Diagnostics for Territorial Indications

Why Diagnose Territorial Indications?

A territorial indication is a measured result, not automatically a reliable explanation of why losses differ. A territory can look unusually costly because it has a different mix of classes, a small number of large claims, or a data or adjustment issue. Diagnostics help an actuary identify these possibilities before treating an indication as evidence of a persistent difference in risk.

The comparisons below assume losses are on a consistent basis for coverage, claim development, and cost level, and that exposures are measured consistently. Where those conditions are not met, differences may reflect the basis of the data rather than territorial risk. The aim is to investigate indications, not to replace the credibility assessment or factor-selection process.

Definition: For territory $t$ and class $c$, let $e_{tc}$ denote earned exposures and $L_{tc}$ the corresponding adjusted aggregate losses. The cell loss cost is $m_{tc}=L_{tc}/e_{tc}$, provided $e_{tc}>0$. The territory crude loss cost is total adjusted losses divided by total exposures. A mix-adjusted comparison recalculates territorial results using the same class weights for each territory. A factor residual compares observed experience with the experience expected under a selected territory factor and a stated base.

Exposure counts are central to these calculations. A territory with more exposures will generally have more total losses even if its loss cost is lower. Comparing aggregate losses without accounting for exposures confounds the volume of business with its average cost.

Crude Results and Class Mix

Let $E_t=\sum_c e_{tc}$ be total territory exposure. The crude loss cost is

m_t=\frac{\sum_c L_{tc}}{E_t} =\sum_c \left(\frac{e_{tc}}{E_t}\right)m_{tc}.

This identity shows why crude territory comparisons can mislead: the weights $e_{tc}/E_t$ vary by territory. If classes have different underlying loss costs, two territories with identical class-specific results can still have different crude loss costs simply because their class mixes differ.

A direct standardization uses a common set of weights $q_c$, where $q_c\geq 0$ and $\sum_c q_c=1$. The standardized loss cost for territory $t$ is $M_t^{\mathrm{std}}=\sum_c q_c m_{tc}$. If $b_c$ is a chosen reference loss cost for class $c$, a corresponding standardized relativity is

R_t^{\mathrm{std}} =\frac{\sum_c q_c m_{tc}}{\sum_c q_c b_c}.

The reference costs and common weights must be stated: changing either can change the standardized result. In particular, $R_t^{\mathrm{std}}$ is not generally the simple weighted average of the class relativities $m_{tc}/b_c$. That average would use weights $q_c$, whereas the displayed standardized relativity weights each class cost by $q_c$ and divides by the common weighted reference cost.

Why standardization works: The crude loss cost weights each class by its share of that territory's exposures. Those shares can differ across territories. Replacing them with a single common set $q_c$ removes that particular source of difference. It does not remove differences within a class, and it does not make an unreliable or incomplete data set reliable.

Worked Example: A Crude Comparison Reversed by Class Mix

A hypothetical insurer compares two territories with low- and high-risk classes. Losses are adjusted to a common basis. Territory A has 900 low-risk exposures with $90,000 of losses and 100 high-risk exposures with $45,000 of losses. Territory B has 100 low-risk exposures with $11,000 of losses and 900 high-risk exposures with $279,000 of losses. Use equal common weights for the two classes and reference loss costs of $100 and $300, respectively. Compare the crude and standardized results.

Method. First calculate cell and crude loss costs using the given exposure counts. Then apply the same 50%/50% class weights to each territory.

For A, the low-risk cell cost is $90{,}000/900=\$100$, and the high-risk cell cost is $45{,}000/100=\$450$. Total losses are $\$135{,}000$ and total exposure is $900+100=1{,}000$, so the crude cost is $\$135{,}000/1{,}000=\$135$.

For B, the low-risk cell cost is $11{,}000/100=\$110$, and the high-risk cell cost is $279{,}000/900=\$310$. Total losses are $\$290{,}000$ and total exposure is $100+900=1{,}000$, so the crude cost is $\$290{,}000/1{,}000=\$290$.

The common-mix standardized costs are $0.5(\$100)+0.5(\$450)=\$275$ for A and $0.5(\$110)+0.5(\$310)=\$210$ for B. The common-mix reference cost is $0.5(\$100)+0.5(\$300)=\$200$. Thus A's standardized relativity is $275/200=1.375$, and B's is $210/200=1.050$. Their standardized comparison is $275/210=1.3095$, to four decimal places.

Check. The crude comparison makes A appear less costly than B ($\$135$ versus $\$290$), while the equal-mix comparison makes A appear more costly ($\$275$ versus $\$210$). This reversal is possible because B has far more high-risk exposures, and the high-risk cell costs are higher in both territories.

Common mistake avoided. Comparing crude costs as if they controlled for class mix would suggest that A is the lower-cost territory. The standardized comparison answers a different question: what would each territory's average cost be under the same stated class distribution?

Investigating Loss Distributions

A territory's aggregate loss cost can be decomposed into claim frequency and average claim severity when claim counts and exposures are available. If $N_t$ is the number of claims, then, for positive claim counts, frequency is $N_t/E_t$, average severity is total losses divided by $N_t$, and their product is the loss cost:

\frac{N_t}{E_t}\times\frac{\sum_c L_{tc}}{N_t} =\frac{\sum_c L_{tc}}{E_t}.

The decomposition indicates whether a high loss cost is associated with more claims per exposure, higher average claim size, or both. It does not by itself establish the cause. Claim-size bands and individual large losses can help locate the source of an unusually high average severity. A large claim may be valid and relevant to the intended measure; it should not be removed merely because it changes the result.

Worked Example: Separating Frequency, Severity, and Large-Loss Effects

Two hypothetical territories each have 400 earned exposures. Territory C has 80 claims: 60 claims below $1,000 total $30,000, 18 claims from $1,000 through $5,000 total $45,000, and 2 claims above $5,000 total $85,000. Territory D has 60 claims: 45 claims below $1,000 total $22,500, 14 claims from $1,000 through $5,000 total $42,000, and 1 claim above $5,000 total $115,500. Compare frequency, severity, loss cost, and the share of losses from claims above $5,000.

Method. Sum the claim bands to obtain total losses, divide claim counts by exposures for frequency, and divide total losses by claim counts for severity.

For C, total losses are $\$30{,}000+\$45{,}000+\$85{,}000=\$160{,}000$. Frequency is $80/400=0.2000$ claims per exposure. Average severity is $\$160{,}000/80=\$2{,}000$, giving a loss cost of $0.2000\times\$2{,}000=\$400$ per exposure. Claims above $5,000 account for $\$85{,}000/\$160{,}000=0.53125$, or 53.13% of C's losses.

For D, total losses are $\$22{,}500+\$42{,}000+\$115{,}500=\$180{,}000$. Frequency is $60/400=0.1500$ claims per exposure. Average severity is $\$180{,}000/60=\$3{,}000$, giving a loss cost of $0.1500\times\$3{,}000=\$450$ per exposure. Claims above $5,000 account for $\$115{,}500/\$180{,}000=0.641666\ldots$, or 64.17% of D's losses.

As a diagnostic only, remove the above-$5,000 band from each total. C then has $\$30{,}000+\$45{,}000=\$75{,}000$, or $\$75{,}000/400=\$187.50$ per exposure. D has $\$22{,}500+\$42{,}000=\$64{,}500$, or $\$64{,}500/400=\$161.25$ per exposure. The ordering reverses under this diagnostic calculation.

Check. D has lower frequency but higher average severity and higher total loss cost. Its single claim above $5,000 contributes 64.17% of its losses. The band-excluded comparison identifies the influence of that band; it is not a revised indication unless the intended measure explicitly excludes such claims.

Common mistake avoided. It would be wrong to conclude that D has persistently worse underlying risk solely because its observed loss cost is higher, or to discard its large claim without a defined treatment. The appropriate investigation is to verify the claim and its adjustments, examine the intended large-loss treatment, and assess whether the experience is sufficiently credible for the question being asked.

Residuals Against Selected Factors

Once territory factors have been selected, a residual can show where observed losses differ from the expected losses on that factor basis. Let $f_t$ be the selected factor for territory $t$, and $b_c$ the base loss cost for class $c$. Under a simple expected-loss benchmark, expected losses are $\sum_c e_{tc}b_cf_t$. The observed-to-expected ratio is

\mathrm{O/E}_t =\frac{\sum_c L_{tc}}{\sum_c e_{tc}b_cf_t}.

An O/E of 1 means observed losses equal this benchmark; a value above 1 means they exceed it. A value below 1 means they are lower. The benchmark must use the same coverage, exposure, loss, and factor basis as the observed data. An O/E ratio is a diagnostic summary, not by itself a statistical significance test or proof that the selected factor is wrong.

Worked Example: Locating a Factor Residual by Class

A hypothetical territory has a selected factor of 0.90. It has 100 low-risk and 100 high-risk exposures, with base loss costs of $100 and $300. Adjusted observed losses are $11,000 for the low-risk class and $31,000 for the high-risk class. Calculate the expected losses, overall O/E, and each class's contribution to the dollar residual.

Method. Calculate expected losses by class under the selected factor, then compare observed with expected totals.

For the low-risk class, expected losses are $100\times\$100\times0.90=\$9{,}000$. For the high-risk class, expected losses are $100\times\$300\times0.90=\$27{,}000$. Total expected losses are $\$9{,}000+\$27{,}000=\$36{,}000$. Total observed losses are $\$11{,}000+\$31{,}000=\$42{,}000$, so $\mathrm{O/E}=42{,}000/36{,}000=1.1667$, to four decimal places.

The low-risk class's dollar residual is $\$11{,}000-\$9{,}000=+\$2{,}000$. The high-risk class's dollar residual is $\$31{,}000-\$27{,}000=+\$4{,}000$. They sum to $\$6{,}000$, which checks against $\$42{,}000-\$36{,}000=\$6{,}000$.

For another check, the selected-factor expected losses before applying the territory factor are $100(\$100)+100(\$300)=\$40{,}000$. Observed losses divided by this base are $\$42{,}000/\$40{,}000=1.05$. Comparing that indication with the selected factor gives $1.05/0.90=1.1667$, matching the O/E calculation.

Common mistake avoided. Dividing observed losses by the selected factor alone does not produce a meaningful residual. The comparison must include the class-specific base expected losses and exposure counts. The class breakdown also shows that the high-risk class contributes twice the dollar residual of the low-risk class, although further investigation is needed to explain either difference.

How to Use a Diagnostic Signal

A useful diagnostic leads to a targeted question, not an automatic factor change. An unusual crude result can prompt a mix-adjusted comparison. A high O/E can prompt verification of exposures, loss adjustments, and the selected-factor base. A severity-driven result can prompt a review of claim sizes and large-loss treatment. Where one class or a small number of claims accounts for most of a residual, examine those components explicitly.

Check the denominator.
Confirm exposure counts, units, and the coverage period. Compare loss costs or frequencies per exposure rather than raw aggregate losses.

Make comparisons on a common basis.
Confirm that losses and exposures are comparable, then use a clearly stated common class mix when class composition differs.

Locate the source of the difference.
Break results down by class, frequency, severity, or claim-size band, as appropriate to the observed signal.

Investigate before interpreting.
Check whether the pattern could reflect data quality, a valid large loss, or the exposure and loss basis. A diagnostic alone does not establish causation.

A frequent exam trap is to use different class weights for different territories while describing the results as mix-adjusted. Another is to average class relativities with exposure weights when the requested comparison uses a common class distribution. State the weights and the denominator explicitly. Also distinguish a sensitivity calculation, such as excluding a loss band, from the result on the defined rating basis.

Key takeaway: A sound territorial diagnostic accounts for exposure volume, makes the comparison basis explicit, and breaks an anomaly into informative components. It identifies what to investigate; it does not, by itself, prove why losses differ or determine the final factor.

Check Your Understanding

For each hypothetical situation, calculate the requested diagnostic and state what it does—and does not—show.

Territory P has 200 exposures and $30,000 of adjusted losses. Territory Q has 500 exposures and $60,000 of adjusted losses. Calculate each crude loss cost and identify which territory has the higher cost per exposure. (Answer: P is $150 per exposure; Q is $120 per exposure; P is higher.)
Two classes have reference costs of $80 and $240. Under equal common weights, a territory's observed class costs are $100 and $300. Calculate its standardized cost and standardized relativity. (Answer: standardized cost is $200; reference cost is $160; standardized relativity is 1.250.)
A territory has 50 claims over 250 exposures and $125,000 of adjusted losses. Calculate frequency, average severity, and loss cost. (Answer: frequency is 0.2000 claims per exposure; average severity is $2,500; loss cost is $500 per exposure.)
A selected factor is 1.10. A territory has 100 exposures in one class with a $200 base loss cost and observed losses of $24,200. Calculate expected losses and O/E. (Answer: expected losses are $22,000; O/E is 1.1000.)
Two territories have the same total exposure. Territory R has 100 losses of $1,000 each. Territory S has 99 losses of $500 each and one loss of $50,000. Calculate each total loss cost per exposure and S's percentage of losses from its largest claim. What is a suitable conclusion? (Answer: R has $100,000 divided by the shared exposure count; S has $99,500 divided by the shared exposure count; S's largest claim is $50{,}000/99{,}500=50.25\%$ of its losses. The result warrants a large-loss and data-basis review, but does not alone establish that S has higher persistent underlying risk.)

What You'll Learn