CFA Level I Exam · Portfolio Risk and Return: Part II

Market Model and Beta Calculation for CFA Level I

Updated 7 October 2026 · Fact-checked

The market model is a single-factor return generating model: Ri = αi + βi·RM + ε. Beta is the regression slope, equal to Cov(Ri, RM) ÷ Var(RM). It measures how much an asset's return moves per 1% move in the market. R-squared shows the share of return variance explained by the market.

Understand Return Generating Models and Beta

A return generating model says what drives an asset's return. The simplest is the single-index model (also called the market model). It uses one factor: the return on a market index.

The model is Ri = αi + βi·RM + ε. Here Ri is the asset return and RM is the market return. Alpha (αi) is the intercept, the return not explained by the market. Beta (βi) is the slope. The error term (ε) is the part of return that is specific to the asset, with an expected value of zero.

You estimate beta by regressing asset returns on market returns over many periods. The ordinary least squares slope is Cov(Ri, RM) ÷ Var(RM). So beta is the asset's covariance with the market, scaled by market variance. A beta of 1.2 means the asset tends to move 1.2% for each 1% move in the market.

R-squared is the share of the asset's return variance explained by market movements. In a single-variable regression it equals the squared correlation. The remainder, 1 − R², is nonsystematic (firm-specific) variance. Beta captures systematic risk only, so it does not tell you total risk.

Beta estimates depend on choices: the index used, the return interval (daily, weekly, monthly), and the length of the sample. A different choice can give a different beta. This is why estimated betas are uncertain and often adjusted.

Key formulas to remember

Single-index (market) model: Ri = αi + βi·RM + ε; One factor. ε has expected value zero and is specific to the asset.
Beta from covariance: β = Cov(Ri, RM) ÷ Var(RM); Use the same type of variance and covariance (both sample or both population). Units cancel in the ratio.
Beta from correlation: β = ρ(i, M) × σi ÷ σM; Useful when you are given correlation and standard deviations.
Alpha (intercept): α = mean(Ri) − β × mean(RM); The regression line passes through the sample means.
R-squared: R² = ρ² = β²·σM² ÷ σi²; Share of asset variance explained by the market. Valid for a one-variable regression.
Systematic and nonsystematic variance: σi² = β²·σM² + σε²; Systematic part is β²·σM². Nonsystematic part is σε² = (1 − R²)·σi².
Adjusted beta (Blume-style): Adjusted β = (2/3) × raw β + (1/3) × 1.0; Pulls beta toward 1. Use only if the question gives these weights or asks for adjustment.

How to solve Return Generating Models and Beta questions

Identify what is given, then choose the shortest link to beta or to variance decomposition.

1Write down what you are given: covariance, variances, standard deviations, correlation, or regression output.
2If you have covariance and market variance, divide: β = Cov ÷ Var(M). Check both use the same basis.
3If you have correlation and standard deviations, use β = ρ × σi ÷ σM. Do not mix up which is the asset and which is the market.
4For R-squared, square the correlation, or use β²·σM² ÷ σi². Remember the answer is a proportion between 0 and 1.
5For variance split, systematic = R² × σi² and nonsystematic = (1 − R²) × σi². Take the square root only if asked for standard deviation.
6If asked for alpha or a predicted return, use α = mean(Ri) − β × mean(RM), then Ri = α + β × RM.
7Sanity check: beta above 1 means more volatile than the market if correlation is positive. Then pick the option that fits.

Quickest way: Beta in one line

When to use it: Any question that gives covariance, variance, correlation or standard deviations and asks for beta or R-squared.

Cov and Var(M) given: divide them. Done.
Correlation and two standard deviations given: ρ × σi ÷ σM.
Need R²: square ρ. Do not square beta.
Need a standard deviation of the unexplained part: σi × √(1 − R²).
Options are listed smallest to largest. Estimate roughly first, then drop the option that is clearly off by a factor of the market volatility or by a squared term.

Common mistakes in Return Generating Models and Beta

Dividing covariance by the asset's variance instead of the market's variance.
The formula looks symmetric, so students pick the wrong denominator.
Fix: Beta measures sensitivity to the market. The market is the independent variable, so Var(RM) goes in the denominator.
Using beta as a measure of total risk.
Beta is called a risk measure, so students treat it like standard deviation.
Fix: Beta captures only systematic risk. Total risk needs σi, which includes nonsystematic variance.
Treating R-squared as the correlation.
Both come from the same data and are often reported together.
Fix: R² = ρ². A correlation of 0.8 gives R² of 0.64. To go back, take the square root.
Mixing sample and population covariance and variance.
Data are given partly as sums, partly as ratios.
Fix: Use the same divisor (n or n − 1) for both. Then the divisors cancel in the ratio.
Taking the square root too early or too late in variance splits.
Variances add; standard deviations do not.
Fix: Split variance first using R². Convert to standard deviation only at the end.
Assuming a high R-squared means a high beta.
Students link a strong fit with a steep slope.
Fix: Slope and fit are separate. A low-volatility stock can have a beta below 1 and still have a high R².

Worked examples

Example 1

A stock has a covariance with the market index of 0.0045. The market return variance is 0.0030. The stock's mean monthly return is 1.4% and the market's mean monthly return is 1.0%. What is the stock's alpha (monthly)? A) −0.10%, B) 0.10%, C) 1.50%.

Show the solution

Beta = Cov(Ri, RM) ÷ Var(RM) = 0.0045 ÷ 0.0030 = 1.5.
Alpha = mean(Ri) − β × mean(RM) = 1.4% − 1.5 × 1.0% = 1.4% − 1.5% = −0.10%.
Check: the regression line passes through the means, so 1.4% = α + 1.5 × 1.0% gives α = −0.10%.

Answer: Alpha is −0.10% per month (option A). The beta used to get there is 1.5.

Example 2

An asset has a standard deviation of returns of 20%. The market index has a standard deviation of 15%. The correlation between them is 0.60. What is the asset's beta? A) 0.45, B) 0.80, C) 1.33.

Show the solution

Beta = ρ × σi ÷ σM = 0.60 × 20% ÷ 15% = 0.60 × 1.3333 = 0.80.
Cross-check with variances: R² = ρ² = 0.36. Systematic variance = β²·σM² = 0.64 × 0.0225 = 0.0144. Total variance = 0.04. 0.0144 ÷ 0.04 = 0.36. Consistent.
Eliminate: 0.45 would come from 0.60 × 15% ÷ 20%, which swaps asset and market. 1.33 ignores the correlation.

Answer: Beta is 0.80 (option B).

Exam tips

Most questions are one-step: Cov ÷ Var, or ρ × σi ÷ σM. Spot which one fits and do it fast.
Read whether the question wants variance or standard deviation. Wrong choice of the two is a common trap.
Eliminate options using logic: beta should have the same sign as the correlation, and R² must lie between 0 and 1.
Conceptual items test that beta is the regression slope, alpha is the intercept, and R² measures explanatory power. Learn these as short statements.
Using the calculator: for regression data on the TI BA II Plus, enter the X and Y values in the data worksheet (2nd, DATA), then open the statistics worksheet (2nd, STAT), set the calculation method to LIN, and read a (intercept) as alpha and b (slope) as beta.

Practice questions from Portfolio Risk and Return: Part II

Return Generating Models and Beta in other exams

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

Actuarial Core Principles (IAI)Security Market Line, Beta and Required Return

Return Generating Models and Beta: frequently asked questions

How do I calculate beta using covariance and variance?

Divide the covariance of the asset's returns with the market's returns by the variance of the market's returns. Use the same sample or population basis for both. The result is the slope of the regression of asset returns on market returns.

What does the beta regression slope mean?

It is the expected change in the asset's return for a one-unit change in the market's return. A beta of 0.8 means the asset is expected to move 0.8% when the market moves 1%, on average. It reflects only market-related movement.

What is the difference between the single-index model and the market model?

At Level I you can treat them as the same idea: asset return is explained by alpha, beta times the market return, and an error term. The key is that there is one factor, the market index.

What does R-squared tell me in a beta regression?

It shows the share of the asset's return variance explained by the market. A higher R-squared means the beta estimate describes more of the asset's behaviour. The rest is firm-specific variance.