C.2. Clark  Reinsurance Pricing
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Difference between insurance and reinsurance pricing
 a reinsurance program is generally tailored more closely to the buyer
 there is no such thing as the average reinsured or average reinsurance price
 pricing paradox: if you can precisely price a contract, the ceding company will not want to buy it

2 main methods by which Reinsurance can be applied
 facultative: designed and purchased separately for each individual risk
 treaty: single contract allowing the reinsurer to cover many risks (price aggregate)

Bases for Reinsurance policies
 risk attaching: all policies that begin or renew during the contract period are covered, regardless of when their losses occur or are reported (policy year / written premium)
 losses occurring: covers all claims that occur during the contract period, regardless of when the policies were incepted or when the losses are reported (accident year / earned premium)

2 Methods to price Reinsurance
 experience rating: use adjusted historical experience to calculate premium (burn cost = unadjusted ratio of past ceded losses to premium
 exposure rating: use the current risk profile and estimated loss distributions to calculate premium; often used in combination with experience rating for nonproportional reinsurance

Proportional treaty
 agreement where the reinsurer assumes a given percent of losses and premium
 quota share: reinsurer receives X% of premium + ceding commission and pays X% of losses
 surplus share: reinsurer assumes x * retained line, translated to a % of losses; this type of treaty allows the reinsured to limit its exposure on any one risk. This is not XS insurance, as we only use the line to calculate the % of risk, which is what's being reinsured on a groundup basis
 fixed and variable quota share: underlying business is XoL, but reinsurer takes a % only

Pricing analysis for proportional treaties
 compile historical experience on treaty (premium & asif losses for 5+ years)
 exclude catastrophe (EQ, hurricane) and shock losses (large settlement on a single policy)
 adjust experience to ultimate level and project to future period
 select the expected noncat LR (using historical data, industry average, ceding company gross LR)
 load for catastrophes (e.g. spread large losses or use engineering models)
 estimate combined ratio given ceding commission, reinsurer's GE and overhead, brokerage fees
 estimate technical ratio = ELR + Commission

Inflation vs Policy limits
 concern is that insureds tend to purchase higher limits over time, so capping losses at historical policy limits and using those to predict future losses may understate the future loss potential
 an alternative is to trend limits and losses, assuming policy limits increase with inflation
 disadvantage of alternative is that historical premium should then be adjusted to reflect higher limit, and that can be difficult to quantify

Special features of proportional treaties
 sliding scale commission: sliding % of premium paid by the reinsurer to the ceding company
 carryforward provision: allows that if the past LR have been above the minimum commission requirement, the excess loss amount may be included in the current year's losses
 profit commission: substracts the actual loss ratio, ceding commission and a margin for expenses from the treaty premium and returns a % of this as additional commission
 loss corridors: ceding company will reassume a portion of the losses if LR > threshold

Property per risk XS treaties
 provides a limit of coverage in excess of retention, attached to a single location
 treaty premium is set as a % of GNEPI or GNWPI (Gross Net Earned/Written Premium Income), which is the premium net of any other reinsurance inuring to the benefit of the per risk treaty, but gross of the per risk treaty being priced

Property per risk XS treaties  Exposure rating
 advantage is that the current risk profile is modeled, not was was written years earlier
 assumes that the exposure curve applies regardless of the size of the insured value
 let R = retention, L = limit, IV = insured value
 Exposure curve = P(p) = ∫_{0 to p*IV} [1  F(x)]dx / E[X] = E[X; p * IV] / E[X]
 Portion of expected loss on risk in treaty layer = exposure factor = P((R+L)/IV)  P(R/IV)
 Note that the table of exposure factor will likely go over 100% as limits profile often don't include business interruption coverage (commercial) or living expenses (homeowner)

Exposure rating and deductible changes
 when deductible increases / decreases, subject premium decreases / increases but E[L] may not change significantly so the old exposure rate would be too low / high
 when deductible decreases, losses below the original deductible may be unknown making it challenging to derive a new exposure curve

Free cover
 experience rating in which no losses trend into the highest portion of the layer being priced
 should use experience rating as a basis for the lower portion and then use relativities in the exposure rating to project the higher layer (where there are no losses)

Credibility
 first measure is # of expected claims (or $ losses based on exposures) during the historical period
 second measure is yeartoyear variation in projected loss cost
 have to balance with the credibility of the complement itself → subjective exercise

Inuring Reinsurance
 occurs when an excess treaty covers losses after a surplus share treaty is applied
 in experience rating we should restate the historical loss experience net of inuring reinsurance
 exposure rating can be applied directly to a risk profile adjusted for reinsurance

Casualty per occurrence XS treaties
 working layer: low layer attachment which is expected to be penetrated (multiple claims per year)
 exposed excess: excess layer which attaches below policy limits (not penetrated each year)
 clash covers: high layer attachment excess (would typically need multiple policies to trigger)

Situations where clash cover is exposed
 extra contractual obligations and excess of policy limits loss
 losses involving 2+ coverages / policies
 ALAE covered outside of policy limits
 WC loss above attachment (WC has no limit)

Data Quality Issues in RAA report
 reporting lag from event occurrence to the establishment of a reinsurer's case varies by company
 mix of attachment points is not cleanly broken out
 RAA requests data exclusive of Asbestos and Environmental claims (insurer compliance issue)
 WC members may not handle the tabular discount on large claims in a consistent manner

Casualty per Occurrence XS Treaties  Exposure Rating
 E[x; L] = E[min(x,L)] = ∫_{0 to L} x f(x) dx + ∫_{L to ∞} L f(x) dx
 ILF_{L, U} = E[x; U] / E[x; L] and ELFL = (E[x]  E[x; L]) / E[x]
 Exposure Factor = [E[x; min(PL,AP + Lim)]  E[x; min(PL, AP)]] / E[x; PL]
 if treaty includes ALAE in proportion to losses, replace AP ( + Lim) with AP ( + Lim) / (1+ e)

Issues with Exposure Rating for Casualty per Occurrence XS Treaties
 assumes that ALAE varies directly with capped indemnity, but ALAE is not a constant precent of loss (e.g. $0 losses still incur expenses); in general ALAE as a % tends to decrease as loss increases
 it assigns an exposure factor of zero for high layers

Including Umbrella policies
 if umbrella policy is above primary policies, consider UL + XS as a single policy with a higher limit
 problems arise when the umbrella policies are above primary policies from other carriers
 Trended Loss = (Loss + Underlying Limit) * (Trend Factor)  Underlying Limit, which still leaves out losses from underlying policy which could pierce the layer after the trend is applied
 Exp Factor = [E[x;min(UL+PL,UL+AP+Lim)]  E[x;min(UL+PL,UL+AP)]] / (E[x;UL+PL]  E[x;UL])
 if there's a drop down coverage with probability ϕ, multiply numerator / denominator term by (1  ϕ), and then repeat terms with UL = 0 multiplied by ϕ

Swing Plans
 actual losses to the layer are loaded for expenses and result is charged back to the ceding company, subject to a maximum and minimum constraints
 Retro Premium = Actual Layer Losses * ELR
 if plan isn’t balanced, actual ELR may be different than the Retro Premium adjustment factor
 in this case we can adjust the loading, maximum or minimum rates to produce an acceptable LR
 we also must be mindful of the provisional rate being well below the swing plan premium rate

Aggregate distribution models options
 empirical distribution
 single distribution model
 recursive calculation of aggregate distribution

Empirical Distribution
 easy to calculate, and should be used at least as a check
 experience does not take into account all possible outcomes
 if the volume or mix of business has been changing, volatility of future may ≠ from current
 if using BF of Cape Cod for LDF, historical may present artificially smooth sequence of LR

Single Distribution
 assumes that the aggregate follows a known CDF (e.g. lognormal);
 easy to use, even with limited data
 no allowance for the loss free scenario (e.g. lognormal isn’t defined for y = 0
 no easy way to reflect the impact of changing per occurrence limits on aggregate losses

Recursive Calculation of Aggregate Distribution
 fit an actual distribution to frequency (e.g. Poisson, Negative Binomial), and use a discrete distribution for severity with equal increments
 simple to work with, and provides an accurate handling of lower frequency scenarios
 for higher expected frequencies, calculation is inconvenient as it involves more scenarios
 only a single severity distribution can be used in the analysis

Disadvantages of more complex collective risk models
 complexity of calculation can lead to black box mentality (should use more than 1 model)
 most models assume that occurrences are independent, and so are frequency and severity
 some models use numerical methods with a large error term for low frequency scenarios
 aggregate distribution reflects the process variance but not the full parameter variance

Traditional Property Catastrophe Covers
 occurrence (EQ, hurricane) may often affect multiple risks and multiple policies
 typically covers the ceding company’s retained exposure net of all other reinsurance
 contracts will usually provide a limited number of reinstatements prorata as to amount/time
 payback period: amount of time it would take for the premium to cover the full limit

4 Components of Typical Catastrophe Models
 event sets: simulate the covered hazard based on simulated frequency and intensity
 calculation of local event intensity for each property within a portfolio
 estimation of damage for each property within a portfolio impacted by a given event
 insured loss estimates based on policies written by the ceding company

Information Required for Catastrophe Models
 measure of exposure: insured values, construction types, occupancies, attachment points
 geographical information: convert addresses into actual coordinates (or aggregate by zip code)
 terms of the insurance policies: including deductibles, coinsurance provisions
 details of inuring reinsurance: occurrence caps, loss corridors, etc.

2 Common Characteristics of Finite Risk Covers
 multiple year features: e.g. 3 year contract cancellable after 1st or 2nd year if premium > losses
 loss sensitive features: profit commissions and additional premium formulas

Conditions for Reinsurance Contract
 the reinsurer assumes significant insurance risk under the reinsured portions of the underlying insurance agreements
 it is reasonably possible that the reinsurer may realize a significant loss from the transaction