Economic Modelling · Measures of investment risk

Variance and Standard Deviation as Measures of Investment Risk

Q: Why is standard deviation preferred to variance for reporting risk?

Standard deviation is in the same units as the return, so it is easier to interpret. Variance is used in calculations because it is additive for independent returns and fits portfolio formulas.

Q: What is the difference between variance and semi-variance?

Variance counts deviations above and below the mean. Semi-variance counts only deviations below a target, usually the mean. It better reflects the investor's concern with losses, but it is harder to use in portfolio calculations.

Q: How do I calculate portfolio variance for two assets?

Use σp² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂. Square the weights, include the covariance term, then take the square root for the standard deviation.

Q: What are the main limitations of standard deviation as a risk measure?

It treats gains and losses the same, ignores skewness and fat tails, and gives no probability of a shortfall. It also depends on estimated inputs that can be unstable. It is most reliable for roughly normal returns.

Updated 11 October 2026 · Fact-checked

Variance is the expected squared deviation of a return from its mean: Var(R) = E[(R − μ)²]. Standard deviation is its square root and is in the same units as return. Both measure the spread of returns, treating gains and losses alike. To solve questions, find the mean first, then the squared deviations, then weight them.

Understand Variance and Standard Deviation as Risk Measures

Investment risk is the chance that the actual return differs from what you expect. The simplest way to measure this is to look at how widely the possible returns are spread around the expected return. That is what variance does.

The variance of a return R is Var(R) = E[(R − μ)²], where μ = E[R]. You square the deviations so that positive and negative deviations do not cancel. The standard deviation σ = √Var(R). It is in the same units as the return (per cent, not per cent squared), so it is easier to interpret and to compare.

Why is variance popular? It is easy to compute and it underpins mean-variance portfolio theory. For a portfolio, variance depends on the individual variances and the covariances between assets, so it captures diversification. If returns are normally distributed, or the investor has quadratic utility, the mean and variance describe preferences completely.

Semi-variance only counts deviations below a chosen target, usually the mean: SV = E[(min(R − μ, 0))²]. It treats only downside outcomes as risk, which matches how most investors feel. Its square root is the semi-standard deviation.

The limitations matter for exams. Variance penalises upside and downside equally. It ignores skewness and fat tails, so it can understate extreme losses when returns are not normal. It says nothing about the probability of falling below a specific level. It needs a reliable estimate of the distribution, and sample estimates can be unstable. It is also not a measure relative to a benchmark or a liability.

Key rules to remember

Expected return (discrete): μ = E[R] = Σ pᵢ rᵢ; pᵢ are probabilities that sum to 1.
Variance: Var(R) = σ² = E[(R − μ)²] = E[R²] − μ²; The second form is usually faster to compute.
Standard deviation: σ = √Var(R); Same units as the return.
Semi-variance (below the mean): SV = Σ pᵢ [min(rᵢ − μ, 0)]²; Outcomes at or above the target contribute zero. State the target used.
Sample variance: s² = Σ (rᵢ − r̄)² ÷ (n − 1); Use when estimating from observed data. Divide by n if the question says to treat the data as the whole population.
Two-asset portfolio variance: σp² = w₁²σ₁² + w₂²σ₂² + 2 w₁ w₂ ρ σ₁ σ₂; w₁ + w₂ = 1 for a fully invested portfolio. ρ is the correlation; Cov = ρσ₁σ₂.
Scaling a return: Var(aR + b) = a² Var(R); Adding a constant does not change variance.

How to solve Variance and Standard Deviation as Risk Measures questions

Use this method for any question on calculating or discussing variance or standard deviation of returns.

1Identify what is given: a probability distribution of returns, historical data, or asset parameters with weights.
2State the assumptions, for example that the probabilities are known or that the sample is representative.
3Calculate the expected return μ = Σ pᵢ rᵢ (or the sample mean).
4Calculate the variance using E[R²] − μ², or the sum of pᵢ(rᵢ − μ)². For a portfolio, use the weights, variances and covariance formula.
5Take the square root for the standard deviation, and keep units consistent (decimals or per cent).
6If semi-variance is asked, set all deviations above the target to zero before squaring and weighting.
7Interpret the result: compare risk between investments and, if asked, comment on limitations such as symmetry and non-normality.

Quickest way: E[R²] − μ² shortcut

When to use it: Use for a discrete return table with 3 to 5 outcomes, when you must find variance quickly.

Work in per cent units if the numbers are whole, and convert at the end only if needed.
Compute μ = Σ pᵢ rᵢ.
Compute E[R²] = Σ pᵢ rᵢ².
Variance = E[R²] − μ². Take the square root for σ.
Check that the variance is not negative and that σ is plausible against the range of returns.

Common mistakes in Variance and Standard Deviation as Risk Measures

Reporting variance when the question asks for standard deviation, or the reverse.
You stop after the main calculation and forget the final step.
Fix: Underline the required measure in the question and finish with the square root if σ is asked.
Forgetting to square the weights in portfolio variance.
You mix up the formula for expected return, where weights are not squared, with the variance formula.
Fix: Write σp² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂ before substituting.
Adding standard deviations of two assets to get the portfolio standard deviation.
It seems natural that risk should add up.
Fix: Add variances and covariance terms, then take the root. Standard deviations add only when ρ = 1.
Calculating semi-variance by dividing by the number of below-mean outcomes only.
You think only the downside outcomes count in the average.
Fix: Include all outcomes in the expectation, with upside deviations set to zero, unless the question defines it otherwise.
Claiming variance is a complete risk measure.
Mean-variance theory is emphasised so much in the course.
Fix: State that it is symmetric, ignores skewness and tail risk, and is fully adequate only under normality or quadratic utility.
Mixing percentages and decimals, such as squaring 8 and then reading it as 0.08².
You change units midway through the working.
Fix: Choose one unit at the start. Variance in per cent units is in (per cent)².

Worked examples

Example 1

An investment has the following return distribution: 20% with probability 0.3, 10% with probability 0.5, and −10% with probability 0.2. Calculate the expected return, variance and standard deviation, and the semi-variance below the mean.

Show the solution

Expected return: μ = 0.3 × 20 + 0.5 × 10 + 0.2 × (−10) = 6 + 5 − 2 = 9%.
E[R²] = 0.3 × 400 + 0.5 × 100 + 0.2 × 100 = 120 + 50 + 20 = 190.
Variance = 190 − 9² = 190 − 81 = 109 (per cent squared).
Standard deviation = √109 ≈ 10.44%.
Deviations from the mean: 11, 1 and −19. Only −19 is below the mean.
Semi-variance = 0.2 × (−19)² = 0.2 × 361 = 72.2 (per cent squared).

Answer: Expected return 9%, variance 109 (%²), standard deviation about 10.44%, semi-variance below the mean 72.2 (%²).

Example 2

Asset A has σ = 20% and asset B has σ = 10%. The correlation is 0.5. A portfolio has 40% in A and 60% in B. Calculate the portfolio standard deviation and explain why it is less than the weighted average of the standard deviations.

Show the solution

Weighted average of standard deviations = 0.4 × 20 + 0.6 × 10 = 8 + 6 = 14%.
Variance terms in decimals: w₁²σ₁² = 0.16 × 0.04 = 0.0064.
w₂²σ₂² = 0.36 × 0.01 = 0.0036.
Covariance term: 2 × 0.4 × 0.6 × 0.5 × 0.20 × 0.10 = 0.0048.
Portfolio variance = 0.0064 + 0.0036 + 0.0048 = 0.0148.
Portfolio standard deviation = √0.0148 ≈ 0.1217, or 12.17%.
This is below 14% because ρ < 1, so the assets do not move perfectly together and some risk is diversified away.

Answer: Portfolio standard deviation ≈ 12.17%, below the 14% weighted average because the correlation is less than 1.

Exam tips

Show the formula, the substitution and the result. Method marks are awarded even if arithmetic slips.
State units clearly and stay in one unit (decimals or per cent) throughout.
For discussion questions, give both a reason variance is used (tractability, diversification, normality) and at least two limitations.
In computer-based questions, check whether R's var() and sd() use n − 1 and say so in your answer.
When asked to compare variance with semi-variance, state the target used and note that for a distribution symmetric about the mean, semi-variance (below the mean) equals half the variance. The two measures therefore rank risks identically in that case.

Practice questions from Measures of investment risk

Variance and Standard Deviation as Risk Measures in other exams

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

Variance and Standard Deviation as Risk Measures: frequently asked questions

Why is standard deviation preferred to variance for reporting risk?

Standard deviation is in the same units as the return, so it is easier to interpret. Variance is used in calculations because it is additive for independent returns and fits portfolio formulas.

What is the difference between variance and semi-variance?

Variance counts deviations above and below the mean. Semi-variance counts only deviations below a target, usually the mean. It better reflects the investor's concern with losses, but it is harder to use in portfolio calculations.

How do I calculate portfolio variance for two assets?

Use σp² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂. Square the weights, include the covariance term, then take the square root for the standard deviation.

What are the main limitations of standard deviation as a risk measure?

It treats gains and losses the same, ignores skewness and fat tails, and gives no probability of a shortfall. It also depends on estimated inputs that can be unstable. It is most reliable for roughly normal returns.