B.7. Miccolis - ILFs and XOL

Card Set Information

Author:
EExam8
ID:
306510
Filename:
B.7. Miccolis - ILFs and XOL
Updated:
2015-09-09 20:02:43
Tags:
Increased Limits Excess Loss Pricing
Folders:

Description:
Increased Limits and Excess of Loss pricing
Show Answers:

Home > Flashcards > Print Preview

The flashcards below were created by user EExam8 on FreezingBlue Flashcards. What would you like to do?


  1. General Pricing Model
    • n = frequency, x = severity, g = loss cost
    • E[g(x)] = ∫0 to ∞ g(x) dF(x) or 0 to ∞ g(x) f(x)dx
    • y = pure premium with E[y] = E[g(x)] * E[n]
  2. Increased Limits Coverage
    • when losses are only covered up until k
    • E[g(x; k)] = ∫0 to k x dF(x) + k[1 - F(k)]
    • I(k) = E[g(x; k)]/E[g(x; b)], b = basic limit
    • I(k) depends only on severity, not frequency
    • can be easier to make judgement on I(k) than working with loss distribution (little data)
  3. Assumptions in determining ILFs
    • all UW expenses & profit are variable and don’t vary by limit (in practice use λ)
    • frequency and severity are independent, and individual claims are independent
    • frequency is the same for all limits (not usually true because of adverse / favourable selection)
  4. Properties of I(k)
    • as k → ∞, F(k) → 1, I'(k) → 0, I(k) → some constant
    • since F(k) is monotonic increasing, I'(k) will be monotonic decreasing
    • consistency test: marginal premium/$1000 of coverage should decrease as k increases
  5. Exception to consistency test
    • adverse selection: higher limit can indicate adverse experience if:
    •    - insureds that expect higher losses are more inclined to purchase higher limits
    •    - liability lawsuits & settlements may be influenced by the size of the limit
    • favourable selection: higher limits can indicate best experience if:
    •    - wealthier insureds may be better risks, protecting their assets with higher limits
    •    - insurers will be more willing to write high limits coverage for better risks
    • if there is anti-selection for a given limit, use only the data from those policies to get I(k)
  6. Excess of Loss Coverage
    • XoL contract would cover losses greater than r up to j (max payment is r + j, not j)
    • h(x;r , j) = 0 if 0 < x < r, x - r if r < x < s, and j if x ≥ s where s = r + j
    • E[h(x; r, j)] = ∫r to s x dF(x) + s[1 - F(s)] - r[1 - F(r)] = E[g(x; s)] - E[g(x; r)]
  7. Trend
    • economic and social inflation can have a substantial effect on pricing ILF and XoL
    • for losses above the basic limits, the trend is entirely in the excess layer
    • losses just under the basic limit are pushed into the layer, creating new losses
    • assume x' is produced from an inflated loss x where x' = α(x), then F(x') = F(x)
    • multiplicative α: x' = ax, E[g(x'; k)] = a * E[g(x; k/a)], I(k') = I(k/a)/I(b/a)
  8. Charge for Risk
    • basic formula don't incorporate an actual charge for the risk of being in insurance business
    • loading incorporated in expenses is usually low for more volatile lines
    • process risk: variation between actual losses and expected losses (stochastic/random)
    • parameter risk: inability to estimate expected losses accurately (not practical to charge)
  9. Variance as a Measure of Risk
    • variance of pure premium is more appropriate than standard deviation
    • it permits the development of risk adjusted ILF from the severity distribution alone
    • Premium = E[y] + λ Var(y), where λ is the judgmentally selected risk charge
    • Var(y) ≈ E[n] * E[g(x)2]
    • Basic Premium = E[n] * (E[h(x; b)] + λ E[x; b]2)
    • E[h(x; r, j)2] = E[g(x; s)2] - E[g(x; r)2] - 2 * r * E[h(x; r, j)]
    • I(k) = (E[g(x; k)] + λ E[g(x; k)2]) / (E[g(x; b) + λ E[g(x; b)2])
  10. Problems with Empirical Distributions
    • loss development has to be estimated (undeveloped, unreported and reopened claims)
    • losses come from different policies with different policy limits, causing bias in distribution
    • may have to fit theoretical distribution to alleviate credibility problems at high values
    • there might be cluster point at round numbers such as $25K, $50K, etc.
  11. Leveraged Effect of Inflation
    • avg incr in loss with fixed upper limit = a * I(k/a) / I(k)
    • avg incr in XS losses with fixed upper limit = a * [I(s/a) - I(r/a)]/[I(s) - I(r)]
    • avg incr in XS losses with no upper limit = a*(E[x] - E[g(x; r/a)])/(E[x] - E[(g(x; r)])
  12. Coinsurance
    • coinsurance: when a piece of property is insured for less than its full value
    • Premium = Coinsurance % * (Coverage @ 100% value / Exposure base) * Coinsurance factor * rate
    • consistency test:
    •    - applied to either premium or product of coinsurance % and coinsurance factor
    •    - test difference in that product / difference in coinsurance %
    •    - marginal premium should decrease as coverage increases
  13. Two-dimensional consistency test
    • consistency test also applies when there is > 1 type of limit (e.g. occurrence + aggregate)
    • test can be applied separately for rows and column
    • in all cases, the marginal rate should decrease as the limit increases
  14. Rosenberg’s consistency test
    • let A2 > A1 = aggregate, O2 > O1 = occurrence
    • I(A2,O2) - I(A1,O2) ≥ I(A2, O1) - I(A1,O1)
    • I(A2,O2) - I(A2,O1) ≥ I(A1, O2) - I(A1,O1)
    • For larger A/O, difference in ILF should increase as O/A increases

What would you like to do?

Home > Flashcards > Print Preview