Actuarial Notes

Exposure Base

Study guides: Exam MAS-II, Exam DISC-RM, Exam 5, Exam GI-101

Exposure Base is the basic unit of risk that underlies the premium — the quantity the rate is charged per. It measures a policy's exposure to loss, which is why it is the basis of the premium: base rates are expressed per exposure, and it is the denominator of the pure premium.

\text{Pure Premium} = \frac{\text{Losses} + \text{LAE}}{\text{Earned Exposures}}

\text{Premium} = \text{Rate} \times \text{Exposures}

Werner & Modlin give three criteria for choosing one:

  • Proportional to expected loss. All else equal, a policy with two exposures should have twice the expected loss of a similar policy with one. Expected loss varies by many other factors too; those become rating or underwriting variables, and the factor with the most direct relationship to losses is chosen as the base.
  • Practical. Objective, and relatively easy and inexpensive to obtain and verify — which also keeps policyholders and producers from manipulating it (declared annual mileage invites more dishonesty than car-years). Where the most proportional measure cannot be tracked, a practical proxy is used: products liability rates on gross sales because products in use cannot be counted.
  • Consistent with historical precedence. A change of base can cause large premium swings for individual insureds, requires a change in the rating algorithm and systems, and may force significant adjustments to the several years of data a ratemaking analysis uses.

Further properties:

  • Common bases: earned car-years (personal auto), earned house-years (homeowners), payroll — in hundreds of dollars in the U.S. — (workers compensation), sales revenue, payroll, square footage or number of units (commercial general liability), amount of insurance (commercial business property).
  • In underwriting (DISC-RM): the premium base for a general liability account follows its classification. IRMI lists payroll, receipts, sales, square footage, area or man-hours for general liability, units for auto and value per $1,000 for property, and a premium audit verifies the base after the policy ends (Commercial General Liability).
  • Exposures come in the same four flavours as premium — written, earned, unearned and in-force; which one is used depends on the analysis, and pure premium is typically calculated on earned exposures.
  • Some bases, such as payroll and sales revenue, are highly influenced by inflation, which is what Exposure Trend measures; count-based bases (car-years, house-years) are not.
  • Where no single base is proportional to loss across the whole book, the shortfall is picked up by rating variables instead: this is the boundary between the exposure base and classification.

Why Workers Compensation Rates on Payroll Two employers in the same class code have identical operations. Employer A employs 10 workers at \$200{,}000 each; Employer B employs 20 workers at \$50{,}000 each.

Compare payroll against headcount as the exposure base.

Answer

\begin{align*}
\text{Payroll}_A &= 10 \times \$200{,}000 = \$2{,}000{,}000 \\
\text{Payroll}_B &= 20 \times \$50{,}000 = \$1{,}000{,}000
\end{align*}

Under headcount, B pays twice A's premium. Under payroll, A pays twice B's.

Payroll is the workers compensation base because it is responsive to changes in exposure: as the number of workers or the hours they work rise or fall, both payroll and the risk of loss move with them. Headcount would miss the hours.

But the comparison also shows where payroll is disputed. A pays twice B's premium for half the workers, purely because its wages are higher. In the 1980s there was pressure to move workers compensation medical coverage to hours worked, to correct perceived inadequacies of payroll for union companies with higher pay scales. The base was not changed — concerns about the transition were a major reason — and the rating variables and algorithm were adjusted instead: historical precedence at work. Payroll is also inflation-sensitive, which is why its trend is measured — see Exposure Trend.

Testing a Proposed Exposure Base A homeowners insurer currently rates on house-years and proposes rating on amount of insurance (AOI) per \$1{,}000 instead. Analysis of its book shows:

AOI band Earned house-years Losses
\$100K–200K 20{,}000 \$12{,}000{,}000
\$200K–400K 15{,}000 \$13{,}500{,}000
over \$400K 5{,}000 \$7{,}000{,}000

Evaluate the proposal.

Answer Pure premium per house-year:

\begin{align*}
\$100\text{K–}200\text{K} &= \frac{12{,}000{,}000}{20{,}000} = \$600 \\[4pt]
\$200\text{K–}400\text{K} &= \frac{13{,}500{,}000}{15{,}000} = \$900 \\[4pt]
\text{over } \$400\text{K} &= \frac{7{,}000{,}000}{5{,}000} = \$1{,}400
\end{align*}

Loss cost rises with amount of insurance, so a single rate per house-year with nothing else would leave small homes subsidizing large ones. But the rise is far from proportional: from the lowest band to the highest, loss per house-year rises by a factor of 1{,}400/600 = 2.3, while the band boundaries quadruple from \$100K to \$400K. Rating per \$1{,}000 of AOI would over-charge large homes.

This is Werner & Modlin's own homeowners illustration: the expected loss for a \$200{,}000 home is higher than for a \$100{,}000 home, but not necessarily twice as high. On the criterion that the exposure base should be the factor most directly proportional to expected loss, house-years remain the exposure base, and amount of insurance is used as a rating variable — an AOI relativity applied to the base rate per house-year, as in Werner's homeowners rating algorithm. The proposal should be rejected; the AOI effect in the table is a case for an AOI relativity, not a new base. (Werner notes that some U.K. homeowners insurers do use AOI or number of bedrooms as the base, adjusting the rating algorithm for the lack of proportionality.)