The spread grid is concave in duration and convex in rating, so a factsheet's two averages misstate the capital in opposite directions.
No. The Solvency II spread charge has to be computed bond by bond, because the Article 176 stress is not a straight line in duration or in rating. On an illustrative €200 million bond fund, charging the fund's average duration of 5.16 years at its average rating of A gives €14.23 million; the line-by-line charge is €18.94 million, 9.47 per cent of market value, so the shortcut understates the capital by €4.71 million, or 24.9 per cent.
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 →
Insurers ask for look-through data because the standard formula charges what a fund holds, not what its factsheet says. Many managers answer with two numbers that look sufficient: the portfolio's modified duration and its average rating. Both are market-value-weighted averages, and the spread grid punishes averaging in opposite directions. The duration error usually overstates the charge; the rating error understates it, and by more.
| Rating | Market value | Modified duration | Stress | Charge |
|---|---|---|---|---|
| AAA | 30 | 1.5 | 1.35% | 0.40 |
| AA | 25 | 6.0 | 6.10% | 1.52 |
| A | 30 | 3.0 | 4.20% | 1.26 |
| A | 35 | 8.0 | 9.10% | 3.19 |
| BBB | 30 | 4.0 | 10.00% | 3.00 |
| BBB | 25 | 11.0 | 21.00% | 5.25 |
| BB | 15 | 2.5 | 11.25% | 1.69 |
| B | 10 | 3.5 | 26.25% | 2.62 |
| Fund | 200 | 5.16 | 9.47% | 18.94 |
Each line is grouped here by rating and duration for clarity; a real look-through file has one row per bond, and the same method applies.
Stress = a + b × (duration − lower bound of the band), with a and b set by the credit quality step and the duration band.
The A line at 8.0 years sits in the five-to-ten band: 7.0% + 0.7% × (8.0 − 5) = 9.10%. The BBB line at 11.0 years sits in the ten-to-fifteen band: 20.0% + 1.0% × 1 = 21.00%. Below five years every rating is a single rate per year of duration: 0.9, 1.1, 1.4, 2.5, 4.5 and 7.5 per cent from AAA to B.
Charge = Σ market value × stress = 18.94, or 9.47 per cent of 200.
Excel, with the grid in a lookup range: =SUMPRODUCT(MV, a + b*(Dur - lb)), where a, b and lb are each returned by INDEX and MATCH on the line's rating and duration band.
Average duration = Σ MV × duration ÷ 200 = 5.16 years. Average credit quality step, weighting AAA as 0 down to B as 5: 2.15, which rounds to A.
A at 5.16 years: 7.0% + 0.7% × 0.16 = 7.11%, a charge of 14.23.
The shortcut misses by €4.71 million. It helps to separate the two averages, because they fail for different reasons.
| Method | Charge | % of MV | Error |
|---|---|---|---|
| Line by line | 18.94 | 9.47% | 0.00 |
| Each rating at the fund's 5.16-year duration | 21.66 | 10.83% | +2.72 |
| Each duration at the average rating, A | 12.45 | 6.22% | −6.49 |
| Both averages | 14.23 | 7.11% | −4.71 |
Duration averaging overstates. The cost of each extra year falls as duration rises: for BBB it is 2.5 points a year up to five years, 1.5 from five to ten, 1.0 from ten to fifteen. The grid is concave in duration, so the stress at the average duration is higher than the average of the stresses whenever the holdings straddle a band boundary.
Rating averaging understates. Each step down the rating scale costs more than the one before: at four years, AAA is charged 3.60 per cent, A 5.60, BBB 10.00, BB 18.00 and B 30.00. The grid is convex in rating, so a fund with some BB and B paper is charged far more than a fund of uniform quality with the same average step. In this fund the BB and B lines are 12.5 per cent of market value and 22.8 per cent of the capital.
Holding the average fixed and changing the spread shows the size of each effect on its own. First, €100 million of BBB with an average duration of 7.0 years:
| Shape | Line-by-line charge | Overstatement by the shortcut |
|---|---|---|
| Bullet at 7 years | 15.50 | 0.00 |
| Half at 4, half at 10 | 15.00 | 0.50 |
| Thirds at 3, 7 and 11 | 14.67 | 0.83 |
| Half at 2, half at 12 | 13.50 | 2.00 |
Then €100 million at four years, every mix averaging BBB:
| Mix | Line-by-line charge | Understatement by the shortcut |
|---|---|---|
| All BBB | 10.00 | 0.00 |
| Half A, half BB | 11.80 | 1.80 |
| Half AA, half B | 17.20 | 7.20 |
If every holding is under five years, the duration shortcut is exact for a single rating, because the first band is linear. The rating shortcut is never exact unless the fund holds a single rating. A fund described as "average BBB" can need 72 per cent more capital than the label implies.
Why it matters to the manager. An insurer that prices the fund on a 6 per cent cost of capital carries €0.28 million a year of capital cost that the factsheet figure hides. When the line-by-line file arrives at the first quarterly report, the fund's return on capital falls, and the manager is blamed for a number it never quoted.
Ask for one row per holding, or at least market value by rating and by duration band, never two averages. The standalone grid is in how the Solvency II spread charge is calculated on a bond, what happens with no look-through at all is in what a missing look-through pack costs, and the Article 176 table as a live formula is in the free workbooks for Insurance Capital for Asset Managers.
It usually overstates it. The cost of each extra year of duration falls in the longer bands, for BBB from 2.5 points a year below five years to 1.5 between five and ten, so a barbell is charged less than a bullet with the same average. Half at 2 years and half at 12 is charged 13.50 per cent; the shortcut at 7 years gives 15.50 per cent.
Not safely. Stresses rise faster than linearly down the rating scale: at four years AAA is charged 3.60 per cent, BBB 10.00 and B 30.00. A fund half AA and half B averages BBB but is charged 17.20 per cent, against 10.00 per cent for the label, 72 per cent more capital.
When every holding has the same rating and a duration below five years, because the first band of the Article 176 grid is a single rate per year of duration: 2.5 per cent for BBB, so 3 and 5 years average to 10.00 per cent either way. Once holdings cross the five-year boundary, or ratings differ, it no longer holds.
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.
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