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MODELING

2.4 Loss Random Variable The Loss random variable (or Ground up Loss) is a r.v. asociated with the amount of a claim. It is usually assumed that X ≥ 0 The amount paid by the insurer may be adjusted, by deductibles or límits, for instance.

Severity distribution It is the distribution of loss amount or insurer's payment Cost per Loss Is the amount paid by the insurer which includes the 0 amount paid by the insurer that can result from some losses (if the loss is below the deductible, for instance)

Cost per Payment It only consider non-zero payments. It would be a conditional distribution, given that the insurrer actually makes a payment.

Frequency distribution It is the distribution of the number of losses, or amounts paid per unit time

2.4.1 Policy Limits The policy limit, u, is the maximum amount paid by the insurer for a single loss. The insurer

pays the full loss amount X for losses up to an amount u, and pays u for losses that are above amount u.

Limited loss r.v. (Cost per Loss) (also called: right censored r.v.)

If there is a policy limit, u, the r.v. of the amount paid by the insurer is given by: Y = X ∧ u = min(X, u) =

The pdf and cfd of Y = X ∧ u are given by:   fX (y) ; y < u SX (u) ; y = u fY (y) =  0 ; y>u



X if X ≤ u u if X > u

FY (y) =



FX (y) ; y ≤ u 1 ; y>u

Note that: Y = X ∧ u has a discrete point of probability at Y = u

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MODELING

Limited Expected value The expected value of the limited loss r.v. is given by:  Ru x · fX (x) dx + u · SX (u) ; X continuous −∞ P E[Y ] = E[X ∧ u] = ; X discrete x≤u x · pX (x) + u · SX (u)

if X ≥ 0, then:

E[Y ] = E[X ∧ u] =

Z

u

SX (x) dx =

Z

0

0

u

1 − FX (x) dx

The k-th moment limited loss r.v. (assuming X>0) is given by: k

k

E[Y ] = E[(X ∧ u) ] =

Z

u

0

xk · fX (x) dx + uk · SX (u)

Integrating by parts, it can be shown that: k

k

E[Y ] = E[(X ∧ u) ] =

listed

The

Z

0

u

k · xk−1 · SX (x) dx

Exam C tables provides the limited expected value for many of the distribution