sample, the fear could justify a high EP.

Why it is unconvincing: In a disaster, the model predicts real interest rates should spike -- but historically they did not. Also, the disaster probability is a free parameter: you can match any EP by picking the right probability, making the theory untestable.

Survivorship Bias

EP: No RF: No

▼

The idea: We study the U.S. because it survived and thrived. Markets that collapsed are not in the data.

Why it fails: In catastrophic episodes, bonds also suffered -- governments defaulted, hyperinflation wiped out bondholders. So the spread between stocks and bonds should not be dramatically affected by looking only at survivors.

Three papers at a glance

Feature	Mehra & Prescott (1985)	Weil (1989)	Kocherlakota (1996)
Type	Original paper	Extension	Survey
Utility	Power utility (CRRA)	Kreps-Porteus / Epstein-Zin	Same as M&P; surveys alternatives
Key parameters	α (risk aversion = 1/IES)	γ (risk aversion), ρ (1/IES), independent	α, β (same framework)
Main finding	Model produces max 0.35% premium with plausible α	Separating γ and ρ does not help the EP; discovers risk-free rate puzzle	Puzzle is robust; comes from 3 assumptions; no fix fully works
Explains EP?	No	No	No (surveys why)
Explains RF puzzle?	Not addressed	Partially (with IES)	Some fixes help
Markets	Complete, frictionless	Complete, frictionless	Surveys relaxations
Data period	1889-1978	Same as M&P	Same + extended samples

More detail

What is the equity premium?

The extra return you get for holding stocks instead of safe government bonds. Over ~100 years in the U.S., stocks returned ~7% per year, T-bills ~1% per year. The 6% gap is the equity premium. Theory says it should reflect how risky stocks are and how much people dislike risk. The puzzle: consumption is fairly smooth, so stocks are not that risky—yet the premium is large.

Why does the model fail?

The equity premium is roughly α times the covariance of consumption growth with stock returns. Consumption growth has low variance and low covariance with stocks. So to get 6%, α must be ~40–50. But micro evidence suggests α is 1–5. And if you set α = 50, the model predicts a risk-free rate of 10%+ instead of 1%.

What did Weil prove?

With i.i.d. dividend and consumption growth, the equity premium does not depend on the intertemporal elasticity of substitution. It depends only on risk aversion. So separating them does not help the equity premium. It can help the risk-free rate puzzle, but often makes it worse when you calibrate realistically.

What fixes have been tried?

Preferences: Epstein-Zin (does not fix equity premium), habit formation (limited), relative consumption. Markets: incomplete markets, idiosyncratic risk, borrowing constraints. Distribution: disaster states, survivorship bias—both criticized. Bottom line: the risk-free rate puzzle has plausible explanations; the equity premium puzzle is still open.

Test your understanding

1. Why is the equity premium a "puzzle"?

The puzzle is that consumption-based models predict a much smaller equity premium than the 6.18% observed. With plausible risk aversion, the model tops out at about 0.35%.

2. What is the root cause of the puzzle?

Consumption barely moves year to year. Since the model prices risk based on how assets co-move with consumption, smooth consumption means stocks look almost as safe as bonds. A small covariance times reasonable risk aversion gives a small premium.

3. What did Weil (1989) discover?

With i.i.d. consumption growth, the equity premium depends only on risk aversion (γ), not on intertemporal substitution. The new parameter helps the risk-free rate but not the EP. Worse, he uncovered the risk-free rate puzzle: why do people save so much at such a low rate?

4. According to Kocherlakota, the puzzle comes from which three assumptions?

Kocherlakota showed the puzzle survives regardless of the consumption process, sample period, or dividend assumptions. It comes from just three things: (1) power utility with reasonable α and β, (2) complete markets, and (3) frictionless trading. Any fix must relax at least one.

5. Why does "habit formation" fail to fully solve the puzzle?

Habit formation amplifies sensitivity to consumption drops, but the formal result shows that when consumption growth is unpredictable (which it is in reality), the amplified risk aversion does not produce a larger equity premium in the model. It does help the risk-free rate, though.

Timeline

1985

Mehra & Prescott

Stocks earn 6% more than bonds. Our best model says they should not.

↓

1989

Weil

Separating risk aversion from time preferences does not help. And now we have a second puzzle: why is the risk-free rate so low?

↓

1996

Kocherlakota

After a decade of attempts, the equity premium remains unexplained. No proposed fix avoids implausible assumptions.

The core tension

Aggregate consumption is too smooth relative to stock returns for any reasonably risk-averse investor to demand a 6% premium for holding stocks over bonds. Every modification to preferences, market structure, or statistical assumptions tried over the first decade of research failed to convincingly change this arithmetic.