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Special Classification

Types

  1. Territorial Ratemaking
  2. Increased Limits Ratemaking
  3. Deductible Pricing
  4. Insurance To Value

Increased Limits Ratemaking

Standard Approach

Assumptions of Standard Approach

  1. UW expenses and profits are variable and don't vary by limit.
  2. Freq and Severity are Independent
  3. Freq is same for all limits1
\[ \text{ILF}(H) = \dfrac{\text{Rate}_{H}}{\text{Rate}_{B}} = \dfrac{\text{PP}_{H}}{\text{PP}_{B}} = \dfrac{\text{Severity}_{H}}{\text{Severity}_{B}} = \dfrac{\text{LAS}(H)}{\text{LAS}(B)} \]
  • \(\text{Rate} = \frac{PP}{1-V-Q}\) but we assume that \(V\) and \(Q\) don't vary by limit, so they get cancelled out.
  • Next, \(PP = \text{Freq} \times \text{Sev}\) but assumption (3) gets the frequencies cancelled.

What is \(\text{LAS}_{H}\)?

  • Limited Average Severity at Limit \(H\).
  • Just the Severity.
    • Assuming every loss is capped at limit \(H\).
    • Regardless of the actual policy limit
  • So...
    • If \(H = \$1000\) and actual policy limit is \(\$500\) then we have an issue of censored data, meaning, we don't know what would the losses be if the limit was \(\$1000\) instead of \(\$500\).
    • Understand that Severity is just the total loss divided by the counts.
  • For a layer of loss, we can find (say) \(\text{LAS}(\$250k \text{ xs } \$500k)\)2 which gives us the severity of losses \(X \ni \$500k < X <\$750k\)
    • Have to figure out how much each of the actual claims (from the data) contribute to this layer.
    • E.g., Claims between 750k and 1000k contribute full 250k per claim to this layer. (Note that this is the \(\$250k\) in the "xs" notation)
    • E.g., claims between 500k and 750k (say their total is given as 160,000k and count as 200) will contribute \(160,000k - (200)500k\). Think about this one.3
    • E.g. Claims between 200k and 500k contribute zero to this layer. (Note that this is the \(\$500k\) in the "xs" notation)
  • And for the "So..." above we have this relation

$$ \begin{align*} \text{LAS}(250k) &= \text{LAS}(100k) + \text{LAS}(150k\text{ xs } 100k) \times Pr(X \geq 100,001) \

\text{LAS}(500k) &= \text{LAS}(250k) + \text{LAS}(250k\text{ xs } 250k) \times Pr(X \geq 250,001) \end{align*} $$

  • The probabilities are based on policies that qualify of having a claim of that minimum size. The denominator should not contain an impossible event of \(X\) having a limit of \(100,000\) and being \(\geq 100,001\).

  1. But in reality, we have favorable selection: Financially secured insureds will have lower frequency but will have more assets to protect and thus will buy higher limits. Adverse Selection: Insureds that expect higher loss potential (high frequency) will buy higher limits. ↩

  2. Call this the "xs" notation ↩

  3. It's like those 200 claims have crossed the 500k mark and are below 750k so, we have 60,000k within the upper and lower bounds, that we obtain after removing (200)500k = 100,000k from the total. ↩

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