Value at Risk & Expected Shortfall
VaR answers one question: how much could we lose, at a given confidence, over a given horizon? Slide the confidence level and watch the cutoff walk into the tail; switch to fat tails and watch TVaR pull away from VaR; then meet the two-loans example that explains why regulators moved to Expected Shortfall.
95.0% VaR
£2.36m
23.6% of the portfolio
95.0% TVaR / Expected Shortfall
£3.11m
mean loss GIVEN you are past the cutoff
TVaR ÷ VaR
1.319
switch to Student-t and watch it grow
Historical simulation vs the Normal formula
250 dealt daily P&L observations (fat-tailed, matched to the inputs above). No distribution assumed: sort the losses and read the quantile straight off the data, then compare with what the parametric Normal formula predicts from the same mean and sd. The gap between the two cutoffs IS the fat-tail correction.
Why VaR is not coherent
Two independent loans, each losing 100 if its borrower defaults. At 95%, with the default probability under 5%, each loan alone reports VaR = 0. Hold both and a default is suddenly likelier than 5%, so the combined VaR jumps. Diversifying INCREASED the reported risk: subadditivity fails. TVaR, which averages the whole tail instead of reading one quantile, never does this.
Read it like this, at 4.0%:
- One loan alone: the chance of losing anything is 4.0%. That is within the 5% of outcomes a 95% VaR ignores, so VaR reports 0: the loss is real but invisible to the quantile.
- Both loans together: P(at least one default) = 1 − (1 − 0.040)² = 7.84%, which is more than 5%, so a loss of 100 is inside the visible part of the distribution and the combined VaR is 100. P(both default) = 0.16%.
- TVaR never plays this game: it averages the worst 5% of outcomes rather than reading the 5% boundary. One loan’s 4.0% chance of losing 100 contributes 0.040 × 100 / 0.05 = 80.0 to the tail average, so the danger is priced in whether or not it crosses the quantile line.
| 95% measure | Loan A | Loan B | A + B held together | Subadditive? |
|---|---|---|---|---|
| VaR | 0 | 0 | 100 | ✗ fails |
| TVaR | 80.0 | 80.0 | 103.2 | ✓ always |
Subadditive means measure(A + B) ≤ measure(A) + measure(B): merging portfolios should never look riskier than the sum of the parts, because diversification cannot create risk. When VaR fails this, a bank could game it in reverse: split a risky book into pieces small enough that each piece’s disaster hides below the quantile, and report near-zero risk overall. Slide the default probability past 5% and watch the failure vanish: once each loan’s loss is likely enough to cross the 95% line on its own, VaR sees it everywhere and subadditivity comes back. The failure lives precisely where losses are individually rare but collectively likely.
Test + worked solutions
Test yourself
Each deal is one of the three exam classics: parametric VaR and TVaR on a portfolio, VaR read off a discrete loss table, or scaling a daily VaR to ten days. Work it on paper, then step through the solution.
One number for the tail
VaR at level p is a quantile of the loss distribution: the loss that only the worst (1−p) of outcomes exceed. A 95% one-year VaR of £2m says one year in twenty loses more than £2m, and says nothing at all about how much more. That silence is the point of TVaR (Tail Value at Risk, also called Expected Shortfall): it averages the whole tail beyond the cutoff, so it is always at least as big as VaR and it grows when the tail fattens even if the quantile itself barely moves. On the chart above, switching from Normal to Student-t at the same volatility hardly shifts the gold VaR line, while the red TVaR line strides out into the tail. Two portfolios can share a VaR and hide completely different disasters behind it.
Three ways to compute it
The parametric (variance-covariance) method assumes a distribution, usually normal, and reads the quantile off a formula: VaR = V(zσ − μ). It is quick and smooth but inherits every flaw of the assumed tails. Historical simulation assumes nothing: sort the actual P&L history and take the empirical quantile, as in the dealt year above. It is honest about tail shape but jumpy, and it can never imagine a loss worse than the window it looks at. Monte Carlo sits between the two: simulate from any model you can code, then read the quantile off the simulated sample exactly as historical simulation does. Exams like the parametric formulas; risk teams tend to run all three.
Coherence, and why the regulators moved on
A coherent risk measure is monotonic, translation-invariant, positively homogeneous and subadditive. VaR passes the first three and fails subadditivity: because it reads a single quantile, pooling risks can push a previously invisible loss over the threshold, making the merged book report MORE risk than the two halves separately. The two-loans panel above lets you watch it happen: below a 5% default probability each loan alone shows VaR = 0, yet the pair together shows 100. TVaR is subadditive always, which is why Solvency-style internal models and the market-risk rules that replaced Basel’s VaR (FRTB) lean on Expected Shortfall. The other exam favourite is the square-root-of-time rule: with independent, identically distributed normal returns and zero drift, a 1-day VaR scales to h days by √h, and every assumption in that sentence is a mark for stating when it breaks.
Make it stick. Risk measures sit in CS2 and CM2 alongside the tails in our extreme value theory tool and the quantiles in the statistical tables; the formulas live on the CM2 cheat sheet. Memori is a flashcard app built by actuarial students, with ready-made CS2 and CM2 sets in the shop. Join the beta.
For education only. Illustrative models, not investment or capital advice.