Pure Premium Method

The pure premium method calculates the indicated rate directly. The indicated rate per exposure equals (projected ultimate losses + LAE) / earned exposures, plus fixed expenses per exposure, all divided by (1 minus variable expense ratio minus profit and contingency factor). This method requires reliable loss and exposure data by class but does not require current rate information. It produces an indicated rate level directly.

The formula can be written as: Indicated Rate = (Pure Premium + Fixed Expense per Exposure) / (1 - V - Q), where Pure Premium = (Projected Ultimate Losses + LAE) / Earned Exposures, V is the variable expense ratio, and Q is the profit and contingency factor. This structure separates the loss component from the expense and profit components, making it clear how each piece contributes to the final rate.

When to Use the Pure Premium Method

The pure premium method is preferred when you have reliable exposure data and can compute loss costs per unit of exposure directly. It is especially useful when setting rates for a new line of business where no current rates exist, or when rates need to be calculated from scratch for a new classification. Since it does not require knowledge of current premium levels, it avoids the complications of on-leveling historical premiums.

For example, suppose a workers compensation class has 500 exposure units (payroll in hundreds), projected ultimate losses plus LAE of 180,000, fixed expenses of 40 per exposure, a variable expense ratio of 22%, and a profit target of 3%. The pure premium is 180,000/500 = 360 per exposure. The indicated rate is (360 + 40)/(1 - 0.22 - 0.03) = 400/0.75 = 533.33 per exposure unit.

Loss Ratio Method

The loss ratio method calculates an indicated rate change relative to current rates. The indicated change factor equals the projected loss and LAE ratio divided by the target (permissible) loss ratio. The target loss ratio equals 1 minus the total expense ratio minus the profit provision.

The formula is: Indicated Rate Change Factor = Actual Loss Ratio / Target Loss Ratio. The indicated percentage change is this factor minus 1. If the indicated change factor is 1.12, rates should increase by 12%. If it is 0.95, rates should decrease by 5%.

On-Leveling Premiums

The loss ratio method requires on-level earned premium, which means adjusting historical premiums to reflect the current rate level. If rates changed during a policy period, different policies within the same experience period were charged different rates. The on-level factor adjusts all premiums to a common (current) rate level.

The parallelogram method handles mid-year rate changes by computing the proportion of earned premium at each rate level using the geometry of a unit square. For a rate change effective partway through the year, the earned premium at the old rate level covers a triangular or trapezoidal region of the parallelogram. The on-level factor equals the current rate level divided by the weighted average rate level for that experience period.

For example, if rates increased by 8% effective July 1 in a calendar year experience period with uniform policy writings, approximately half the earned premium is at the old level and half at the new level. The average rate level factor for that year is 0.5 * 1.00 + 0.5 * 1.08 = 1.04. If the current level is 1.08, the on-level factor is 1.08/1.04 = 1.0385, meaning historical premiums must be multiplied by 1.0385 to bring them to the current rate level.

Loss Development and Trend

Both methods require projecting losses to ultimate and trending them to the future policy period. Loss development adjusts reported losses for claims that have not yet been fully settled, using development factors from loss triangles. Loss trend adjusts for inflation and other systematic changes in loss levels over time. Frequency and severity trends may be applied separately, with the total trend factor being the product of (1 + frequency trend)^n and (1 + severity trend)^n, where n is the number of years from the historical period to the future effective period.

Comparing the Two Methods

The pure premium method and loss ratio method are algebraically equivalent when applied to the same data with consistent assumptions. The pure premium method produces an indicated rate directly, while the loss ratio method produces an indicated rate change. In practice, the loss ratio method is more commonly used because it leverages existing rate structures and avoids the need for detailed exposure counts by class.

The pure premium method is preferred when exposure data is more reliable than premium data, when no current rates exist, or when rates are being established for a new product or classification. The loss ratio method is preferred when premium data is readily available and exposure data may be inconsistent or difficult to obtain.

Common Pitfalls

A frequent error is forgetting to on-level premiums before computing loss ratios. Using historical premiums at varying rate levels distorts the loss ratio and produces incorrect rate indications. Another mistake is double-counting trend by applying it to both losses and premiums. Trend should be applied to losses (or loss ratios) but not to already on-leveled premiums.

Students also sometimes confuse the target loss ratio with the actual loss ratio. The target loss ratio is a function of expense and profit provisions; the actual loss ratio is computed from historical data. The ratio of actual to target drives the indicated change.

Study Tips

Exam STAM problems often require adjusting historical premiums to current rate levels using parallelogram or other on-leveling techniques before applying the loss ratio method. Practice computing on-level factors for various rate change scenarios. Work through problems that require both loss development and trending before computing the final rate indication. Make sure you can go from raw historical data to a final indicated rate or rate change using either method.

Rate Making: Pure Premium and Loss Ratio Methods