Constructing a Severity Exposure Curve

Purpose and Definition

A severity exposure curve summarizes how expected limited loss changes as a limit changes. Each value is relative to a stated base limit, so the curve is useful only when the severity model, coverage terms, and loss basis are specified consistently. A ratio without that context is difficult to interpret and may not be suitable for comparing limits.

The construction begins with a severity random variable $X$ and a payment function that states what the insurer pays for a loss at each limit. For ground-up losses with no deductible, the payment per loss at limit $u$ is $Y_u^L=\min(X,u)$. If a deductible applies and $u$ is the insurer’s maximum payment above the deductible, the payment is $Y_u^L=\min((X-d)_+,u)$, where $(z)_+=\max(z,0)$. The limit convention must be stated: a limit measured from ground up requires a different payment function.

Definition: For a specified severity distribution, coverage basis, and payment function $Y_u^L$, let $L(u)=E[Y_u^L]$. The severity exposure curve relative to base limit $u_0$ is $$ C(u;u_0)=\frac{L(u)}{L(u_0)}, \qquad L(u_0)>0. $$ It is a loss-only ratio. At the base limit, $C(u_0;u_0)=1$.

The curve’s values are dimensionless, but the expected losses in the numerator and denominator must be on the same basis. State whether $X$ is ground-up or already adjusted, whether a deductible is applied, how the limit is measured, and the limit range for which values are being reported. If the expected loss at the base limit is zero, the ratio is undefined.

Constructing the Curve From a Severity Model

For a discrete severity distribution, calculate the insurer’s payment at each candidate limit for each possible loss, multiply each payment by its probability, and add the results. Divide each resulting expected limited loss by the expected limited loss at the selected base. The same probabilities and coverage terms are used throughout.

State the basis.
Identify the severity variable, deductible, and whether the policy limit is a cap on insurer payment or a ground-up loss amount.

Calculate limited expected losses.
For each limit in the intended range, apply the stated payment function and take its expectation under the same severity distribution.

Normalize to the base.
Divide every expected limited loss by the expected limited loss at $u_0$, checking that the base value is 1 and the denominator is positive.

Worked Example: A Ground-Up Curve at Three Limits

A hypothetical coverage has ground-up loss $X$, measured in thousands of dollars, with $P(X=10)=0.50$, $P(X=40)=0.30$, and $P(X=100)=0.20$. There is no deductible. Construct curve values at limits of $30,000, $60,000, and $100,000, using $30,000 as the base.

Method and basis. The limit is the maximum payment per loss, and $Y_u^L=\min(X,u)$. The base expected limited loss must be calculated first.

At $u=\$30,000$, the payments, in thousands, are 10, 30, and 30. Thus $L(30)=0.50(10)+0.30(30)+0.20(30)=5+9+6=20$, or $20,000.

At $u=\$60,000$, the payments are 10, 40, and 60. Therefore $L(60)=0.50(10)+0.30(40)+0.20(60)=5+12+12=29$, or $29,000.

At $u=\$100,000$, the payments are 10, 40, and 100. Therefore $L(100)=0.50(10)+0.30(40)+0.20(100)=5+12+20=37$, or $37,000.

Normalize each value by the base expected loss of 20: $C(30;30)=20/20=1.0000$, $C(60;30)=29/20=1.4500$, and $C(100;30)=37/20=1.8500$. The computed limited expected losses rise with the limits, and the base value is exactly 1. The common mistake avoided here is using the full severity $X$ in the numerator instead of the payment capped at each limit.

Why Normalization Works

The curve expresses each limited expected loss in units of the base expected loss. If $L(u)=kL(u_0)$, then by definition the curve value at $u$ is $k$. This provides a scale-free description of relative loss, while retaining the meaning of the chosen base. It does not make two curves comparable if they use different deductibles, severity populations, or loss bases.

C(u;u_0)=\frac{L(u)}{L(u_0)} \quad\Longrightarrow\quad C(u_0;u_0)=\frac{L(u_0)}{L(u_0)}=1

For the discrete model in the first example, the expectation at any limit is obtained from the definition of expectation: $$ L(u)=\sum_j P(X=x_j)\min(x_j,u). $$ Dividing this sum by the corresponding sum at $u_0$ yields the normalized curve. This explains why the probabilities, severity units, and payment convention must be identical in both calculations: changing any of them changes one side of the ratio’s intended comparison.

Worked Example: A Deductible With a Limit Above the Deductible

A hypothetical policy has deductible $d=\$10,000$. The stated limit $u$ is the insurer’s maximum payment above the deductible. Ground-up loss $X$, in thousands of dollars, is 5 with probability 0.50, 25 with probability 0.30, and 70 with probability 0.20. Find the exposure curve at insurer-payment limits of $20,000 and $40,000, using $20,000 as the base.

Method and basis. Apply $Y_u^L=\min((X-d)_+,u)$. At $u=20$, the payments for losses 5, 25, and 70 are respectively 0, 15, and 20. Hence $L(20)=0.50(0)+0.30(15)+0.20(20)=0+4.5+4=8.5$, or $8,500.

At $u=40$, the payments are 0, 15, and 40. Hence $L(40)=0.50(0)+0.30(15)+0.20(40)=0+4.5+8=12.5$, or $12,500.

The normalized values are $C(20;20)=8.5/8.5=1.0000$ and $C(40;20)=12.5/8.5=1.4706$, rounded to four decimal places. The expected losses and their ratio use thousands of dollars consistently.

Common mistake and correction. Treating the $40,000 insurer-payment limit as a $40,000 ground-up cap would produce payments of 0, 15, and 30, because the deductible reduces the covered layer below that ground-up amount. That is not the stated contract. Here the $40,000 cap applies after the deductible, so the largest payment is 40, not 30. If a problem instead defines $U$ as the ground-up policy limit, the payment is $\min((X-d)_+,(U-d)_+)$; the convention must be read before calculating.

State the Range and Keep the Basis Fixed

A curve should be presented with its base and the limits it covers. A statement such as “the curve is 1.45” is incomplete unless it identifies the comparison limit, base limit, severity basis, and coverage terms. For example, the first example’s 1.4500 means that expected limited loss at a $60,000 limit is 1.45 times expected limited loss at a $30,000 limit, under the stated ground-up model and no-deductible basis.

As a consistency check, when the same nonnegative severity distribution and payment basis are used, increasing a per-loss payment limit cannot reduce the payment for any individual loss. Therefore it cannot reduce expected limited loss or the normalized curve value. This check does not authorize mixing bases: a sequence can fail the check if one point uses a different deductible or definition of limit. The selected range should also be explicit; a value at an unexamined limit is not obtained merely by naming the endpoints.

Expenses Are Not Part of a Loss-Only Curve

The exposure curve defined here normalizes expected limited losses, not premiums. Fixed expenses and variable expenses can change the ratio of indicated premiums even when the loss-only curve is unchanged. This distinction matters whenever the question asks for a premium comparison rather than a relative loss measure.

For illustration, suppose indicated premium $P$ is set to cover expected loss $L$, fixed expense $F$, and variable expense equal to fraction $v$ of premium, with no other provision. The balance equation is $P=L+F+vP$, so $P=(L+F)/(1-v)$, provided $v<1$. If $F$ and $v$ are the same at two limits, they still need not cancel from the premium ratio because the fixed expense is added to each expected loss.

Worked Example: Loss-Only Curve Versus Premium Ratio

At two limits, a hypothetical policy has expected limited losses of $18,000 and $27,000; the $18,000 value is at the stated base limit. At both limits, fixed expense is $3,000 per policy and variable expense is 25% of premium. Find the loss-only curve value at the higher limit and the ratio of indicated premiums, higher limit to base.

Method. First construct the loss-only ratio. Then solve the premium balance at each limit using $P=L+F+vP$, with $F=\$3,000$ and $v=0.25$.

The loss-only curve value is $27,000/18,000=1.5000$. For the base limit, $P_0=(18,000+3,000)/(1-0.25)=21,000/0.75=\$28,000$. For the higher limit, $P_1=(27,000+3,000)/(1-0.25)=30,000/0.75=\$40,000$. Thus the indicated premium ratio is $40,000/28,000=1.4286$, rounded to four decimal places.

The calculations check because $28,000=18,000+3,000+0.25(28,000)$, and $40,000=27,000+3,000+0.25(40,000)$. The common mistake avoided is calling the premium ratio 1.5000: that is the loss-only curve value, not the premium ratio when the same fixed expense is added at each limit.

Common Construction Checks

Check the base. The curve at $u_0$ must equal 1, and $L(u_0)$ must be positive.
Check the payment definition. Distinguish a cap on insurer payment from a ground-up policy limit, particularly when a deductible applies.
Check the basis. Numerator and denominator must use the same severity distribution, coverage terms, and units.
Check the requested measure. A loss-only curve excludes expenses; a premium ratio requires the applicable premium assumptions.
Check the range. Name the base and the limits represented. Do not imply that a stated set of values establishes the curve outside that range.

Constructing the curve carefully makes its interpretation transparent: each value answers a specific relative-loss question under a defined set of terms. Keeping the severity basis and limit convention explicit is essential whether the curve is used for analysis or as an input to later limit comparisons. This treatment is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Check Your Understanding

For each problem, state or use the specified basis before forming the ratio. Money values in thousands are identified as such.

A no-deductible ground-up severity is 10 with probability 0.40 and 50 with probability 0.60, in thousands of dollars. Calculate the curve value at limit 50 using limit 20 as the base. (Answer: $2.1250$)
A policy has deductible $d=5$, and its stated limit is the maximum insurer payment above the deductible. Losses, in thousands, are 2 with probability 0.25, 12 with probability 0.50, and 40 with probability 0.25. Calculate the curve at limit 30 using limit 10 as the base. (Answer: $1.8333$)
On one consistent basis, expected limited losses are $24,000 at the base limit and $36,000 at a higher limit. What is the higher-limit loss-only curve value? (Answer: $1.5000$)
Expected limited losses at the base and higher limits are $16,000 and $28,000. Fixed expense is $2,000 at both limits, and variable expense is 20% of premium at both limits. Find the loss-only curve value and the higher-to-base indicated premium ratio. (Answer: Loss-only curve $=1.7500$; premium ratio $=1.6667$)
A discrete no-deductible severity has losses 20 and 80, in thousands, with probabilities 0.60 and 0.40. Using limit 40 as the base, calculate the expected limited losses at limits 40 and 80, then the curve value at 80. (Answer: $L(40)=28$ thousand, $L(80)=44$ thousand; curve value $=1.5714$)

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