Strategic Financial Management · Portfolio Theory and Practice

Single Index Model and Beta Estimation Explained

Updated 11 October 2026 · Fact-checked

The single index model says a security's return depends on one common factor, the market return, plus a firm-specific random part: Ri = αi + βi·Rm + ei. You estimate beta as the regression slope, Cov(i, m) ÷ σm², and alpha as Ȳ − βX̄. Then split total variance into systematic and unsystematic risk.

Understand Single Index Model and Beta Estimation

Estimating a full Markowitz model for many securities needs a covariance for every pair. The single index model (Sharpe) cuts this work. It assumes securities move together only because they all respond to the market. Apart from that common link, their firm-specific surprises are unrelated.

The model is a straight-line regression: Ri = αi + βi·Rm + ei. Ri is the security return, Rm is the market (index) return, and ei is the random error with an expected value of zero. Beta is the slope: how many percentage points the security moves for a one-point move in the market. Alpha is the intercept: the return the security earns when the market return is zero. A positive alpha is return not explained by the market.

Total risk splits into two parts. Systematic risk is βi²·σm², the part caused by the market, which diversification cannot remove. Unsystematic risk is σei², the firm-specific part, which diversification reduces. So σi² = βi²σm² + σei².

The ratio of systematic variance to total variance is R², the coefficient of determination. An R² of 0.80 means 80% of the security's variance is explained by the market.

For a portfolio, beta and alpha are simple weighted averages. Unsystematic risk shrinks as you add securities, because the weights are squared in Σwi²σei². This is why the model is useful for large portfolios.

Key rules to remember

Single index (market) model: Ri = αi + βi·Rm + ei; Expected return: E(Ri) = αi + βi·E(Rm), since E(ei) = 0.
Beta by covariance: βi = Cov(i, m) ÷ σm² = ρim·σi ÷ σm; Use the same basis (population or sample) for covariance and variance.
Beta by regression sums: β = [nΣXY − ΣX·ΣY] ÷ [nΣX² − (ΣX)²]; X is market return, Y is security return. Do not swap them.
Alpha: α = Ȳ − β·X̄; Ȳ and X̄ are mean security and mean market returns.
Total risk split: σi² = βi²·σm² + σei²; Systematic part is βi²σm²; unsystematic part is σei².
Coefficient of determination: R² = βi²σm² ÷ σi² = ρim²; Share of total variance explained by the market. 1 − R² is the unsystematic share.
Portfolio beta and alpha: βp = Σwi·βi ; αp = Σwi·αi; Weights are market-value proportions and sum to 1.
Portfolio variance: σp² = βp²·σm² + Σwi²·σei²; Valid under the model's assumption that the errors are uncorrelated across securities.

How to solve Single Index Model and Beta Estimation questions

Use this order for any single index model question, whether the data is raw returns or summary statistics.

1Identify X (market return) and Y (security return). Beta is always the slope of Y on X.
2If raw returns are given, compute the means, then deviations from the means. Find Σ(x−x̄)(y−ȳ), Σ(x−x̄)² and Σ(y−ȳ)².
3Beta = Σ(x−x̄)(y−ȳ) ÷ Σ(x−x̄)². The n or n−1 cancels, so beta is the same on either basis.
4Alpha = Ȳ − β·X̄. Keep the sign.
5Find variances on one consistent basis: σm² = Σ(x−x̄)² ÷ n and σi² = Σ(y−ȳ)² ÷ n (or both on n−1).
6Systematic risk = β²σm². Unsystematic risk = σi² − β²σm². R² = systematic ÷ total.
7If asked for expected return or a portfolio, apply E(Ri) = α + β·E(Rm), or take weighted averages of α and β.
8State the interpretation in one line: aggressive or defensive, and how much risk is diversifiable.

Quickest way: Deviation method with mean-centred data

When to use it: Raw return data for 4 to 6 periods, no calculator tricks needed, and you want fewer arithmetic slips.

Write the means first, then a column of x − x̄ and y − ȳ.
Compute three sums only: Sxy, Sxx, Syy.
β = Sxy ÷ Sxx. α = ȳ − β·x̄.
R² = Sxy² ÷ (Sxx × Syy). You get the risk split without computing variances separately.
Check: Σ(x−x̄) and Σ(y−ȳ) must each be zero. If not, a mean is wrong.

Common mistakes in Single Index Model and Beta Estimation

Regressing the market on the security instead of the security on the market.
The question lists the security first and students assign X to it.
Fix: Market return is always X (independent), security return is Y. Beta = Cov(i, m) ÷ σm².
Mixing population variance for the market with sample covariance, or the reverse.
Students divide by n in one place and n−1 in another.
Fix: Use the deviation sums Sxy and Sxx. The divisor cancels in beta. Use one basis for all variances.
Adding β·σm instead of β²·σm² for systematic risk.
Beta is applied to standard deviation and variance without squaring.
Fix: Systematic variance = β²σm². Take the square root only at the end if standard deviation is asked.
Treating alpha as the average return of the security.
Alpha is called the 'intercept' and students confuse it with the mean.
Fix: Alpha = Ȳ − βX̄. It is the return when the market return is zero, not the security's average return.
Calling unsystematic risk the residual standard deviation without squaring when subtracting.
Students subtract β²σm² from σi (not σi²).
Fix: Subtract in variance terms only: σei² = σi² − β²σm².
Weighting portfolio unsystematic risk linearly.
Students copy the weighted-average rule used for beta and alpha.
Fix: Portfolio residual variance is Σwi²σei². Weights are squared.

Worked examples

Example 1

The returns (%) of the market index (X) and Security A (Y) for five years are: Market 2, 4, 6, 8, 10; Security A 3, 7, 9, 13, 18. Find beta, alpha, R² and the split of Security A's total variance into systematic and unsystematic risk (use n as the divisor).

Show the solution

Means: X̄ = 30 ÷ 5 = 6; Ȳ = 50 ÷ 5 = 10.
Deviations x − 6: −4, −2, 0, 2, 4. Deviations y − 10: −7, −3, −1, 3, 8.
Sxy = 28 + 6 + 0 + 6 + 32 = 72.
Sxx = 16 + 4 + 0 + 4 + 16 = 40. Syy = 49 + 9 + 1 + 9 + 64 = 132.
Beta = 72 ÷ 40 = 1.8.
Alpha = 10 − 1.8 × 6 = 10 − 10.8 = −0.8%.
σm² = 40 ÷ 5 = 8. σA² = 132 ÷ 5 = 26.4.
Systematic variance = 1.8² × 8 = 3.24 × 8 = 25.92.
Unsystematic variance = 26.4 − 25.92 = 0.48.
R² = 25.92 ÷ 26.4 = 0.9818 (about 98.2%).

Answer: Beta = 1.8, alpha = −0.8%, R² ≈ 0.982. Of total variance 26.4, systematic risk is 25.92 and unsystematic risk is 0.48. The security is aggressive and almost entirely market-driven.

Example 2

Security B has α = 1.5%, β = 1.2 and residual standard deviation σe = 6%. The market is expected to return 12% with a standard deviation of 15%. Find B's expected return, total variance, total standard deviation, and the proportion of risk that is systematic.

Show the solution

Expected return = α + β × E(Rm) = 1.5 + 1.2 × 12 = 1.5 + 14.4 = 15.9%.
Systematic variance = β²σm² = 1.44 × 225 = 324.
Unsystematic variance = σe² = 6² = 36.
Total variance = 324 + 36 = 360.
Total standard deviation = √360 ≈ 18.97%.
Systematic proportion = 324 ÷ 360 = 0.90, so 90%.

Answer: Expected return = 15.9%. Total variance = 360 (%²), standard deviation ≈ 18.97%. Ninety per cent of the risk is systematic and 10% is unsystematic.

Exam tips

Read carefully whether variances are given as σ or σ². Square standard deviations before using the risk split.
Show the deviation table. Marks are given for Sxy, Sxx, beta and alpha even if the final risk split has an arithmetic slip.
Always write the interpretation: beta above 1 is aggressive, below 1 is defensive, and R² shows how much risk diversification cannot remove.
For portfolios, compute weighted beta and alpha first, then the variance with squared weights on residual variances.
In MCQs, check the answer by the split: systematic + unsystematic must equal total variance.

Practice questions from Portfolio Theory and Practice

Single Index Model and Beta Estimation 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 and Beta Estimation: frequently asked questions

What is the difference between the single index model and CAPM?

The single index model is a statistical regression of security returns on market returns. It describes past return behaviour and splits risk. CAPM is an equilibrium pricing theory that says expected return depends on beta and the risk-free rate. Alpha in the index model is not forced to be zero; in CAPM equilibrium it is expected to be zero.

How do I calculate beta using regression in the exam?

Take market return as X and security return as Y. Compute the deviation sums Sxy and Sxx. Beta = Sxy ÷ Sxx. Then alpha = mean of Y − beta × mean of X.

Does it matter whether I use n or n−1 for variance?

For beta it does not, because the divisor cancels between covariance and variance. It matters when you compute the risk split. Use the same divisor for the market and security variances, and follow what the question states.

Why does the single index model reduce the number of inputs needed?

It needs only each security's alpha, beta and residual variance, plus the market's expected return and variance. The Markowitz approach needs a covariance for every pair of securities. The index model gets those covariances from the common link to the market.