Scaling and aggregating risk

RiskAtoms Bank's four trading books report one-day 99% VaRs of £1.45m (Equities), £0.88m (Rates), £0.64m (FX and commodities), £0.49m (Credit). Add them up and you get £3.46m. The desk's VaR, measured on all eight positions together, is £1.94m. Where did £1.52m go? And when the board asks for a 10-day figure, can the desk just multiply by √10? This lesson answers both questions with the desk's own numbers.

From one day to ten

L2.5.06: Value at risk: the idea met the square-root-of-time ruleThe rule that an h-day VaR is √h times the one-day VaR. Exact only for independent, identically distributed, normal daily P&L with a mean of zero and unchanged positions.: a 10-day VaR is about √10 = 3.162 times the one-day VaR. For the desk that turns £1.94m into £6.15m. The rule comes from adding variances (M-C11: Combining random variables): ten independent days with the same standard deviation have ten times the variance, so √10 times the standard deviation. It needs four things to be true, and each can fail on a real desk.

Autocorrelation is the one assumption with a tidy correction. If the lag-k autocorrelation of daily P&L is ρᵏ, the variance of a 10-day sum is not 10σ² but σ² times a larger or smaller number (the formal section derives it). Move the sliders to see how fast the error grows:

Checking √10 on the desk's history

Historical simulation lets you test the rule instead of assuming it. Take every run of ten consecutive days in the 750-day history (741 overlapping windowsWindows of h days that start on consecutive days, so neighbouring windows share h − 1 days. They give more windows but are far from independent.), add up each run's factor moves and revalue today's positions in full. The 99% VaR is the 8th worst of those 741 outcomes. Compare three numbers:

Here √10 overstates. Going from the first row to the second is the fat-tail effect: the worst single days of the stressed spell do not line up ten in a row. Going from the second to the third is the puts. A 10% fall of the UK index in one go costs the desk £2.17m; ten separate 1% falls, each from today's position, cost £3.89m. The puts' gains grow faster than the fall, so a convexA P&L that is curved in the factor move, like an option's: a large move is not a multiple of a small one. book does not scale like a straight line. Keep the caveat in mind: 741 overlapping windows contain only 75 independent ones, so the 10-day historical figure is itself noisy.

From four books to one desk

Each book's stand-alone VaRA book's VaR measured on its own, as if the rest of the desk did not exist. ignores the other three. The desk VaR does not. If the books' P&Ls are jointly normal, the desk's variance is the variance of a sum, so the desk VaR is the stand-alone VaRs combined with the correlations between the books:

The correlations come from the factor covariance matrix: two books are correlated because they load on correlated factors. For the desk's books:

Equities and Rates are slightly negatively correlated: when equities fall, rates tend to fall too, and the desk's long gilt gains. Rates and Credit are positively correlated (0.41), because both own bond duration. With these correlations √(v′Rv) gives back exactly the desk's £1.94m, and the diversification benefitThe amount by which the sum of the parts' stand-alone VaRs exceeds the VaR of the whole: the risk that offsets between parts. is £1.52m, 43.8% of the sum.

Historical simulation measures diversification without assuming any correlations: revalue each book on each of the 750 days and take the 8th worst of each. The books' historical VaRs sum to £4.39m; the desk's is £2.27m, a benefit of £2.12m. The books' worst days fall on different dates, and that is all diversification is.

In a crisis, volatilities rise and correlations shift. Under the stressed spell's covariance (factor volatilities × 2.5, crisis correlations) the books' VaRs sum to £8.50m and the desk VaR is £4.72m. For this desk the benefit stays near 44%: the crisis correlations push some pairs of books together (Equities and FX and commodities, from 0.14 to 0.28) and others apart (Equities and Credit, from 0.10 to −0.05). The rise in VaR comes from the volatilities. That is a property of this book, not a law: a desk whose books all load on one risk factor loses its diversification just when it needs it.

The h-day variance with autocorrelation. Let daily P&L have standard deviation $\sigma$ and lag-k correlation $\rho_k$. Among h days, exactly $h - k$ pairs are k days apart, so (M-C11: Combining random variables)

With $\rho = 0$ the root is $\sqrt h$. With $\rho = 0.1$ and $h = 10$ it is 3.461, not 3.162.

Aggregation. The desk's P&L is the sum of the books' P&Ls, $P = \sum_i P_i$. Its variance is a quadratic form in the books' standard deviations:

The book correlations come from the factor covariance matrix $\Sigma$ and each book's sensitivity vector $\delta_i$: $\rho_{ij} = \delta_i^\top\Sigma\,\delta_j / (\sigma_i\sigma_j)$, with $\sigma_i = \sqrt{\delta_i^\top\Sigma\,\delta_i}$. Then $\sum_{ij}\delta_i^\top\Sigma\,\delta_j = \delta^\top\Sigma\,\delta$, the desk's variance from L3.2.09: Parametric VaR for a portfolio, so the two routes agree exactly.

Two books in full. Equities and Rates alone:

Start with the correlations the factors imply. Then switch to one common correlation and slide it to 1 (the sum) and to 0. Untick a book to see its incremental VaRThe change in the desk VaR when one book is removed entirely.: how much the desk VaR falls without it. Untick Rates and notice that the desk VaR falls by less than the Rates book's own VaR.

Both methods agree on the story, if not the level: about half of the books' stand-alone risk offsets at desk level. One-day, 99%, pounds, illustrative data.

The 1996 Market Risk Amendment asked for a 10-day 99% VaR and allowed banks to scale up a one-day figure by √10 (L2.5.06: Value at risk: the idea). The Fundamental Review of the Trading Book keeps a 10-day base horizon for expected shortfall but lengthens it for less liquid risk factors, and it limits how much diversification a bank may claim across risk classes. The detail is in M4.3: Market risk capital (FRTB) and CVA. Aggregating risk across a whole firm, with economic capital and correlations between risk types, comes in M7.1: Enterprise risk and economic capital.

• VaR scales by √h only for independent, identically distributed, normal days with unchanged positions.
• With lag-k autocorrelation ρᵏ the factor is √(h + 2Σ(h − k)ρᵏ): trending P&L makes √h too small, mean reversion too large.
• On the desk's history √10 overstates the frozen book's 10-day VaR (£7.19m against £4.92m): fat-tailed days average out and the puts are convex.
• Desk VaR = √(v′Rv) for normal P&L. The sum of book VaRs is an upper bound reached only when every correlation is 1.
• The desk's books sum to £3.46m; the desk VaR is £1.94m, a diversification benefit of 44%.

Square-root-of-time rule

The rule that an h-day VaR is √h times the one-day VaR. Exact only for independent, identically distributed, normal daily P&L with a mean of zero and unchanged positions.

Autocorrelation

The correlation of a series with its own past values. Lag-one autocorrelation is the correlation between today's P&L and yesterday's.

Overlapping windows

Windows of h days that start on consecutive days, so neighbouring windows share h − 1 days. They give more windows but are far from independent.

Convex P&L

A P&L that is curved in the factor move, like an option's: a large move is not a multiple of a small one.

Stand-alone VaR

A book's VaR measured on its own, as if the rest of the desk did not exist.

Diversification benefit

The amount by which the sum of the parts' stand-alone VaRs exceeds the VaR of the whole: the risk that offsets between parts.

Incremental VaR

The change in the desk VaR when one book is removed entirely.