Why Exposure Units Matter
A frequency model relates claim counts to both the risk characteristics being rated and the amount of time the risk was exposed. That exposure may be recorded in policy-years, policy-months, or another consistent unit. Changing the unit changes the numerical exposure and the numerical rate, but it should not change the expected number of claims for the same insured risk and coverage period.
This is a parameterization issue, not a change in the underlying risk. For example, a rate of 0.48 claims per policy-year is equivalent to 0.04 claims per policy-month when a policy-year is defined as 12 policy-months and exposure accumulates proportionally. The challenge is to adjust the model so its predictions retain that equivalence.
For the usual frequency interpretation, assume that expected claims are proportional to exposure, the same risk characteristics apply over the exposure period, and exposure is converted consistently for every observation. In particular, treating 12 policy-months as one policy-year assumes that no within-year seasonality or other time-varying pattern is being modeled. The conversion does not change the observed claim counts or the underlying coverage.
The Unit-Conversion Rule
Let \(E_j^{(y)}\) denote exposure in policy-years, and let \(E_j^{(m)}\) denote the same exposure in policy-months. Under the conversion in this tutorial,
A rate per policy-month must therefore be one-twelfth of the corresponding rate per policy-year. If \(r_j^{(y)}\) is the annual-unit fitted rate, the equivalent monthly-unit rate is
In a log-link model with an intercept, this means subtracting \(\log(12)\) from the intercept while leaving the other coefficients unchanged. The offset also changes from the logarithm of policy-year exposure to the logarithm of policy-month exposure.
This rule assumes that the model includes an intercept that can absorb the shift. It also assumes the same observations, claim counts, rating variables, and model structure are used before and after conversion. If there is no intercept, or if exposure is not simply multiplied by the same conversion factor for every observation, this particular adjustment need not preserve predictions.
Why the Expected Counts Stay the Same
The result follows directly from the model’s mean. Write the annual-unit linear predictor as \(\eta_j^{(y)}=\beta_0+z_j^\mathsf{T}\gamma\), where \(z_j\) contains the rating variables other than the intercept. Then
For a conversion factor \(c\), set \(E_j^{(new)}=cE_j^{(y)}\), \(\beta_0^{(new)}=\beta_0-\log(c)\), and \(\gamma^{(new)}=\gamma\). Substitution gives
The cancellation works because \(\exp(-\log(c))=1/c\). Thus every observation has the same fitted expected count after conversion. Dividing that expected count by the new exposure gives a rate \(1/c\) as large as before, exactly as required for a rate expressed per larger unit.
For a Poisson frequency model, the same argument also explains why fitting the model does not inherently change the fitted claim counts. The Poisson log-likelihood for observation \(j\) is \(Y_j\log(\mu_j)-\mu_j-\log(Y_j!)\). Since the conversion gives the same \(\mu_j\) for every observation, it gives the same log-likelihood for every possible set of counts. The parameter values correspond through the intercept shift, so the equivalent maximized fit is preserved.
Worked Example: Converting a Single Risk Profile
A hypothetical portfolio model estimates 0.48 claims per policy-year for a reference risk. A rating factor of 1.25 applies to a particular profile with 2.5 policy-years of exposure. Convert the model to policy-month units, compute the expected claim count in both parameterizations, and verify the intercept shift. Use natural logarithms and round displayed logarithms to four decimal places.
Method and assumptions. Use a log-link frequency model, proportional exposure, and the same rating factor in either unit. The 2.5 policy-years correspond to \(2.5\times12=30\) policy-months.
Annual-unit calculation. The profile’s annual-unit rate is \(0.48\times1.25=0.60\) claims per policy-year. Its expected claim count is \(2.5\times0.60=1.50\).
The reference-risk intercept is \(\log(0.48)\approx-0.7340\). After conversion, the reference rate is \(0.48/12=0.04\) claims per policy-month, so the new intercept is \(\log(0.04)\approx-3.2189\). The intercept difference is \(-3.2189-(-0.7340)\approx-2.4849\), which is \(-\log(12)\approx-2.4849\) to four decimal places. The rating coefficient remains \(\log(1.25)\); only the intercept and exposure unit change.
Monthly-unit check. The profile’s rate is \(0.04\times1.25=0.05\) claims per policy-month. Its expected count is \(30\times0.05=1.50\), matching the annual-unit result.
Check and common mistake avoided. Both calculations give 1.50 expected claims. Keeping the annual intercept after changing exposure to 30 months would incorrectly give \(30\times0.60=18.00\) expected claims: the intercept must be reduced by \(\log(12)\).
Applying the Rule to Multiplicative Rating Factors
With a log link, rating coefficients often represent multiplicative factors on the rate. Changing the exposure unit does not alter those relativities. It changes the rate’s unit and shifts the intercept, while each risk’s relative position remains the same.
Worked Example: Keeping Rating Relativities Fixed
A hypothetical insurer uses a log-link frequency model with annual-unit base rate 0.30 claims per policy-year. A risk has two factors, 1.40 and 0.80, and 3.75 policy-years of exposure. Convert the model to policy-month units and calculate its expected claim count. Round rates to three significant figures and expected counts to the nearest cent.
Method. Multiply the rating factors to obtain the risk’s relativity, calculate the annual-unit rate and expected count, then divide the base rate by 12 and multiply exposure by 12.
Annual-unit rate. The combined relativity is \(1.40\times0.80=1.12\). The risk’s annual-unit rate is \(0.30\times1.12=0.336\) claims per policy-year. Its expected count is \(3.75\times0.336=1.260\), or 1.26 claims.
Intercept and monthly-unit rate. The annual intercept is \(\log(0.30)\). The monthly-unit base rate is \(0.30/12=0.0250\), so the new intercept is \(\log(0.0250)=\log(0.30)-\log(12)\). The factors are unchanged, and the risk’s monthly-unit rate is \(0.0250\times1.12=0.0280\) claims per policy-month.
Monthly-unit expected count. Exposure is \(3.75\times12=45\) policy-months. Thus the expected count is \(45\times0.0280=1.260\), or 1.26 claims to the nearest cent.
Check and common mistake avoided. The two expected counts agree, and the factor product remains 1.12. A common error is to divide the final rate by 12 but also divide the exposure by 12; that would apply the conversion twice. Here, the original exposure is converted to 45 policy-months exactly once.
Estimating Rates After Conversion
A unit conversion changes the numerical value of a fitted rate, but it does not create a different risk pattern. This is especially easy to see in a model that estimates a separate constant rate for each class. Under a Poisson count model with exposure \(E_j\), class rate \(r\), and mean \(\mu_j=E_jr\), the maximum-likelihood estimate for the class rate is total claims divided by total exposure. The same ratio in a different exposure unit is rescaled by the conversion factor.
Worked Example: Checking Fitted Class Rates from Counts
A hypothetical two-class frequency study has 12 claims over 30 policy-years in Class A and 6 claims over 15 policy-years in Class B. Suppose each class has its own constant Poisson claim rate. Find the fitted rates in policy-year and policy-month units, the corresponding intercepts, and the fitted total claim counts. Round rates to four decimal places and logarithms to four decimal places.
Method and assumptions. For each class, estimate the rate as claims divided by exposure. Treat the class counts as independent Poisson observations with exposure-proportional means. Convert all exposures by multiplying by 12.
Annual-unit estimates. Class A has rate \(12/30=0.4000\) claims per policy-year, and Class B has rate \(6/15=0.4000\) claims per policy-year. In a log-link model with these equal class rates, the common fitted intercept is \(\log(0.4000)\approx-0.9163\), with no class rate difference needed.
Monthly-unit estimates. Class A exposure becomes \(30\times12=360\) policy-months, and Class B exposure becomes \(15\times12=180\) policy-months. Their fitted rates are \(12/360=0.0333\) and \(6/180=0.0333\) claims per policy-month (unrounded, each is \(1/30\)). The new common intercept is \(\log(1/30)\approx-3.4012\). Its difference from the annual intercept is \(-3.4012-(-0.9163)=-2.4849\), equal to \(-\log(12)\) to four decimal places.
Fitted-count check. In annual units, the fitted counts are \(30\times0.4000=12\) for Class A and \(15\times0.4000=6\) for Class B. In monthly units they are \(360\times(1/30)=12\) and \(180\times(1/30)=6\). The fitted total is 18 claims under either representation.
Common mistake avoided. The monthly rate is not 12 times the annual rate. Because a policy-month is one-twelfth of a policy-year, the numerical rate per policy-month is one-twelfth of the rate per policy-year.
Practical Checks and Exam Traps
A sound conversion check compares both the rate and the expected count. The rate must change inversely with the exposure-unit conversion; the expected count for unchanged coverage must not change. The coefficient shift provides a third check: multiplying exposure by \(c\) requires an intercept change of \(-\log(c)\).
| Quantity | After exposure is multiplied by \(c\) | For policy-years to policy-months |
|---|---|---|
| Exposure | Multiply by \(c\) | Multiply by 12 |
| Rate per exposure unit | Divide by \(c\) | Divide by 12 |
| Intercept | Subtract \(\log(c)\) | Subtract \(\log(12)\) |
| Other rating coefficients | Unchanged under the same model | Unchanged |
| Expected claim count | Unchanged | Unchanged |
Several errors are worth checking explicitly:
- Changing the offset but not the intercept. Multiplying every exposure by 12 while leaving the fitted rate unchanged multiplies each expected count by 12.
- Using the wrong sign. Increasing the numerical exposure by 12 requires reducing the intercept by \(\log(12)\), not increasing it.
- Confusing rate units. A rate per policy-month is smaller numerically than the equivalent rate per policy-year; the expected count remains the same when paired with the matching exposure.
- Changing the data’s meaning. The conversion is appropriate only when policy-month exposure is genuinely equivalent to one-twelfth of policy-year exposure for the model’s purpose. Seasonal effects or changing risk characteristics may require a different specification.
- Dropping the intercept condition. The simple intercept-shift rule relies on an intercept being available to absorb the constant offset change.
The key audit is compact: verify \(E^{(new)}=cE^{(old)}\), verify that the new fitted rate is the old rate divided by \(c\), and multiply them to confirm that the fitted mean count is unchanged. This dimensional check catches many offset and unit errors before a model’s coefficients are interpreted.
Check Your Understanding
For each problem, distinguish the exposure-unit rate from the expected claim count. Assume proportional exposure and a log link unless stated otherwise.
- A risk has an annual-unit rate of 0.72 claims per policy-year and 1.5 policy-years of exposure. What are its monthly-unit rate, monthly exposure, and expected claim count? (Answer: 0.06 claims per policy-month; 18 policy-months; 1.08 expected claims.)
- A log-link model has annual-unit intercept \(\log(0.25)\) and one rating factor of 1.6. State the monthly-unit intercept and calculate the profile’s rate per policy-month. (Answer: intercept \(\log(0.25/12)=\log(1/48)\approx-3.8712\); rate \(0.25\times1.6/12=0.0333\) claims per policy-month.)
- A hypothetical account has a fitted annual-unit rate of 0.36 and 7.25 policy-years of exposure. Calculate the equivalent monthly exposure and rate, then verify the expected claim count. (Answer: 87 policy-months; 0.03 claims per policy-month; 2.61 expected claims.)
- A modeler multiplies every exposure by 12 but leaves the intercept unchanged. By what factor do fitted expected counts change, assuming all other terms remain fixed? (Answer: They are multiplied by 12.)
- A Poisson class has 9 claims over 22.5 policy-years. Find its fitted rate and intercept in policy-month units, using four decimal places for the rate and logarithm. (Answer: Rate \(9/270=0.0333\) claims per policy-month; intercept \(\log(1/30)\approx-3.4012\).)