There's a problem with the way many actuaries handle risk in reinsurance prices today

Conventional Wisdom:
	"Reinsurance Rates-on-Line look a lot like probabilities ...
	ROL(no coverage) = 0
	ROL(certain loss) = 1
	ROL(A+B) = ROL(A) + ROL(B), if disjoint, i.e., separate limits,
	... So let's treat them like they are probabilities."

Conventional Wisdom in Use

Dual Triggers: ROL(A&B) = ROL(A) • ROL(B) (independent case)

Second-Event Covers: ROL(2nd Event) = ROL(1st Event)²

Price with Reinstatement @ 100%:
ROL(1@100%) = ROL(w/o RI) - ROL(w/o RI)²/2

< ROL(w/o RI)

But does this make sense with what we know about pricing one contract at a time?

General Economic Theory:
Marginal Revenue = (at equilibrium) Marginal Cost

Insurance Pricing:
Marginal Profit = (we hope) Cost of Marginal Risk

The standard application in reinsurance today is Kreps' model, which assumes that each reinsurer maintains a constant ratio of surplus to standard deviation of results.

Marginal Cost of Risk
	Standard Deviation of Constract's Results
	so
	Reinsurance Premium = µ + Rσ
	or
	ROL = Frequency + Rσ / limit

Let's use Kreps to compare the value of a layer with and without mandatory reinstatement.

Layers Price Without Reinstatement
Poisson Frequency - Kreps' Profit Model

Covers Without Reinstatement

Traditionally, Cat and Clash covers have occurrence/aggregate limits. After one full hit the limit is exhausted, unless it is reinstated, often for additional premiums.

Reinstatement at 100%

Many covers today provide an additional limit, which is provided on a mandatory basis, reducing the loss collection by an additional premium for the reinstated limit. The premium is equal to the initial premium times the amount of loss (as a portion of the limit), but not reduced for the portion of the contract which has expired:
100% with respect to time

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