Exposure Rating From Ground-Up Loss Costs

From Ground-Up Loss Costs to Layer Loss

Exposure rating estimates an excess layer’s expected loss by applying a severity curve to an estimate of the portfolio’s ground-up losses. The calculation has two distinct parts: first estimate expected ground-up losses from the loss cost and the subject portfolio’s exposure; then apply the supplied curve’s layer share. Getting the exposure base right is as important as multiplying by the correct share.

For example, a ground-up loss cost stated per vehicle-year cannot be multiplied directly by payroll, and a share of ground-up expected losses is not the probability that a claim reaches the attachment. The share represents the expected fraction of ground-up loss dollars that the specified layer pays, under the curve and assumptions supplied.

Definition: For a per-occurrence layer attaching at $a$ with limit $L$, the payment on a ground-up loss $X$ is $Y=\min((X-a)_+,L)$, where $(z)_+=\max(z,0)$. The layer share is $q=E[Y]/E[X]$, when $E[X]$ is finite and positive. If expected ground-up losses for the subject portfolio are $G$, the exposure-rated expected layer loss is $qG$, provided the curve’s share and $G$ are on compatible bases.

The limit $L$ caps the amount paid by the layer on one loss. The layer is often described as “$L$ excess of $a$,” or “$L$ xs $a$.” The supplied layer share may already reflect the attachment, limit, and assumed severity distribution. Use the share for the specified layer rather than trying to infer it from the attachment alone.

Establish the Ground-Up Loss Total

A loss cost is expected loss per unit of exposure. If the ground-up loss cost is $c$ dollars per exposure unit and the subject portfolio has $E$ such units, its expected ground-up losses are $G=cE$. The exposure unit might be a vehicle-year, payroll amount, insured value, or another defined basis. Use the stated unit exactly.

The ground-up loss cost should represent losses before the layer’s attachment and limit are applied. It should also be prospective for the period being rated and consistent with the supplied curve’s loss definition, currency, and portfolio mix. If the data are already limited, excess-only, or otherwise adjusted, they are not automatically a ground-up basis.

G=cE,\qquad \widehat{L}_{\text{layer}}=qG=qcE

If a portfolio has multiple classes with different loss costs or severity curves, calculate expected ground-up losses and layer losses by class, then add them. A single portfolio-wide share is appropriate only when it represents the portfolio’s applicable severity mix and layer.

\widehat{L}_{\text{layer,total}}=\sum_{j=1}^{k}q_jc_jE_j

Why Multiplying by the Layer Share Works

Suppose the portfolio has $N$ ground-up claims with severities $X_1,\ldots,X_N$, and each is subject to the same per-occurrence layer. The total ground-up loss is the sum of the severities, while the total layer payment is the sum of the individual layer payments. By linearity of expectation, expected totals are the expected number of claims multiplied by the corresponding mean amount per claim, under the usual frequency-severity setup.

E[\text{ground-up total}]=E[N]E[X],\qquad E[\text{layer total}]=E[N]E[Y]

Taking the ratio cancels the expected claim count. Since $q=E[Y]/E[X]$, the expected layer total is $q$ times the expected ground-up total. This explains why a severity-curve share can be applied directly to a ground-up loss cost total. It is an expectation identity, not a claim that every individual loss or every year’s layer loss equals that fraction of ground-up losses.

This argument assumes that the severity mean used in the curve is appropriate for the subject losses. In a frequency-severity model where severity depends on frequency, the same cancellation is valid when the curve share correctly represents the claim-weighted severity relationship for the subject portfolio. A curve for a materially different mix may not be appropriate merely because its layer boundaries match.

Read the loss-cost basis.
Record the dollar amount per exposure unit and confirm it is ground-up and prospective for the rating period.

Measure the subject exposure.
Express the portfolio exposure in the same units used in the loss cost. Convert units before multiplying if needed.

Obtain the matching layer share.
Confirm the share corresponds to the specified attachment and limit and to a suitable severity curve.

Calculate and check.
Multiply ground-up expected losses by the share. The result should be nonnegative and no greater than ground-up expected losses when the share is between zero and one.

Worked Applications

Worked Example: Vehicle-Year Loss Cost

A hypothetical insurer expects ground-up losses of $425 per vehicle-year for a book of 2,400 vehicle-years. For a layer of $500,000 excess of $500,000, the supplied severity curve gives a layer share of 7.5%. Calculate the expected layer loss.

Method. Convert the ground-up loss cost to a total expected ground-up loss, then apply the supplied layer share. The exposure unit is one vehicle-year, so the stated portfolio has 2,400 units.

Step 1: Ground-up expected losses.

G=\$425\text{ per vehicle-year}\times 2{,}400\text{ vehicle-years} =\$1{,}020{,}000

Step 2: Apply the share. The share is $7.5\%=0.075$.

\widehat{L}_{\text{layer}}=0.075\times \$1{,}020{,}000=\$76{,}500

Check. The layer estimate is positive and below the $1,020,000 ground-up total. The calculation avoids treating 7.5% as the probability that a claim penetrates the attachment; it is the expected layer payment divided by expected ground-up loss under the supplied curve.

Worked Example: Two Classes With Different Curves

A hypothetical equipment portfolio has two classes. Class A has 800 equipment-years and a ground-up loss cost of $310 per equipment-year. Class B has 500 equipment-years and a ground-up loss cost of $520 per equipment-year. For the same $500,000 excess of $500,000 layer, the supplied curve shares are 8% for class A and 13% for class B. Find the portfolio’s expected layer loss.

Method. Calculate each class separately because both its loss cost and its applicable severity share differ.

Class A ground-up losses:

G_A=800\times \$310=\$248{,}000

Class A layer losses: $q_A=0.08$, so $0.08\times \$248{,}000=\$19{,}840$.

Class B ground-up losses:

G_B=500\times \$520=\$260{,}000

Class B layer losses: $q_B=0.13$, so $0.13\times \$260{,}000=\$33{,}800$.

Total: Add the class indications.

G_{\text{total}}=\$248{,}000+\$260{,}000=\$508{,}000

\widehat{L}_{\text{layer,total}}=\$19{,}840+\$33{,}800=\$53{,}640

Check. The weighted portfolio share implied by these class results is $\$53{,}640/\$508{,}000\approx 10.56\%$. It lies between the two class shares, as expected for a weighted average. Applying either class’s share to the entire portfolio would ignore the stated class differences.

Worked Example: Converting a Payroll Exposure Basis

A hypothetical commercial portfolio has $4,800,000 of payroll. Its prospective ground-up loss cost is $6.25 per $100 of payroll. For a $1,000,000 excess of $2,000,000 layer, the supplied curve gives a share of 18%. Calculate the expected layer loss.

Method. Convert payroll to the same $100 units used by the loss cost before calculating ground-up losses.

Step 1: Convert the exposure.

E=\frac{\$4{,}800{,}000}{\$100}=48{,}000\text{ units of \$100 payroll}

Step 2: Calculate ground-up expected losses.

G=48{,}000\times \$6.25=\$300{,}000

Step 3: Apply the layer share.

\widehat{L}_{\text{layer}}=0.18\times \$300{,}000=\$54{,}000

Check. The units cancel correctly: payroll units multiplied by dollars per payroll unit give dollars. Multiplying $4,800,000 directly by $6.25 would be wrong because $6.25 is charged per $100, not per dollar of payroll.

Common Mistakes and Scope Checks

The arithmetic is usually short; the difficult work is deciding whether its inputs belong together. A share is specific to a layer and a severity basis. The ground-up loss cost is specific to an exposure unit and loss basis. A correct multiplication cannot repair a mismatch between them.

Using the wrong exposure unit. If the rate is per $100 of payroll, divide payroll by 100 before multiplying. Do not treat the rate as dollars per dollar of payroll.
Applying the share to exposure directly. The share applies to expected ground-up loss dollars, not to the number of policies, payroll, or insured values by themselves.
Confusing share with probability. The share is an expected-dollar ratio. It is not the probability a claim exceeds the attachment and is not generally equal to that probability.
Using an unmatched curve. A layer share from a different attachment, limit, or severity mix may not represent the subject layer. Confirm the supplied curve’s intended basis.
Using losses that are not ground-up. Do not label a limited or excess-only loss cost as ground-up without an appropriate basis adjustment. The formula requires compatible ground-up expected losses.

A frequent wrong calculation is to multiply the exposure amount by the layer share and then by the loss cost without converting the exposure unit. In the payroll example, this would be $0.18\times \$4{,}800{,}000\times \$6.25$, producing an absurdly large result because it treats a per-$100 rate as a per-dollar rate. The correction is to calculate $48{,}000$ units of $100 payroll first, obtain $300,000 of ground-up expected losses, and then apply 18%, producing $54,000.

Key takeaway: Exposure rating from ground-up loss costs follows a disciplined sequence: align the loss and exposure bases, calculate the subject portfolio’s ground-up expected loss total, and multiply that total by the applicable layer share. Segment the calculation when the relevant costs or shares differ by class.

Check Your Understanding

For each problem, use the supplied prospective ground-up loss cost and layer share. Assume the share is appropriate for the stated portfolio and layer.

A hypothetical fleet has 950 vehicle-years at a ground-up loss cost of $180 per vehicle-year. The applicable layer share is 6%. What is the expected layer loss? (Answer: $10,260)
A hypothetical portfolio has $2,400,000 payroll and a ground-up loss cost of $8.40 per $100 payroll. The layer share is 11%. Calculate expected layer loss. (Answer: $22,176)
A hypothetical book has 600 units in class A at $240 ground-up loss cost per unit and 900 units in class B at $360 per unit. Their layer shares are 7% and 10%, respectively. Calculate total expected layer loss. (Answer: $42,480)
A hypothetical subject portfolio has $750,000 expected ground-up losses. Its supplied share for the specified layer is 14%. Calculate expected layer loss and state whether the answer can exceed ground-up losses. (Answer: $105,000; no)
A hypothetical group has 1,200 exposure units and a ground-up loss cost of $275 per unit. A curve gives a 9% share for the specified layer. Calculate the expected layer loss, and identify what the 9% measures. (Answer: $29,700; the expected layer payment as a fraction of expected ground-up losses, not the probability of reaching the attachment)

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