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CAS Exam 5 · Loss and Exposure Trending · Tutorial 163 of 989

Worked Projection From Historical Loss Cost

Calculate a historical pure premium, determine the correct time interval, and compound the selected annual trend to project the cost level.

Intermediate 10 min read

What You'll Learn

  • Calculate historical pure premium from aggregate losses and exposure units
  • Measure the interval between historical and prospective average dates
  • Apply a selected effective annual trend over whole and fractional years
  • Interpret the direction and size of a positive or negative trend
  • Identify common errors in compounding and choosing the trend interval

Projecting Historical Pure Premium

A historical pure premium is the average loss cost per exposure unit observed at a historical average date. Ratemaking often requires expressing that cost at a later prospective average date. A trend projection makes this adjustment by applying a selected annual loss-cost trend over the time between the two dates.

The central calculation is simple, but its result depends on two choices being handled correctly: the historical pure premium must use losses and exposure units that belong together, and the elapsed time must run from the historical average date to the prospective average date. A trend rate is not applied merely for the number of calendar years touched by the two periods.

Definition: If aggregate losses \(L\) are observed over \(E\) exposure units, the historical pure premium is \(L/E\). Given a selected constant effective annual trend rate \(t\) and an elapsed interval of \(n\) years from the historical average date to the prospective average date, the projected pure premium is
$$ \text{Projected pure premium} = \frac{L}{E}(1+t)^n. $$
Here, \(t\) is expressed as a decimal, and \(n\) may be fractional. A positive \(t\) increases the projected cost; a negative \(t\) decreases it. This formula assumes the selected trend is applicable and constant over the interval, and that the historical pure premium is an appropriate starting cost level. It does not by itself adjust for changes in coverage, exposure mix, claim development, or rate level.

A pure premium is already a loss amount divided by exposure. For this projection, the historical exposure count is used to calculate the starting pure premium; the projected amount remains a cost per exposure unit. The purpose of the trend factor is to move that unit cost from one average date to another.

Why the Trend Factor Compounds

An effective annual trend rate describes a multiplicative change over one year. If the cost at one date is \(C\), then one year later it is \(C(1+t)\), and after a second year it is \(C(1+t)^2\). Repeating this annual change for \(n\) years gives the factor \((1+t)^n\). Fractional-year projection uses the same compound model, with the fractional exponent representing the corresponding portion of the annual compounding interval.

$$ C_1=C_0(1+t),\qquad C_2=C_1(1+t)=C_0(1+t)^2,\qquad C_n=C_0(1+t)^n. $$

This is a justification under the assumption that the same effective annual rate applies throughout the interval. It is not a claim that actual costs change smoothly or uniformly in every period; the selected annual rate is the projection assumption. If the assumed annual rate changes by year, the appropriate factor is the product of the year-specific factors rather than one constant rate raised to \(n\).

Worked Projections

Worked Example: A Positive Trend Over Eighteen Months

A hypothetical portfolio of specialty service policies has $1,080,000 of historical losses on 9,000 policy-years. The historical average date is June 30, 2024, and the prospective average date is December 31, 2025. The selected effective annual loss-cost trend is 3.0%. Calculate the projected pure premium per policy-year. State the method and identify a calculation error to avoid.

Method. Calculate the historical pure premium and multiply it by the compound trend factor over the elapsed time between the average dates.

Step 1: Calculate the historical pure premium.

$$ \frac{\$1{,}080{,}000}{9{,}000\text{ policy-years}} = \$120.00\text{ per policy-year}. $$

Step 2: Find the time interval. June 30, 2024, to December 31, 2025, is 18 months, or \(18/12=1.5\) years.

Step 3: Calculate the factor and projection.

$$ (1+0.030)^{1.5}=1.03^{1.5}\approx1.0453358, $$ $$ \$120.00(1.0453358)=\$125.4403\approx\$125.44 \text{ per policy-year}. $$

The projected pure premium is $125.44 per policy-year, rounded to the nearest cent. The check is directionally consistent: a positive trend over a positive interval makes the projected value greater than $120.00.

Common mistake avoided. Applying the rate for two years because the dates occur in different calendar years would overstate the interval. The elapsed time between the specified average dates is 1.5 years, not two years.

Worked Example: A Two-and-a-Half-Year Projection

A hypothetical equipment-protection program has $840,000 of historical losses on 6,000 covered-unit years. Its historical average date is April 1, 2023; the prospective average date is October 1, 2025. The selected effective annual trend is 4.0%. Find the projected pure premium per covered-unit year.

Method. Divide losses by exposure to get historical pure premium, then compound the annual trend for the exact interval between average dates.

Step 1: Calculate the historical pure premium.

$$ \frac{\$840{,}000}{6{,}000\text{ covered-unit years}} = \$140.00\text{ per covered-unit year}. $$

Step 2: Determine the interval. April 1, 2023, to October 1, 2025, is 30 months, so \(n=30/12=2.5\) years.

Step 3: Calculate the trend factor.

$$ 1.04^{2.5}=1.04^2\sqrt{1.04} =1.0816(1.0198039) \approx1.1030199. $$

Step 4: Project the pure premium.

$$ \$140.00(1.1030199)=\$154.4228 \approx\$154.42\text{ per covered-unit year}. $$

The projected pure premium is $154.42 per covered-unit year, rounded to the nearest cent. It exceeds the historical amount, as expected under a positive trend.

Common mistake avoided. A 4.0% effective annual trend must be compounded. The simple calculation \(1+0.04(2.5)=1.10\) would produce \(\$140(1.10)=\$154.00\), which is not the effective annual compound factor. The correct factor is approximately 1.1030199.

Worked Example: Projecting With a Negative Trend

A hypothetical building-coverage program has $1,440,000 of historical losses on 8,000 building-years. The historical average date is March 31, 2025, and the prospective average date is December 31, 2026. The selected effective annual trend is \(-2.0\%\). Calculate the prospective pure premium and check whether its direction is reasonable.

Method. Use the same pure-premium and compound-factor procedure, entering the negative trend as a factor of \(1-0.02=0.98\).

Step 1: Calculate the historical pure premium.

$$ \frac{\$1{,}440{,}000}{8{,}000\text{ building-years}} = \$180.00\text{ per building-year}. $$

Step 2: Determine the time interval. March 31, 2025, to December 31, 2026, is 21 months, so \(n=21/12=1.75\) years.

Step 3: Calculate the factor and projected amount.

$$ 0.98^{1.75}\approx0.96526296, $$ $$ \$180.00(0.96526296)=\$173.7473 \approx\$173.75\text{ per building-year}. $$

The projected pure premium is $173.75 per building-year, rounded to the nearest cent. It is below the historical $180.00, which is consistent with a negative trend over a positive interval.

Common mistake avoided. Do not change a negative trend into a positive factor: \(-2.0\%\) gives an annual factor of 0.98, not 1.02. Using 1.02 would project an increase and reverse the intended direction.

A Reliable Calculation Sequence

The examples use the same sequence. Keeping the steps distinct makes it easier to catch a wrong exposure denominator, a reversed date interval, or a misapplied trend rate before reporting the result.

1
Confirm the starting unit cost.
Divide the losses by the associated exposure units. Retain the resulting unit in the answer, such as dollars per policy-year.
2
Measure the date interval.
Count from the historical average date forward to the prospective average date. Convert months to years by dividing by 12 when the dates give an exact number of months.
3
Form the annual factor.
For an annual effective trend \(t\), use \(1+t\). A negative trend therefore produces a factor below 1.
4
Project and check.
Multiply the historical pure premium by \((1+t)^n\), round only the final monetary result, and compare the direction with the sign of the trend.

Interpretation and Common Traps

The projected pure premium is an estimated future loss cost per exposure unit under the selected trend assumption. It is not a forecast of aggregate losses unless it is multiplied by a suitable prospective exposure estimate. The projection also does not establish whether the selected trend is appropriate; it applies that selection to the stated historical cost and time interval.

Several checks help prevent avoidable errors:

  • Use average dates, not period labels alone. “Historical year” and “prospective year” do not necessarily imply a one-year interval. Use the specific average dates provided.
  • Use the right elapsed time. If the interval is 18 months, use \(n=1.5\), not 18 or 2. The exponent is measured in years because the trend rate is annual.
  • Compound an effective annual rate. For a constant annual effective rate, use \((1+t)^n\), including for a fractional year. Do not substitute \(1+nt\).
  • Keep the sign of the trend. A negative trend corresponds to a factor below 1. For a positive interval it lowers, rather than raises, the projected cost.
  • Keep units and scope clear. State the projected amount per the same type of exposure unit used to calculate the historical pure premium.
Key takeaway: A historical pure premium is moved to a future average date by multiplying it by the selected annual effective trend factor raised to the elapsed time in years. Verify the starting unit cost, dates, exponent, and direction before rounding the final result. This calculation is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Check Your Understanding

For each hypothetical portfolio, calculate the historical pure premium first, then project it to the stated prospective average date. Round final dollar amounts to the nearest cent.

  1. A specialty repair portfolio has $684,000 of losses on 4,000 policy-years. The historical average date is September 30, 2024, and the prospective average date is March 31, 2026. The selected effective annual trend is 2.5%. What is the time interval and projected pure premium? (Answer: 1.5 years; $177.45 per policy-year)
  2. A hypothetical rental-equipment plan has $990,000 of losses on 5,500 covered-unit years. Its historical average date is January 1, 2023, and its prospective average date is July 1, 2024. The selected effective annual trend is 3.0%. Calculate the projected pure premium and check its direction. (Answer: 1.5 years; $188.16 per covered-unit year, above the historical $180.00)
  3. A hypothetical watercraft portfolio has $1,125,000 of losses on 7,500 boat-years. Project from January 1, 2024, to April 1, 2026, using a selected effective annual trend of 4.0%. State the elapsed time and projected pure premium. (Answer: 2.25 years; $163.84 per boat-year)
  4. A hypothetical property program has $2,400,000 of losses on 12,000 building-years. The historical average date is June 30, 2025, and the prospective average date is December 31, 2026. The selected effective annual trend is \(-1.5\%\). Calculate the projected pure premium and state whether it is above or below the historical amount. (Answer: 1.5 years; $195.52 per building-year, below the historical $200.00)
  5. A hypothetical device-protection portfolio has $1,140,000 of losses on 12,000 insured-device years. The historical average date is April 1, 2022, and the prospective average date is October 1, 2024. With a selected effective annual trend of 6.0%, calculate the interval and projected pure premium. (Answer: 2.5 years; $109.90 per insured-device year)

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