Skip to content

FRM Exam Part I · Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM)

Single-Index Model, Market Model and Regression Explained

Updated 11 October 2026 · Fact-checked

The single-index model says a security's excess return equals alpha plus beta times the market's excess return plus a random error. You estimate alpha and beta by regressing security returns on market returns. Beta is the slope, R-squared is the share of variance that is systematic, and the rest is idiosyncratic.

Understand Single-Index Model, Market Model and Regression

A stock moves for two broad reasons. Some of its movement comes from the market as a whole. The rest comes from news about that one firm. The single-index model writes this split as one equation, using one common factor: the market index.

The equation is Rᵢ − R_f = αᵢ + βᵢ(R_M − R_f) + εᵢ. Beta measures how strongly the stock responds to the market. Alpha is the average return not explained by the market. The error term εᵢ is the idiosyncratic (firm-specific) part. It has mean zero and is uncorrelated with the market return. In the basic model, the errors of different stocks are also uncorrelated with each other. That is what makes the model simple.

The market model is the same idea written with raw returns: Rᵢ = αᵢ + βᵢR_M + εᵢ. It is a statistical relationship you estimate from data. CAPM is an equilibrium theory that predicts alpha is zero in excess-return form. The regression itself does not assume that. Alpha from the regression is what you test against zero.

You estimate the model by ordinary least squares (OLS). Beta equals Cov(Rᵢ, R_M) ÷ Var(R_M), which is the regression slope. Variance then splits cleanly: Var(Rᵢ) = βᵢ² Var(R_M) + Var(εᵢ). The first term is systematic risk. It cannot be diversified away. The second is idiosyncratic risk. It shrinks in a large portfolio.

R-squared is the systematic share of total variance: R² = β²σ²_M ÷ σ²ᵢ. In a simple regression with one explanatory variable, it also equals the squared correlation between the stock and the market. A high R² means the market explains most of the stock's variation. A low R² means firm-specific risk dominates.

Key formulas to remember

Single-index model (excess returns)
Rᵢ − R_f = αᵢ + βᵢ(R_M − R_f) + εᵢ
E(εᵢ) = 0 and Cov(εᵢ, R_M) = 0. In the basic model, Cov(εᵢ, εⱼ) = 0 for i ≠ j.
Market model (raw returns)
Rᵢ = αᵢ + βᵢR_M + εᵢ
Statistical regression. It does not assume CAPM holds.
Beta as slope
βᵢ = Cov(Rᵢ, R_M) ÷ Var(R_M) = ρᵢ,M × σᵢ ÷ σ_M
Same as the OLS slope in a simple regression.
Alpha (intercept)
α̂ = mean(Rᵢ) − β̂ × mean(R_M)
With excess returns, use mean excess returns.
Variance decomposition
σ²ᵢ = βᵢ²σ²_M + σ²(εᵢ)
First term is systematic. Second is idiosyncratic.
R-squared
R² = βᵢ²σ²_M ÷ σ²ᵢ = ρ²ᵢ,M = 1 − SSR ÷ TSS
Equals squared correlation only with one regressor.
Portfolio beta and idiosyncratic variance
β_P = Σ wᵢβᵢ ; σ²(ε_P) = Σ wᵢ² σ²(εᵢ)
The second form needs uncorrelated errors. It falls as the portfolio gets larger.
Covariance between two stocks
Cov(Rᵢ, Rⱼ) = βᵢβⱼσ²_M
Valid under the single-index assumptions for i ≠ j.

How to solve Single-Index Model, Market Model and Regression questions

Use this order for any question on the single-index model, the market model or regression output.

  1. 1Identify what you are given: betas, volatilities, correlation, regression output, or raw return data.
  2. 2Check whether returns are excess returns or raw returns, and use the matching form of the equation.
  3. 3If you need beta, compute Cov(Rᵢ, R_M) ÷ Var(R_M), or ρ × σᵢ ÷ σ_M. Do not mix variance and standard deviation.
  4. 4If you need alpha, use mean return of the stock minus beta times mean market return.
  5. 5Split variance: systematic = β²σ²_M, idiosyncratic = total variance minus systematic.
  6. 6Get R² as systematic variance ÷ total variance. For a simple regression, it also equals ρ².
  7. 7For portfolios, average the betas with weights, and add squared weights times idiosyncratic variances.
  8. 8Take square roots only at the end if the question asks for volatility, and check the answer is reasonable.

Quickest way: Work in variances, convert at the end

When to use it: Use for decomposition and R-squared questions with volatilities given as percentages.

  1. Square every volatility first to get variances in decimals.
  2. Compute β²σ²_M and subtract from σ²ᵢ to get idiosyncratic variance.
  3. Divide systematic variance by total variance for R².
  4. Cross-check: R² should equal ρ² and must lie between 0 and 1.
  5. Take square roots only if asked for risk in volatility terms.

Common mistakes in Single-Index Model, Market Model and Regression

  • Adding standard deviations instead of variances when splitting risk

    Volatility is the number quoted, so it feels natural to work with it.

    Fix: Only variances add. Use σ²ᵢ = β²σ²_M + σ²(ε), then take the square root.

  • Treating alpha as always zero

    CAPM predicts zero alpha, so students carry that into the regression.

    Fix: In the regression, alpha is estimated from data. Zero alpha is a hypothesis to test, not an assumption of the model.

  • Confusing R² with beta

    Both describe the link between stock and market.

    Fix: Beta is sensitivity (slope). R² is the share of variance explained. A stock can have a high beta and a low R².

  • Using the portfolio idiosyncratic variance formula with correlated errors

    The formula Σw²σ²(ε) is memorised without its condition.

    Fix: State the assumption that errors are uncorrelated across stocks. Without it, covariance terms are needed.

  • Calling idiosyncratic risk non-diversifiable

    The words systematic and idiosyncratic get swapped.

    Fix: Systematic (market) risk stays. Idiosyncratic risk shrinks as you add more stocks.

  • Using excess returns for some inputs and raw returns for others

    Questions switch between the two forms without warning.

    Fix: Read the problem and convert everything to one basis before computing alpha.

Worked examples

Example 1

A stock has beta 1.2 and total volatility 30%. The market volatility is 20%. Find the stock's systematic variance, idiosyncratic volatility, and R².

Show the solution
  1. Total variance = 0.30² = 0.09.
  2. Market variance = 0.20² = 0.04.
  3. Systematic variance = 1.2² × 0.04 = 1.44 × 0.04 = 0.0576.
  4. Idiosyncratic variance = 0.09 − 0.0576 = 0.0324.
  5. Idiosyncratic volatility = √0.0324 = 0.18, or 18%.
  6. R² = 0.0576 ÷ 0.09 = 0.64.

Answer: Systematic variance 0.0576, idiosyncratic volatility 18%, R² = 64%.

Example 2

From monthly data, a stock has mean return 1.4% and the market has mean return 1.0%. Cov(stock, market) = 0.0030 and Var(market) = 0.0025. Using the market model, estimate beta and alpha.

Show the solution
  1. Beta = Cov ÷ Var(market) = 0.0030 ÷ 0.0025 = 1.2.
  2. Alpha = mean stock return − beta × mean market return.
  3. Alpha = 1.4% − 1.2 × 1.0% = 1.4% − 1.2% = 0.2%.

Answer: Beta = 1.2 and alpha = 0.2% per month.

Exam tips

  • Always check whether the question gives variance or standard deviation before plugging into β²σ²_M.
  • Expect conceptual questions on what is diversifiable. Idiosyncratic risk falls with more stocks. Systematic risk does not.
  • Read regression output carefully. The slope coefficient is beta, the intercept is alpha, and R² is the fit.
  • Remember that R² equals ρ² only when there is a single explanatory variable.
  • If a question asks about a test of alpha, compare the t-statistic (estimate ÷ standard error) with the critical value.

Practice questions from Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM)

Single-Index Model, Market Model and Regression in other exams

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

Single-Index Model, Market Model and Regression: frequently asked questions

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

Both regress a stock's return on the market's return. The single-index model is usually written with excess returns and used to build portfolio covariances. The market model is the plain statistical regression on returns, and it does not assume CAPM holds.

How do you estimate beta by regression?

Regress the stock's returns on the market's returns using OLS. The slope coefficient is beta. It equals the covariance of the stock and the market divided by the variance of the market.

What does R-squared tell you in this model?

It tells you the proportion of the stock's return variance explained by the market. The remainder, 1 − R², is idiosyncratic variance. A higher R² means more of the risk is systematic.

Why does the single-index model simplify portfolio analysis?

It lets you write each covariance as βᵢβⱼσ²_M, so you only need each stock's beta, its idiosyncratic variance and the market variance. This replaces a full covariance matrix of estimates.