What you are actually buying when you lower a retention
An insurance premium is expected loss plus expenses plus a profit provision, and the expenses and profit do not shrink in proportion to the layer being insured. Buying the first $250,000 of every claim therefore costs meaningfully more than the losses in that layer, because that layer is where the claims are and the loading is applied to all of them. That is the whole case for a higher retention.
The case against it is that the loss you retain is not the average loss. It is a random variable with a long right tail, and the year in which it is unusually large is exactly the year the business can least afford it. A retention decision is therefore a trade between a known reduction in expected cost and an unknown increase in volatility, and any analysis that reports only the expected saving has answered half the question.
This calculator reports both. The expected side is λ × E[min(X, d)] — frequency multiplied by the average amount of a claim that falls inside the retention. The volatility side is the same aggregate distribution taken at a percentile you choose. And the collateral cost, which buyers routinely forget, is the annual charge on the letter of credit or trust the insurer will require to secure losses you have promised to pay but have not yet paid.
The same arithmetic drives large-deductible workers compensation programmes, where the retention decision sits alongside the experience modifier and the manual premium calculation that experience rating multiplies.
The limited expected value, and why it is not the average claim
The quantity that matters is E[min(X, d)]: the average amount of a claim that falls at or below the retention d. Actuaries call it the limited expected value, and it is not the average claim, not the average of the claims below d, and not d times the probability of exceeding d. Every claim contributes something — small ones contribute their whole size, large ones contribute exactly d.
For a lognormal severity with unlimited mean μX and coefficient of variation CV, set σ² = ln(1 + CV²) and μ = ln(μX) − σ²/2. Then the limited expected value has a closed form, given in the formula block above: the first term is the contribution of claims below the retention and the second is d multiplied by the probability of exceeding it.
The behaviour is the important part. At a retention of zero the value is zero. As the retention rises without limit it approaches the unlimited mean and stops, because you cannot retain more than the whole claim. In between it rises steeply and then flattens, and where it flattens is where extra retention stops buying much premium credit. With a mean of $45,000 and a CV of 2, E[min] is $16,931 at a $25,000 retention, $32,728 at $100,000, $40,233 at $250,000 and $44,458 at $1,000,000 — so quadrupling the retention from $250,000 to $1,000,000 captures only another $4,225 per claim, against the $23,302 captured by the earlier tenfold step from $25,000 to $250,000.
For the bad year, treat aggregate retained loss as compound Poisson: mean λ·E[min(X,d)] and variance λ·E[min(X,d)²]. The second moment has the same closed form with 2σ in place of σ and d² in place of d. A normal approximation then gives the percentile. That approximation is adequate at moderate claim counts and understates the tail badly when there are only a handful of claims a year, which is why the calculator warns below five.
Worked example: moving from a $25,000 to a $250,000 retention
A company expects 8 claims a year with a mean severity of $45,000 and a coefficient of variation of 2. Its current programme carries a $25,000 retention at a premium of $620,000; the alternative is a $250,000 retention at $310,000. Collateral costs 3% a year and the bad year is measured at the 90th percentile.
- Fit the severity. σ² = ln(1 + 4) = 1.609438, so σ = 1.268636 and μ = ln(45,000) − 0.804719 = 9.909699.
- Limited expected values. E[min(X, 25,000)] = $16,930.97; E[min(X, 250,000)] = $40,233.30.
- Expected retained loss. 8 × $16,930.97 = $135,447.78 at the low retention; 8 × $40,233.30 = $321,866.38 at the high one.
- Collateral. 3% of each: $4,063 and $9,656.
- Total expected cost. $620,000 + $135,448 + $4,063 = $759,511 at the low retention; $310,000 + $321,866 + $9,656 = $641,522 at the high one.
- Expected advantage. $759,511 − $641,522 = $117,989 a year in favour of the higher retention.
Check where that came from, because it is not obvious. The premium credit is $620,000 − $310,000 = $310,000. The extra expected retained loss is $321,866 − $135,448 = $186,419. The extra collateral cost is $5,593. So $310,000 − $186,419 − $5,593 = $117,989 — the credit exceeds the extra expected loss by 66%, which tells you the insurer was charging a great deal more than expected loss for that layer.
Now the other side. At the 90th percentile, aggregate retained loss is $204,322 at the low retention and $562,672 at the high one, so total bad-year cost is $828,385 against $882,328. The higher retention costs $53,942 more in a bad year. That is the trade in one line: save $117,989 in a normal year, pay $53,942 more one year in ten. Whether that is a good deal depends on whether the business can absorb $882,328 without disruption, which is a balance-sheet question rather than an actuarial one.
How to read the comparison
Start with the expected cost advantage. Positive means the higher retention is cheaper in an average year; negative means the premium credit does not cover the extra expected loss and the higher retention should be rejected on cost grounds alone. Insurers do sometimes quote a credit smaller than the expected loss transferred, particularly at renewal on a deteriorating account, and the arithmetic here catches it.
Then look at the bad-year figures, and read them as an affordability test rather than as a cost. The relevant question is not whether the bad-year cost is larger — at a higher retention it will be — but whether the business can pay it without breaching a covenant, cancelling capital expenditure, or explaining an earnings miss. If it cannot, the expected saving is irrelevant.
The retention table shows where the curve flattens. Beyond that point additional retention buys progressively less premium credit while adding volatility at the same rate, and the marginal deal gets worse. That flattening point is a property of your severity distribution and it is the single most useful output for structuring a programme.
Two things routinely omitted from this comparison deserve a place in the decision. Retained claims still have to be adjusted, and a third-party administrator charges for it — usually per claim, which makes frequency more expensive than this model shows. And retained losses are paid over years while the premium credit arrives at once, so the credit is worth more than the nominal comparison suggests. Both effects favour the higher retention; the collateral cost, which is included here, runs the other way.
Expected and bad-year retained loss by retention level
| Retention | E[min(claim, retention)] | Expected annual retained | Share of unlimited mean |
|---|---|---|---|
| $25,000 | $16,931 | $135,448 | 37.6% |
| $50,000 | $24,911 | $199,290 | 55.4% |
| $100,000 | $32,728 | $261,826 | 72.7% |
| $250,000 | $40,233 | $321,866 | 89.4% |
| $500,000 | $43,190 | $345,520 | 96.0% |
| $1,000,000 | $44,458 | $355,665 | 98.8% |
The last column is E[min(X,d)] ÷ $45,000, the share of the ground-up loss cost that the retention captures. It reaches 89% at a $250,000 retention and only 99% at $1,000,000 — which is why excess layers above a few hundred thousand dollars are comparatively cheap.
Collateral is the cost buyers forget, and it compounds
When you retain losses, the insurer fronting the programme carries the statutory obligation to pay claims and will require security — normally a letter of credit, sometimes a trust or cash. That security is sized to your outstanding retained liabilities, not to one year of them, so it grows for several years as unpaid claims from successive years accumulate before the oldest years run off. A programme in its fifth year can be collateralising three or four years of retained losses at once. This calculator applies the collateral rate to a single year of expected retained loss, which understates the mature-state cost. Multiply it by the number of years of retained liabilities you expect to be carrying at steady state, and remember that a letter of credit usually reduces your borrowing capacity as well as costing a fee.
What this model leaves out
- Claims handling cost. A third-party administrator charges per claim or as a percentage of paid losses. Because that cost scales with frequency, it makes a high-frequency programme less attractive at a high retention than this comparison shows.
- The time value of retained losses. The premium credit is received at inception; retained losses are paid over years. Discounting the payout pattern makes the higher retention look better than the undiscounted comparison here.
- Aggregate stop-loss. Many retention programmes are bought with an annual aggregate cap, which truncates the bad-year figure. If yours has one, the bad-year cost cannot exceed premium plus the aggregate.
- The lognormal assumption. Real severity distributions may be heavier-tailed than lognormal. Where the tail matters — liability with high limits, for example — fit the distribution rather than assuming it.
- The normal approximation in the tail. Aggregate loss is right-skewed, so a normal approximation understates upper percentiles. The effect is small at high claim counts and large below about five claims a year.
- Frequency uncertainty. Frequency is treated as a known Poisson mean. Parameter uncertainty in that mean adds variance the calculation does not capture, which again understates the bad year.
Retentions, deductibles and captives
The three structures differ in mechanics rather than in economics. Under a deductible programme the insurer pays the claim from the first dollar and bills you back for the deductible portion, which keeps the insurer in control of claim handling and keeps the policy limits measured from ground up. Under a self-insured retention you pay the layer directly and the insurer's obligation begins above it, which gives you control of the claims but also the duty to defend within the retention. A captive is the same retention decision with an insurance company you own sitting in the middle, which changes the tax and accounting treatment and adds real fixed costs — capitalisation, fronting fees, actuarial and audit work — that must be recovered before the arrangement is worth having.
Whichever structure you choose, the analysis is the one on this page: expected retained loss, plus the frictional costs of retaining it, against the premium credit for doing so. What changes between structures is the size of the frictional costs, not the shape of the trade.
Two inputs deserve more care than the rest. Frequency and severity should come from your own developed loss history, not from raw reported figures — use the IBNR reserve calculator to develop them to ultimate first, since undeveloped losses systematically understate the expected retained loss and make every retention look better than it is. And where your own history is thin, blend it against a class benchmark with the credibility weighting calculator rather than treating a handful of years as fact. The premium side, meanwhile, is a market quote: get real numbers at each retention rather than assuming a credit, which is why this calculator asks for two quoted premiums instead of modelling one. The structure of those quotes is worth understanding through the pure premium calculator.