The standard formula correlation matrix, the interest rate switch that changes three of its entries, and what a new fund really costs once diversification is shared out.
The market risk charge is not the sum of its sub-modules. It is the square root of Σ Corrij × SCRi × SCRj, with the correlations set in the standard formula. On an illustrative insurer whose six sub-module charges sum to €840.0 million, the aggregate is €679.1 million, a diversification benefit of 19.2 per cent. If the interest rate charge comes from the upward shock rather than the downward one, three correlations drop from 0.5 to zero and the charge falls to €619.0 million.
Worked in full in Insurance Capital for Asset Managers by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The insurer and its charges are illustrative, in euro millions. Each sub-module charge has already been computed on the asset and liability side, the interest rate charge as the larger of the upward and downward shocks. The correlations are those of Article 164 of Commission Delegated Regulation (EU) 2015/35, the standard formula. The parameter A, between interest rate risk and equity, property and spread, is 0.5 when the downward interest rate shock is the binding one and 0 when the upward shock is. Many life insurers, with liabilities longer than their assets, are bound by the downward shock, so the base case uses 0.5.
| Sub-module | Charge | Int. | Eq. | Prop. | Spr. | Cur. | Conc. |
|---|---|---|---|---|---|---|---|
| Interest rate | 120.0 | 1 | A | A | A | 0.25 | 0 |
| Equity | 300.0 | A | 1 | 0.75 | 0.75 | 0.25 | 0 |
| Property | 90.0 | A | 0.75 | 1 | 0.5 | 0.25 | 0 |
| Spread | 260.0 | A | 0.75 | 0.5 | 1 | 0.25 | 0 |
| Currency | 50.0 | 0.25 | 0.25 | 0.25 | 0.25 | 1 | 0 |
| Concentration | 20.0 | 0 | 0 | 0 | 0 | 0 | 1 |
| Sum | 840.0 |
Square every charge and add the cross terms, each pair counted twice and weighted by its correlation. The squares come to 183,000. The cross terms come to 278,150, and one pair dominates: equity with spread, 2 × 0.75 × 300 × 260 = 117,000. Interest rate with equity adds 36,000 and with spread 31,200, both at the 0.5 of the downward scenario.
SCRmkt = √( Σi Σj Corrij × SCRi × SCRj )
= √(183,000 + 278,150) = √461,150 = 679.1
Excel, charges in a column S and the matrix in M: =SQRT(MMULT(TRANSPOSE(S),MMULT(M,S))), entered as an array formula
The benefit is 840.0 − 679.1 = 160.9, or 19.2 per cent of the undiversified sum. Concentration contributes almost nothing to the cross terms because its correlation with everything is zero; currency contributes little because 0.25 is a weak link.
A is a switch, not a parameter to estimate. If the 120.0 had come from the upward shock, A would be zero and the three interest rate cross terms would vanish:
Upward scenario, A = 0: SCRmkt = 619.0, which is 60.1 less, or 8.8 per cent of the downward-scenario figure
This is one of the few places in the standard formula where the direction of a risk, not just its size, changes the capital. An insurer that shortens its liabilities or lengthens its assets until the upward shock binds can gain twice: a smaller interest rate charge and a lower correlation on it.
For an asset manager, the standalone charge on a fund is the wrong figure to quote; what the insurer pays is the charge after diversification. The Euler allocation splits the 679.1 exactly among the sub-modules: each charge times its correlated exposure to the others, divided by the total.
Allocatedi = SCRi × Σj Corrij × SCRj / SCRmkt
| Sub-module | Standalone | Allocated | Kept | Marginal cost of +50 |
|---|---|---|---|---|
| Interest rate | 120.0 | 80.8 | 67.4% | 34.6 |
| Equity | 300.0 | 280.5 | 93.5% | 47.0 |
| Property | 90.0 | 68.6 | 76.2% | 38.8 |
| Spread | 260.0 | 230.7 | 88.7% | 44.7 |
| Currency | 50.0 | 17.9 | 35.7% | 19.4 |
| Concentration | 20.0 | 0.6 | 2.9% | 3.3 |
| Market risk | 840.0 | 679.1 | 80.8% |
Equity and spread keep almost all of their standalone charge, because they are the two largest positions and correlate at 0.75 with each other. A further €50 million of spread charge, a new credit fund for instance, raises the market risk charge by 44.7: the insurer gets 10.5 per cent diversification on it, not the 19.2 per cent the book average suggests. The same 50 of currency charge costs 19.4, and of concentration 3.3. The ranking of mandates changes once capital is measured this way, and the order depends on what the insurer already holds.
The first mistake is quoting the portfolio-average benefit to every asset: 19.2 per cent is what the whole book gets, while the marginal spread exposure gets 10.5. The second is using a convenient matrix. A flat 0.5 everywhere gives 666.6, and independence gives 427.8; neither is the regulation. The third is trusting the obvious checks. An aggregate below the sum, above the largest charge and with a positive benefit will pass on almost any symmetric matrix of positive entries, including a mistyped one. The check that catches a wrong matrix is comparing every coefficient with the published value and testing that the matrix is positive semi-definite.
The Solvency II review adopted in 2025, applying from 2027, comes with amendments to the standard formula calibration. Check the version of the delegated regulation in force at the reporting date before reusing any factor here.
The sub-module charges themselves come from the rules in how to calculate the spread risk charge and how the symmetric adjustment moves the equity charge.
The full matrix, the principal-minor check and the upward scenario are live in the free workbook for this case.
0.75 in the standard formula market risk matrix, the same as equity with property; spread with property is 0.5 and currency is 0.25 with the four main risks. On the illustrative insurer here the equity-spread pair alone contributes 117,000 of the 278,150 of cross terms, which is why those two sub-modules keep 93.5 and 88.7 per cent of their standalone charge after diversification.
It is the correlation between interest rate risk and equity, property and spread risk. It is 0 when the capital requirement for interest rate risk comes from the upward shock and 0.5 when it comes from the downward shock. On the illustrative charges here, switching A from 0.5 to 0 lowers the market risk charge from €679.1 million to €619.0 million.
Its marginal charge, which depends on what the insurer already holds. Here €50 million of extra spread charge raises the aggregate market risk charge by €44.7 million, a diversification benefit of 10.5 per cent, while the same €50 million of currency charge costs €19.4 million and of concentration charge €3.3 million.
This article is one calculation from Insurance Capital for Asset Managers. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
Get the book on Amazon →Free companion files
Also on Amazon UK · Amazon Germany · Amazon France · Amazon Canada
Reading guide: regulation, compliance and banking → · All 453 articles →
If this book helped, or didn’t, a few lines on Amazon are worth more than they look: they are what the next reader goes on. Write a review. The workbook stays free either way.