Case 020Market risk limits and VaRHard

A bank has a Rs 100 crore firm-wide VaR budget across rates, credit and equities desks whose standalone VaRs add to Rs 120 crore. Compute the diversified VaR and allocate the budget by component VaR.

The situation

Tolvane Bank's board has set a firm-wide 99% one-day VaR budget of Rs 100 crore for trading. The three desks' standalone VaRs are Rs 50 crore for rates, Rs 40 crore for credit and Rs 30 crore for equities, adding to Rs 120 crore. Each desk head argues that the total is over budget and wants a cut to fall on someone else.

The risk team estimates correlations between desk P&Ls of 0.3 between rates and credit, 0.1 between rates and equities and 0.5 between credit and equities. Treat VaR as scaling with standard deviation, so the usual correlation formula applies.

Your task

Is the firm over its budget, how would you allocate the Rs 100 crore across the desks, and what could make the allocation fail?

Quick check

What is Tolvane's firm-wide VaR on today's positions?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

Firm VaR is about Rs 87.7 crore, inside the Rs 100 crore budget, although standalone VaRs add to Rs 120 crore. Component VaRs are 37.0 for rates, 31.9 for credit and 18.8 for equities, adding exactly to the total. Allocating the budget in those proportions gives Rs 42.2, 36.4 and 21.4 crore. The allocation fails if correlations rise: at stressed levels the same positions reach Rs 102.6 crore.

Step 1Why do the standalone VaRs not add up?

Three friends each risk losing their jobs in a bad year, but not always in the same year. The chance they all lose together is smaller than the sum of each one's risk. Firm VaR adds desk risks through their correlations, so unless the desks always lose together, the whole is smaller than the sum of the parts. Here the variance is 50 squared plus 40 squared plus 30 squared, plus twice each product times its correlation: 7,700, so firm VaR is Rs 87.75 crore.

Step 2How do you split the firm total fairly across desks?

Use component VaREach position or desk share of the total VaR: its own VaR times its correlation-weighted link to the whole portfolio, so that the shares add exactly to the firm figure.: each desk's VaR times the sum of its correlations with every desk's VaR, divided by the firm VaR. Rates contribute 37.0, credit 31.9 and equities 18.8, which add to 87.7. Equities, the smallest desk, is also the most diversifying relative to rates, so its share, 21.4%, is below its 25% share of the standalone total.

Standalone VaRs add to 120; the firm's real risk is 87.7, and components add to itRates 50.0Credit 40.0Equities 30.0120.0Sum of standalone VaRsRates 37.0Credit 31.9Equities 18.887.7Diversified, by componentBudget Rs 100 croreHeadroom 12.3under the budgetOver by 20 ifrisks simply added
Tolvane's standalone VaRs add to Rs 120 crore, over the Rs 100 crore budget, but its diversified VaR is Rs 87.7 crore, and the component VaRs of 37.0, 31.9 and 18.8 add exactly to it.
The relationship
CVaRi=VaRi⋅∑jρij VaRjVaRfirm∑iCVaRi=VaRfirmCVaR_i = VaR_i\cdot\frac{\sum_j \rho_{ij}\,VaR_j}{VaR_{\text{firm}}} \qquad \sum_i CVaR_i = VaR_{\text{firm}}
VaR_ithe desk's standalone VaR
\rho_{ij}the correlation between desks i and j
VaR_{firm}the diversified firm VaR, here 87.7
What it says in wordsA desk's component VaR is its own risk weighted by how it moves with the whole firm, and the components always add to the firm figure.
DeskStandalone VaRComponent VaRShareBudget allocatedStandalone limit
Rates5037.0442.2%42.257.0
Credit4031.9136.4%36.445.6
Equities3018.8021.4%21.434.2
Firm12087.75100.0%100.0136.8
Rs crore. Allocating the Rs 100 crore budget by component share gives Rs 42.2, 36.4 and 21.4 crore; expressed as standalone limits the same budget is Rs 57.0, 45.6 and 34.2 crore, adding to Rs 136.8 crore.
Step 3What could make the allocation fail?

The correlations. If rates-credit, rates-equities and credit-equities correlations rise to 0.6, 0.4 and 0.8, as they tend to in a sell-off, the same positions give a firm VaR of Rs 102.6 crore, over budget without a single new trade. A budget allocated on calm correlations is spent in the stress it exists for. So keep a reserve: allocate perhaps 85% of the budget to desks and hold the rest centrally, and check the stressed-correlation VaR alongside the normal one.

Same positions, higher correlations: the budget breaks without a tradeToday: 0.3, 0.1, 0.587.7Stress: 0.6, 0.4, 0.8102.6All correlations 1120.0Rs 100 crore budgetFirm VaR, Rs crore
On unchanged positions, Tolvane's firm VaR is Rs 87.7 crore at today's correlations, Rs 102.6 crore at stressed correlations, over the Rs 100 crore budget, and Rs 120 crore if every desk moved together.

Close with the incentive point the desk heads were arguing about. Component VaR rewards a desk for diversifying the firm and charges it for adding to the firm's main risk, which is the behaviour the board wants. Its limitation is that a desk's component changes when another desk trades, so allocations need a regular reset and a rule for who gets headroom that appears.

Where candidates lose it

Candidates add the standalone VaRs, declare the firm Rs 20 crore over budget and start cutting. That ignores diversification and would force the bank to shrink positions that do not add to firm risk.

The opposite miss is allocating limits on today's correlations with no reserve. Correlations rise in exactly the markets where the budget matters, so an allocation that uses the whole budget in calm conditions is a breach waiting for a stress.

What the interviewer asks next

  • The equities desk doubles its position. What happens to each desk's component VaR?
  • Why might marginal VaR be a better guide than component VaR for approving a single new trade?
  • How would you allocate the budget if VaR were replaced by expected shortfall?

Company names and figures are illustrative.