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IAI Actuarial Core Principles · Economic Modelling · Ruin theory

A company's aggregate claims follow a compound Poisson process with λ = 2 per year. Each claim is ₹1 lakh with probability 0.5 and ₹2 lakh with probability 0.5 (units in lakhs). Premium rate c = 4 lakh per year. Which equation, in r, determines the adjustment coefficient, and what is true of the root?

The equation is 2 + 4r = e^r + e^{2r}, since λ times the claim MGF is 2 × (0.5e^r + 0.5e^{2r}). Checking r=0.5 gives the right side larger, so the positive root lies between 0 and 0.5.

  1. A2 + 4r = 1 e^{r} + 1 e^{2r}; R is the positive root, found by substituting y = e^{r} gives a quadratic with R = ln 2
  2. B2 + 4r = e^{r} + e^{2r}; the positive root is found by trial and lies between 0 and 0.5Correct
  3. C2 + 4r = 2 e^{1.5 r}; R = 0
  4. D4 + 2r = e^{r} + e^{2r}; R = ln 2
  5. 2 + 4r = 0.5 e^{r} + 0.5 e^{2r}; R = 0

Explanation

M_X(r) = 0.5e^r + 0.5e^{2r}, so λM_X(r) = e^r + e^{2r}. The equation is 2 + 4r = e^r + e^{2r}. At r=0 both sides equal 2; slopes are 4 versus 3 (mean claim 1.5 times 2), so the line is above initially. At r=0.5: LHS 4, RHS 1.6487+2.7183=4.367, so RHS exceeds LHS and the root lies between 0 and 0.5. Option 1 is not a quadratic solution and ln 2 gives LHS 3.77 vs RHS 2+4=6, wrong.

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