Why Organize Experience by Risk and Period?
Bühlmann-Straub credibility uses experience observed for multiple risks across multiple periods, where the amount of exposure can differ between observations. Before any credibility calculation, the data must make clear which risk generated each observation, which period it belongs to, and how much exposure was at risk during that period. A loss amount without its matching exposure can be misleading: $30,000 of loss could be high for a small exposure and moderate for a large one.
The basic observation is a risk-period pair. Its total loss is the aggregate loss recorded for that risk in that period. Its loss cost is the total loss divided by the exposure for that same risk and period. The exposure is also called an exposure weight because, in later Bühlmann-Straub calculations, it indicates how much information an observation contributes. Here, the focus is on preparing and interpreting those observations correctly, not on calculating a credibility estimate.
Choosing and Recording Exposure Weights
An exposure weight measures the amount of insured exposure represented by an observation. It might be earned vehicle-years, insured building-years, payroll in thousands of dollars, or another measure appropriate to the coverage. The unit must be defined explicitly. If the weight is 25 earned vehicle-years, a loss cost of $1,400 means $1,400 per earned vehicle-year.
An exposure weight is not automatically the number of policies. A policy in force for half a year generally contributes half as much policy-year exposure as a policy in force for a full year, if policy-years are the chosen basis. Conversely, a coverage with exposure measured by payroll or insured value should use that basis rather than a policy count. The data definition, not the record’s format, determines the weight.
A useful working table has one row per risk-period observation and separate columns for risk identifier, period, total loss, exposure weight, and loss cost. If a risk has no exposure in a period, it has no valid loss-cost observation for that period: dividing by zero is undefined. A zero loss with positive exposure is valid and gives a loss cost of zero.
| Risk | Period | Total loss | Exposure weight | Loss cost |
|---|---|---|---|---|
| Risk R1 | Period 1 | Recorded aggregate | Matching earned exposure | Total loss divided by exposure |
| Risk R1 | Period 2 | Recorded aggregate | Matching earned exposure | Total loss divided by exposure |
| Risk R2 | Period 1 | Recorded aggregate | Matching earned exposure | Total loss divided by exposure |
The periods need not have identical exposure. One risk-period row might represent a full year and another a partial year. What matters is that each row’s loss and exposure cover the same interval. It is also important to state whether losses are paid, incurred, or measured on another defined basis, and to use that basis consistently. Dividing a paid loss amount by an exposure measure from a different period does not produce a meaningful period loss cost.
Worked Example: Building a Risk-Period Table
A hypothetical equipment insurer measures exposure in earned machine-years. For two risks, the recorded experience is: Risk A has total losses of $31,200 on 24 machine-years in Period 1 and $45,000 on 30 machine-years in Period 2. Risk B has total losses of $17,000 on 10 machine-years in Period 1 and $27,900 on 18 machine-years in Period 2. Construct the loss-cost entries and distinguish each cost from its total loss.
Method. Treat each risk-period pair as a separate observation. Divide each row’s total loss by its own exposure weight.
For Risk A, Period 1: $$ C_{A1}=\frac{\$31{,}200}{24\text{ machine-years}}=\$1{,}300\text{ per machine-year}. $$ For Risk A, Period 2: $$ C_{A2}=\frac{\$45{,}000}{30\text{ machine-years}}=\$1{,}500\text{ per machine-year}. $$ For Risk B, Period 1: $$ C_{B1}=\frac{\$17{,}000}{10\text{ machine-years}}=\$1{,}700\text{ per machine-year}. $$ For Risk B, Period 2: $$ C_{B2}=\frac{\$27{,}900}{18\text{ machine-years}}=\$1{,}550\text{ per machine-year}. $$
The resulting table has four observations: (A, Period 1, $31,200, 24, $1,300), (A, Period 2, $45,000, 30, $1,500), (B, Period 1, $17,000, 10, $1,700), and (B, Period 2, $27,900, 18, $1,550), where each tuple lists risk, period, total loss, exposure, and loss cost. As a check, multiplying each loss cost by its exposure recovers the row’s total loss: for example, \(24\times \$1{,}300=\$31{,}200\) and \(18\times \$1{,}550=\$27{,}900\).
Common mistake avoided: reporting $31,200 as Risk A’s Period 1 loss cost confuses the total loss with loss per exposure unit. The period loss cost is $1,300 per machine-year; $31,200 remains the total loss.
Partial Periods and Exposure Units
For partial periods, calculate exposure in the same time units used to define the full exposure unit. If a policy-year is the unit, six months of covered exposure is one-half policy-year, subject to the exposure measurement convention specified for the data. Do not divide by the count of policies when the denominator should reflect time in force.
Worked Example: Accounting for a Half-Year of Coverage
A hypothetical fleet has 80 vehicles insured for six months during a period. The covered losses are $52,000. A separate fleet has 40 vehicles insured for a full year and incurs $60,000 of loss. Calculate each loss cost in dollars per earned vehicle-year. Assume exposure accrues uniformly over the stated coverage durations.
Method. Convert the vehicle counts and coverage durations into earned vehicle-years, then divide each total loss by that exposure.
For the first fleet, each of 80 vehicles contributes \(6/12=0.5\) vehicle-year. Thus its exposure is $$ 80\times 0.5=40\text{ earned vehicle-years}. $$ Its loss cost is $$ \frac{\$52{,}000}{40}=\$1{,}300\text{ per earned vehicle-year}. $$ For the second fleet, each of 40 vehicles contributes one vehicle-year, so its exposure is $$ 40\times 1=40\text{ earned vehicle-years}. $$ Its loss cost is $$ \frac{\$60{,}000}{40}=\$1{,}500\text{ per earned vehicle-year}. $$
The first fleet has 80 insured vehicles but only 40 earned vehicle-years; the second has 40 insured vehicles and also 40 earned vehicle-years. As a check, multiplying each calculated cost by 40 earned vehicle-years gives $52,000 and $60,000, respectively.
Common mistake avoided: dividing $52,000 by 80 gives $650 per vehicle, not the requested cost per earned vehicle-year. It ignores the six-month duration. The correction is to use 40 earned vehicle-years as the denominator.
Combining Exposure Segments Within a Risk-Period
A risk-period can sometimes contain several exposure segments, such as separately recorded locations or groups of insured units. If those segments belong to the same risk-period and have compatible loss and exposure definitions, the combined total loss is the sum of segment losses and the combined exposure is the sum of segment exposures. The combined loss cost is then the combined loss divided by the combined exposure.
This identity follows directly from the definition. For each segment, \(L_k=w_kC_k\). Therefore the sum of the segment losses is \(\sum_k w_kC_k\). Dividing that sum by the total exposure, \(\sum_k w_k\), gives the combined loss per exposure unit. The identity also explains why simply averaging segment loss costs can be wrong: it gives each segment equal weight even when their exposures differ.
Worked Example: Combining Unequal Segments
A hypothetical property risk has two location segments in the same period. Segment 1 has exposure of 12 building-years and total loss of $18,000. Segment 2 has exposure of 8 building-years and total loss of $20,000. Calculate the combined risk-period loss cost and compare it with the unweighted average of the segment costs.
Method. First calculate each segment’s cost, then combine by adding losses and exposures before dividing.
Segment 1’s loss cost is $$ \frac{\$18{,}000}{12}=\$1{,}500\text{ per building-year}. $$ Segment 2’s loss cost is $$ \frac{\$20{,}000}{8}=\$2{,}500\text{ per building-year}. $$ The combined total loss is $$ \$18{,}000+\$20{,}000=\$38{,}000, $$ and the combined exposure is $$ 12+8=20\text{ building-years}. $$ Thus the combined loss cost is $$ \frac{\$38{,}000}{20}=\$1{,}900\text{ per building-year}. $$ The unweighted average of the two segment costs is \((\$1{,}500+\$2{,}500)/2=\$2{,}000\). That is not the combined risk-period cost because the segments have different exposures. The exposure-weighted calculation checks as \((12\times \$1{,}500+8\times \$2{,}500)/20=(\$18{,}000+\$20{,}000)/20=\$1{,}900\).
Common mistake avoided: averaging $1,500 and $2,500 equally treats 12 and 8 building-years as if they represented equal amounts of experience. Add the loss and exposure totals, or equivalently weight each segment cost by its exposure.
Keep Units and Records Internally Consistent
The numerical loss cost depends on the exposure unit. If payroll is measured in thousands of dollars, the result is dollars of loss per $1,000 of payroll, not dollars per dollar of payroll. A unit conversion error can change the reported cost by a factor of 1,000 even when the division itself is correct.
Worked Example: Payroll Measured in Thousands
A hypothetical contractor’s covered losses for a period are $24,600. Payroll for the same period is $1,200,000, and the stated exposure unit is $1,000 of payroll. Calculate the loss cost in dollars per $1,000 of payroll. Then check the result by reconstructing the total loss.
Method. Convert payroll to the stated exposure unit before dividing. The exposure weight is $$ \frac{\$1{,}200{,}000}{\$1{,}000}=1{,}200\text{ units of \$1,000 payroll}. $$ The loss cost is $$ \frac{\$24{,}600}{1{,}200}=\$20.50\text{ per \$1,000 of payroll}. $$ To check, multiply the cost by the exposure weight: $$ \$20.50\times 1{,}200=\$24{,}600. $$
Common mistake avoided: dividing $24,600 by $1,200,000 produces $0.0205 per dollar of payroll, which is a different unit. It is equivalent to $20.50 per $1,000 of payroll, but it must be reported with that conversion made explicit.
Data Checks Before Credibility Analysis
The risk-period table is the foundation for subsequent Bühlmann-Straub work. Before using it, check that identifiers and period labels are unambiguous, exposures are positive for rows with a calculated loss cost, and the loss and exposure bases match. If a purported observation has zero exposure and positive loss, do not force a loss-cost calculation: investigate whether the loss was assigned to the wrong period, the exposure was omitted, or the exposure definition is unsuitable.
- Check the unit: label each exposure weight, such as earned vehicle-years or thousands of dollars of payroll.
- Check the period: match losses and earned exposure to the same time interval.
- Check the denominator: use actual exposure, including partial-period exposure where appropriate, rather than an unadjusted policy count.
- Check the aggregation: combine compatible segments by adding losses and exposures, not by taking an unweighted average of their costs.
- Check the interpretation: retain both the total loss and the loss cost; they answer different questions.
A well-prepared table does not by itself determine how much credibility to assign to an observation. It ensures that the inputs have a coherent interpretation before credibility methods are applied. This organization is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
Check Your Understanding
For each problem, identify the appropriate exposure denominator and distinguish the calculated loss cost from the total loss.
- A hypothetical insurer records $45,900 of loss for 36 earned car-years in one risk-period. What is the loss cost per earned car-year? (Answer: $1,275 per earned car-year.)
- A risk has 20 policies, each covered for nine months, with total period losses of $24,000. Assuming policy-year exposure accrues uniformly, calculate the loss cost per policy-year. (Answer: 15 policy-years of exposure and $1,600 per policy-year.)
- A hypothetical bicycle program covers 30 bikes for three months and records $10,875 of loss. Assuming uniform exposure accrual, calculate the cost per bike-year. (Answer: 7.5 bike-years and $1,450 per bike-year.)
- Two compatible segments of one risk-period have exposures of 12 and 8 building-years and losses of $18,000 and $16,000, respectively. Find the combined loss cost and the unweighted average of the segment costs. (Answer: Combined cost $1,700 per building-year; unweighted average $1,750 per building-year.)
- A risk-period record shows zero earned exposure and $5,000 of recorded loss. Can a loss cost be calculated from these entries, and what should be done next? (Answer: No; division by zero is undefined. Investigate the exposure and loss-period assignment or the exposure definition before using the record.)