Strategic Financial Management · Asset Pricing Theories

Beta and Systematic Risk Measurement for CMA Final

Updated 11 October 2026 · Fact-checked

Beta measures how much a security's return moves with the market's return, so it captures systematic risk only. Calculate it as Cov(stock, market) ÷ Variance of market, or as the regression slope. Portfolio beta is the weighted average of the individual betas, using investment weights.

Understand Beta and Systematic Risk Measurement

Total risk of a share has two parts. Systematic risk comes from things that hit the whole market: interest rate changes, inflation, policy shifts, recessions. You cannot remove it by holding more shares. Unsystematic risk is specific to one company or industry: a strike, a product failure, a management change. A well-spread portfolio largely cancels it out.

Because diversification removes unsystematic risk, the market rewards only systematic risk. That is why CAPM prices a share on beta and not on total standard deviation.

Beta (β) measures the sensitivity of a security's return to the market's return. The market portfolio has a beta of 1. A share with β = 1.5 is expected to move about 1.5% for each 1% move in the market. A share with β = 0.6 is defensive. A share with β below 0 moves against the market, which is rare.

You find beta by comparing the share's returns with the market index returns over the same periods. The slope of the line through those points is beta. Algebraically it is covariance divided by market variance. It also equals correlation × (σ of stock ÷ σ of market).

Because variance is squared, systematic variance of a stock is β² × σm². The rest of its total variance is unsystematic. For a portfolio, beta is simply the weighted average of the betas, which makes it easy to adjust a portfolio to a target beta.

Key rules to remember

Beta from covariance: β = Cov(Rs, Rm) ÷ σm²; σm² is the variance of market returns. Use the same type of variance (sample or population) in both numerator and denominator.
Beta from correlation: β = r(s,m) × σs ÷ σm; Use when correlation and standard deviations are given instead of covariance.
Regression (characteristic line): Rs = α + β × Rm + e; β is the slope. The error term e represents unsystematic return.
Beta from paired data: β = [nΣXY − ΣXΣY] ÷ [nΣX² − (ΣX)²]; X = market return, Y = stock return. Do not swap them.
Systematic and unsystematic variance: Systematic variance = β² × σm²; Unsystematic variance = σs² − β² × σm²; Also systematic share of total risk = r² (coefficient of determination).
Portfolio beta: βp = Σ (wi × βi); Weights are based on market value invested and must add up to 1. Include a risk-free asset with β = 0 if held.
CAPM expected return: E(Ri) = Rf + βi × (Rm − Rf); Gives the required return for a given beta.

How to solve Beta and Systematic Risk Measurement questions

Use this order for any beta or systematic risk question. It works whether data are given as returns, covariances or portfolio holdings.

1Read what is given: raw returns, covariance and variance, correlation and standard deviations, or a list of holdings.
2Choose the formula: covariance ÷ market variance, r × σs ÷ σm, the regression slope, or the weighted average for a portfolio.
3If standard deviations are given, square the market's value to get variance. Do not divide by σm directly in the covariance formula.
4For raw return data, compute averages, then deviations, then covariance and variance. Use the same divisor (n or n − 1) for both.
5Compute weights as value of each holding ÷ total value. Check that they add up to 1.
6Calculate beta and state its meaning: more or less volatile than the market, and by how much.
7If asked, split risk: systematic variance = β²σm², unsystematic = total variance − systematic.
8If asked for required return or a market-move impact, apply CAPM or multiply beta by the market move, and write a one-line conclusion.

Quickest way: Shortcut for beta and portfolio beta

When to use it: Use when the question gives covariance, correlation or holdings and you have little time.

If covariance and market variance are given, divide once. Done.
If correlation and standard deviations are given, multiply r by σs and divide by σm.
For portfolio beta, write weight × beta in one column and add. Use rupee amounts divided by total.
To change portfolio beta, remember that beta is additive: adding a zero-beta asset lowers beta in proportion to its weight.
For risk split, find r² first. Systematic share of total variance equals r² and unsystematic equals 1 − r².

Common mistakes in Beta and Systematic Risk Measurement

Dividing covariance by market standard deviation instead of variance.
Students see σ given in the question and use it directly.
Fix: Square σm first. Beta = Cov ÷ σm². Check that the units work: % squared over % squared.
Using equal weights in portfolio beta.
Students average the betas out of habit.
Fix: Weight each beta by the amount invested in that security divided by total portfolio value.
Treating standard deviation as beta.
Both are called measures of risk.
Fix: Standard deviation is total risk. Beta is only systematic risk. A share can have a high standard deviation and a low beta.
Swapping the market and stock in the regression formula.
X and Y are mixed up in paired-data calculations.
Fix: Market return is always X (independent). Stock return is Y (dependent).
Saying beta of 1.5 means the share earns 1.5 times the market return.
Beta is read as a return ratio.
Fix: Beta is the ratio of movements, not returns. Expected return comes from CAPM with Rf included.
Computing unsystematic risk as σs − βσm in standard deviation terms.
Students subtract standard deviations instead of variances.
Fix: Subtract in variance terms: σs² − β²σm². Take the square root only if asked for standard deviation.

Worked examples

Example 1

The covariance between the returns of Tata Motors and the Nifty 50 is 54 (%²). The standard deviation of Nifty returns is 6% and the standard deviation of Tata Motors returns is 12%. The risk-free rate is 7% and the expected market return is 13%. Calculate (a) beta, (b) required return, (c) systematic and unsystematic variance of the stock. Treat the data as given.

Show the solution

Market variance = 6² = 36.
Beta = 54 ÷ 36 = 1.5.
Required return = 7% + 1.5 × (13% − 7%) = 7% + 9% = 16%.
Total variance of stock = 12² = 144.
Systematic variance = β² × σm² = 2.25 × 36 = 81.
Unsystematic variance = 144 − 81 = 63.
Check: correlation = 54 ÷ (12 × 6) = 0.75, r² = 0.5625, and 0.5625 × 144 = 81, which matches.

Answer: Beta = 1.5. Required return = 16%. Systematic variance = 81 and unsystematic variance = 63 (%²). The stock is more volatile than the market, and 56.25% of its variance is systematic.

Example 2

Meera holds a portfolio worth ₹10,00,000: Stock A ₹4,00,000 (β 1.2), Stock B ₹3,00,000 (β 0.8), Stock C ₹2,00,000 (β 1.5) and Stock D ₹1,00,000 (β 0.5). The risk-free rate is 6% and the market return is 12%. Find the portfolio beta, the required return and the expected loss in value if the market falls by 10%.

Show the solution

Weights: A = 0.4, B = 0.3, C = 0.2, D = 0.1. They add up to 1.
Weighted betas: 0.4 × 1.2 = 0.48; 0.3 × 0.8 = 0.24; 0.2 × 1.5 = 0.30; 0.1 × 0.5 = 0.05.
Portfolio beta = 0.48 + 0.24 + 0.30 + 0.05 = 1.07.
Required return = 6% + 1.07 × (12% − 6%) = 6% + 6.42% = 12.42%.
Expected fall = 1.07 × 10% = 10.7%.
Loss in value = 10.7% × ₹10,00,000 = ₹1,07,000.

Answer: Portfolio beta = 1.07. Required return = 12.42%. If the market falls 10%, the portfolio is expected to fall 10.7%, a loss of about ₹1,07,000.

Exam tips

Look for the data type first. MCQs usually give covariance, correlation or weights, and you can finish in one line.
In written answers, always add one sentence interpreting beta (aggressive, defensive or neutral). Examiners reward the conclusion.
Show the weights column in portfolio beta problems. Marks are given for the method even if arithmetic slips.
When variance and standard deviation are both mentioned, mark which is which before you start.
Link beta to CAPM when asked about required return. Beta questions often continue into expected return or security valuation.

Practice questions from Asset Pricing Theories

Beta and Systematic Risk Measurement: frequently asked questions

How do I calculate the beta of a stock?

Divide the covariance of the stock's returns with the market's returns by the variance of the market's returns. You can also use the slope of a regression of stock returns on market returns. Both give the same value if the data are treated consistently.

What is the difference between systematic and unsystematic risk?

Systematic risk affects the whole market and cannot be removed by diversification. Unsystematic risk is specific to a firm or industry and can be reduced by holding many different securities. Beta measures only systematic risk.

How do I find portfolio beta?

Multiply each security's beta by its weight in the portfolio and add the results. Weights come from the market value invested in each security divided by the total value. A risk-free asset counts as beta zero.

Can beta be negative or greater than 2?

Yes. A negative beta means the security tends to move opposite to the market. Betas above 2 are possible for very volatile securities. Neither is common, but the formulas still apply.