Worked Example of Loss Cost Relativities

Why Loss Cost Relativities Need Careful Normalization

Territorial experience can be summarized as a loss cost: losses per exposure unit. Comparing those costs shows how the observed loss levels differ among territories. To use the comparisons as rating factors, an actuary typically expresses them as relativities and normalizes them against the exposure distribution of the portfolio being rated.

The order matters. First calculate each territory’s loss cost on a consistent basis. Then divide each cost by a benchmark cost to obtain raw relativities. Finally, normalize the relativities using the relevant exposure weights. Normalization changes the overall scale of the factors, but not the ratios between territories.

Definition: For territory $t$, the loss cost is the relevant aggregate loss amount divided by the corresponding exposure units. A raw loss cost relativity compares that territory’s loss cost with the loss cost in a selected benchmark territory. A normalized territory factor is a scaled relativity, conventionally chosen so its exposure-weighted average across the territories is one.

Set the Basis Before Calculating

A loss cost comparison is meaningful only when the territory results are comparable. Before calculating, specify the exposure unit and ensure that the losses across territories are measured on a consistent basis. For example, the hypothetical examples below treat the stated losses as already adjusted to a common coverage and loss level. No additional trend, development, or expense adjustment is being calculated here.

The exposure weights should reflect the population for which the territory factors are being normalized. If territory $t$ has $E_t$ exposure units and loss amount $L_t$, its loss cost is $C_t=L_t/E_t$. If $b$ is the chosen benchmark territory, its raw relativity is $r_t=C_t/C_b$, so $r_b=1$. Let $E=\sum_t E_t$ be total exposure and $w_t=E_t/E$ be territory $t$’s exposure share.

C_t=\frac{L_t}{E_t}, \qquad r_t=\frac{C_t}{C_b}, \qquad w_t=\frac{E_t}{\sum_j E_j}.

The normalized factor is the raw relativity divided by its exposure-weighted average. This gives each territory a factor on a common scale while retaining the relative differences indicated by the loss costs.

\bar r_w=\sum_t w_t r_t, \qquad F_t=\frac{r_t}{\bar r_w}.

Why the Normalization Works

The normalization is not an arbitrary rescaling. It ensures that applying the factors to a common base loss cost reproduces the portfolio’s overall loss cost, when the weights are the same exposure shares used in the normalization.

To see this, define the overall loss cost as $C_{\text{all}}=\sum_t L_t/\sum_t E_t$. Since $L_t=E_tC_t$, this is the exposure-weighted average of the territorial loss costs. Also, $C_t=C_b r_t$. Substituting that relationship into the overall loss cost gives $C_{\text{all}}=C_b\sum_t w_t r_t=C_b\bar r_w$. Therefore, the normalized factor can also be written as $F_t=C_t/C_{\text{all}}$.

C_{\text{all}} =\frac{\sum_t E_tC_t}{\sum_t E_t} =\sum_t w_tC_t =C_b\sum_t w_tr_t =C_b\bar r_w, \qquad F_t=\frac{r_t}{\bar r_w} =\frac{C_t}{C_{\text{all}}}.

The factors have an exposure-weighted average of one because $\sum_t w_tF_t=\sum_t w_tr_t/\bar r_w=1$. They also preserve the raw relativities: for any two territories $s$ and $t$, $F_s/F_t=r_s/r_t=C_s/C_t$. The benchmark choice affects the scale of the raw relativities, but not the normalized factors, provided the benchmark loss cost is positive.

Calculate territory loss costs.
Divide each territory’s comparable loss amount by its exposure units.

Calculate raw relativities.
Divide each territory loss cost by the selected benchmark loss cost.

Calculate the weighted average relativity.
Multiply each raw relativity by its exposure share and sum the products.

Normalize and verify.
Divide each raw relativity by the weighted average. Check that the weighted factor average is one and that pairwise factor ratios match the raw relativity ratios.

Worked Calculations

In each example, the data are hypothetical. Loss amounts are assumed to be on the same basis, and the listed exposures are the exposure units used for both the loss costs and normalization. Intermediate calculations are retained at full precision; displayed factors are rounded only at the end.

Worked Example: Normalizing Three Territory Loss Costs

A hypothetical insurer has the following territory experience. Calculate the loss costs, use Territory R as the benchmark, and normalize the raw relativities using the listed exposure distribution.

Territory	Exposure units	Losses
R	1,000	$125,000
S	3,000	$300,000
T	2,000	$350,000

Step 1: Calculate loss costs. For R, $C_R=125{,}000/1{,}000=\$125$. For S, $C_S=300{,}000/3{,}000=\$100$. For T, $C_T=350{,}000/2{,}000=\$175$.

Step 2: Calculate raw relativities to R. The benchmark has $r_R=125/125=1$. The other relativities are $r_S=100/125=0.8$ and $r_T=175/125=1.4$.

Step 3: Calculate exposure shares and their weighted relativity. Total exposure is $1{,}000+3{,}000+2{,}000=6{,}000$. The weighted relativity is $(1{,}000(1)+3{,}000(0.8)+2{,}000(1.4))/6{,}000=(1{,}000+2{,}400+2{,}800)/6{,}000=6{,}200/6{,}000=31/30$.

Step 4: Normalize. Divide each raw relativity by $31/30$. Thus $F_R=30/31$, $F_S=24/31$, and $F_T=42/31$. Rounded to four decimal places, the factors are 0.9677, 0.7742, and 1.3548, respectively.

Check. The exposure-weighted factor average is $(1{,}000(30/31)+3{,}000(24/31)+2{,}000(42/31))/6{,}000 =(30{,}000+72{,}000+84{,}000)/(31(6{,}000))=1$. Also, $F_S/F_R=(24/31)/(30/31)=0.8$, and $F_T/F_R=(42/31)/(30/31)=1.4$, matching the raw relativities. The overall loss cost is $(125{,}000+300{,}000+350{,}000)/6{,}000=\$129.1667$, which gives $129.1667F_R=\$125$, $129.1667F_S=\$100$, and $129.1667F_T=\$175$, up to rounding. The method is exposure-weighted normalization; it avoids treating the three territories as if they had equal exposure.

Worked Example: A Different Benchmark, the Same Normalized Factors

A hypothetical program has these loss costs: Territory A has 800 exposures and $96,000 in losses; B has 1,500 exposures and $210,000 in losses; C has 700 exposures and $126,000 in losses. Calculate normalized factors first using B as the benchmark, then verify the result using the portfolio average loss cost.

Step 1: Calculate loss costs. $C_A=96{,}000/800=\$120$, $C_B=210{,}000/1{,}500=\$140$, and $C_C=126{,}000/700=\$180$.

Step 2: Calculate raw relativities to B. $r_A=120/140=6/7$, $r_B=1$, and $r_C=180/140=9/7$.

Step 3: Find the weighted average relativity. Total exposure is $800+1{,}500+700=3{,}000$. Hence $\bar r_w=(800(6/7)+1{,}500(1)+700(9/7))/3{,}000=(4{,}800/7+10{,}500/7+6{,}300/7)/3{,}000=21{,}600/(7(3{,}000))=36/35$.

Step 4: Normalize. $F_A=(6/7)/(36/35)=5/6$, $F_B=1/(36/35)=35/36$, and $F_C=(9/7)/(36/35)=5/4$. Rounded to four decimal places, these are 0.8333, 0.9722, and 1.2500.

Check. The weighted factor average is $(800(5/6)+1{,}500(35/36)+700(5/4))/3{,}000=(666.6667+1{,}458.3333+875)/3{,}000=1$, with the exact fractions summing to 3,000 before division. The overall loss cost is $432{,}000/3{,}000=\$144$. Multiplying by the factors gives $\$120,\$140,\$180$, respectively. This check uses the overall average as the base and shows why the normalized factors do not depend on choosing B as the benchmark.

Worked Example: Checking Factors Against a Portfolio Average

A hypothetical insurer records 450 exposures and $81,000 of losses in X, 900 exposures and $126,000 in Y, and 650 exposures and $130,000 in Z. Use Y as the benchmark. Calculate the normalized factors, then check their relationship to the overall loss cost.

Step 1: Calculate loss costs. $C_X=81{,}000/450=\$180$, $C_Y=126{,}000/900=\$140$, and $C_Z=130{,}000/650=\$200$.

Step 2: Calculate raw relativities. Relative to Y, $r_X=180/140=9/7$, $r_Y=1$, and $r_Z=200/140=10/7$.

Step 3: Calculate the weighted average relativity. Total exposure is $450+900+650=2{,}000$. Therefore $\bar r_w=(450(9/7)+900+650(10/7))/2{,}000=(4{,}050/7+6{,}300/7+6{,}500/7)/2{,}000=16{,}850/14{,}000=337/280$.

Step 4: Normalize. $F_X=(9/7)/(337/280)=360/337$, $F_Y=280/337$, and $F_Z=400/337$. Rounded to four decimal places, the factors are 1.0682, 0.8309, and 1.1869.

Check. The weighted factor sum is $(450(360)+900(280)+650(400))/337=(162{,}000+252{,}000+260{,}000)/337=674{,}000/337=2{,}000$, so its average is one. Total losses are $337,000, giving an overall loss cost of $337{,}000/2{,}000=\$168.50$. Multiplying $168.50 by the exact factors gives $180, $140, and $200. The method is direct exposure-weighted normalization; the check confirms that the normalized factors recover each territory’s indicated loss cost from the portfolio average.

Common Mistakes and Exam Traps

A frequent error is to calculate a simple average of the raw relativities instead of an exposure-weighted average. In the first example, the simple average would be $(1+0.8+1.4)/3=1.0667$. That is not the correct normalizing divisor for exposures of 1,000, 3,000, and 2,000. The correct divisor is $31/30=1.0333$. Using the simple average would not make the exposure-weighted average of the resulting factors equal one.

Another trap is to confuse the benchmark relativity with the normalized benchmark factor. The benchmark’s raw relativity is one by definition, but after normalization its factor generally is not one. In the first example, R’s normalized factor is $30/31$, while its raw relativity remains one. Normalization scales every factor, not just the non-benchmark territories.

Do not round the weighted average or raw relativities early. Keep exact fractions or sufficient precision through normalization, then round the displayed factors. Finally, verify both properties: the exposure-weighted factor average is one, and the factor ratios agree with the raw relativity ratios. These checks catch arithmetic errors that a plausible-looking set of factors can conceal.

Key takeaway: Calculate loss costs on a consistent basis, form raw relativities against a benchmark, and divide every relativity by the exposure-weighted average relativity. This produces factors with a weighted average of one while preserving the territorial loss cost comparisons.

Check Your Understanding

For each problem, use the stated exposure units as the normalization weights. Retain exact values until the final factor calculation.

A hypothetical book has 400 exposures with $36,000 of losses in A and 600 exposures with $90,000 in B. Use A as benchmark. Find the loss costs and normalized factors, rounded to four decimal places. (Answer: Loss costs are $90 and $150; factors are 0.7143 for A and 1.1905 for B.)
Territory D has 500 exposures and $40,000 in losses, E has 1,000 exposures and $100,000, and F has 500 exposures and $80,000. Use E as benchmark. Calculate and normalize the raw relativities. (Answer: Loss costs are $80, $100, and $160; normalized factors are $8/11$, $10/11$, and $16/11$, or 0.7273, 0.9091, and 1.4545.)
A hypothetical portfolio has loss costs of $110 in J and $165 in K, with exposures of 1,200 and 800, respectively. Use J as benchmark. Find the raw relativities, the normalizing divisor, and the normalized factors. (Answer: Raw relativities are 1 and 1.5; the divisor is 1.2; normalized factors are 0.8333 and 1.2500.)
Three territories have loss costs of $75, $100, and $125, with exposures of 500, 1,000, and 500. Using the middle territory as benchmark, calculate normalized factors and check their exposure-weighted average. (Answer: Raw relativities are 0.75, 1, and 1.25; the divisor is 1; normalized factors are 0.7500, 1.0000, and 1.2500; their exposure-weighted average is 1.)
A territory has a raw relativity of 1.2 and an exposure share of 0.25. The remaining territories have a combined exposure share of 0.75 and a combined exposure-weighted raw relativity contribution of 0.90. Find the normalizing divisor and the territory’s normalized factor. (Answer: The divisor is $0.25(1.2)+0.90=1.20$; the normalized factor is $1.2/1.20=1.0000$.)

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