B.7. Miccolis  ILFs and XOL
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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]

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)

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)

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

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 antiselection for a given limit, use only the data from those policies to get I(k)

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)]

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)

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)

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}])

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.

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)])

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

Twodimensional 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

Rosenberg’s consistency test
 let A_{2} > A_{1} = aggregate, O_{2} > O_{1} = occurrence
 I(A_{2},O_{2})  I(A_{1},O_{2}) ≥ I(A_{2}, O_{1})  I(A_{1},O_{1})
 I(A_{2},O_{2})  I(A_{2},O_{1}) ≥ I(A_{1}, O_{2})  I(A_{1},O_{1})
 For larger A/O, difference in ILF should increase as O/A increases