2.4.2. The Exposure Clause

The exposure clause has two parts: exposures and an optional layers sub-clause described in The Limits Sub-Clause. It specifies the volume of insurance. There are five forms:

  1. Expected loss

  2. Premium and loss ratio

  3. Exposure and rate

  4. Claim count

  5. Using the dfreq keyword to enter the frequency distribution directly

Examples:

1000 loss
1000 premium at 0.7 lr
5 exposure at 2000 rate
123 claims
dfreq [1 2 3] [3/4 3/16 1/16]

  • 1000 loss directly specifies expected loss. The claim count is derived from average severity. It is typical for an actuary to estimate the loss pick and select a severity curve, and then derive frequency.

  • 1000 premium at 0.7 lr directly specifies premium and a loss ratio. Expected losses equal the product. The claim count is again derived from severity. Actuaries often take plan premiums and apply loss ratio picks to determine losses, rather than starting with a loss pick. This idiom supports that approach.

  • 5 exposure at 2000 rate directly specifies exposure and a loss rate. It is analogous to the loss ratio form. Actuaries often know exposure and unit rates (per vehicle, per 100 insured value, per location). This idiom supports that approach.

  • 123 claims directly specifies the expected claim count; the last letter s on claims is optional, allowing 1 claim. Expected losses equal claim count times average severity.

  • dfreq [1 2 3] [3/4 3/16 1/16] specifies frequency outcomes and probabilities directly. It is described in Non-Parametric Frequency Distributions.

All values in the first three specifications can be vectorized, see Vectorization: Limit Profiles and Mixed Severity.

2.4.2.1. Determining Expected Claim Count

Variables are used in the following order to determine overall expected losses.

  • If count is given it is used and loss is derived from severity.

  • Else if loss is given, then count is derived from the severity.

  • Else if either pp premium at xx lr or ee exposure at rr rate is given, then the loss is derived by multiplication and counts from severity.

  • In all cases, if premium is given the loss ratio is computed

These choices present no ambiguity when using DecL. But the input arguments could conflict if you create the object directly.

By default, claim count is conditional on a loss to the layer, but severity can have a mass at zero. The severity can be specified to be unconditional, see Unconditional Severity.

Details.

In terms of exp_en, exp_el, exp_premium, and exp_lr the second and third steps are:

exp_el = np.where(exp_el > 0, exp_el, exp_premium * exp_lr)

i.e., expected losses are used if given and premium times loss ratio used if not. All these values default to 0. At this point the object must know either loss or claim count:

assert np.all( exp_el > 0 or exp_en > 0 )

Then

  • If exp_en is input, it determines the expected claim count; expected losses determined from expected severity

  • Else if exp_el > 0 then it is used as expected loss and claim count determined from severity

Finally,

  • If exp_prem > 0 then the the loss ratio is computed

  • Else if exp_lr > 0 the premium is computed

Thus, if only exp_en or exp_loss is entered, the object knows loss, but not premium or loss ratio.