IAI Actuarial Core Principles · Economic Modelling · Ruin theory
In the classical compound Poisson risk model with premium loading, the adjustment coefficient R is defined as the positive solution of which equation? (Let c be the premium income rate, λ the Poisson claim rate, and M_X the moment generating function of an individual claim X.)
R is the positive root of λ + cr = λ M_X(r). This comes from requiring that exp(−rU(t)) has constant expectation, so the premium rate, claim rate and claim moment generating function balance at r = R.
- Aλ + cr = λ M_X(r)Correct
- Bλ − cr = λ M_X(−r)
- Cλ M_X(r) = c + r
- Dλ + cr = λ M_X(−r)
- cr = λ M_X(r)
Explanation
The adjustment coefficient satisfies λ + cr = λ M_X(r), the condition that e^{-rU(t)} is a martingale. Using M_X(-r) would correspond to the wrong sign of the exponent. The other options do not come from the exponential martingale condition.
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