Insurer asset-liability management

RiskAtoms Insurance SE owes €50.9bn to policyholders, some of it to annuitants who will be paid for another four decades. Its €54bn of bonds have almost the same duration as those promises: own funds move just €0.69m per basis point. Yet if rates fell 100bp, the solvency ratio would drop from 186% to 165%. Matching duration is where insurer ALM starts, not where it ends.

Asset-liability managementAsset-liability management: choosing and managing assets so that they meet the liabilities as they fall due, and so that the balance sheet holds its value when markets move. in a bank is about deposits that can leave and loans that reprice (L3.6.04: Net interest income sensitivity, L3.6.05: Economic value of equity sensitivity). In an insurer it starts from the liabilities: promises whose timing is fairly well known, often for decades, with options attached. The assets are chosen to meet them.

The cash flows

The bonds pay €63,205m in all, bunched in the first five years, and the last one matures in 2045. The liabilities pay €62,225m, running to 2069. The spikes in 2033 and 2037 are savings contracts maturing (€7,410m and €5,754m). €3,739m of payments (present value €1,845m) fall after the last bond has repaid: they will be met from money reinvested at rates no one knows today. That is reinvestment riskThe risk that liability payments due after the assets mature must be met from money reinvested at rates nobody knows today., and it is the reason cash-flow matchingCash-flow matching: buying assets whose payments arrive when, and in the amounts that, the liabilities fall due. is the gold standard for annuities. Perfect matching is rarely possible: very long bonds are limited in supply, and the liability flows are themselves uncertain.

Duration and PV01, book by book

The usual compromise is to match value sensitivity rather than each cash flow. Revalue every book on the curve moved up and down a little: that gives an effective durationDuration measured by revaluing on the curve moved up and down a little: the percentage change in value per unit change in rates. It works for cash flows that themselves depend on rates. that works even for the savings book, whose cash flows themselves depend on rates.

Annuities are the long book (9.24 years). Savings are short (3.25 years) because policies mature or are surrendered, but they are as convex as the annuities: that is the guarantee. Term assurance has a negative best estimate (future premiums exceed claims), so its value rises when rates rise and its duration is negative. On the whole balance sheet, the bonds' duration is 5.03 years and the liabilities' 5.48 years; the PV01sThe change in value for a one basis point rise in rates. Here measured on the whole curve, as value lost. are €27.19m and €27.88m per basis point, and the duration gapDuration gap: D_A − (L/A)·D_L. Zero means own funds do not change, to first order, when rates move in parallel. is −0.13 years. By the measure of L3.6.05: Economic value of equity sensitivity, that is close to matched.

Matched, and still exposed

Own funds fall whichever way rates move: −€447m at −100bp and −€158m at +100bp. The liabilities are 2.9 times as convexThe curvature of value in the rate: how much the duration itself changes as rates move. as the bonds (124.6 against 43.5). Long annuities explain part of it. The rest is the savings guaranteesA minimum crediting rate promised to savers. When rates fall below it, the insurer pays the difference: an option it has sold. (L3.8.08: Lapse and policyholder behaviour): when rates fall, more cohorts' guarantees bite and the liability grows faster than any bond; when rates rise, the crediting rate follows them up and the liability barely falls. The insurer has sold options, and sold options lose money in big moves either way. The capital requirement moves too: at −100bp the SCR rises by €353m, so the ratio falls to 165%.

The regulatory shocks

The standard formula does not use parallel shifts. It moves each zero rate by a relative amount that falls with maturity (Delegated Regulation 2015/35, Arts 166–167), with an up move of at least one percentage point and no down move on negative rates:

Both shocks are of similar size in basis points on this curve, and largest at the short end where the bonds are. Bonds lose €3,084m, best estimates fall €2,691m, so own funds lose €393m. The down shock costs only €87m. The interest rate charge is the larger, €393m, with the up scenario binding: the input to the market risk module in L3.8.12: Insurer investment risk and the ORSA.

Rebalancing: which target?

Suppose the investment team moves money from the 2-year government bonds into the 20-year ones. About €600m closes the PV01 gap (+€0.06m per bp), yet it raises the interest charge to €458m, because the up shock binds. Going the other way, €1,500m from 20-year into 2-year bonds, gives the smallest charge, €237m, with the up and down losses balanced (€231m and €237m), but own funds would then lose €657m if rates fell 100bp. The target depends on what the board wants to protect: the regulatory charge under the prescribed shocks, own funds in parallel moves, or the cash to pay annuities in thirty years. The Try it lets you explore the trade.

The matching adjustment, in outline

An insurer that holds bonds to maturity against predictable annuities is not exposed to their price swings in the way a trader is: it is exposed to default. The matching adjustmentMatching adjustment: an addition to the risk-free rate used to discount predictable liabilities backed by a ring-fenced portfolio of matching assets, equal to the assets' spread less a fundamental spread for default and downgrade. ( Directive 2009/138/EC, Arts 77b–77c) recognises this. With supervisory approval, it adds to the risk-free rate the assets' spread less a fundamental spreadFundamental spread: the part of a bond's spread that pays for expected default and downgrade losses. It stays with the insurer and is not part of the matching adjustment. for default and downgrade. The conditions are strict: an assigned portfolio of fixed-cash-flow assets that replicates the liabilities, managed and ring-fenced separately, no future premiums, only longevity, expense, revision and mortality risk, and no policyholder options beyond a capped surrender value. A firm that breaks the conditions and cannot restore them within two months loses the adjustment for 24 months.

On the book, a spread of 1.20% less a fundamental spread of 0.45% gives 75bp. Discounting the annuities at the risk-free rate plus 75bp cuts their best estimate from €19,928m to €18,625m, €1,304m more own funds before tax. RiskAtoms does not use it in the base: its annuities are not backed by a ring-fenced matched portfolio, and its bond portfolio is far shorter than the annuities.

A bank's ALM and an insurer's

On the book. Durations and convexities measured with ±10bp moves (M-B03: Taylor expansion), A = €54,000m, L = €50,922m:

The liabilities' duration here, 5.51, is the average of the slopes up and down; measured on a rise alone it is 5.48. The difference is the guarantee again: the liability is steeper on the way down. That is also why the Taylor estimate gets only part of the −100bp loss: convexity measured at today's rates understates how fast the options gain value as rates keep falling. Second order is a good guide for small moves and for bonds; for books with options, revalue.

A run-off as a geometric series. If a book pays $C_1$ next year and then a fixed fraction $q$ of the year before, its value on a flat rate $y$, with $v = 1/(1+y)$, is a geometric series (M-A07: Geometric series), and so is its duration:

using $\sum_{t\ge1} t\,x^{t-1} = 1/(1-x)^2$, the derivative of the geometric series. Choose $q$ so the series pays the same total as the book, $q = 1 - C_1/\sum C_t$:

For the short-tail non-life claims, which settle by a roughly constant fraction each year, the shortcut is within 0.1% on value and 0.03 years on duration. For annuities it fails: deaths accelerate with age, so payments shrink slowly at first and then fast, and a geometric tail puts too much weight far out (11.3 years against 9.6). That is why insurers project annuity cash flows life by life.

Move money between the 2-year and the 20-year government bonds. Watch the PV01 of own funds go through zero, the binding shock switch from up to down, and the own-funds curve tilt. Can you find a position that is good on all three?

All figures from the book, € millions, before tax.

21 points of ratio from a 100bp move, with a duration gap of −0.13 years. The fall in own funds is a convexity loss; the rise in the SCR is the guarantees and longevity costing more at lower rates.

Solvency II values liabilities on a risk-free curve published by EIOPA, which can include a volatility adjustment, or, with approval, a matching adjustment (Directive 2009/138/EC, Arts 77b–77d). A firm using either must assess in its ORSA whether it meets its capital requirements with and without it (Art 45(2a)). The interest rate charge is the worse of the up and down shocks of Delegated Regulation 2015/35, Arts 166–167. The 2025 review, applying from 30 January 2027 (Directive (EU) 2025/2 and Delegated Regulation (EU) 2026/269), recalibrates the interest rate shocks so that they also apply when rates are negative, limited by a floor that rises with maturity. The detail is in M4.6: Solvency II and insurance regulation; banks' rate risk rules are in M4.5: Liquidity and IRRBB regulation.

• Insurer ALM starts from long, fairly predictable liabilities: RiskAtoms' run to 2069, its bonds to 2045, leaving €3,739m of payments to reinvestment.
• Effective duration and PV01 by book: annuities 9.2 years, savings 3.3, term −1.5, non-life 2.3; overall 5.03 against 5.48, a gap of −0.13.
• ΔOF ≈ −(D_A A − D_L L)Δ + ½(C_A A − C_L L)Δ²: with the gap closed, more convex liabilities (annuities and sold guarantees) make own funds fall both ways, −€447m at −100bp and −€158m at +100bp.
• The Solvency II shocks are relative and not parallel; on this curve the up shock binds with €393m, and matching PV01 does not minimise it.
• A run-off shrinking by q a year is a geometric series: PV = C₁v/(1 − qv), D = 1/(1 − qv). Good for short-tail claims, poor for annuities.
• The matching adjustment discounts matched, ring-fenced annuities at risk-free plus spread less fundamental spread: 75bp would add €1,304m to own funds here.

Asset-liability management (ALM)

Asset-liability management: choosing and managing assets so that they meet the liabilities as they fall due, and so that the balance sheet holds its value when markets move.

Cash-flow matching

Cash-flow matching: buying assets whose payments arrive when, and in the amounts that, the liabilities fall due.

Reinvestment risk

The risk that liability payments due after the assets mature must be met from money reinvested at rates nobody knows today.

PV01

The change in value for a one basis point rise in rates. Here measured on the whole curve, as value lost.

Effective duration

Duration measured by revaluing on the curve moved up and down a little: the percentage change in value per unit change in rates. It works for cash flows that themselves depend on rates.

Duration gap

Duration gap: D_A − (L/A)·D_L. Zero means own funds do not change, to first order, when rates move in parallel.

Convexity

The curvature of value in the rate: how much the duration itself changes as rates move.

Guaranteed crediting rate

A minimum crediting rate promised to savers. When rates fall below it, the insurer pays the difference: an option it has sold.

Matching adjustment

Matching adjustment: an addition to the risk-free rate used to discount predictable liabilities backed by a ring-fenced portfolio of matching assets, equal to the assets' spread less a fundamental spread for default and downgrade.

Fundamental spread

Fundamental spread: the part of a bond's spread that pays for expected default and downgrade losses. It stays with the insurer and is not part of the matching adjustment.

Liability-driven investment (LDI)

Liability-driven investment: a pension scheme strategy that hedges the rate sensitivity of the liabilities, often with leverage through swaps and repos that need collateral.