CFA Level I Exam · Portfolio Risk and Return: Part I

Variance, Standard Deviation, Covariance and Correlation Explained

Updated 7 October 2026 · Fact-checked

Variance is the average squared deviation of returns from their mean, and standard deviation is its square root. Covariance measures how two assets' returns move together. Correlation is covariance divided by the product of the two standard deviations, so it lies between -1 and +1. Use them to compute portfolio risk.

Understand Variance, Standard Deviation, Covariance and Correlation

Variance measures how spread out returns are around their mean. You take each return, subtract the mean, square the gap, and average the squares. Squaring stops positive and negative gaps from cancelling. The cost is that variance is in squared units (%²), which is hard to read.

Standard deviation fixes that. It is the square root of variance, so it is in the same units as returns. In portfolio theory, standard deviation is the usual measure of total risk.

Covariance extends the idea to two assets. For each period, multiply asset A's deviation by asset B's deviation, then average. A positive value means the assets tend to be above or below their means together. A negative value means they tend to move in opposite directions. The size of covariance depends on the units and volatility, so it is hard to compare across pairs.

Correlation removes that problem. Divide covariance by σA × σB. The result lies between -1 and +1. A value of +1 means a perfect positive linear relationship, -1 a perfect negative one, and 0 means no linear relationship. Correlation measures only linear association, and it does not prove cause and effect.

This matters for portfolios. Portfolio risk depends on the weights, each asset's standard deviation and the covariance between them. The lower the correlation, the more risk falls when you combine assets. That is the source of diversification benefits.

Key formulas to remember

Sample variance: s² = Σ(Rᵢ − R̄)² ÷ (n − 1); Use n − 1 when the data are a sample. For a whole population use σ² = Σ(Rᵢ − μ)² ÷ N.
Standard deviation: s = √s²; Same units as returns. Never average variances and call it a standard deviation.
Variance with probabilities: σ² = Σ P(i) × [Rᵢ − E(R)]²; Use when you are given scenarios with probabilities. E(R) = Σ P(i) × Rᵢ.
Sample covariance: Cov(A,B) = Σ(RA,ᵢ − R̄A)(RB,ᵢ − R̄B) ÷ (n − 1); With probabilities, use Σ P(i)(RA,ᵢ − E(RA))(RB,ᵢ − E(RB)). Cov(A,A) equals the variance of A.
Correlation: ρ(A,B) = Cov(A,B) ÷ (σA × σB); Rearranged: Cov(A,B) = ρ × σA × σB. Always between -1 and +1.
Two-asset portfolio variance: σp² = wA²σA² + wB²σB² + 2wAwB ρ σA σB; The last term equals 2wAwB Cov(A,B). Portfolio standard deviation is the square root of this.

How to solve Variance, Standard Deviation, Covariance and Correlation questions

Use this order for any question on dispersion, covariance, correlation or two-asset portfolio risk.

1Identify what is asked: variance, standard deviation, covariance, correlation or portfolio risk. Note whether the data are a sample, a population or a probability table.
2Compute the mean (or expected return) first.
3Compute deviations from the mean, then square them (variance) or multiply the two assets' deviations (covariance).
4Divide by n − 1 for a sample, N for a population, or weight by probabilities for a scenario table.
5Take the square root for standard deviation. Convert covariance to correlation by dividing by σA × σB.
6For a portfolio, put weights and standard deviations into the two-asset variance formula in decimals, then take the square root last.
7Sanity check: correlation must be within -1 to +1, and portfolio standard deviation must not exceed the weighted average of the two standard deviations.
8Match your answer to the options and check units (% versus %², decimals versus percent).

Quickest way: Calculator shortcut and bounds check

When to use it: Use it when you have a short list of returns and need standard deviation or correlation fast, or when a portfolio question has options you can screen by logic.

On the BA II Plus, press 2nd DATA, enter the returns as X values (and Y values for a second asset in LIN mode), then press 2nd STAT and select the mode.
Scroll to Sx for the sample standard deviation or σx for the population one. In LIN mode, scroll to r for correlation.
Square Sx if the question asks for variance.
For portfolios, first compute the weighted average of the standard deviations. Portfolio risk is at most this value when correlation is below +1, so you can usually eliminate the option that equals or exceeds it.
If ρ = +1, portfolio standard deviation is exactly the weighted average. If ρ = -1, it can fall to zero for the right weights.

Common mistakes in Variance, Standard Deviation, Covariance and Correlation

Dividing by n instead of n − 1 for sample data.
Students remember the plain average and forget that a sample is used to estimate the population.
Fix: Check for the words sample or population. Default to n − 1 for a list of historical returns unless told otherwise. Read Sx versus σx carefully on the calculator.
Reporting variance when the question asks for standard deviation, or the reverse.
The last square-root step is skipped under time pressure.
Fix: Underline the requested measure. Variance is in %² and standard deviation is in %.
Treating covariance as bounded or comparing covariances across pairs.
Covariance and correlation sound similar.
Fix: Covariance has no fixed range and depends on units. Only correlation is bounded between -1 and +1 and is comparable across pairs.
Forgetting the 2wAwB cross term, or using covariance where correlation belongs.
Students memorise the first two terms and drop the third, or put Cov into a formula that already has σA σB.
Fix: Write the formula in full each time. If you are given Cov, use 2wAwB Cov. If you are given ρ, use 2wAwB ρ σA σB.
Assuming zero correlation means the assets are independent.
Zero feels like no relationship at all.
Fix: Correlation captures only linear association. Zero correlation means no linear relationship, nothing more.
Mixing percentages and decimals in portfolio variance.
Using 20 in one term and 0.20 in another gives a wildly wrong result.
Fix: Convert everything to decimals before squaring, then convert the final standard deviation back to a percentage.

Worked examples

Example 1

An asset earned annual returns of 10%, 4%, -2% and 8% over four years. Treating these as a sample, what is the standard deviation of returns? A. 4.58% B. 5.29% C. 28.00%

Show the solution

Mean = (10 + 4 − 2 + 8) ÷ 4 = 20 ÷ 4 = 5%.
Deviations: 5, -1, -7, 3.
Squared deviations: 25, 1, 49, 9. Sum = 84.
Sample variance = 84 ÷ (4 − 1) = 28 (%²).
Sample standard deviation = √28 = 5.29%.
Option A (4.58%) comes from dividing by n: √(84 ÷ 4) = √21. Option C is the variance, not the standard deviation.

Answer: B. 5.29%

Example 2

A portfolio holds 60% in Asset A (standard deviation 20%) and 40% in Asset B (standard deviation 10%). The correlation between them is 0.25. What is the portfolio standard deviation? A. 1.84% B. 13.56% C. 16.00%

Show the solution

Convert to decimals: wA = 0.6, wB = 0.4, σA = 0.20, σB = 0.10, ρ = 0.25.
wA²σA² = 0.36 × 0.04 = 0.0144.
wB²σB² = 0.16 × 0.01 = 0.0016.
Cross term = 2 × 0.6 × 0.4 × 0.25 × 0.20 × 0.10 = 0.48 × 0.25 × 0.02 = 0.0024.
Portfolio variance = 0.0144 + 0.0016 + 0.0024 = 0.0184.
Portfolio standard deviation = √0.0184 ≈ 0.1356 = 13.56%.
Check: the weighted average of the standard deviations is 0.6 × 20 + 0.4 × 10 = 16%. Option C equals this, which would only hold if ρ = +1. Since ρ is below 1, risk must be lower. Option A is the variance misread as a percentage.

Answer: B. 13.56%

Exam tips

Expect at least one portfolio risk calculation. Practise the two-asset formula until the three terms come automatically.
Use the weighted-average ceiling to eliminate options: unless ρ = +1, portfolio standard deviation is below the weighted average of the two standard deviations.
Watch the wording: sample versus population, variance versus standard deviation, covariance versus correlation. Wrong options are built from these slips.
Conceptual items often ask what happens to portfolio risk when correlation falls, or whether correlation of 0 means independence. Know the answers cold.
Options run from smallest to largest, so a very small or very large value is often a variance or an unsquared slip. Check units before you choose.

Practice questions from Portfolio Risk and Return: Part I

Variance, Standard Deviation, Covariance and Correlation in other exams

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

Variance, Standard Deviation, Covariance and Correlation: frequently asked questions

What is the difference between covariance and correlation?

Covariance shows the direction in which two assets' returns move together, but its size depends on the units and volatility of the assets. Correlation is covariance divided by the product of the standard deviations, so it is standardised between -1 and +1. That makes correlation easier to interpret and compare.

How do you calculate portfolio standard deviation for two assets?

Compute the variance as wA²σA² + wB²σB² + 2wAwB ρ σA σB using decimals. Then take the square root. Portfolio risk is not the weighted average of the two standard deviations unless the correlation is +1.

Do I use n or n − 1 for variance in the CFA exam?

Use n − 1 when the returns are a sample, which is the usual case for historical data. Use N only when the data are the whole population or when the question says so. For probability-weighted scenarios, weight each squared deviation by its probability.

What does a correlation of 0 or -1 mean for a portfolio?

A correlation of 0 means there is no linear relationship, and combining the assets reduces risk meaningfully. A correlation of -1 gives the strongest diversification, and with suitable weights portfolio risk can reach zero. A correlation of +1 gives no risk reduction beyond the weighted average.