CFA Level I Exam · The Capital Asset Pricing Model, Market Model, and Other Factor-Based Equity Models

Beta Estimation and the Market Model Regression

Updated 7 October 2026 · Fact-checked

The market model is a regression of a security's return on the market's return: Ri = αi + βi Rm + ei. The slope is beta, estimated as Cov(Ri, Rm) ÷ Var(Rm). Adjusted beta pulls the raw estimate toward 1: 2/3 × raw beta + 1/3 × 1. Analysts use beta for CAPM cost of equity.

Understand Beta Estimation and Market Model

Beta measures how sensitive a security's return is to the market's return. A beta of 1.2 means that, on average, the security moves 1.2% when the market moves 1%. It captures only systematic risk, the part you cannot diversify away.

The market model is how you estimate beta from data. You collect periodic returns for the security and for a market index, such as weekly or monthly returns over several years. Then you run a simple linear regression: security return as the dependent variable, market return as the independent variable. The slope is the beta. The intercept is alpha, the average return not explained by the market. The residual is the part of return specific to the security.

The market model is a statistical description of past returns. CAPM is an equilibrium pricing model about expected returns. CAPM says E(Ri) = Rf + βi[E(Rm) − Rf], and it predicts that alpha is zero once you use excess returns. The market model does not need that assumption. It can be run on raw returns, and its intercept can be nonzero. Beta is the link between the two: you estimate it with the market model and then plug it into CAPM.

Raw betas from regressions contain estimation error, and betas tend to drift toward 1 over time. So analysts often use adjusted beta, a weighted average of the raw beta and 1. The common Blume-type adjustment uses weights of 2/3 and 1/3. High raw betas are adjusted down and low raw betas are adjusted up.

The quality of the estimate depends on choices you make: the market index, the return interval, the length of the data window, and whether you use excess returns. R-squared tells you how much of the security's return variation the market explains. The standard error of beta tells you how reliable the estimate is.

Key formulas to remember

Market model: Ri = αi + βi Rm + ei; Regression of security return on market return. Intercept is alpha, slope is beta, e is the residual with expected value zero.
Beta from covariance: βi = Cov(Ri, Rm) ÷ Var(Rm); Same as the OLS slope. Use the same sample or population basis for numerator and denominator.
Beta from correlation: βi = ρ(i,m) × σi ÷ σm; Useful when you are given correlation and standard deviations.
Adjusted beta (Blume-type): Adjusted β = (2/3) × raw β + (1/3) × 1.0; Pulls raw beta toward 1. Weights of 2/3 and 1/3 are the commonly used ones.
Alpha from regression: αi = mean(Ri) − βi × mean(Rm); The regression line passes through the sample means.
CAPM expected return: E(Ri) = Rf + βi × [E(Rm) − Rf]; Use beta from the market model to find required return or cost of equity.
Systematic variance share: R² = βi² σm² ÷ σi²; For a simple regression, R² equals the squared correlation between security and market.

How to solve Beta Estimation and Market Model questions

Most questions give you either regression output or summary statistics. Decide which, then follow these steps.

1Identify what is given: regression output (slope, intercept), covariance and variance, or correlation and standard deviations.
2Pick the matching beta formula: slope as given, Cov ÷ Var(market), or ρ × σi ÷ σm.
3Check that the market is the denominator. Beta is not Cov ÷ Var(security).
4If the question asks for adjusted beta, compute 2/3 × raw + 1/3 × 1 after you have the raw beta.
5If the question asks for expected or required return, put the right beta (raw or adjusted, as stated) into CAPM with the given risk-free rate and market return.
6If asked for alpha, use mean security return minus beta times mean market return, or read the intercept.
7Sanity check: a beta above 1 means more volatile than the market; adjusted beta must lie between the raw beta and 1.
8Eliminate options that fall on the wrong side of 1 or ignore the adjustment.

Quickest way: Three-option elimination for beta questions

When to use it: When you have about 90 seconds and the question asks for a beta, adjusted beta or required return.

Estimate roughly first: is raw beta above or below 1?
Adjusted beta lies between raw beta and 1, and is closer to raw beta. Remove any option outside that range.
Compute adjusted beta as 1 + (2/3)(raw − 1). For raw 1.30 this gives 1 + 0.2 = 1.20.
For CAPM, calculate Rf + β × premium. Remove options that are below Rf when beta is positive.
On the BA II Plus, you can key 2 ÷ 3 × 1.3 + 1 ÷ 3 = to get 1.2000 quickly. Use parentheses carefully.

Common mistakes in Beta Estimation and Market Model

Dividing covariance by the security's variance instead of the market's variance.
Students recall 'Cov over Var' but forget which variance.
Fix: Beta measures sensitivity to the market, so the market's variance always goes in the denominator.
Treating the market model and CAPM as the same thing.
Both use beta and a market return.
Fix: The market model is a regression describing historical returns with a free intercept. CAPM is an equilibrium model for expected returns, with no alpha.
Applying the adjustment as 2/3 × raw beta only.
Students forget the 1/3 × 1.0 term.
Fix: Adjusted beta = 2/3 × raw + 1/3 × 1. Check that the result falls between raw and 1.
Mixing sample and population statistics in the beta ratio.
Covariance and variance are quoted from different sources.
Fix: Use consistent bases. If the n−1 or n divisors cancel, the ratio is the same, but mismatching them distorts it.
Assuming a high R-squared means beta is high.
Confusing fit with sensitivity.
Fix: R-squared shows how much return variation the market explains. Beta is the slope. A security can have a low beta and a high R-squared.
Using the regression intercept as the CAPM alpha without regard to the data used.
In a raw-return regression, the intercept may contain a risk-free-rate component, so it is not necessarily a pure alpha.
Fix: Read the question: if returns are excess returns, the intercept is alpha. If they are raw returns and CAPM held, the intercept would equal Rf × (1 − β), so do not read it as alpha without adjusting for Rf.

Worked examples

Example 1

A stock's monthly returns are regressed on a market index. The covariance of stock and market returns is 0.0045 and the variance of market returns is 0.0030. The risk-free rate is 3%, and the expected market return is 8%. What is the stock's CAPM required return using adjusted beta? A) 8.00% B) 9.67% C) 10.50%

Show the solution

Raw beta = 0.0045 ÷ 0.0030 = 1.50.
Adjusted beta = (2/3)(1.50) + (1/3)(1) = 1.00 + 0.3333 = 1.3333.
Market risk premium = 8% − 3% = 5%.
Required return = 3% + 1.3333 × 5% = 3% + 6.6667% = 9.6667%, which rounds to 9.67%.
Option C, 10.50%, is the trap: it uses the raw beta of 1.50, giving 3% + 1.50 × 5% = 10.50%.

Answer: B) 9.67%

Example 2

A security has a standard deviation of returns of 24%, the market's standard deviation is 16%, and the correlation between them is 0.60. What is the security's adjusted beta (Blume-type, weights 2/3 and 1/3)? A) 0.60 B) 0.90 C) 0.93

Show the solution

Raw beta = ρ × σi ÷ σm = 0.60 × 24 ÷ 16.
24 ÷ 16 = 1.5, and 0.60 × 1.5 = 0.90.
Adjusted beta = (2/3)(0.90) + (1/3)(1) = 0.60 + 0.3333 = 0.9333, which rounds to 0.93.
Option C matches this value.
Option B, 0.90, is the trap of forgetting the adjustment: it is the raw beta.
Option A comes from inverting the ratio: raw beta 0.60 × 16 ÷ 24 = 0.40, and its adjusted value is (2/3)(0.40) + 1/3 = 0.60.

Answer: C) 0.93 (raw beta 0.90).

Exam tips

Read whether the question asks for raw or adjusted beta before calculating. The wrong one is usually an option.
Adjusted beta always sits between raw beta and 1. Use this to remove two options fast.
Know the conceptual contrast: market model is a regression with an intercept, CAPM is an equilibrium model about expected returns.
Questions on regression output may ask about R-squared or the standard error of beta. R-squared is the share of variance explained by the market.
With no penalty for wrong answers, never leave a question blank. Eliminate and guess.

Practice questions from The Capital Asset Pricing Model, Market Model, and Other Factor-Based Equity Models

Beta Estimation and Market Model in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Beta Estimation and Market Model: frequently asked questions

How do you calculate beta from a regression?

Regress the security's returns on the market's returns. The slope coefficient is beta. It equals the covariance of the two return series divided by the variance of the market returns.

What is the adjusted beta formula?

Adjusted beta = 2/3 × raw beta + 1/3 × 1.0. It moves the raw estimate toward 1 because estimated betas tend to drift toward the market average over time.

What is the difference between the market model and CAPM?

The market model is a statistical regression of a security's return on the market return, with an intercept that can be nonzero. CAPM is an equilibrium model that links expected return to beta and predicts zero alpha. You estimate beta with the market model and use it in CAPM.

What does alpha mean in the market model?

Alpha is the regression intercept. It is the average return the market does not explain. With excess returns, a positive alpha means the security earned more than its beta would predict over the sample.