Rational expectations theory and the efficient markets hypothesis: formula sheet

Key formulas

Rational expectation

X(e, t+1) = E[X(t+1) | I(t)]

The forecast equals the conditional expected value given all information I(t) available at time t.

Forecast error

ε(t+1) = X(t+1) − E[X(t+1) | I(t)]

Under rational expectations, E[ε(t+1) | I(t)] = 0, so the error is unbiased and unpredictable from I(t).

Adaptive expectations

X(e, t+1) = X(e, t) + λ × (X(t) − X(e, t)), 0 < λ ≤ 1

Forecast is revised by a fraction λ of the last error. It uses only past values of the variable.

Orthogonality property

Cov(ε(t+1), Y) = 0 for any Y in I(t)

No known variable can predict the error. A correlation would mean information was not used fully.

Weak form

Information set = past prices and trading data

Technical (chartist) analysis should not earn abnormal returns.

Semi-strong form

Information set = all public information (includes the weak-form set)

Fundamental analysis on public data should not earn abnormal returns. Prices react quickly to announcements.

Strong form

Information set = all public and private information (includes the semi-strong set)

Even insider information should not earn abnormal returns.

Nesting of forms

Strong ⇒ Semi-strong ⇒ Weak

If a stronger form holds, the weaker ones hold. The reverse is not true.

Abnormal return

Abnormal return = actual return − return expected for the risk taken

Efficiency says this should be zero on average when you trade on the information in the set.

Abnormal return

ARₜ = Rₜ − E(Rₜ)

Actual return minus expected return from a model such as the market model or CAPM.

Market model expected return

E(Rₜ) = α + β × R_mₜ

α and β are estimated over a period before the event, called the estimation window.

Cumulative abnormal return

CAR = Σ ARₜ over the event window

Sum of abnormal returns from the start to the end of the window.

Average abnormal return across N firms

AARₜ = (1 ÷ N) × Σ ARᵢₜ

Averaging across firms reduces noise unrelated to the event.

Autocorrelation at lag k

ρₖ = Cov(Rₜ, Rₜ₋ₖ) ÷ Var(Rₜ)

Under a random walk with stationary returns, ρₖ = 0 for k ≥ 1. Compare with the approximate 95% band ±1.96 ÷ √n.

Random walk

Pₜ = Pₜ₋₁ + εₜ, with εₜ independent

Price changes are unpredictable from past changes. This is consistent with weak form efficiency but is not identical to it.

Excess (abnormal) return

Abnormal return = Actual return − Required (expected) return

Efficiency is tested against a stated risk model, such as CAPM. This is the joint hypothesis problem: you test the market and the model together.

Active return

Active return = Portfolio return − Benchmark return

Under efficiency, expected active return before costs is about zero, and negative after costs.

Net return after costs

Net return = Gross return − Fees and trading costs

Used to compare active and passive. Active needs a gross edge larger than its extra costs.

Forms of EMH

Weak ⊂ Semi-strong ⊂ Strong (information sets)

Strong efficiency implies semi-strong, which implies weak. Technical analysis fails under weak form; fundamental analysis fails under semi-strong; insider information fails under strong.

Quick revision

• Rational expectations: forecasts use all available information efficiently, so forecast errors are unpredictable and have zero mean on average.
• Adaptive expectations rely on past values only, so errors can be systematic.
• EMH: market prices fully reflect the relevant information set.
• Weak form: prices reflect all past price and trading information, so technical analysis should not earn abnormal returns.
• Semi-strong form: prices reflect all public information, so fundamental analysis should not earn abnormal returns.
• Strong form: prices reflect all information, including private, so even insiders should not earn abnormal returns.
• Each form includes the one before it: strong implies semi-strong, which implies weak.
• Efficiency means prices react quickly to new information, so price changes follow new information and are hard to predict.
• Tests of weak form look at patterns in past returns; event studies look at price reaction to public announcements.
• Testing efficiency is a joint test with an asset pricing model, so a rejection may reflect a faulty model.
• Evidence is mixed: markets look largely efficient in the weak form, with some anomalies found in other forms.
• For investors, efficiency favours low-cost passive strategies; for actuaries, it supports using market prices in valuation.

Common mistakes

• Saying rational expectations means forecasts are always correct. Fix: Say forecasts are correct on average. Errors exist but are random, with zero mean and unpredictable.
• Saying adaptive expectations use all available information. Fix: Adaptive expectations use only past values of the variable. Rational expectations use all information and the model structure.
• Saying the weak form means prices are weakly related to information or that markets are inefficient. Fix: Weak refers only to a small information set: past prices. A market can be weak-form efficient and fully rational.
• Thinking semi-strong efficiency rules out profit from insider information. Fix: Semi-strong covers public information only. Insiders may still profit if the strong form fails.
• Saying an abnormal return proves the market is inefficient. Fix: State that every test is a joint test of efficiency and the pricing model.
• Treating a price rise before an announcement as always showing inefficiency. Fix: Say the event may have been expected, and judge efficiency by whether drift continues after the news becomes public.
• Saying efficient markets mean prices never change or are always correct. Fix: Say prices reflect available information and change when new information arrives. Errors are random, not predictable.
• Claiming that under efficiency no one should invest actively or that investing is pointless. Fix: Say active management has no expected net advantage. Asset allocation, diversification and liability matching remain important.

Exam tips

• Learn a clean definition with notation: E[X(t+1) | I(t)], and state the error properties in one sentence.
• For adaptive versus rational questions, compare on three points: information used, predictability of errors and bias.
• In discussion questions, give both assumptions and criticisms. Examiners reward balance.
• Link to the efficient markets hypothesis in one line when the question is about markets, and to policy when it is about macroeconomics.
• In numerical parts, show the forecast error calculation step by step so method marks are secured.
• Always name the information set first. Examiners award marks for the link between the form and the information.
• Use the nesting rule in any question asking what a result implies for the other forms.
• In written answers, mention the joint hypothesis problem when discussing evidence.