Value at Risk (VaR)

VaR summarizes downside risk as the loss threshold breached only with a small target probability, but it ignores the severity of losses beyond that threshold.

1. The idea

Value at Risk at confidence level $1 - α$ over horizon $T$ is the loss level such that the probability of exceeding it is $α$ : $P (Loss > V a R_{α}) = α$ . Under a normal returns assumption with mean $μ$ and volatility $σ$ , parametric VaR is $V a R_{α} = - (μ + z_{α} σ)$ for portfolio value $V$ , often simplified by ignoring $μ$ over short horizons.

For a 95% VaR, $z_{0.05} = 1.645$ ; for 99%, $z_{0.01} = 2.326$ . Scaling to different horizons uses $σ_{T} = σ_{1} T$ under the assumption of i.i.d. returns (the square-root-of-time rule).

VaR's key limitation is that it says nothing about the magnitude of losses beyond the threshold (tail severity), is not subadditive in general (can penalize diversification), and its normal-distribution version underestimates risk when actual returns have fat tails — a shortfall exposed dramatically in market crises. Expected Shortfall (average loss beyond VaR) is often used to address the tail-severity gap.

2. Why it matters

VaR is a standard regulatory and desk-level risk metric, and interviewers probe both the computation and a critical understanding of its well-known shortcomings.

3. Core mathematics

Parametric (normal) VaR

V a R_{α} = V (z_{α} σ - μ)

loss expressed as a positive number

Square-root-of-time scaling

σ_{T} = σ_{1} T

Common z-values

z_{0.05} = 1.645, z_{0.01} = 2.326

4. How to recognize it

Think of this when…

'Find the 1-day (or 10-day) 95% (or 99%) VaR of a portfolio'
Any question about scaling volatility across horizons
Critiques asking 'what does VaR fail to capture' (tail severity, fat tails, subadditivity)

5. Worked example

Worked example

A $10 million portfolio has daily volatility 2% and negligible mean return. Find the 1-day 99% VaR.

1
Use the 99% z-value.
$z_{0.01} = 2.326$
2
Apply the parametric VaR formula (mean ≈ 0).
$V a R = V \cdot z σ = 10, 000, 000 \times 2.326 \times 0.02$
3
Compute.
$V a R \approx $465, 200$
Answer: 1-day 99% VaR ≈ $465,200

6. Common mistakes

Confusing the confidence level with the tail probability (95% VaR uses $z = 1.645$ , not $z = 0.95$ )
Scaling volatility linearly with time instead of with the square root of time
Treating VaR as a worst-case loss, when by definition losses beyond VaR are excluded

7. Quick check

The z-value for 95% parametric VaR is approximately:

A key limitation of VaR is that it:

8. Practice

16 QuantSolve questions use this concept.

Practice this concept

9. Related methods

This lesson is conceptual; see the topic hub for methods.

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