CAS Exam 9 Spring 2019, Question 1
Study guide: Exam 9
A startup is looking to enter the insurance business. Consider the following information about the company:
- The company has $10 million of capital
- The company is considering two lines of business, A and B
- Management would like to allocate capital using an approach that captures tail risk
- Underwriting results are uncorrelated with investment results
- There is no reinsurance
- There are no other expenses or taxes
The insurer ran 1,000 simulations of the returns on premium each line of business would generate, as well as the returns on investment a AAA Bond would generate. Below are selected results from this simulation, sorted from highest to lowest returns:
| N | \text{LOB}_A |
\text{LOB}_B |
AAA Bond |
|---|---|---|---|
| 991 | 0% | -50% | 3% |
| 992 | 0% | -50% | 3% |
| 993 | 0% | -50% | 3% |
| 994 | 0% | -50% | 3% |
| 995 | 0% | -50% | 3% |
| 996 | 0% | -50% | 3% |
| 997 | -400% | -50% | 3% |
| 998 | -450% | -50% | 3% |
| 999 | -450% | -50% | 3% |
| 1000 | -500% | -50% | 3% |
\text{LOB}_A |
\text{LOB}_B |
AAA Bond | |
|---|---|---|---|
| E(r) | 10% | 8% | 3% |
| SD | 33% | 20% | 0% |
\text{Cov}(R_A, R_B) = 5\%
Part a (2 points)
Justify a risk measure and determine which line of business is a more efficient use of capital from a risk-adjusted return on capital perspective.
Part b (1 point)
Identify and briefly describe two arguments for why management should not invest solely in the more efficient line of business.
Part c (1.25 points)
The company plans to write $10 million of gross written premium in its first year of operation. Determine the optimal amount of premium they should write in each of the above lines of business.
Part d (2.25 points)
The board of directors has given the company the following investment constraint:
The company may invest a portion of its capital in Stock C provided that it reduces the amount of insurance business across the combined lines of business A and B by the amount it chooses to invest in Stock C.
Stock C produces an expected return of 8% with a standard deviation of 30%, and its expected returns are uncorrelated with the expected returns of the insurer's underwriting portfolio.
Including underwriting and investments, determine the total return of the insurer's optimal risky portfolio as a percentage of its capital.
No current syllabus covers this question; it is kept for the record.
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