Why Selecting a Frequency Trend Requires Judgment
A frequency trend describes how claims per unit of exposure change over time. It is not the trend in claim counts alone: counts can rise because there are more insured exposure units, even when claims are becoming less frequent per unit. Selecting a trend therefore requires a consistent frequency definition, comparable periods and exposure, and a reasoned choice among candidate estimates.
The earlier material on frequency-severity analysis defines frequency as claims divided by a matching exposure measure. This tutorial applies that idea to selecting a frequency trend from experience. The trend selected here concerns frequency only; when using it to illustrate loss costs, severity is held fixed so the frequency effect can be isolated.
Make the Experience Comparable Before Measuring Trend
A numerical trend is meaningful only if the quantities being compared are sufficiently comparable. Check that the claim definition, exposure unit, coverage, and period boundaries are consistent. For example, claim counts for earned vehicle-years should be divided by earned vehicle-years in each period, not by policy counts in one period and vehicle-years in another. Counts and exposures should also refer to matching experience periods.
Use period midpoints to measure elapsed time when the experience covers full intervals and claims and exposure are distributed reasonably evenly within them. For adjacent annual periods, midpoint-to-midpoint time is one year. For a three-year gap between midpoints, the observed change must be annualized over three years. If periods are irregular, use their actual elapsed midpoint times rather than treating each observation as one year apart.
A standard candidate is the compound annual trend between two observed frequencies. It assumes a constant proportional change per year over the selected interval. If \(f_1\) and \(f_2\) are the frequencies at period midpoints separated by \(k\) years, the annual trend rate \(r_f\) is:
The corresponding annual trend factor is \(1+r_f\). A negative \(r_f\) means frequency is declining. This endpoint calculation summarizes the net change between the chosen observations; it does not establish that frequency changed smoothly or for the same reason throughout the interval.
Why the Annualized Formula Works
Under a constant compound trend, the frequency after one year is multiplied by \(1+r_f\), after two years by \((1+r_f)^2\), and after \(k\) years by \((1+r_f)^k\). Thus, if \(f_1\) is the starting frequency, the model implies \(f_2=f_1(1+r_f)^k\). Dividing by \(f_1\) and taking the \(k\)th root gives the annualized formula above. The formula is not appropriate when the starting frequency is zero, because the ratio is undefined; nor does it make a volatile or structurally changing series into a stable trend merely by annualizing it.
Worked Example: Comparing a Recent Trend with a Longer-Period Trend
A hypothetical insurer records the following annual claim counts and earned exposures for a product. All periods use the same claim definition and exposure unit. Assume annual exposure is distributed evenly through each year, so the midpoint of each period is halfway through that year.
| Year | Claims \(N_t\) | Exposure \(E_t\) | Frequency \(f_t\) |
|---|---|---|---|
| 2019 | 700 | 10,000 | 0.070 |
| 2020 | 990 | 15,000 | 0.066 |
| 2021 | 1,080 | 18,000 | 0.060 |
| 2022 | 1,260 | 20,000 | 0.063 |
Method. Calculate each annual frequency first, then compare annualized endpoint candidates. The 2019 and 2022 midpoints are three years apart. Their frequencies are \(700/10{,}000=0.070\) and \(1{,}260/20{,}000=0.063\). The full-period annual trend is:
This candidate is an annual decrease of about 3.451%. For the 2020-to-2022 comparison, the midpoint gap is two years. The endpoint ratio is \(0.063/0.066=21/22\), so:
This candidate is an annual decrease of about 2.299%. Suppose, for this hypothetical decision, the insurer has independent documentation that the 2019 experience predates a product change, while the 2020–2022 periods share the current product design. On that stated basis, the recent-period estimate is more relevant, though the year-to-year pattern still deserves review. The selected annual factor is approximately \(0.97701\).
To show the effect on projected costs, start with the latest frequency, \(0.063\) claims per exposure, and assume a constant severity of $4,000. One year forward, projected frequency is \(0.063(0.97701)=0.0615516\), rounded here to seven decimal places. The corresponding projected loss cost is \(0.0615516(4{,}000)=\$246.21\), rounded to the nearest cent. Without a frequency trend, the loss cost would be \(0.063(4{,}000)=\$252.00\). The selected trend therefore reduces this illustrative one-year projected loss cost by $5.79.
Common mistake avoided: Treating the full-period candidate as automatically preferable because it uses more years. The stated product change makes that interval less comparable; a trend decision should reflect both the calculated rates and the relevance of the underlying experience.
Using Multiple Periods to Assess a Candidate
An endpoint estimate can be sensitive to either endpoint. A multi-period method helps assess whether the pattern supports a steady trend or instead shows a change in direction. One option is a linear regression of log frequency on time, with a constant proportional trend as its model. If \(x_t\) is time in years and the fitted slope is \(b\), the model is \(\log(f_t)=a+bx_t\). It implies a frequency multiplier of \(e^b\) per year and an annual trend rate \(r_f=e^b-1\).
This regression is a descriptive method for log frequencies. Ordinary least squares gives each observation equal weight unless weights are specified; that may not be suitable when periods have very different exposures or frequency estimates have substantially different precision. A count-based alternative models \(N_t\) directly. Under an independent Poisson count model, assume \(N_t\) has mean \(\lambda_t=E_t\exp(a+bx_t)\), where exposure is the offset and \(\exp(a+bx_t)\) is the expected frequency. The model assumes the expected count is proportional to exposure and that frequency changes exponentially at a constant annual rate. Its estimated trend is \(e^b-1\). Excess variability or dependence can make the Poisson model’s uncertainty estimates inappropriate unless addressed.
Worked Example: Checking a Constant Trend Across Four Periods
A hypothetical portfolio has four consecutive annual periods with equal exposures of 25,000 units. The observed claim counts are 2,000, 2,200, 2,420, and 2,662. Use an ordinary least-squares regression of log frequency on time, with period indices \(x=0,1,2,3\), to evaluate the constant-trend candidate.
Method and assumption. Because exposures are equal, each frequency has the same denominator. For this illustrative regression, treat the four log frequencies as equally weighted observations and use the log-linear model. This is not a claim that such a model is always the best choice; it is a transparent candidate to compare with period-specific changes.
The frequencies are \(2{,}000/25{,}000=0.08000\), \(2{,}200/25{,}000=0.08800\), \(2{,}420/25{,}000=0.09680\), and \(2{,}662/25{,}000=0.10648\). Each frequency is 1.10 times the preceding one. Therefore the log frequencies increase by \(\ln(1.10)\approx0.095310\) each period.
For a direct regression check, the mean time is \(\bar{x}=1.5\), and the centered times are \(-1.5,-0.5,0.5,1.5\). Since the log frequencies rise by \(0.095310\) for every one-unit rise in time, their centered values are \(-0.142965,-0.047655,0.047655,0.142965\). The regression slope is:
The annual trend rate is \(e^{0.095310}-1=1.10-1=0.10\), or 10.0%. The calculation checks against the four observations: each adjacent frequency ratio is exactly \(1.10\). With severity held constant, a two-year frequency projection uses a cumulative factor of \(1.10^2=1.21\), not merely 1.10.
Common mistake avoided: Reporting the log-regression slope, \(0.095310\), as the trend rate. The slope is on the logarithmic scale; convert it to a proportional annual rate using \(e^b-1\). Here the exact conversion is 10.0%.
Exposure Growth Is Not Frequency Growth
Claim counts combine the amount of exposure with the frequency per exposure. When exposure changes, compare \(N_t/E_t\), not \(N_t\) alone. A raw count increase does not establish an increase in frequency, and a raw count decrease does not establish a decrease in frequency. The exposure denominator must be measured consistently, but a denominator alone cannot correct for changes in the definition or composition of the insured risk.
Worked Example: Projecting Counts and Loss Cost After Exposure Growth
A hypothetical insurer observes 900 claims on 15,000 exposure units in 2021, 1,100 claims on 20,000 units in 2022, and 1,200 claims on 24,000 units in 2023. Assume the claim definition and exposure basis are unchanged, and use the 2021 and 2023 period midpoints to estimate a constant annual trend. The midpoint gap is two years.
Method. The frequencies are \(900/15{,}000=0.0600\), \(1{,}100/20{,}000=0.0550\), and \(1{,}200/24{,}000=0.0500\). The endpoint annual trend is:
The estimated annual decrease is 8.713%. For one year beyond 2023, apply the factor \(\sqrt{5/6}\approx0.912871\) to the latest frequency: \(0.0500(0.912871)=0.0456436\) claims per exposure. At a hypothetical future exposure of 30,000 units, projected claims are \(0.0456436(30{,}000)=1{,}369.31\), rounded to two decimal places. A fractional expected count is appropriate as a projection; it is not a claim that a fraction of a claim will occur.
If average severity is held at $3,200, the projected loss cost is \(0.0456436(3{,}200)=\$146.06\) per exposure, rounded to the nearest cent. Without a frequency trend, the loss cost at the 2023 frequency would be \(0.0500(3{,}200)=\$160.00\). The trend affects projected claims and loss cost through frequency; it does not change the assumed severity.
Common mistake avoided: Using the increase from 900 to 1,200 claims as a frequency increase. Counts rose by 33.3%, but exposure rose by 60.0%; frequency fell from 0.0600 to 0.0500. The raw count comparison would give the wrong direction.
A Practical Selection Process
No single calculation decides which candidate is appropriate. Use a disciplined sequence: first verify comparability, then calculate and inspect candidate estimates, and finally select a rate that is supportable for the projection purpose. A short-term estimate may respond quickly to recent changes but can be volatile; a longer interval can stabilize a rate while including experience that no longer reflects current conditions.
Specify the claim count, exposure unit, period boundaries, and frequency definition. Confirm that the counts and exposures match.
Divide each claim count by its corresponding exposure. Inspect the sequence for unusual changes, not just the endpoints.
Calculate annualized endpoint rates for defensible intervals. Where useful, compare them with a multi-period trend model.
Explain why the selected periods and method represent the projection. Apply the annual factor once per year of projection, then keep the frequency and severity effects distinct.
A useful check is to compare the selected trend’s projected frequency with the observed values and the intended projection horizon. If the projection is implausible relative to the experience or depends on a single unusual period, revisit the period selection and assumptions. Do not select a rate solely because it produces a convenient loss cost.
Check Your Understanding
For each problem, use the stated assumptions and distinguish claim counts from frequency.
- A hypothetical book has 550 claims on 10,000 exposure units in one year and 660 claims on 12,000 units the following year. What is the annual frequency trend? (Answer: 0.0%)
- Annual frequencies in three consecutive periods are 0.0400, 0.0440, and 0.0484. Calculate the endpoint annual trend using the first and third period midpoints. (Answer: 10.0% per year)
- A book’s latest frequency is 0.055 claims per exposure. If the selected annual frequency trend is \(-5.0\%\), what expected claim count is projected for 20,000 future exposure units one year later? (Answer: 1,045 claims)
- A constant frequency trend is estimated from endpoints two years apart: 0.0500 and 0.0605 claims per exposure. What is the annualized trend? (Answer: 10.0% per year)
- A selected annual frequency trend is 4.0%. Current frequency is 0.080 claims per exposure, average severity is $2,500, and severity is assumed unchanged. What is the projected loss cost after two years? (Answer: $216.32 per exposure)