B.7. Miccolis - ILFs and XOL

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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
•    - 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

Card Set Information

 Author: EExam8 ID: 306510 Filename: B.7. Miccolis - ILFs and XOL Updated: 2015-09-10 00:02:43 Tags: Increased Limits Excess Loss Pricing Folders: Description: Increased Limits and Excess of Loss pricing Show Answers:

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