Actuarial Notes

Pure Premium Analysis

Study guide: Exam 5

Pure Premium Analysis is the examination of loss and LAE per exposure — decomposed into frequency and severity — to test rate adequacy, select trends, set classification relativities and diagnose reserve results.

\text{Pure Premium} = \frac{\text{Losses} + \text{LAE}}{\text{Earned Exposures}} = \text{Frequency} \times \text{Severity}

\text{Relativity}_i = \frac{\text{Pure Premium}_i}{\text{Pure Premium}_{\text{base}}}

  • Pure premium is the premium-free view of cost: because it never divides by premium, it needs no on-levelling and is unaffected by rate history. That makes it the natural measure for a new book, a re-tiered book, or any analysis where premium data is unreliable.
  • Comparing the projected pure premium with the rate-implied pure premium (current rate \times PLR) tests adequacy directly; a ratio above 1.0 means the rate is inadequate.
  • For classification work, pure premiums by class or territory give relativities directly. They must be credibility-weighted against the overall average, and — because a univariate pure premium picks up the effect of every correlated variable — checked against a multivariate (GLM) result before use.
  • The frequency/severity split is what makes the analysis actionable: a frequency problem calls for underwriting, classification and risk-control responses; a severity problem calls for rate, limits, deductibles and claims handling.
  • In reserving, projected pure premium is a reasonableness test: multiply the reserving actuary's ultimate losses by nothing at all — just divide by exposures — and compare the resulting cost per exposure across accident years. A series that jumps in a way trend and rate changes cannot explain points at the estimate, not the world.

Territory Relativities from Pure Premiums Three territories:

Territory Losses Exposures
Urban \$1{,}500{,}000 5{,}000
Suburban \$1{,}760{,}000 8{,}000
Rural \$450{,}000 3{,}000

Compute pure premiums and relativities to rural.

Answer

Territory Pure premium Relativity
Urban \$1{,}500{,}000/5{,}000 = \$300 2.00
Suburban \$1{,}760{,}000/8{,}000 = \$220 1.47
Rural \$450{,}000/3{,}000 = \$150 1.00

Before these become rating factors, two checks are mandatory:

  1. Credibility. Rural has 3{,}000 exposures. If that falls short of the full-credibility standard, its indicated relativity must be weighted toward the statewide average or a complement.
  2. Correlation. If urban policyholders also skew toward younger drivers or higher limits, part of the 2.00 belongs to those variables rather than to geography. A univariate analysis assigns the whole effect to territory and would double-count when the other variables are rated too — the problem multivariate methods exist to solve.

Testing Rate Adequacy by Class A commercial property book, with projected pure premiums and current rates per \$1{,}000 of insured value. The permissible loss ratio is 62\%.

Class Projected PP Current rate
A \$1.86 \$3.20
B \$2.79 \$4.00
C \$1.24 \$1.80

Which classes are mispriced, and by how much?

Answer The rate-implied pure premium is \text{rate} \times \text{PLR}:

Class Implied PP Projected PP Indicated change
A 3.20 \times 0.62 = \$1.984 \$1.86 1.86/1.984 - 1 = -6.2\%
B 4.00 \times 0.62 = \$2.480 \$2.79 2.79/2.480 - 1 = +12.5\%
C 1.80 \times 0.62 = \$1.116 \$1.24 1.24/1.116 - 1 = +11.1\%

Class A is over-priced by 6\%; B and C are under-priced by around 12\% and 11\%.

The overall indication is the exposure-weighted combination of these, and the class changes must then be off-balanced against it so that correcting the relativities does not accidentally deliver an overall increase the analysis did not call for — see Rate Change and Classification Ratemaking.