Lesson 12 of 25 · 7 min

Aggregation: why two risk numbers do not add

Under a bell curve two exposures combine to less than their sum. Add a jump and they combine to more.

A percentile is a point on a curve, and points on curves do not add. That is the whole content of this lesson, and it cuts in both directions depending on the shape of the curve.

Start with the familiar direction. A position of $1,840,000 at 1.9 percent daily has a dollar standard deviation of $34,960; one of $960,000 at 1.55 percent has $14,880. Their individual 99 percent figures are $81,329 and $34,616, summing to $115,945. At a correlation of 0.35 the combined standard deviation is √(34,960² + 14,880² + 2 × 0.35 × 34,960 × 14,880) = $42,518, so the combined figure is 2.3263 × 42,518 = $98,911. The sum overstated the pair by $17,034, and that gap is the diversification benefit.

Now the direction nobody expects. Two positions of $500,000 each, independent, each with a 0.8 percent chance of losing $200,000 in a jump and otherwise losing $5,000. Taken alone, 99.2 percent of outcomes are the $5,000 loss, so the 99 percent figure for each is $5,000 and the two sum to $10,000.

Taken together, the chance that at least one jumps is 1 − 0.992² = 1.594 percent, which is greater than 1 percent. So the 99th percentile of the combined loss is no longer the quiet outcome; it is one jump plus one quiet position, $200,000 + $5,000 = $205,000. Combining two positions multiplied the reported figure by 20.5 while genuinely reducing nothing. The threshold moved because a percentile jumped across a step in the distribution, not because the positions became riskier.

Two facilities at two lenders, each with comfortable headroom, do not give you the sum of the two headrooms. Whether they give more or less depends on whether the collateral moves smoothly or in steps.

Key terms

Subadditivity: The property that a combined risk figure is no larger than the sum of the parts. Value at risk has it under a normal assumption and can lose it when losses arrive in jumps.
Diversification benefit: The amount by which combining exposures reduces a risk figure below the sum of the separate figures. Here $17,034 in the first case and negative in the second.
Jump risk: Loss that arrives as a discrete step rather than a gradual move. It puts a cliff in the distribution, and percentiles behave badly near cliffs.

Worked example

The same operation, two shapes

First: two smooth positions at a 0.35 correlation. Second: two independent positions with a 0.8 percent chance of a $200,000 jump each. Illustrative.

Smooth case, separate figures: $81,329 + $34,616 = $115,945
Smooth case, combined: $98,911
Benefit from combining: $17,034
Jump case, separate figures: $5,000 + $5,000 = $10,000
Chance at least one jumps: 1.594%
Jump case, combined: $205,000

One arithmetic operation on two pairs of positions produced a $17,034 saving and a twenty-fold increase. The difference was entirely the shape.

Check yourself

Two independent positions each have a 0.8 percent chance of a $200,000 loss and otherwise lose $5,000. Each reports a 99 percent value at risk of $5,000. Why is the combined figure $205,000?

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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.