Financial Management and Business Data Analytics · Risk and Return

Expected Return, Variance and Standard Deviation Explained

Updated 10 October 2026 · Fact-checked

Expected return is the probability-weighted average of possible returns. Variance is the probability-weighted average of squared deviations from that mean. Standard deviation is the square root of variance and measures total risk. Coefficient of variation divides standard deviation by expected return, giving risk per unit of return for comparing investments.

Understand Measuring Expected Return, Variance and Standard Deviation

A future return is uncertain. Instead of one number, you have several possible outcomes, each with a probability. The expected return is the average of these outcomes, with each outcome weighted by its probability. It is a long-run average, not a promise of what you will earn.

Two investments can have the same expected return and very different risk. To measure risk, you check how far outcomes spread around the expected return. Take each outcome's deviation from the mean and square it, so that positive and negative deviations do not cancel. Then weight the squares by probability. That gives the variance.

Variance is in squared units (%²), which is hard to interpret. So you take its square root. That gives the standard deviation (σ), in the same unit as the return. A higher σ means a wider spread of outcomes and higher total risk.

Standard deviation is an absolute measure. It works well when comparing investments with the same expected return. When expected returns differ, use the coefficient of variation (CV) = σ ÷ expected return. It shows risk per unit of return. A lower CV means less risk for each unit of return. So neither measure is always better: use σ for equal returns and CV for unequal returns.

This topic covers total risk (systematic plus unsystematic). Beta, which measures only market risk, is a separate topic.

Key rules to remember

Expected return: E(R) = Σ (pᵢ × Rᵢ); Probabilities must add up to 1 (or 100%). Rᵢ is the return in outcome i.
Variance: σ² = Σ pᵢ × (Rᵢ − E(R))²; Unit is %² if returns are in %. Do not forget to weight by probability.
Variance shortcut: σ² = Σ (pᵢ × Rᵢ²) − [E(R)]²; Faster when E(R) is not a round number. Gives the same answer as the main formula.
Standard deviation: σ = √σ²; Same unit as the return. This is the usual measure of total risk.
Coefficient of variation: CV = σ ÷ E(R); Risk per unit of expected return. Lower CV is better. Use when expected returns differ.
Equal-probability case: E(R) = ΣR ÷ n; Use only when the question says all outcomes are equally likely.

How to solve Measuring Expected Return, Variance and Standard Deviation questions

Use a column table. It keeps the working tidy and earns step marks even if you make an arithmetic slip.

1List each outcome with its probability and return. Check the probabilities add up to 1.
2Compute p × R for each row and add them. This is the expected return E(R).
3Find the deviation (R − E(R)) for each row.
4Square each deviation, then multiply by the probability. Add the column. This is the variance.
5Take the square root of the variance to get the standard deviation. Keep the unit as %.
6If the question asks for CV, divide σ by E(R). Use the same unit for both, such as % ÷ %.
7If asked to compare or choose, state the decision: for equal returns, pick the lower σ; for unequal returns, pick the lower CV. Add one line of interpretation.

Quickest way: Shortcut using Σ pR² − [E(R)]²

When to use it: Use when the expected return is a decimal and deviations would be awkward to square, or when you have many outcomes and little time.

Compute E(R) = Σ pR first.
Add a column for p × R² and total it.
Subtract the square of E(R) from that total. This is the variance.
Take the square root for σ, then divide by E(R) for CV.
Check: the variance cannot be negative. If it is, you made an error.

Common mistakes in Measuring Expected Return, Variance and Standard Deviation

Forgetting to weight squared deviations by probability, and just averaging them.
Students remember the sample-variance formula from statistics and divide by n.
Fix: In risk and return with probabilities, always multiply each squared deviation by its probability and add. Do not divide by n.
Reporting the variance as the standard deviation.
The student stops one step early.
Fix: Always take the square root as the last step. Write σ = √variance explicitly.
Squaring the probabilities or the returns before subtracting the mean, then using the shortcut wrongly.
Confusion between the main formula and the shortcut formula.
Fix: In the shortcut, square each return only, weight by p, add, then subtract the square of the mean E(R).
Dividing the expected return by σ instead of σ by the expected return in CV.
Mixing up with the Sharpe-type ratio, which puts return over risk.
Fix: Remember CV is risk per unit of return, so σ goes on top.
Choosing the investment with the lower σ when expected returns differ.
Students treat σ as the only risk measure.
Fix: If E(R) differs, compare CV and pick the lower. State why in one line.
Mixing percentages and decimals, such as using 0.12 for one return and 12 for another.
Rushing through the table.
Fix: Convert everything to one form before you start. Percent is usually easier, but then variance is in %².

Worked examples

Example 1

The returns on the shares of Kaveri Textiles Ltd. under three economic conditions are: Boom (probability 0.30) 25%; Normal (probability 0.50) 12%; Recession (probability 0.20) −10%. Calculate the expected return, variance, standard deviation and coefficient of variation.

Show the solution

Check probabilities: 0.30 + 0.50 + 0.20 = 1.00.
Expected return = (0.30 × 25) + (0.50 × 12) + (0.20 × −10) = 7.5 + 6.0 − 2.0 = 11.5%.
Deviations from 11.5%: Boom 25 − 11.5 = 13.5; Normal 12 − 11.5 = 0.5; Recession −10 − 11.5 = −21.5.
Squared deviations: 182.25; 0.25; 462.25.
Weighted: 0.30 × 182.25 = 54.675; 0.50 × 0.25 = 0.125; 0.20 × 462.25 = 92.45.
Variance = 54.675 + 0.125 + 92.45 = 147.25 (%²).
Check by shortcut: Σ pR² = 0.30 × 625 + 0.50 × 144 + 0.20 × 100 = 187.5 + 72 + 20 = 279.5. Then 279.5 − 11.5² = 279.5 − 132.25 = 147.25. Matches.
Standard deviation = √147.25 ≈ 12.14%.
CV = 12.14 ÷ 11.5 ≈ 1.06.

Answer: Expected return = 11.5%; variance = 147.25 (%²); standard deviation ≈ 12.14%; coefficient of variation ≈ 1.06.

Example 2

Project X gives returns of 10%, 20% and 30% with probabilities 0.25, 0.50 and 0.25. Project Y has an expected return of 30% and a standard deviation of 9.6%. Which project has the lower risk per unit of return? Which would you prefer, and why?

Show the solution

Project X expected return = (0.25 × 10) + (0.50 × 20) + (0.25 × 30) = 2.5 + 10 + 7.5 = 20%.
Deviations: −10, 0, +10. Squared: 100, 0, 100.
Variance of X = (0.25 × 100) + (0.50 × 0) + (0.25 × 100) = 25 + 0 + 25 = 50 (%²).
Standard deviation of X = √50 ≈ 7.07%.
CV of X = 7.07 ÷ 20 ≈ 0.35.
CV of Y = 9.6 ÷ 30 = 0.32.
Expected returns differ (20% against 30%), so standard deviation alone is not a fair comparison. Y has the higher σ (9.6% against 7.07%), but its CV is lower.

Answer: Project Y has the lower risk per unit of return (CV 0.32 against about 0.35 for X). Because the expected returns differ, compare using CV, so Y is preferred, even though its standard deviation is higher.

Exam tips

Draw the column table every time. Even if the final figure is wrong, you earn step marks for the correct method.
Write the unit of each answer: % for expected return and σ, %² for variance, and no unit for CV.
In MCQs, check whether the question asks for variance or standard deviation. Examiners often put the variance as one of the wrong options for a σ question.
When the question says compare or choose, always finish with a one-line decision and reason. Use σ when expected returns are equal and CV when they differ.
Verify your variance with the shortcut formula if time permits. It is a quick check on your arithmetic.

Practice questions from Risk and Return

Measuring Expected Return, Variance and Standard Deviation in other exams

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

Measuring Expected Return, Variance and Standard Deviation: frequently asked questions

How do you calculate the coefficient of variation?

First find the expected return and the standard deviation using the probability table. Then divide the standard deviation by the expected return. Both must be in the same form, such as both in %. A lower CV means less risk per unit of return.

Coefficient of variation vs standard deviation: which is better?

Neither is always better. Standard deviation is fine when the investments have the same expected return. When expected returns differ, CV gives a fairer comparison because it measures risk per unit of return.

Do I divide by n when calculating variance with probabilities?

No. When probabilities are given, you weight each squared deviation by its probability and add them. The probabilities already act as the weights. Divide by n only when the question says all outcomes are equally likely and you use the simple average.

Can variance or standard deviation be negative?

No. Variance is a sum of squared terms weighted by non-negative probabilities, so it cannot be negative. Standard deviation is its non-negative square root. If you get a negative value, recheck your working.