NISM-Series-XV: Research Analyst · Fundamentals of Risk and Return

Measures of Risk: Variance, Standard Deviation and Beta

Updated 11 October 2026 · Fact-checked

Variance and standard deviation measure how widely an asset's returns spread around their average. Covariance, correlation and beta measure how returns move with another asset or the market. To solve questions, find the mean, square deviations, divide by n or n−1, take the square root, and use β = Cov(i, m) ÷ Var(m).

Understand Measures of Risk: Variance, Standard Deviation, Beta

Risk in investing means returns may differ from what you expect. Statistics gives us numbers for this uncertainty. The more widely returns are scattered around their average, the riskier the asset.

Variance is the average of the squared deviations of returns from the mean return. Squaring removes negative signs and gives extra weight to large deviations. Its unit is squared percent, which is hard to interpret. Standard deviation is the square root of variance. It is in the same unit as returns, so a standard deviation of 12% is easy to read as the typical spread around the mean.

Covariance measures whether two assets' returns move together. Positive covariance means they tend to move in the same direction. Negative means opposite directions. Its size has no fixed limit, so it is hard to compare. Correlation fixes this by dividing covariance by the product of the two standard deviations. It always lies between −1 and +1. A value of +1 is perfect positive movement, −1 is perfect negative movement and 0 means no linear relationship.

Beta measures systematic risk: how sensitive a stock's returns are to market returns. It equals the covariance of the stock with the market divided by the variance of the market. The market itself has a beta of 1. A beta above 1 means the stock moves more than the market. A beta below 1 means it moves less. Standard deviation captures total risk, while beta captures only market risk.

The coefficient of variation divides standard deviation by mean return. It gives risk per unit of return and helps compare assets with different average returns.

Key formulas to remember

Mean return: R̄ = ΣR ÷ n; Add all returns and divide by the number of observations.
Variance (population): σ² = Σ(R − R̄)² ÷ n; Use when the data covers the whole population.
Variance (sample): s² = Σ(R − R̄)² ÷ (n − 1); Use when the data is a sample. Read the question for which one applies.
Standard deviation: σ = √variance; Same unit as returns. Measures total risk.
Expected return and variance with probabilities: E(R) = Σ p × R; σ² = Σ p × (R − E(R))²; Use when scenarios have given probabilities.
Covariance: Cov(A, B) = Σ(RA − R̄A)(RB − R̄B) ÷ n (or n − 1 for a sample); Sign shows direction of joint movement.
Correlation: ρ(A, B) = Cov(A, B) ÷ (σA × σB); Always between −1 and +1.
Beta: β = Cov(stock, market) ÷ Var(market) = ρ × σstock ÷ σmarket; Market beta is 1. Measures systematic risk.
Coefficient of variation: CV = σ ÷ mean return; Lower CV means less risk per unit of return.

How to solve Measures of Risk: Variance, Standard Deviation, Beta questions

Use this order for any question on risk measures. Read first what is asked and whether data is a sample or population.

1Identify the measure asked: variance, standard deviation, covariance, correlation, beta or CV.
2Check whether the returns are equally likely observations or have probabilities attached.
3Compute the mean (or expected) return first.
4Find each deviation from the mean and square it (for covariance, multiply the two deviations).
5Sum the values and divide by n, n − 1 or use the probability weights, as the question indicates.
6Take the square root for standard deviation. Do not stop at variance if SD is asked.
7For correlation, beta or CV, plug the results into the relevant formula.
8Sanity check: correlation within −1 to +1, variance not negative, and SD in the same unit as returns.

Quickest way: Shortcut using relationships between measures

When to use it: Use when the question gives some measures and asks for another, so no full calculation is needed.

If SD is given and variance is asked, square it. If variance is given, take the square root.
If correlation and both SDs are given, covariance = ρ × σA × σB.
If covariance and market variance are given, beta = covariance ÷ market variance.
If correlation, stock SD and market SD are given, beta = ρ × σstock ÷ σmarket.
For a quick market move estimate, multiply beta by the market change. Beta 1.2 and a market rise of 10% suggest about 12%.
Eliminate options that break rules: correlation above 1, negative variance, or SD larger than the range allows.

Common mistakes in Measures of Risk: Variance, Standard Deviation, Beta

Reporting variance when standard deviation is asked.
Students stop after summing the squared deviations and dividing.
Fix: Always finish with the square root and check the question wording.
Dividing by n when the data is a sample.
Population and sample formulas look alike.
Fix: Use n − 1 for sample data unless the question says population or gives probabilities.
Using standard deviation of the market in the beta denominator instead of variance.
Confusing beta with the correlation-based form.
Fix: Beta = covariance ÷ market variance. Only the ρ form uses the ratio of standard deviations.
Treating beta as total risk.
Beta is often called the risk number of a stock.
Fix: Beta captures only systematic risk. Standard deviation captures total risk.
Reading correlation of 0 as 'no relationship of any kind'.
Overstating what the number means.
Fix: Correlation measures linear co-movement only. Zero means no linear relationship.
Choosing the asset with the lower SD without checking returns.
Ignoring that assets have different average returns.
Fix: Compare using coefficient of variation when means differ.

Worked examples

Example 1

A stock gave returns of 10%, 12%, 14% and 8% in four years. Treating these as the full population, find the variance and standard deviation.

Show the solution

Mean = (10 + 12 + 14 + 8) ÷ 4 = 44 ÷ 4 = 11%.
Deviations: −1, +1, +3, −3.
Squared deviations: 1, 1, 9, 9. Sum = 20.
Variance = 20 ÷ 4 = 5 (%²).
Standard deviation = √5 ≈ 2.24%.

Answer: Variance = 5 (%²); standard deviation ≈ 2.24%.

Example 2

A stock has a standard deviation of 30%, the market has a standard deviation of 20%, and their correlation is 0.8. Find the covariance and the beta of the stock.

Show the solution

Convert to decimals: σstock = 0.30, σmarket = 0.20.
Covariance = ρ × σstock × σmarket = 0.8 × 0.30 × 0.20 = 0.048.
Market variance = 0.20² = 0.04.
Beta = 0.048 ÷ 0.04 = 1.2.
Check: ρ × σstock ÷ σmarket = 0.8 × 0.30 ÷ 0.20 = 1.2.

Answer: Covariance = 0.048; beta = 1.2, so the stock is more volatile than the market.

Exam tips

Expect direct numerical questions: mean, SD, covariance or beta from a short data set. Keep arithmetic clean with small numbers.
Know the interpretation of beta: above 1 aggressive, below 1 defensive, equal to 1 moves with the market.
Remember that correlation lies between −1 and +1 and covariance has no such limit. Option traps often break this.
Note that wrong answers carry negative marking of 25% of the marks for the question, so do not guess blindly on long calculations. Eliminate options using the rules first.
Questions often ask which risk a measure captures. Link SD to total risk and beta to systematic risk.

Practice questions from Fundamentals of Risk and Return

Measures of Risk: Variance, Standard Deviation, Beta in other exams

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

Measures of Risk: Variance, Standard Deviation, Beta: frequently asked questions

How do I calculate standard deviation of returns?

Find the mean return, subtract it from each return, and square the results. Add them up and divide by n (population) or n − 1 (sample) to get variance. Then take the square root.

What is beta in the stock market?

Beta shows how much a stock's returns move relative to the market. A beta of 1.5 suggests the stock tends to move 1.5 times as much as the market. It is calculated as covariance with the market divided by market variance.

What is the difference between covariance and correlation?

Covariance shows the direction of joint movement but its size is hard to interpret. Correlation scales covariance by both standard deviations, so it always lies between −1 and +1.

When is the coefficient of variation used?

Use it to compare risk per unit of return across assets with different average returns. It is standard deviation divided by mean return. A lower value means less risk for each unit of return.