The volatility adjustment worked on an annuity book: the discounting, the release at four levels, the duration shortcut that overstates it, and why it matters to the manager of the assets.
The volatility adjustment is added to the risk-free rate used to discount technical provisions, so what it releases is the difference between the liabilities discounted at the risk-free rate and at the risk-free rate plus the adjustment. On an illustrative 3,200m annuity book with a 12-year duration and a 3.0 per cent risk-free rate, a 50 basis point adjustment releases 180.66m, 5.65 per cent of the provisions. The shortcut of adjustment times duration times liabilities says 192.00m and overstates it by 6.28 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 →
Article 77d of the Solvency II Directive lets insurers add a volatility adjustment (VA) to the relevant risk-free curve when they discount their best estimate liabilities. It exists because an insurer holding bonds to maturity is not forced to sell when spreads widen, so the fall in asset values overstates its loss. EIOPA computes and publishes the VA every month, per currency, from the spreads of a reference portfolio less a risk correction, multiplied by an application ratio, 65 per cent in the original text and increased by the recent review of the Directive. The insurer does not calculate the VA. It calculates what the VA does to its balance sheet, which is the calculation below.
The matching adjustment, its stricter cousin, depends on the insurer's own assigned portfolio; it is worked in how the Solvency II matching adjustment is calculated. An insurer uses one or the other on a given block of liabilities, not both.
| Input | Value |
|---|---|
| Best estimate liabilities at the risk-free rate | 3,200 |
| Duration of the liabilities | 12 years |
| Risk-free rate, flat | 3.0% |
| Volatility adjustment tested | 10 to 50 bp |
To keep the arithmetic visible, the liabilities are treated as a single payment at their 12-year duration, as in the book's workbook. A real annuity book is a run-off profile discounted on a full curve; the shape of the result is the same.
Liability cash flow = BEL × (1 + r)D
BEL with VA = cash flow / (1 + r + VA)D
Release = BEL × [1 − ((1 + r) / (1 + r + VA))D]
In Excel: =B1*(1-((1+B3)/(1+B3+B4))^B2) with BEL in B1, duration B2, rate B3, VA B4.
Step 1, the cash flow. 3,200m at 3.0 per cent for twelve years implies a payment of 4,562.43m in year twelve.
Step 2, discount it again. At 3.0 per cent plus 50 bp, 4,562.43m / 1.03512 = 3,019.34m.
Step 3, the release. 3,200m less 3,019.34m = 180.66m, 5.65 per cent of the provisions. It goes straight to own funds, before tax and before any change in the risk margin or the SCR.
| VA | Discount rate | Provisions | Release | Share | VA × D × BEL | Overstated |
|---|---|---|---|---|---|---|
| 10 bp | 3.10% | 3,162.95 | 37.05 | 1.16% | 38.40 | 3.65% |
| 20 bp | 3.20% | 3,126.37 | 73.63 | 2.30% | 76.80 | 4.30% |
| 30 bp | 3.30% | 3,090.24 | 109.76 | 3.43% | 115.20 | 4.96% |
| 50 bp | 3.50% | 3,019.34 | 180.66 | 5.65% | 192.00 | 6.28% |
The release grows a little less than in proportion to the adjustment, because discounting is convex. Scaling the 10 bp answer by five gives 185.24m, still 2.54 per cent too high, and the duration shortcut is worse at every level.
| Duration | Release | Share of provisions |
|---|---|---|
| 8 years | 49.28 | 1.54% |
| 12 years | 73.63 | 2.30% |
| 16 years | 97.80 | 3.06% |
| 20 years | 121.77 | 3.81% |
Long liabilities gain most. That is why annuity writers and long-tail life books care about the VA, and why a short general insurance book barely notices it.
The VA moves with spreads. When credit spreads widen, the insurer's bond portfolio falls and the published VA usually rises, releasing provisions at the same reporting date. In an illustrative case, a 50 bp widening on 3,000m of bonds with a 6.0-year spread duration costs about 90.0m. If the VA rises from 10 to 30 bp over the same month, provisions fall by 72.71m, offsetting 80.8 per cent of the loss and leaving 17.29m. The offset depends on how closely the insurer's portfolio resembles EIOPA's reference portfolio: a manager whose credit book has much wider spreads, or a different currency mix, gives the insurer a loss the VA will not cover. That basis risk, not the headline release, is the conversation to have with an insurer's investment team. Its interaction with the rate shock is covered in the Solvency II interest rate risk charge.
Using VA × duration × liabilities, the figure many presentations print, and then adding it to the insurer's ratio without asking whether the VA is permitted for that business, in that currency, and whether it is already in the reported base. The shortcut overstates the release by 6.28 per cent at 50 bp on this book; double counting a VA already in the base overstates it by all of it.
Treat the VA as an input published by EIOPA, not as an assumption. A pitch that relies on the VA staying at its current level is a pitch on credit spreads.
The release at 10, 20, 30 and 50 bp, beside the matching adjustment on a private credit sleeve, is computed in the free workbooks for the book.
EIOPA, not the insurer. It is published each month for each currency, from the spreads of a reference portfolio less a risk correction, multiplied by an application ratio, 65 per cent in the original text. The insurer only chooses to apply it, where permitted, and discounts its liabilities at the risk-free rate plus the published figure.
Approximately, and always on the high side, because the true effect is convex. On 3,200m of liabilities with a 12-year duration, 50 bp times 12 years gives 192.00m against a discounted release of 180.66m, 6.28 per cent too much. Scaling a 10 bp result of 37.05m by five gives 185.24m, still 2.54 per cent high.
Because it moves with credit spreads and offsets part of the fall in the insurer's bond portfolio. In this illustration a 50 bp widening costs about 90.0m on 3,000m of bonds with a 6.0-year spread duration; if the VA rises from 10 to 30 bp, provisions fall by 72.71m, 80.8 per cent of the loss.
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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