Value at Risk and Tail Value at Risk (VaR and TVaR)

Understand Value at Risk and Tail Value at Risk

Key rules to remember

Value at Risk

VaR_α = smallest x such that P(L ≤ x) ≥ α

L is the loss. For a continuous distribution, P(L ≤ VaR_α) = α. Always state α and the time horizon.

VaR for a normal loss

VaR_α = μ + z_α × σ

L ~ N(μ, σ²). z_α is the standard normal value with Φ(z_α) = α. Common values: z = 1.645 (95%), z = 2.326 (99%).

Tail Value at Risk

TVaR_α = (1 ÷ (1 − α)) × ∫ from α to 1 of VaR_u du

Average loss in the worst (1 − α) of probability (expected shortfall). For a continuous loss this equals E[L | L ≥ VaR_α]. For a discrete loss, E[L | L ≥ VaR_α] can differ from it (see the worked example).

TVaR for a normal loss

TVaR_α = μ + σ × φ(z_α) ÷ (1 − α)

φ is the standard normal density, φ(z) = e^(−z²/2) ÷ √(2π). Applies to a normal loss.

Scaling over time

σ over n periods = σ × √n, mean over n periods = n × μ

Valid only if period returns are independent and identically distributed with the same variance.

Coherence conditions

Monotonicity; ρ(X + c) = ρ(X) + c; ρ(kX) = kρ(X) for k > 0; ρ(X + Y) ≤ ρ(X) + ρ(Y)

All four must hold. VaR fails subadditivity in general. TVaR defined as expected shortfall satisfies all four. The E[L | L ≥ VaR] form is coherent for continuous losses but can fail subadditivity for discrete losses.

How to solve Value at Risk and Tail Value at Risk questions

1. 1Define the loss L clearly. Loss is positive when money is lost. If you are given returns, the loss is the negative of the return times the portfolio value.
2. 2Note the confidence level α and the time horizon. Convert the mean and standard deviation to that horizon first, using n × μ and σ√n if returns are independent.
3. 3Identify the distribution. If it is normal, use VaR = μ + zσ. If it is discrete or empirical, find the smallest value whose cumulative probability reaches α.
4. 4Find the correct z-value or cumulative probability from tables. Check that you are using the upper tail for losses.
5. 5For TVaR, average the losses at or beyond the VaR. For a discrete case, weight each tail loss by its probability and divide by the tail probability. For a normal, use μ + σφ(z) ÷ (1 − α).
6. 6Check that TVaR ≥ VaR. If not, there is an error.
7. 7For discussion parts, comment on the assumptions, tail behaviour, coherence and the data limits. Link comments to the numbers in your answer.

Quickest way: Normal VaR and TVaR in three lines

Common mistakes in Value at Risk and Tail Value at Risk

• Treating a return as a loss and getting the sign wrong.

  Questions often give returns, but VaR is defined on losses.

  Fix: Write L = −(return) × value first. For a normal return R, the loss is −R, and VaR at 95% is −μ_R + 1.645σ_R times the value.

• Forgetting to scale mean and standard deviation to the horizon.

  Students plug in one-day figures for a ten-day question.

  Fix: Multiply the mean by n and the standard deviation by √n before using z. State the independence assumption.

• Using the wrong tail, for example z = −1.645 for a loss VaR.

  Mixing up the left tail of returns with the right tail of losses.

  Fix: For losses, use the upper-tail z with Φ(z) = α, which is positive for α above 0.5.

• Saying VaR is the maximum possible loss.

  The word 'risk' suggests a worst case.

  Fix: Say VaR is a threshold exceeded with probability 1 − α. Losses can be far larger.

• Claiming VaR is always non-coherent or always coherent.

  Memorising a slogan without the condition.

  Fix: Say VaR fails subadditivity in general, though it is subadditive for some distributions such as the normal (for α ≥ 0.5). TVaR is coherent.

• Computing a discrete TVaR by averaging only the loss values, ignoring probabilities.

  Treating tail outcomes as equally likely.

  Fix: Weight each tail loss by its probability and divide by the total tail probability.

Worked examples