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Questions from the 1993 Exam:
46. (2 Points)
Answer the following question according to Miccolis in On the Theory of Increased Limits and Excess of Loss Pricing. You are given the following information:
Frequency of loss (n) is distributed Poisson with mean equal to 0.10.
Severity (X) is distributed as follows:
| Limit (C) |
F(X) |
E(X:C) |
Var (X;C) |
| 25,000 |
0.98427 |
2,663 |
20,289,725 |
| 50,000 |
0.99551 |
2,875 |
33,920,831 |
| 100,000 |
0.99895 |
2,986 |
48,577,626 |
| 500,000 |
0.99998 |
3,052 |
72,387,267 |
Based on the above information, fill in the missing information in the following increased limits table:
| Limit (C) |
Expected
Value Pure
Premium |
ILF
No Risk
Load |
Risk
Load |
ILF
with Risk
Load |
| 25,000 |
|
1.000 |
56 |
1.000 |
| 50,000 |
|
|
|
|
| 100,000 |
|
|
|
|
| 500,000 |
|
|
|
|
PLEASE REMEMBER TO RECREATE THE ABOVE TABLE ON YOUR ANSWER SHEET WHEN ANSWERING THE QUESTION. |
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Solutions to questions from 1993 Exam:
Question 46.
Step 1: Calculate the Expected Value Pure Premium:
Expected Value Pure Premium = E[n]*E[X;C]
| Limit (C) |
E[n] |
E(X:C) |
Expected
Value Pure
Premium |
| 25,000 |
.10 |
2,663 |
266.3 |
| 50,000 |
.10 |
2,875 |
287.5 |
| 100,000 |
.10 |
2,986 |
298.6 |
| 500,000 |
.10 |
3,052 |
305.2 |
Step 2: Calculate the ILFs without risk load:  =  .
| |
Expected |
|
| |
Value Pure |
ILF No Risk |
| Limit (C) |
Premium |
Load |
| 25,000 |
266.3 |
1.000 |
| 50,000 |
287.5 |
(287.5/266.3) = 1.080 |
| 100,000 |
298.6 |
(298.6/266.3) = 1.121 |
| 500,000 |
305.2 |
(305.2/266.3) = 1.146 |
Step 3: Calculate l based on the dollar amount of risk load given: Risk Load = l*Var[y] = l*E[n]*{Var(X;b) + (E[X;b])2} l = Risk Load / Var[y] = Risk Load / E[n]*{Var(X;b) + (E[X;b])2}.
Note: Since a Poisson frequency is given, Var[y] can be computed using the simplified formula: Var[y] = E[n]*E[g(x)2] (eq. 29). Therefore:
l = Risk Load / E[n]*{E[g(x;b)2] = Risk Load / E[n]*{Var(X;b) + (E[X;b])2}
l = {56 / .10*[20,289,725 + (2,663)2]}= 2.05*(10)-5.
Step 4: Calculate the risk loads for the other limits: Risk Load = l*E[n]*{Var(X;C) + (E[X;C])2}
| Limit (C) |
Risk Load |
| 50,000 |
2.05*(10)-5*.10*[33,920,831 + (2,875)2] = 86
|
| 100,000 |
2.05*(10)-5*.10*[48,577,626 + (2,986)2] = 118
|
| 500,000 |
2.05*(10)-5*.10*[72,387,267 + (3,052)2] = 167
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Step 5: Calculate the risk adjusted ILFs:
| Limit (C) |
Expected
Value Pure
Premium |
Risk Load |
ILFs with Risk Load |
| 25,000 |
266.3 |
56 |
1.000 |
| 50,000 |
287.5 |
86 |
[(287.5+86)/(266.3+56)] = 1.160 |
| 100,000 |
298.6 |
118 |
[(298.6+118)/(266.3+56)] = 1.292 |
| 500,000 |
305.2 |
167 |
[(305.2+167)/(266.3+56)] = 1.467 |
Return to Exam 9 |
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