Why Class Frequency Relativities Matter
A class frequency relativity summarizes how expected claim counts per exposure differ between a class and a selected base class. It is a multiplicative comparison: a relativity of 1.25 indicates a frequency 25% higher than the base frequency, while a relativity of 0.80 indicates one 20% lower. These comparisons can help describe differences among rating classes, provided the counts and exposures are measured on a consistent basis.
The calculation has two parts: estimate each class’s frequency, then divide each class frequency by the frequency of the base. Counts alone are not enough. A class with more claims may simply have more exposure. The denominator must represent comparable exposure, and records combined for one class must be pooled by adding their counts and exposures—not by taking an unweighted average of their individual frequencies.
A frequency comparison is meaningful only if the claim-count definition, exposure unit, coverage, and observation basis are consistent across classes. For example, counts per vehicle-year cannot be compared directly with counts per policy-month. If experience periods need adjustment to a common basis, apply the selected frequency trend before comparing frequencies; the calculations below focus on the class comparison itself.
Estimating Frequency and Relativity
The direct estimate of a class’s frequency is its claim count divided by its exposure. When experience from several records belongs to the same class and is on a comparable basis, add the counts and exposures first. The resulting estimate weights each record according to its exposure.
A standard model-based justification uses independent Poisson claim counts. Assume that, within class \(j\), the count \(N_j\) has a Poisson distribution with mean \(e_j\lambda_j\), where \(\lambda_j\) is the class’s underlying claim frequency per exposure unit. This model assumes a constant frequency within the class and observation basis, and that counts are conditionally independent across the records being modeled. Under this model, the maximum-likelihood estimate is the total count divided by total exposure.
Here \(r\) indexes records or periods for class \(j\). This is also the natural pooled empirical frequency. It is not generally the same as the unweighted average of the record-level frequencies, because the records may have different exposures.
The relativity is dimensionless when both frequencies use the same exposure unit. Its expected-count interpretation follows directly: for any equal exposure amount \(e\), the estimated counts are \(e\widehat{\lambda}_j\) and \(e\widehat{\lambda}_b\). Their ratio is
Thus the relativity compares expected counts for equal exposure, not total observed counts from classes with possibly different exposure volumes. It describes a multiplicative difference in frequency; it does not by itself establish why the classes differ or how precise the estimate is.
Worked Class Comparisons
Worked Example: Comparing Three Vehicle Classes
Suppose a hypothetical insurer observes the following claim counts and earned vehicle-years for three vehicle-use classes. Calculate each class frequency per 1,000 vehicle-years and its relativity to the rural class. Then interpret the urban relativity.
| Class | Claims | Vehicle-years |
|---|---|---|
| Urban | 420 | 6,000 |
| Suburban | 300 | 7,500 |
| Rural | 180 | 6,000 |
Method and assumptions. Use claim count divided by vehicle-years for each class. The comparison assumes the claim definition, coverage, and observation basis are consistent among classes. No frequency trend adjustment is needed for this hypothetical data because the classes are presented on the same basis.
Urban frequency. The rate per vehicle-year is \(420/6{,}000=0.0700\). Per 1,000 vehicle-years, this is \(0.0700\times1{,}000=70.0\) claims.
Suburban frequency. The rate per vehicle-year is \(300/7{,}500=0.0400\). Per 1,000 vehicle-years, this is \(0.0400\times1{,}000=40.0\) claims.
Rural frequency. The rate per vehicle-year is \(180/6{,}000=0.0300\). Per 1,000 vehicle-years, this is \(0.0300\times1{,}000=30.0\) claims.
Relativities to rural. Divide each class rate by the rural rate: urban \(0.0700/0.0300=2.3333\), suburban \(0.0400/0.0300=1.3333\), and rural \(0.0300/0.0300=1.0000\). Rounded to four decimal places, the urban relativity is 2.3333 and the suburban relativity is 1.3333.
Interpretation and check. For equal exposure, the estimated urban count is about 2.3333 times the rural count. For 1,000 vehicle-years, the estimated counts are 70.0 urban claims and 30.0 rural claims; \(70.0/30.0=2.3333\), as required. A common mistake is to compare the raw counts \(420\) and \(180\) and report \(420/180=2.3333\) without checking exposure. That ratio happens to agree here because urban and rural exposures are equal; it would not generally do so.
Worked Example: Converting Policy-Months Before Comparing
In a hypothetical portfolio, Class A has 54 claims over 900 policy-months, and Class B has 84 claims over 1,200 policy-months. Convert both frequencies to claims per policy-year and calculate the relativity of Class B to Class A. Interpret the result.
Method and assumptions. One policy-year is 12 policy-months, so convert the exposure denominator to policy-years before calculating the rates. Assume the claim definitions and observation bases are otherwise comparable.
Class A exposure and frequency. The exposure is \(900/12=75\) policy-years. Its frequency is \(54/75=0.72\) claims per policy-year.
Class B exposure and frequency. The exposure is \(1{,}200/12=100\) policy-years. Its frequency is \(84/100=0.84\) claims per policy-year.
Relativity. Using Class A as the base, the Class B relativity is \(0.84/0.72=1.1667\), rounded to four decimal places.
Interpretation and check. For equal policy-year exposure, Class B is estimated to have 1.1667 times Class A’s expected claim count, or about 16.67% higher frequency. For 50 policy-years, the estimates are \(50(0.72)=36\) Class A claims and \(50(0.84)=42\) Class B claims; \(42/36=1.1667\). Dividing the raw counts, \(84/54=1.5556\), would be wrong because the exposure amounts differ. The conversion also matters: the denominator is policy-years, not policy-months.
Worked Example: Pooling Records Within Each Class
A hypothetical insurer has two experience records for each of two equipment-insurance classes. Record 1 has 24 claims and 800 exposure units for Class Basic, and 18 claims and 400 exposure units for Class Plus. Record 2 has 30 claims and 1,200 exposure units for Class Basic, and 27 claims and 600 exposure units for Class Plus. Calculate each pooled frequency and the Plus relativity to Basic.
Method and assumptions. The records are on the same exposure and claim basis within and across classes. Pool by adding counts and exposures within each class, then divide the pooled frequency for Plus by that for Basic.
Pool Basic. Basic has \(24+30=54\) claims and \(800+1{,}200=2{,}000\) exposure units. Its frequency is \(54/2{,}000=0.0270\) claims per exposure unit.
Pool Plus. Plus has \(18+27=45\) claims and \(400+600=1{,}000\) exposure units. Its frequency is \(45/1{,}000=0.0450\) claims per exposure unit.
Relativity and check. The Plus-to-Basic relativity is \(0.0450/0.0270=1.6667\), rounded to four decimal places. For 100 exposure units, the estimated counts are 4.50 for Plus and 2.70 for Basic, and \(4.50/2.70=1.6667\). An unweighted average of Basic’s record frequencies would be \((24/800+30/1{,}200)/2=(0.0300+0.0250)/2=0.0275\), which is not its pooled frequency of 0.0270. The correct pooled estimate gives each record weight proportional to its exposure.
Calculation Sequence and Interpretation
A reliable calculation can be organized as a short sequence. Keeping the base class explicit helps avoid reversing the ratio, and checking the result against expected counts gives the relativity a direct interpretation.
Check that claim counts, exposure units, coverage, and observation bases align. Convert units or adjust periods to a common basis when appropriate.
Add counts and exposures separately for records that belong to that class and share the same basis.
Divide each pooled count by its pooled exposure, retaining enough precision for the ratio.
Divide every class frequency by the frequency of the stated base. The base’s own relativity must equal 1.
Describe the relativity as a multiplicative difference in expected counts for equal exposure, and verify it using a common exposure amount.
The base is a reference point, not necessarily the class with the lowest frequency or the largest exposure. A relativity greater than 1 means the class frequency estimate exceeds the base estimate; a relativity below 1 means it is lower. If the base changes, the numerical relativities change even though the underlying class frequencies do not. For example, reversing the comparison from Plus-to-Basic to Basic-to-Plus gives the reciprocal, \(1/1.6667\), approximately 0.6000.
Common Mistakes and Limits
- Using claim counts without exposure. Counts describe volume, not frequency, unless exposures are equal. Calculate count per exposure before comparing classes.
- Mixing exposure units. Convert all denominators to the same unit. A rate per policy-month and a rate per policy-year differ by a factor of 12.
- Averaging record-level rates without weights. When exposures differ, add counts and exposures before dividing. An unweighted average gives a small record the same influence as a large one.
- Reversing the ratio. For a class-to-base relativity, divide the class frequency by the base frequency. The base class must have relativity 1.
- Confusing frequency with total expected claims. A relativity compares expected counts for equal exposure. It does not say which class has the most total claims when the exposure amounts differ.
- Over-interpreting an observed ratio. The calculated relativity describes the estimates from the stated experience. It does not, by itself, account for sampling uncertainty, establish causation, or prove that the difference will persist.
Check Your Understanding
For each problem, calculate frequencies on the stated basis before forming the requested relativity. Answers are rounded only where indicated.
- A hypothetical fleet has 63 claims over 2,100 vehicle-years in Class M and 48 claims over 2,400 vehicle-years in Class N. Calculate both frequencies per vehicle-year and the M-to-N relativity. (Answer: M = 0.0300, N = 0.0200 claims per vehicle-year; M-to-N relativity = 1.5000.)
- Class Base has 50 claims over 2,000 exposure units. Class R has 42 claims over 1,200 exposure units. Calculate the R-to-Base relativity and state whether R is higher or lower in frequency. (Answer: Base = 0.0250, R = 0.0350 claims per exposure unit; relativity = 1.4000, so R is 40% higher.)
- A hypothetical group has 32 claims in 800 policy-months for the base class and 45 claims in 900 policy-months for Class Q. Convert to policy-years, then calculate Q’s relativity to the base. (Answer: base exposure = 66.6667 policy-years and frequency = 0.4800; Q exposure = 75 policy-years and frequency = 0.6000 claims per policy-year; relativity = 1.2500.)
- Class A has two records: 16 claims over 500 exposure units and 30 claims over 1,500 units. The base Class B has 27 claims over 1,500 units and 18 claims over 1,000 units. Pool each class, calculate its frequency, and find A’s relativity to B to four decimal places. (Answer: A = 46/2,000 = 0.0230; B = 45/2,500 = 0.0180 claims per exposure unit; A-to-B relativity = 1.2778.)
- The estimated base frequency is 0.0400 claims per exposure unit, and Class C’s relativity to the base is 2.25. For 800 exposure units, calculate Class C’s frequency and expected claim count. (Answer: Class C frequency = 0.0900 claims per exposure unit; expected count = 72 claims.)