CFA Level II Exam · Measuring and Managing Market Risk

Value at Risk (VaR) Concepts for CFA Level II

Updated 7 October 2026 · Fact-checked

Value at risk (VaR) is the minimum loss you expect to be equalled or exceeded with a given probability over a set time horizon. A 5% one-day VaR of ₹10,00,000 means a loss of at least that size is expected on about 1 day in 20. Read the confidence level, horizon and currency first.

Understand Value at Risk (VaR) Concepts

Value at risk (VaR) answers one question: how bad can losses get under normal market conditions? It gives a single money figure for a portfolio, a desk or a firm, so risk can be compared and limited.

A VaR statement has three parts: a loss amount, a probability and a time horizon. Example: "The one-day 5% VaR is ₹10,00,000." This means that, on a given day, there is a 5% chance of a loss of ₹10,00,000 or more. Equivalent wording: a 95% confidence level. Be careful with the wording. VaR is the minimum loss in the worst 5% of outcomes. It is not the maximum loss.

The two main parameters are the confidence level and the horizon. A higher confidence level (99% instead of 95%) pushes you further into the tail, so VaR rises. A longer horizon gives more time for prices to move, so VaR usually rises. The exception is a large positive mean return, which grows with T and can offset the wider spread. The choice depends on use. Trading desks use short horizons such as one day because they can close positions fast. Pension funds or institutions with illiquid assets may use longer horizons. Regulators often use high confidence levels.

VaR has real strengths. It is simple, gives one number across asset classes, is easy to communicate, can be used for risk limits and capital allocation, and can be compared across time and across desks. It also forces a firm to measure its risks.

It also has clear limits. VaR says nothing about the size of the loss beyond the cutoff. It depends on the estimation method and its inputs, so different methods give different numbers. It can be hard to estimate for portfolios with options or illiquid assets. It can understate risk if returns have fat tails or correlations rise in a crisis. It does not capture liquidity risk well, and it can create a false sense of safety. Because of these limits, it is used with stress tests and scenario analysis. You see the estimation methods (parametric, historical, Monte Carlo) in the related topic.

Key formulas to remember

Interpreting VaR: Probability(loss ≥ VaR) = 1 − confidence level, over the stated horizon; A 95% confidence VaR is the same as a 5% VaR. The loss is a minimum in the tail, not a maximum.
Parametric VaR (normal returns, return form): VaR = [−(μ − z × σ)] = z × σ − μ, where z is the critical value; Common one-tailed z values: 1.645 for 5% and 2.33 for 1%. Multiply by portfolio value for a money figure.
Money VaR: VaR (₹) = VaR (%) × portfolio value; Use the starting portfolio value unless the vignette says otherwise.
Scaling the horizon: VaR over T periods ≈ VaR over 1 period × √T; Assumes returns are independent and identically distributed and the mean is close to zero. State this when you use it.
Converting periods for the mean and volatility: μ over T = μ × T; σ over T = σ × √T; Use this when the mean is not negligible. The mean scales with T, volatility with √T. With a large positive mean, VaR may not rise as the horizon lengthens.

How to solve Value at Risk (VaR) Concepts questions

Use this order for any VaR concept or calculation question in an item set.

1Find the stated confidence level or tail probability, the horizon and the currency in the vignette. Convert 95% to a 5% tail, or 99% to a 1% tail.
2Check whether the question asks for a calculation, an interpretation, or a strength or limitation.
3For an interpretation, restate it as: with probability (tail), the loss over the horizon is at least the VaR. Do not call it a maximum loss.
4For a calculation, get the return mean and standard deviation for the right horizon, pick the correct z value, then compute z × σ − μ and multiply by portfolio value.
5If the horizon changes, scale volatility by √T (and the mean by T) before applying z.
6Check direction: a higher confidence level should raise VaR. With a zero or negligible mean, a longer horizon also raises VaR. With a positive mean it may not, because the mean grows with T. If your answer moved the wrong way, recheck.
7For method or limitation items, match the point to the vignette: tail loss ignored, fat tails, option positions, illiquidity or correlation breakdown.

Quickest way: Tail, horizon, z, scale

When to use it: Use for numeric VaR items when time is short and returns are assumed normal with a negligible mean.

Write the tail probability and pick z: 1.645 for 5%, 2.33 for 1%.
Compute VaR (%) = z × σ for the stated horizon, using σ × √T if the horizon is longer than the volatility period.
Multiply by portfolio value.
Sanity check: 99% VaR must be larger than 95% VaR, and with a negligible mean a 10-day VaR is about 3.16 times a 1-day VaR.

Common mistakes in Value at Risk (VaR) Concepts

Calling VaR the maximum loss.
The word "risk" suggests a worst case.
Fix: Say VaR is the minimum loss expected in the worst tail of outcomes. Losses beyond it can be much larger.
Mixing up confidence level and tail probability.
95% confidence and 5% VaR sound like different things.
Fix: Tail probability = 1 − confidence. Use the tail for z values and for the "1 day in 20" reading.
Scaling volatility by T instead of √T.
Students scale everything linearly.
Fix: Volatility scales with √T. Only the mean scales with T. Do not scale VaR by T.
Using the wrong z value, such as two-tailed values.
Memorised 1.96 for 95% from hypothesis testing.
Fix: VaR is a one-tailed measure. Use 1.645 for 95% and 2.33 for 99%.
Treating VaR as a complete measure of risk.
One tidy number looks complete.
Fix: State that VaR ignores losses beyond the cutoff, depends on method and inputs, and should be paired with stress tests and scenario analysis.
Subtracting the mean with the wrong sign.
Formula is applied to returns, but losses are quoted as positive.
Fix: Loss at the cutoff = −(μ − z × σ). A positive mean reduces VaR.

Worked examples

Example 1

A portfolio is worth ₹50,00,00,000. Daily return volatility is 1.2% and the mean daily return is assumed to be zero. Returns are normal. (1) What is the one-day 5% VaR? (2) What is the one-day 1% VaR? (3) Which interpretation of the 1% VaR is correct: A) the portfolio will not lose more than this amount on any day; B) a loss of at least this amount is expected on about 1 day in 100; C) the average loss on bad days is this amount?

Show the solution

Part 1: z for a 5% one-tailed tail is 1.645. VaR % = 1.645 × 1.2% = 1.974%.
Money VaR = 1.974% × ₹50,00,00,000 = ₹98,70,000.
Part 2: z for 1% is 2.33. VaR % = 2.33 × 1.2% = 2.796%.
Money VaR = 2.796% × ₹50,00,00,000 = ₹1,39,80,000.
Part 3: VaR is a minimum loss in the tail with probability equal to the tail, so B is correct. A treats it as a maximum. C describes expected tail loss, not VaR.

Answer: (1) ₹98,70,000; (2) ₹1,39,80,000; (3) B.

Example 2

A risk manager reports a one-day 5% VaR of $2.0 million for a fund. Returns are assumed independent with zero mean. (1) Estimate the 10-day 5% VaR. (2) The manager then moves to a 1% level for one day. If the 5% z is 1.645 and the 1% z is 2.33, estimate the one-day 1% VaR. (3) Which is a valid limitation: A) VaR cannot be computed for a diversified portfolio; B) VaR gives no information on losses beyond the cutoff; C) VaR falls when the confidence level rises?

Show the solution

Part 1: scale by √10. √10 = 3.1623. VaR = $2.0 million × 3.1623 = $6.32 million.
Part 2: the VaR scales with z. Ratio = 2.33 ÷ 1.645 = 1.4164. VaR = $2.0 million × 1.4164 = $2.83 million.
Part 3: A is false, since VaR can be computed for diversified portfolios. C is false, since VaR rises with confidence. B is true: VaR is a cutoff and says nothing about how large losses in the tail are.

Answer: (1) about $6.32 million; (2) about $2.83 million; (3) B.

Exam tips

Read the vignette for the tail probability and horizon before anything else. Many wrong answers come from using 95% as if it were the tail.
Expect interpretation questions with three similar-sounding options. Eliminate any option that says "maximum" or "will not exceed".
For a longer horizon, scale volatility by √T and, with a zero or negligible mean, check the answer rises. If the mean is large and positive, VaR may not rise. State the assumption of independent returns if the question asks why scaling may fail.
For strengths and limitations, tie your pick to the vignette: options in the book, illiquid assets, or a crisis with rising correlation point to VaR understating risk.
There is no penalty for wrong answers, so answer every question even if you must guess between two options.

Value at Risk (VaR) Concepts in other exams

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

Value at Risk (VaR) Concepts: frequently asked questions

What does a 95% one-day VaR mean?

It means that, under the assumptions used, there is a 5% chance of a loss at least as large as the VaR over one day. In practice, about 1 day in 20 would be expected to breach it. It does not say how large the loss on that day could be.

Why does VaR increase with a higher confidence level or longer horizon?

A higher confidence level moves the cutoff further into the tail, where losses are larger. A longer horizon allows prices more time to move, so the spread of outcomes grows. Under independent returns and a negligible mean, VaR scales with the square root of time. With a large positive mean, the growing expected return can offset this, so VaR may not rise.

What are the main advantages of VaR?

It is simple, summarises risk in one money figure, works across asset classes, and is easy to communicate to management and regulators. It also supports risk limits and capital allocation. It makes comparisons across desks and periods straightforward.

What are the main limitations of VaR?

It ignores the size of losses beyond the cutoff and depends on the method and inputs used. It can understate risk when returns have fat tails or correlations rise in stress, and it is hard to estimate for options and illiquid positions. It should be used with stress tests and scenario analysis.