**This section of sample problems and solutions is a part of** **The Actuary’s Free Study Guide for Exam 5, authored by Mr. Stolyarov. This is Section 112 of the Study Guide. See an index of all sections by following the link in this paragraph.**

This section of the study guide is intended to provide practice problems and solutions to accompany the pages of *Basic Ratemaking*, cited below. Students are encouraged to read these pages before attempting the problems. This study guide is entirely an independent effort by Mr. Stolyarov and is not affiliated with any organization(s) to whose textbooks it refers, nor does it represent such organization(s).

Some of the questions here ask for short written answers based on the reading. This is meant to give the student practice in answering questions of the format that will appear on Exam 5. Students are encouraged to type their own answers first and then to compare these answers with the solutions given here. Please note that the solutions provided here are not necessarily the only possible ones.

**Source:**

Werner, Geoff and Claudine Modlin. *Basic Ratemaking**.* Casualty Actuarial Society. 2009. Chapter 14, pp. 259-267.

**Original Problems and Solutions from The Actuary’s Free Study Guide**

**Problem S5-112-1.** If an insurance company’s current rates are not adequate to pay for expected losses, expenses, and the company’s profit provision, name four non-pricing solutions that the company can implement in an attempt to correct this. (Assume that the company may not change its rates.)

**Solution S5-112-1.** This question is based on the discussion in Werner and Modlin, pp. 259-260.

The following are non-pricing solutions for inadequate rates:

1. Reducing the marketing budget (this reduces expenses);

2. Reducing staff levels (this reduces expenses);

3. Imposing more stringent underwriting requirements (this may reduce losses to a greater extent than premium would be reduced);

4. Non-renewing particularly risky policies (this may reduce losses to a greater extent than premium would be reduced);

5. Narrowing or excluding coverage for previously covered perils (this may reduce losses payable by the insurer);

6. Instituting superior loss control procedures (this may reduce losses);

7. Reducing the company’s profit provision and waiting until the external situation improves.

Any four of the above suffice as an answer. Other valid answers may also be possible.

**Problem S5-112-2.** An insurance company has a policy fee built into its rating algorithm. The average fixed expense per policy is $70. The company also has a profit provision of 7% and variable expenses of 20% of premium. Based on the information above, what should be the magnitude of the policy fee charged by the company?

**Solution S5-112-2.** This question is based on the discussion in Werner and Modlin, p. 262. The variable expense percentage and profit percentage are distributed throughout all components of the rate, including the fixed policy fee. This is expressed by Werner and Modlin via the formula AP = E-F/(1.0 – V – QT), where AP is the policy fee, E-F is the fixed expense amount, V is the variable expense percentage, and QT is the profit percentage. Here, AP = 70/(1 – 0.2 – 0.07) = 95.89041096 = **$95.89.**

**Problem S5-112-3.** An insurance company uses the premium-based projection method, under which it estimates fixed expenses as being 5% of the premium for a policy. The projected average premium per exposure is $440, and the company’s variable expense percentage is 18%. The company has also selected a profit provision of 6%. An insurance company has a policy fee built into its rating algorithm. Based on the information above, what should be the magnitude of the policy fee charged by the company?

**Solution S5-112-3.** This question is based on the discussion in Werner and Modlin, pp. 262-263.

Where the premium-based projection method is used, the formula AP = E-F/(1.0 – V – QT) still applies, but E-F must be estimated as (Fixed Expense Ratio)*(Average Premium Per Exposure) = 0.05*440 = $22 in this case. Here, V = 0.18, and QT = 0.06. Thus, AP = 22/(1 – 0.18 – 0.06) = 28.94736842 = **$28.95.**

**Problem S5-112-4.** An insurance company is using the extension of exposures method to change its base rate in order to achieve an average premium of $600. The company’s rating structure incorporates a fixed policy fee of $50. To begin the determination of what the base rate should be, the company uses a “seed” base rate of $200.

The company’s current book of business contains 5 insureds, for whom the amounts of premium, if the “seed” base rate were used, would be as follows: $400, $500, $215, $736, $121.

Using the extension of exposures method, what should the company’s proposed base rate be?

**Solution S5-112-4.** This question is based on the discussion in Werner and Modlin, pp. 263-265.

Using the “seed” base rate, the average premium would be (400 + 500 + 215 + 736 + 121)/5 = 394.4. The company wishes to achieve an average premium of $600.

We use the formula BP = BS*(P-P – AP)/(P-S – AP), where BP is the desired base rate, BS is the “seed” base rate, P-P is the desired average premium, P-S is the average premium given the “seed” base rate, and AP is the fixed policy fee. Thus, here, BP = 200*(600 – 50)/(394.4 – 50) = **BP = 319.3960511**.

**Problem S5-112-5.** An insurance company is using the loss ratio method to achieve an average premium decrease of 8% from its current average premium of $900. The company’s rating structure incorporates a fixed policy fee of $120. To begin the determination of what the base rate should be, the company uses a “seed” base rate of $400.

The company’s current book of business contains 4 insureds, for whom the amounts of premium, if the “seed” base rate were used, would be as follows: $801, $410, $610, $1243.

Using the extension of exposures method, what should the company’s proposed base rate be?

**Solution S5-112-5.** This question is based on the discussion in Werner and Modlin, p. 267.

Using the “seed” base rate, the average premium would be (801 + 410 + 610 + 1243)/4 = $766.

The company desires to decrease its average premium to 900*(1 – 0.08) = $828.

BP = BS*(P-P – AP)/(P-S – AP), where BP is the desired base rate, BS is the “seed” base rate, P-P is the desired average premium, P-S is the average premium given the “seed” base rate, and AP is the fixed policy fee. Thus, here, BP = 400*(828 – 120)/(766 – 120) = **BP = 438.3900929**.

**See other sections of The Actuary’s Free Study Guide for Exam 5.**