Economic Modelling · Mean-variance portfolio theory

Limitations of Mean-Variance Theory and Alternative Risk Measures

Updated 11 October 2026 · Fact-checked

Mean-variance theory ranks portfolios using only expected return and variance. It works well if returns are normal or investors have quadratic utility. Real returns are skewed and fat-tailed, and variance penalises gains as much as losses. Alternatives such as semi-variance, shortfall probability, VaR and TVaR focus on downside outcomes.

Understand Limitations of Mean-Variance Theory and Alternative Risk Measures

Mean-variance portfolio theory (Markowitz) says an investor looks only at two numbers for each portfolio: the expected return and the variance (or standard deviation) of return. A portfolio is efficient if no other portfolio gives a higher expected return for the same variance, or a lower variance for the same expected return.

This is only justified under certain conditions. Either the investor has a quadratic utility function, or returns follow a multivariate normal distribution. In both cases the mean and variance carry all the information the investor needs. If neither holds, two portfolios with the same mean and variance can have very different shapes, and a sensible investor might prefer one.

The main criticisms are these:

Variance treats upside and downside the same. A large gain raises variance just as a large loss does. Most investors are happy about gains and only dislike losses.
Returns are not normal. Real returns are often skewed and have fat tails, so extreme losses happen more often than a normal model suggests. Variance says little about tail risk.
Quadratic utility has flaws. It implies increasing absolute risk aversion, so the investor would hold less in risky assets as wealth rises. It also has a satiation point beyond which more wealth lowers utility.
Single period and inputs. The model is one period only. It needs expected returns, variances and covariances as inputs, and these are estimated with error. Small input errors can change the optimal portfolio a lot.
Liabilities and other assumptions. It ignores liabilities, which matters for actuaries. It also assumes no transaction costs, and that investors share the same views.

The alternatives measure downside only. Semi-variance averages squared shortfalls below a target. Shortfall probability is the chance return falls below a target. Value at Risk and Tail Value at Risk describe the size of losses in the tail. Each fixes some weakness but has its own: they are harder to compute and less neat mathematically, and they need a chosen target or confidence level.

Note that if returns are symmetric, such as normal, and the target is set at the mean, semi-variance is half the variance and gives the same ranking. With a target other than the mean, the ranking can differ even for symmetric returns. The alternatives add most when returns are skewed.

Key rules to remember

Variance of return: σ² = E[(R − μ)²]; Counts deviations above and below the mean equally.
Semi-variance below a target: SV(T) = E[(min(R − T, 0))²]; Only outcomes below the target T contribute. Common choices of T are the mean or a required return. Check which definition the question uses.
Downside deviation: √SV(T); Square root of semi-variance, in the same units as return.
Shortfall probability: P(R < T); Chance of falling below target T. It ignores how large the shortfall is.
Quadratic utility: U(w) = w − b·w², b > 0, for w < 1 ÷ (2b); Expected utility depends only on mean and variance of wealth. Absolute risk aversion rises with wealth.
Value at Risk at level α: P(Loss > VaR) = 1 − α; For example, 95% VaR is the loss exceeded with 5% probability.
Tail Value at Risk: TVaR = E[Loss | Loss ≥ VaR]; Average loss in the tail beyond VaR, for a continuous loss distribution. For a discrete distribution the definitions of VaR and TVaR need more care. It shows how bad losses are when they occur.

How to solve Limitations of Mean-Variance Theory and Alternative Risk Measures questions

Use this method for both discussion questions and calculation questions on the limits of mean-variance theory.

1Read what is asked: explain a limitation, compare measures, or calculate a measure from given returns.
2State the assumption mean-variance theory needs: quadratic utility or normally distributed returns, and one period.
3For a discussion question, name the limitation and explain why it matters in one or two sentences. Link it to skewness, fat tails, upside versus downside, or estimation error.
4For a calculation, list outcomes and probabilities, then find the mean first.
5Compute the required measure using the stated target. For semi-variance, include only outcomes below the target and square the shortfalls, weighting by probability.
6Compare the measures. Say which investment looks riskier under each and why they may disagree.
7Add a closing remark: say whether the alternative measure fixes the weakness, and mention its own drawback.

Quickest way: Table method for comparing variance and semi-variance

When to use it: Use when you are given a small discrete distribution of returns and asked for variance, semi-variance or shortfall probability.

Write a table with columns: return, probability, return minus target.
Compute the mean with Σ p·R.
For variance, square each deviation from the mean and weight by p.
For semi-variance, set any deviation above the target to zero, square the rest and weight by p.
For shortfall probability, add the probabilities of outcomes below the target.
Quick check: for a symmetric distribution with target equal to the mean, semi-variance equals half the variance.

Common mistakes in Limitations of Mean-Variance Theory and Alternative Risk Measures

Saying variance is wrong because it measures only the spread around the mean.
Students remember that variance is criticised but not the precise reason.
Fix: Say the problem is that it treats gains and losses alike and ignores skewness and tail behaviour. Variance is fully adequate if returns are normal.
Claiming semi-variance always gives a different ranking from variance.
Students assume a downside measure must change the answer.
Fix: For symmetric returns with the target at the mean, semi-variance is half the variance, so rankings agree. Differences arise with skewed returns.
Including upside outcomes in semi-variance.
Students copy the variance procedure and forget to set gains to zero.
Fix: Take min(R − T, 0) first. Gains above the target contribute zero but their probability is still part of the average.
Dividing semi-variance by the probability of the downside only.
Confusion with conditional expectations.
Fix: Use the full probability weights over all outcomes unless the question says otherwise.
Listing only 'returns are not normal' as a limitation.
It is the best-known point, so students stop there.
Fix: Give several: quadratic utility flaws, one period, estimation error in inputs, ignoring liabilities, and equal treatment of upside and downside.
Treating VaR or shortfall probability as a complete fix.
Alternatives are presented as improvements.
Fix: Note that shortfall probability ignores the size of losses and VaR ignores losses beyond the cutoff. TVaR addresses this but is harder to estimate.

Worked examples

Example 1

Portfolio A returns 2% or 12% with equal probability. Portfolio B returns 7% for certain. Explain why a risk-averse mean-variance investor prefers B. Then give one reason why mean-variance ranking could mislead when comparing two risky portfolios with equal mean and variance.

Show the solution

Mean of A = 0.5×2 + 0.5×12 = 7%. Mean of B = 7%. The means are equal.
Variance of A: deviations are −5 and +5, so variance = 0.5×25 + 0.5×25 = 25 (%²). Variance of B = 0.
A risk-averse mean-variance investor prefers the lower variance for the same mean, so prefers B.
For two risky portfolios with equal mean and variance, the investor is indifferent under mean-variance theory.
But their distributions can differ in skewness. One may have a small chance of a very large loss and the other a small chance of a very large gain.
Most investors would prefer the positively skewed one, but mean and variance cannot show this. A downside measure such as semi-variance or TVaR would separate them.

Answer: B is preferred because it has the same mean of 7% and zero variance versus 25 (%²). Mean-variance can mislead because it ignores skewness and tail shape, so equal mean and variance portfolios can differ in downside risk.

Exam tips

Link every limitation to its cause: utility assumption, distribution assumption, or the choice of risk measure. Marks go to the reasoning, not just a list.
In calculations, state the target used for semi-variance and show the min(R − T, 0) step so method marks are safe.
Always mention that for normal returns variance is sufficient. This shows you understand when the theory holds.
For each alternative measure, give one advantage and one drawback. Examiners often ask for both.
In MCQs, watch for the words 'always' and 'only'. Statements like 'semi-variance always ranks differently' are false.

Practice questions from Mean-variance portfolio theory

Limitations of Mean-Variance Theory and Alternative 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.

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Limitations of Mean-Variance Theory and Alternative Risk Measures: frequently asked questions

Why is variance a poor measure of risk?

Variance counts gains and losses equally, although investors dislike only losses. It also says little about skewness and fat tails. It is fine when returns are normal, but real returns often are not.

What is the difference between variance and semi-variance?

Variance averages squared deviations from the mean in both directions. Semi-variance averages squared shortfalls below a chosen target only, so upside outcomes add nothing. With symmetric returns and the target at the mean, semi-variance is half the variance.

When is mean-variance theory exactly justified?

With quadratic utility, expected utility depends only on the mean and variance of wealth, whatever the return distribution. This holds only below the satiation point, where utility is still increasing. With jointly normal returns, any risk-averse expected-utility maximiser with an increasing concave utility function ranks portfolios by mean and variance. Outside these cases it is an approximation.

Which alternative risk measure is best?

No single measure is best. Semi-variance and shortfall probability focus on downside, VaR gives a loss threshold, and TVaR shows average tail loss. The choice depends on the investor's objective and the data available.