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Credibility

Three Necessary criteria

  1. Should be \(0 \leq Z \leq 1\)
  2. \(\dfrac{dZ}{dN} \gt 0\)
  3. \(\dfrac{d}{dn}(Z / n) \lt 0\)

Calculation

  • Partial Credibility
    • \(Z = \min\left(\sqrt{ \dfrac{n}{N_{0}} }, 1\right)\)
    • Estimate = \(Z \times \text{Observed} + (1-Z) \times \text{Related}\)
  • Buhlmann Credibility
    • \(K = \dfrac{EPVP}{VHM}\) (Echoing/Valley)
      • lower the expected Process variance \(\implies\) more homogenous (differences within the group are low) \(\implies\) more credibility
      • More the variance between the the groups \(\implies\) more discrete clusters \(\implies\) more credibility
      • Thus, small numerator, large denominator \(\implies\) small \(K\)
    • \(Z = \dfrac{N}{N+K}\)
      • Smaller the \(K\) \(\implies\) More the credibility
    • Estimate = \(Z \times \text{Observed} + (1-Z) \times \text{Prior Mean}\)
  • Bayesian Analysis
    • Bayes Theorem
    • Complex (unlikely to be tested)

Complement of Credibility

Six desirable qualities

  1. Accurate (close to target)
  2. Unbiased (=target, on average)
  3. Statistically Independent from base statistic (else errors compound)
  4. Available (else impractical)
  5. Easy to compute (else difficult to justify)
  6. Logical relationship to base statistic (else difficult to justify)

ABI, ACL: Abhi Asal mein…

Comparison of Credibility Complement Methods

This table evaluates five common methods for determining the complement of credibility against key actuarial criteria. A checkmark (✅) indicates the criterion is generally met, a cross (❌) indicates it is generally not met, and a question mark (❓) indicates it is conditional or variable.

Method Accurate Unbiased Independent Available Easy to Compute Logical Relationship
Competitor's Rates Biased if competitor makes different assumptions Yes Difficult to Obtain Yes Yes
Loss Costs of a Larger Related Group ❓ Possibly ❌ Usually biased ✅ Yes ✅ Yes ✅ Yes ❓ If chosen reasonably
Loss Costs of a Larger Group (including subject) ✅ Yes (due to stability) ❌ Biased ❌ No ✅ Yes ✅ Yes ✅ Yes
Rate Changes from Larger Group Applied to Present Rates ✅ Yes ❓ Better (reduces bias of the above) ❓ Yes, if subject data is excluded ✅ Yes ❓ Slightly harder ✅ Yes
Harwayne's Method ✅ Yes ✅ Yes ✅ Mostly ✅ Yes ❌ Harder ✅ Yes (but harder to explain)
Trended Present Rates ❓ Depends on the stability of indications ✅ Yes ❓ May or may not be ✅ Yes (ALWAYS) ✅ Yes ✅ Yes
  • #Trended Present Rates is always available, if the company has written any policy in the past (as they need a rate to do it.)

Complements in First Dollar Ratemaking

Please note that the first 3 are are credibility weighting the LOSS COST! not the final indication.

LC: larger group including subject

  • E.g. regional, countrywide etc
  • Pros
    • Accuracy (stability, larger data)
    • Available
    • Easy to compute
    • May have logical connection
    • Independent if subject experience excluded
  • Cons
    • Biased (there is a reason why subject group has been separated from larger group)
  • neighbor
  • Cons
    • Biased
  • Pros
    • Available
    • Independent
    • Possibly accurate
    • May have logical connection (if reasonably selected)

Rate changes from larger group applied to present rates

  • Complement = Current LC of Subject \(\times \dfrac{\text{Larger Group Ind LC}}{\text{Larger Group Curr Avg LC}}\)
  • Adjusted version of #LC larger related group to reduce bias

Harwayne's Method

This is for class ratemaking, not for overall indication.

Good problems

  • Spring 2017 Exam 5 - Q9

Say we are trying to find a complement for State A, class 1

  1. Wtd Avg PP of State A
  2. Using State A's exposure distribution, Wtd Avg PP for B,C
  3. Adjustment factors \(\dfrac{\text{State A}}{\text{State B}}\) and \(\dfrac{\text{State A}}{\text{State C}}\)
  4. Adjusted PP
    • State B, class 1 adjusted = SBC1 PP \(\times\) Adj Factor
    • State C, class 1 adjusted = SCC1 PP \(\times\) Adj Factor
  5. Complement = average of adjusted PP, weighted on their exposure

Trended Present Rates

Use when there is no larger group.

  • Accuracy depends on stability of (past) indications
  • Unbiased (done by the actuary)
  • May or may not be independent (done by the same actuary?)
  • Readily available (apna hi company hai boss)
  • Easy to compute
  • Logical relationship to subject's experience (apna hi company hai boss)

Solve an example

Variable Value Comment
Total Number of Claims in Historical Period 3,612
Number of Claims for Full Credibility 1,082
Credibility 100.0% Square root rule
Latest Indicated Rate Change 13.2% Calculated by the previous Actuary, not necessarily implemented
Last Rate Change Effective Date 1/1/2016 From [[One-time|On levelling Premiums]], last rate change date
Last Rate Change Taken 5.0% From [[One-time|On levelling Premiums]], last rate change
Residual Indication 7.8% The amount of change that couldn't be implemented. \(\dfrac{1.132}{1.05}-1\)
Projected Loss Trend 0.5% From Loss trending
Projected Premium Trend 2.0% From premium trending
Net Trend -1.5% The loss ratio trend
Trend Period 1.0 From last rate Change effective date to the future effective date in quesiton.
Trended Present Rates Indication 6.2% Project the residual indication to the future. =(1+B15)*(1+B19)^B21-1

Pure Premium Method

Hands-on explanation. Take this example

$200 Present average rate
10% Annual loss trend
20% Rate change requested in last filing
15% Rate change approved with last filing
January 1, 2006 Effective date requested in last filing
June 1, 2006 Actual effective date of last change
January 1, 2008 Proposed effective date of next change
  • We need to calculate the complement of credibility using trended present rates approach.
  • What are we trying to find now?

What would be the present rate had the requested rate filing, WOULD HAVE BEEN APPROVED NOW?
- The actuary 2 years ago, would have calculated on Jan 1, 2006, that the rate change should be 20%
- But after some implementation protocol, they approved 15% rate change.

  • Let's look at the solution and break down the answer for ourselves
    • =B8*(1+B10)/(1+B11)*(1+B9)^2
\[ \text{Present Avg Rate} \times \dfrac{\text{Requested RL chg}}{\text{Approved RL chg}} \times \text{Loss Trend}^{\text{Trend Period}} \]
  • We are adjusting the \(\text{Present Avg Rate}\)
  • By removing the effect of the \(\text{Aproved RL chg}\) and applying the \(\text{Requested RL chg}\) to it, this takes care of the part "HAD THE requested rate BE APPROVED NOW".
  • Apply the \(\text{Loss Trend}\) because times have changed.
  • The \(\text{Trend Period}\) would be:
    • From: The date for which the previous actuary actually performed the calculation
    • To: The future policy period when the rates will be in effect.
    • Hence, it should be 2 years (Jan 1, 2006 \(\to\) Jan 1, 2008)

Calculation:

\[ \$200 \times \dfrac{1+20\%}{1+ 15\%} \times (1+10\%)^2 = \$252.52 \]

Loss Ratio method

\[ C = \dfrac{\text{Loss Trend Factor}}{\text{Prem Trend Factor}} \times \dfrac{\text{Prior Indicated Rate chg Factor}}{\text{Prior \textbf{Implemented} Rate chg Factor}} \]
  • Basically, undo the effect of implementation
  • Apply the actual calculation…

Complements in Excess Ratemaking

Complement when estimating excess losses.

NOTE: These are complements of losses. If you are asked to find the complement of loss cost in the question, then ensure that you divide it by earned exposures before submitting your answer. Give what is asked for!
- Estimate of losses
- Estimate of loss cost

Increased Limits Analysis

When ground-up loss data up to attachment point, \(A\) is available.

For \(L\text{ xs }A\),

\[ C = \text{Losses capped at A} \times \dfrac{ILF_{A+L} - ILF_{A}}{ILF_{A}} \]
  • Dividing by \(ILF_{A}\), makes losses capped to basic limit
  • Then we multiply by the differential that lets us know the losses in the layer L xs A

So, essentially it is,

\[ C = \text{Losses Capped at A} \times \dfrac{\text{Expected losses in L xs A}}{\text{Expected Losses capped at A}} \]

Note the distinction between actual and expected in the formula above.

  • Notes
    • If expected values come from a different size of loss distribution than subject experience
      • \(\implies\) Biased complement
    • If data available \(\implies\) Independent and practical
    • but biased
    • inaccurate due to low volume of data

Lower Limits Analysis

Use data at a lower limit \(d\) than the attachment point, \(A\). \(d \lt A\)

Intuitively,

\[ C = \text{Losses Capped at d} \times \dfrac{\text{Expected losses in L xs A}}{\text{Expected losses capped at d}} \]

so, using \(ILF\)s,

\[ C = \text{Losses Capped at d} \times \dfrac{ILF_{A+L}-ILF_{A}}{ILF_{d}} \]
  • more bias, but also more accuracy

Limits Analysis

Further generalization of #Increased Limits Analysis

\[ C = \text{Expected Loss Ratio} \times \sum_{d \gt A} \text{Premium}_{d} \times \dfrac{ILF_{\min(d,A+L)}- ILF_{A}}{ILF_{d}} \]

assets/images/2025/10/Credibility-1760605762557.webp

Limit Contribution to complement
\(d_{1}\) Doesn't contribute so \(d\geq A\)
\(d_{2}\) \(ILF_{A}- ILF_{A}=0\) so \(d\gt A\)
\(d_{3}\) \(\dfrac{ILF_{d_{3}}-ILF_{A}}{ILF_{d_{3}}} \times E(\text{Loss capped }d_{3})\) \(\checkmark\)
\(d_{4}\) \(\sum_{d_{4}\gt A+L} \dfrac{ILF_{A+L} - ILF_{A}}{ILF_{d_{4}}}\times\) \(E(\text{Loss capped }d_{4})\) Just like #Increased Limits Analysis but allows data from higher limits.
  • Biased and inaccurate
  • Assumes loss ratio doesn't vary by limit
  • Used by reinsurers who don't have ground-up loss data