We illustrate this study procedure below. We choose alternative input figures for an asset share exhibit. These figures follow the framework of the study note examples, but the values are different, to provide practice in working up the exhibits.
Suppose that first year premium is $1,000, and the expected first year losses (discounted to the beginning of the year) are $800. [In exhibit 3, the first year premium is $800, and the expected first year losses (discounted) are $656.]
The insurer is a direct writer, and underwriting expenses are much higher in the first policy year than in renewal years. Assume the following expense ratios:
| New Policies | Renewal Policies | |||
| Fixed Expense Provision |
Variable Expense Provision |
Fixed Expense Provision |
Variable Expense Provision | |
| Agency Commissions | 0.0% | 20.0% | 0.0% | 4.0% |
| Advertising and Other Acq | 8.0 | 0.0 | 0.0 | 0.0 |
| General Expenses | 10.0 | 2.0 | 2.0 | 1.0 |
| Premium Tax | 0.0 | 2.0 | 0.0 | 2.0 |
| Taxes, Licenses, and Fees | 0.8 | 0.2 | 0.8 | 0.2 |
| Total | 18.8% | 24.2% | 2.8% | 7.2% |
These figures differ slightly from those in Exhibit 2 of the syllabus reading for the first three rows.
Trends:
The expected loss cost trend is 8% per annum,
The expected fixed expense trend is 4% per annum, and
The expected premium rate increases are 7% per annum.
These rates are similar to those in the study note example, which uses 10% for loss costs, 5% for fixed expenses, and 9% for premium rate increases.
Persistency: We assume that the retention rate is 85% the first year, increasing by 1% each year, until it levels off at 95%.
Loss Cost Improvement: We assume that the average loss cost on any policy improves by 3% a year since policy inception, after adjusting for inflation. [This is the same assumption as in the study note.]
The discount rate for this company is 10% per annum. [The study note uses 12% per annum.]
The company has a two to one (2:1) premium to surplus ratio.
On the May exam, you will probably be asked to reproduce only a few rows or entries from an asset share exhibit. It is good practice to replicate the entire exhibit, so we use a 15 year exhibit in this practice problem.
Given the assumptions listed above, what are the present values of premiums and profits in the first three years?
Using a 15 year asset share pricing model, what is the ratio of the present value of the policy's profits to the present value of the policy's premiums?
What is the return on surplus for this policy?
Completing the Exhibit
Begin with a blank 13 column asset share exhibit, of the format used in the syllabus reading. For most entries, the entire column can be completed directly from the assumption about the individual element. This is true for the premium, loss, expense, persistency, and discount factor columns. For the profit column and the present value columns, one must combine two or more other columns.
Complete the exhibit in the following manner. (See the completed exhibit as well as the
Excel spreadsheet in the download library.)
Column 2: Place the $1,000 premium in the first row ("policy year 1"). Each subsequent row is increased by 7%, the expected annual premium rate increase.
Column 3: Place the $800 of discounted losses in the first row ("policy year 1"). Each subsequent row is increased by the loss cost trend of 8% and decreased by the "loss cost improvement" of 3%. For instance, $800 * 1.08 * 0.97 = $838.
Column 4 has only one entry: the variable expense ratio for the initial policy year times the premium in the initial policy year, or 24.2% * $1,000 = $242.
Column 5 has the variable expense ratio for renewal years, which is 7.2%. For instance, in the second row, the premium of $1,070 * 7.2% = $77.
Column 6 has only one entry: the fixed expense ratio for the initial policy year, or $1,000 * 18.8% = $188.
Column 7 has the variable expense ratio for renewal years. In the initial policy year, the "renewal fixed expenses" would be 2.8% * $1,000, or $28. This amount is increased by 4% each year, so it is $29 in the second policy year.
[If this is confusing, think of the calculations in the following manner: Suppose the initial policy year in the asset share exhibit is 1996. Then a policy first issued in some previous year, but renewed in 1996 for a premium of $1,000, would have "fixed expenses" of $28 in 1996.]
Column 8 shows the retention rate each year. In this example, 85% of policyholders in the first year persist into the second year; 86% of these continuing policyholders persist into the third year; and so forth.
Column 9 shows the cumulative persistency. This is the downward product of the individual retention rates. For instance, the 73% in the third row is the product of 100% * 85% * 86%.
Column 10 shows the profit in each year. Since losses are discounted to the beginning of the policy year, and premiums and expenses are collected and paid at the beginning of the policy year, the profit is being valued as of the beginning of each policy year.
The profit is (premiums - losses - expenses) * (cumulative persistency). In the first policy year, the profit is
In the second policy year, the profit is
Column 11: The discount factor uses the discount rate of 10% between the inception of the first policy year and the inception of the policy year under consideration. For instance, since the inception of the third policy year is two years past the inception of the first policy year, the discount rate is (1.10)2 = 1.21.
Column 12 shows the present value of profits. Since the profits have already been adjusted for persistency, column 12 is column 10 divided by column 11. For instance, in the second row, $107 ÷ 1.10 = $97.
Column 13 shows the present value of premiums. The premiums in column 2 have not yet been adjusted for persistency. Rather, column 12 is calculated as
column 13 = column 2 * column 9 ÷ column 11.
For instance, in the second row, $1,070 * 85% ÷ 1.10 = $827.
This completes the work needed for Part (a) of the practice problem. Complete the worksheet to make sure you can fill in all the entries, and compare your completed worksheet with the worksheet in this study aid.
Columns 12 and 13 show present values at the beginning of the first policy year. We sum all the figures in these rows to get the present value of all profits and the present value of all premiums. These figures are $802 and $6,397, respectively. The ratio of present value of profits to present value of premiums is 12.535%.
This is the solution to Part (b) of the practice problem.
The company uses a two to one premium to surplus ratio. Think of this as follows: if the company collects $6,397 of present value premiums, it must hold (6,397÷2) = 3,198.50 "dollar-years" of surplus. The return on surplus is
This is the solution to Part (c) of the practice problem.