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CAS Exam 5 · Territory Rating Methods · Tutorial 442 of 989

Normalizing Relativities to a Statewide Average

Derive statewide-normalized territory factors from indicated relativities and exposure weights, and check that the resulting weighted average matches its intended target.

Intermediate 9 min read

What You'll Learn

  • Distinguish a reference-territory relativity from a factor normalized to a statewide average.
  • Select exposure weights that represent the intended statewide averaging population.
  • Derive and apply the normalization multiplier for a target weighted average.
  • Verify normalized factors using the original weights.
  • Recognize why an unweighted average or arbitrary reference factor can give the wrong result.

Why Normalize Territory Relativities?

Territory indications usually describe relative differences: one territory is indicated to have a higher or lower rate need than another. A rating plan, however, also needs factors on a defined scale. Normalizing sets that scale—for example, by requiring the exposure-weighted average territory factor to equal 1.000 or another specified statewide target.

Normalization changes the overall scale of the factors, not their relative pattern. If one raw indication is 1.5 times another, multiplying both by the same normalization constant preserves that ratio. The choice of weights matters: it determines which statewide population the normalized factors average over. This tutorial focuses on that calculation and its interpretation, not on deriving the original territory indications from experience.

Definition: Let \(r_t\) be the unnormalized indicated relativity for territory \(t\), and let \(w_t\) be its nonnegative weight, with \(\sum_t w_t=1\). A set of normalized factors \(F_t\) has target weighted average \(M\) when \(\sum_t w_t F_t=M\). For a statewide average factor of 1, \(M=1\).

From a Reference Territory to a Statewide Average

Indicated relativities are often first expressed relative to a reference territory. If territory A is selected as the reference, its raw relativity may be set to 1, with other territories shown relative to it. That convention is useful for comparison, but it does not generally make the exposure-weighted statewide average equal to 1.

Suppose the raw relativities are \(r_t\). To preserve their pattern, assume the final factors have the form \(F_t=c r_t\), where \(c\) is one common positive scaling constant. Requiring a target weighted average \(M\) gives:

$$ \sum_t w_t F_t=M \quad\Longrightarrow\quad c\sum_t w_t r_t=M \quad\Longrightarrow\quad c=\frac{M}{\sum_t w_t r_t}. $$

Therefore the normalized factor for territory \(t\) is:

$$ F_t=r_t\frac{M}{\sum_j w_jr_j}. $$

For a statewide average factor of 1, use \(M=1\). The denominator is the weighted average of the raw relativities. Dividing each raw relativity by that denominator makes the weighted average of the resulting factors exactly 1 before rounding.

Condition: The weights must be defined on the intended averaging basis and sum to 1. If territory exposure counts are \(E_t\), one common choice is \(w_t=E_t/\sum_j E_j\), so \(M\) is the exposure-weighted average factor. That identity alone does not guarantee premium neutrality for every multivariable rating plan; it describes the weighted average on the basis used to define \(w_t\).

Worked Calculations

Worked Example: Normalizing Three Indications to a Statewide Average of 1

A hypothetical insurer has three territories. Its selected raw relativities are 0.80, 1.10, and 1.50. The corresponding earned exposures are 1,000, 2,000, and 1,000 risk-years. Normalize the relativities to an exposure-weighted statewide average of 1. Report factors to three decimal places, but use unrounded values for the check.

Method. Convert exposures to weights by dividing each territory’s exposure by total exposure:

$$ E_{\text{total}}=1{,}000+2{,}000+1{,}000=4{,}000, \qquad (w_1,w_2,w_3)=(0.25,0.50,0.25). $$

The weighted average of the raw relativities is:

$$ \bar r_w=(0.25)(0.80)+(0.50)(1.10)+(0.25)(1.50) =0.200+0.550+0.375=1.125. $$

For target \(M=1\), the normalization multiplier is \(c=1/1.125=0.888888\ldots\). Apply it to each raw relativity:

$$ F_1=\frac{0.80}{1.125}=0.711111\ldots,\quad F_2=\frac{1.10}{1.125}=0.977777\ldots,\quad F_3=\frac{1.50}{1.125}=1.333333\ldots. $$

Rounded to three decimals, the factors are 0.711, 0.978, and 1.333. Check with unrounded values:

$$ (0.25)(0.711111\ldots)+(0.50)(0.977777\ldots) +(0.25)(1.333333\ldots) =0.177777\ldots+0.488888\ldots+0.333333\ldots=1. $$

The factors preserve the raw ratios; for example, \(F_3/F_1=1.50/0.80=1.875\). The common mistake avoided here is to assume that the reference scale already has statewide average 1. The raw weighted average was 1.125, so every raw relativity had to be scaled down.

Worked Example: Changing the Reference Does Not Change Normalized Factors

A hypothetical rating analyst records raw territory indices of 1.00, 1.40, and 0.70, using Territory A as the reference. Exposures are 100, 300, and 600 risk-years, respectively. Normalize to a statewide exposure-weighted average of 1. Then re-express the raw indices using Territory B as the reference and verify that the normalized results are unchanged.

Method using Territory A as reference. The exposure weights are \(0.10, 0.30, 0.60\). The raw weighted average is:

$$ (0.10)(1.00)+(0.30)(1.40)+(0.60)(0.70) =0.10+0.42+0.42=0.94. $$

The normalization multiplier is \(1/0.94=50/47\). The normalized factors are:

$$ F_A=1.00\left(\frac{50}{47}\right)=1.063829\ldots,\quad F_B=1.40\left(\frac{50}{47}\right)=1.489361\ldots,\quad F_C=0.70\left(\frac{50}{47}\right)=0.744680\ldots. $$

Method using Territory B as reference. Divide all raw indices by 1.40. The new raw indices are \(5/7,\ 1,\ 1/2\). Their weighted average is:

$$ (0.10)\left(\frac57\right)+(0.30)(1)+(0.60)\left(\frac12\right) =\frac1{14}+0.30+0.30=\frac{47}{70}. $$

Normalizing these new indices means dividing each by \(47/70\). For Territory A, for instance:

$$ \frac{5/7}{47/70}=\frac{50}{47}=1.063829\ldots. $$

Likewise, Territory B becomes \(70/47=1.489361\ldots\), and Territory C becomes \(35/47=0.744680\ldots\). These match the first calculation. Changing the reference multiplies every raw index by the same constant; subsequent normalization cancels that change. The trap is to conclude that the reference territory must have final factor 1. It has factor 1 only if the normalization target and weights happen to make that so.

Worked Example: Normalizing to a Target Other Than 1

A hypothetical statewide rate review requires the territory factors to have an exposure-weighted average of 1.03, rather than 1. The raw indications for Territories North, Central, and South are 0.80, 1.00, and 1.40. Their exposures are 2,000, 5,000, and 3,000 risk-years. Find the normalized factors to three decimals and verify the target.

Method. Total exposure is \(2{,}000+5{,}000+3{,}000=10{,}000\), so the weights are 0.20, 0.50, and 0.30. The raw weighted average is:

$$ \bar r_w=(0.20)(0.80)+(0.50)(1.00)+(0.30)(1.40) =0.16+0.50+0.42=1.08. $$

The multiplier that changes this average from 1.08 to 1.03 is \(c=1.03/1.08=103/108=0.953703\ldots\). Thus:

$$ F_N=(0.80)\left(\frac{103}{108}\right)=0.762962\ldots,\quad F_C=(1.00)\left(\frac{103}{108}\right)=0.953703\ldots,\quad F_S=(1.40)\left(\frac{103}{108}\right)=1.335185\ldots. $$

The factors rounded to three decimals are 0.763, 0.954, and 1.335. Before rounding, the weighted-average check follows directly:

$$ (0.20)F_N+(0.50)F_C+(0.30)F_S =\frac{1.03}{1.08}\big[(0.20)(0.80)+(0.50)(1.00)+(0.30)(1.40)\big] =\frac{1.03}{1.08}(1.08)=1.03. $$

This checks both the factor calculation and the selected target. A common error is to divide by the raw average when the target is 1.03. That would produce an average of 1, not 1.03; the correct multiplier is target divided by raw weighted average.

Choosing and Checking the Weights

For exposure-weighted normalization, the exposure definition must be consistent across territories. If one territory’s exposure is earned risk-years and another’s is policy count, the resulting weights do not represent a coherent statewide average. Confirm the exposure base and valuation period before calculating weights. Also verify that all included territories are represented in both the indications and the exposure totals.

The weights answer a substantive question: what mix of territory exposures should the statewide average represent? If the intended population is the observed portfolio, exposure shares from that portfolio are a natural basis. If the purpose is to normalize to a specified target or another explicitly defined population, use the weights and target associated with that purpose. Do not silently substitute equal weights just because there are several territories.

A useful normalization check is to calculate \(\sum_t w_tF_t\) with unrounded factors and confirm it equals \(M\). If it does not, investigate the weights, the denominator, and the application of the common multiplier. Small differences after rounding are expected; the unrounded result should satisfy the identity.

1
Set the scale.
State the intended weighted average \(M\), such as 1.000 for a statewide average factor of one.
2
Define the population weights.
Calculate each \(w_t\) on one consistent basis and check that \(\sum_t w_t=1\).
3
Find the raw weighted average.
Calculate \(\sum_t w_tr_t\) from the unnormalized indications.
4
Scale and verify.
Set \(F_t=r_tM/\sum_jw_jr_j\), then check \(\sum_t w_tF_t=M\) before rounding.

Common Exam Traps and Interpretation

  • Using an unweighted mean. Equal territory weights are appropriate only when equal weighting is the intended averaging basis. With unequal exposures, use the specified exposure shares.
  • Confusing the reference factor with the statewide average. A raw reference index of 1 is a scale convention. It does not imply that the weighted average of all raw indices is 1.
  • Normalizing to the wrong target. Dividing by the raw weighted average forces the result to average 1. For a different target \(M\), multiply by \(M/\sum_t w_tr_t\).
  • Rounding too early. Keep full precision through the weighted average, multiplier, and verification. Round only the final displayed factors.
  • Over-interpreting the result. A weighted average of 1 means the factors average to 1 on the chosen weighting basis. It does not establish that every territory is adequately priced or that the total premium under all other rating variables is unchanged.

The core principle is straightforward: choose the intended statewide average and its weights, then apply one common scaling constant to all indicated relativities. That preserves the relative territory pattern while placing the factors on the required statewide scale. Normalization is therefore a scale adjustment, not a revision to the underlying indicated differences.

Check Your Understanding

For each problem, retain full precision until the final factor calculation and check the weighted average against the stated target.

  1. A hypothetical program has raw territory relativities 0.75 and 1.25, with exposures of 600 and 400 risk-years. Normalize to a weighted average of 1. Give factors to three decimals. (Answer: 0.789 and 1.316.)
  2. Three territories have raw indices 0.90, 1.20, and 1.80, with exposure proportions 0.20, 0.30, and 0.50. Find the factors normalized to a weighted average of 1, to three decimals. (Answer: 0.625, 0.833, and 1.250.)
  3. Raw indications are 0.80, 1.00, and 1.40 for territories with exposures in the ratio 2:5:3. Normalize to a weighted average of 1.06. Give factors to three decimals. (Answer: 0.785, 0.981, and 1.374.)
  4. A hypothetical analyst uses Territory A as reference, with raw indices 1.00, 1.40, and 0.70 for A, B, and C. Exposures are 100, 300, and 600. Normalize to an exposure-weighted average of 1 and report factors to three decimals. (Answer: A = 1.064, B = 1.489, C = 0.745.)

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