Applying Frequency Trends to Future Costs

Why Frequency Trends Need an Explicit Time Interval

A historical claim frequency describes claims per unit of exposure for a particular period or representative date. A frequency trend projects that rate to a different time. The arithmetic may be short, but a sound projection depends on defining when the historical frequency applies, when the target period applies, and how much time separates them. An unclear interval can make a correctly applied trend rate produce the wrong result.

This tutorial focuses on projecting frequency to a future cost period. It does not project severity or combine frequency and severity trends. The frequency-severity framework treats expected claim count as frequency multiplied by exposure; therefore, the frequency must be expressed per the same exposure unit used for the target period. The method is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: Frequency is the number of claims divided by a stated exposure base, such as claims per vehicle-year. A frequency trend is a selected rate of change in frequency over time. A projected frequency is the historical frequency adjusted from its representative date to the target period’s representative date using the selected trend assumptions.

For the calculations below, an annual trend rate \(r\) is interpreted as a multiplicative change per year. Thus, a 4% annual trend means multiplying frequency by \(1.04\) for each year of projection, not adding 0.04 claims per exposure unit. For a fractional year, this tutorial assumes annual compounding, so a duration of \(t\) years uses the factor \((1+r)^t\). No interest rate, mortality assumption, or claim-count probability distribution is required for these deterministic projections of expected frequency and expected count.

Projecting Frequency Across One or More Trend Periods

Let \(f_0\) be the historical frequency at its representative date. If one annual trend rate \(r\) applies for \(t\) years, the projected frequency is \(f_0(1+r)^t\). If the applicable rate changes during the projection interval, apply each rate to its own duration and multiply the factors. The durations must be measured in years and must span the interval from the historical representative date to the target representative date.

f_{\text{target}} = f_0(1+r)^t

f_{\text{target}} = f_0\prod_{j=1}^{k}(1+r_j)^{t_j}, \qquad \sum_{j=1}^{k}t_j=t

Here \(r_j\) is the annual trend rate for segment \(j\), and \(t_j\) is the length of that segment in years. A negative trend is represented by a negative \(r_j\), provided \(r_j>-1\). The product is multiplicative: later trend rates act on the frequency already changed by earlier rates.

The historical and target representative dates are often the midpoints of the respective experience and cost periods. Using midpoints is a convention for representing each period by one date; it does not mean that every claim actually occurs at that date. Once the dates are specified, calculate their separation and align the trend segments to that separation. Do not assume that a period described as “two years of experience” necessarily requires two years of trending: the interval depends on the representative dates and the target date.

Establish the historical frequency.
Confirm the claim count, exposure denominator, and frequency basis. For pooled data, calculate claims divided by total exposure.

Set the dates and durations.
Identify the historical and target representative dates. Express each intervening trend segment as a duration in years.

Apply the selected trend factors.
Multiply the historical frequency by each annual factor raised to its segment duration.

Calculate expected target count, if requested.
Multiply the unrounded projected frequency by target-period exposure on the same basis.

The exposure denominator is not itself trended by this calculation. A change in the amount of target exposure changes expected count, while the frequency trend changes the rate per unit of exposure. Keeping these effects separate makes the projection interpretable.

Worked Frequency Projections

Worked Example: Two Annual Trend Rates Over a Fractional Interval

A hypothetical insurer estimates historical frequency of 0.06 claims per insured unit-year at a representative date. A 4% annual frequency trend applies for the first year, followed by a 2% annual trend for the next 18 months. The target period has 6,000 insured-unit-years of exposure. Find the projected frequency and expected claim count.

Method and assumptions. This is a piecewise compound frequency projection. The second duration is \(18/12=1.5\) years. The trend intervals total \(1+1.5=2.5\) years from the historical representative date to the target representative date. No interest rate, mortality model, or claim-count distribution is used.

Step 1: Calculate the piecewise trend factor. The first-year factor is \(1+0.04=1.04\). The next 1.5-year factor is \(1.02^{1.5}\), approximately \(1.0301495\). Therefore,

1.04(1.02)^{1.5} =1.04(1.0301495) \approx 1.0713555

Step 2: Project the frequency.

f_{\text{target}} =0.06(1.0713555) \approx 0.0642813 \text{ claims per insured unit-year}

Step 3: Calculate the expected count. Use the unrounded projected frequency and the target exposure in insured-unit-years.

E[N_{\text{target}}] =0.0642813(6{,}000) \approx 385.688 \text{ claims}

Answer and check. The projected frequency is \(0.0643\) claims per insured unit-year to four decimal places, and the expected count is \(385.7\) claims to one decimal place. The factor exceeds 1, so the projected frequency is higher than 0.06, as expected for two positive trend rates. The common mistake avoided here is reporting 18 months as 1.8 years or rounding frequency too early before multiplying by exposure.

Worked Example: Deriving Historical Frequency Before Trending

A hypothetical specialty vehicle program recorded 84 claims over 2,400 vehicle-years of exposure. Its historical frequency is to be projected 2.5 years from the historical representative date to the target representative date, using a 3% annual trend throughout. The target period is expected to have 3,200 vehicle-years. Find the historical frequency, projected frequency, and expected target count.

Method and assumptions. First calculate pooled frequency as total claims divided by total exposure. Then compound the selected 3% annual rate for 2.5 years. The 3,200 vehicle-years are target exposure, not an additional trend duration. No interest rate, mortality model, or claim-count distribution is used.

Step 1: Calculate historical frequency.

f_0=\frac{84}{2{,}400} =0.035 \text{ claims per vehicle-year}

Step 2: Calculate the trend factor.

(1.03)^{2.5} =(1.03)^2\sqrt{1.03} \approx 1.0766959

Step 3: Project frequency and expected count.

f_{\text{target}} =0.035(1.0766959) \approx 0.0376844 \text{ claims per vehicle-year}

E[N_{\text{target}}] =0.0376844(3{,}200) \approx 120.590 \text{ claims}

Answer and check. The projected frequency is \(0.03768\) claims per vehicle-year to five significant figures, and the expected count is \(120.6\) claims to one decimal place. Since the trend is positive, the projected frequency exceeds the historical 0.035. The common mistake avoided is treating the historical count of 84 as a frequency; the exposure denominator must first be used to obtain claims per vehicle-year.

Worked Example: A Trend Rate That Changes During the Projection

A hypothetical property portfolio has historical frequency of 0.08 claims per insured location-year. The selected annual frequency trend is \(-5\%\) during the first year after the historical representative date and \(+10\%\) during the following year. The target period has 2,500 insured location-years of exposure. Calculate the target frequency and expected count.

Method and assumptions. This is a two-segment compound projection, with one year at each selected rate. The later positive trend is applied to the frequency after the first segment; the rates are not added together. No interest rate, mortality model, or claim-count distribution is used.

Step 1: Apply the first-year trend.

$$ 0.08(1-0.05) =0.08(0.95) =0.076 $$

Step 2: Apply the next year’s trend.

f_{\text{target}} =0.076(1+0.10) =0.076(1.10) =0.0836 \text{ claims per insured location-year}

Step 3: Calculate expected count.

E[N_{\text{target}}] =0.0836(2{,}500) =209 \text{ claims}

Answer and check. The projected frequency is \(0.0836\) claims per insured location-year, and the expected count is 209 claims. The overall factor is \(0.95(1.10)=1.045\), so the final frequency is 4.5% above the historical frequency. The common mistake avoided is adding \(-5\%\) and \(10\%\) to claim a net 5% trend. Sequential multiplicative changes produce a 4.5% net increase instead.

Why the Projection Formula Works

The compound formula follows from the meaning of an annual multiplicative trend. If the frequency at the start of a one-year interval is \(f_0\), a trend rate \(r\) gives frequency \(f_0(1+r)\) at the end of that year. Applying the same change for a second year gives \(f_0(1+r)(1+r)=f_0(1+r)^2\). Repeating this multiplication for \(n\) years gives \(f_0(1+r)^n\). Under the stated annual-compounding convention, the corresponding factor over a fractional duration \(t\) is \((1+r)^t\).

For piecewise rates, the same reasoning applies segment by segment. After the first segment, frequency is \(f_0(1+r_1)^{t_1}\); applying the second segment multiplies that result by \((1+r_2)^{t_2}\). Continuing through \(k\) segments yields the product formula. The order does not change the numerical product, but the rates must still be paired with their correct durations and chronological segments.

Expected count follows from the definition of frequency. If the expected number of claims per unit of exposure is \(f_{\text{target}}\), then for \(E_{\text{target}}\) units of exposure the expected count is

E[N_{\text{target}}] =f_{\text{target}}E_{\text{target}}

This calculation gives an expected count, which may be noninteger. It does not assert that the realized count must equal that expectation. Nor does the formula specify a probability distribution for the count.

Common Errors and Practical Checks

Frequency trends are easy to misapply when time and exposure are not kept distinct. Before accepting a result, check the interval, units, and direction of change.

Using the wrong time unit. Convert months to years when the rate is annual: 18 months is 1.5 years, not 18 years or 1.8 years.
Adding trend percentages. Sequential rates multiply. For example, a 5% decrease followed by a 10% increase has factor \(0.95(1.10)=1.045\), not a 1.05 factor.
Applying trend to exposure. Frequency is trended; target exposure is then multiplied by projected frequency. Changing exposure changes expected count, not the frequency rate itself.
Ignoring representative dates. The interval is the elapsed time between the historical and target representative dates, not automatically the full length of either data period.
Rounding too early. Retain sufficient precision in trend factors and frequency until the expected count has been calculated; round only the reported result.

Key takeaway: State the historical frequency basis, historical and target representative dates, each trend rate and duration, and the target exposure basis. Then compound the rates over their respective intervals and multiply the resulting frequency by target exposure only if an expected count is required.

Check Your Understanding

For each problem, state the trend interval explicitly and retain unrounded values until the final answer.

A hypothetical insurer’s frequency is 0.045 claims per equipment-year at the historical representative date. A 4% annual trend applies for three years, and target exposure is 2,000 equipment-years. Find the projected frequency and expected count. (Answer: frequency 0.05062 claims per equipment-year; expected count 101.2 claims)
A portfolio recorded 120 claims over 3,000 vehicle-years. Project the frequency 18 months using a 2% annual trend. If target exposure is 5,000 vehicle-years, find the projected frequency and expected count. (Answer: frequency 0.04121 claims per vehicle-year; expected count 206.0 claims)
A hypothetical book has frequency 0.10 claims per policy-year. A \(-3\%\) annual trend applies for one year, followed by a \(+2\%\) annual trend for one year. Target exposure is 2,400 policy-years. Find the projected frequency and expected count. (Answer: frequency 0.09894 claims per policy-year; expected count 237.5 claims)
A historical period has 63 claims over 1,800 exposure units. From its representative date to the target representative date, a 6% annual trend applies for six months and a 2% annual trend applies for the following 18 months. Target exposure is 4,000 units. Find the projected frequency and expected count. (Answer: frequency 0.03712 claims per exposure unit; expected count 148.5 claims)
A hypothetical coverage has frequency 0.03 claims per insured-year at the historical representative date. The target representative date is two years later, and the selected annual frequency trend is 5% throughout. Find the projected frequency and expected count for 7,000 insured-years. (Answer: frequency 0.03308 claims per insured-year; expected count 231.5 claims)

What You'll Learn