**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 11 of the Study Guide. See an index of all sections by following the link in this paragraph.**

The problems in this section all refer to the following excerpt from a fictional workers’ compensation insurance rating manual from Srekrow Insurance Company:

**Class Rating Table**

**Class…………………………..Rate per $100 of payroll**

Hippopotamus trainers………………….1.35

Asteroid miners…………………………3.56

Writers of actuarial study guides………0.45

All other workers………………………1.00

**Premium Credits/Debits**

**Item………………………….Credits(-) or Debits(+)**

Presence of blue sofas…………………-2%

Presence of green sofas………………..+5%

Rounded corners on walls…………….-10%

Spikes sticking out of walls……………+50%

**Expense Constant**

The expense constant is $95 per policy.

**Minimum Premium**

The minimum premium for any policy is $1000.

**Rating Algorithm**

Total Premium = Higher of

(i=1NΣ(Class rate * $Payroll for class /100) where N = number of classes

*(1.0 – Blue Sofa Credit)

*(1.0 + Green Sofa Debit)

*(1.0 – Rounded Corners Credit)

*(1.0 + Spikes Debit)

+ Expense Constant)

and

Minimum Premium specified in the rating manual ($1000 in this

example).

There is to be no rounding at any step of the algorithm.

**Source:**

Werner, Geoff and Claudine Modlin. *Basic Ratemaking.* Casualty Actuarial Society. 2009. Chapter 2, pp. 29-32.

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

**Problem S5-11-1.** Hippopotamus Actuarial and Asteroid Mining Services employs the following individuals:

6 hippopotamus trainers who are paid $56000 each per year.

2 asteroid miners who are paid $100000 each per year.

9 writers of actuarial study guides who are paid $30000 each per year.

The company does not own any sofas and has ordinary sharp corners on the walls of its offices. There are also no spikes sticking out of the walls. Find the total premium that the company will have to pay to Srekrow Insurance Company to insure all of its employees.

**Solution S5-11-1.** We first find the **manual premium** or

i=1NΣ(Class rate * $Payroll for class /100). Here, N = 3.

For hippopotamus trainers, Class rate * $Payroll for class /100 = 1.35*6*56000/100 = 4536.

For asteroid miners, Class rate * $Payroll for class /100 = 3.56*2*100000/100 = 7120.

For writers of actuarial study guides, Class rate * $Payroll for class /100 = 0.45*9*30000/100 = 1215.

Thus, the manual premium is 4536 + 7120 + 1215 = 12871.

Total Premium = Higher of

i=1NΣ(Class rate * $Payroll for class /100) (12871)

*(1.0 – Blue Sofa Credit) (1 – 0 = 1)

*(1.0 + Green Sofa Debit) (1 + 0 = 1)

*(1.0 – Rounded Corners Credit) (1 – 0 = 1)

*(1.0 + Spikes Debit) (1 + 0 = 1)

+ Expense Constant (95)

and

Minimum Premium specified in the rating manual ($1000 in this

example).

12871 + 95 = 12966 > 1000, so the premium for this policy is **$12,966.**

**Problem S5-11-2.** Hippopotamus Actuarial and Asteroid Mining Services employs the following individuals:

6 hippopotamus trainers who are paid $56000 each per year.

2 asteroid miners who are paid $100000 each per year.

9 writers of actuarial study guides who are paid $30000 each per year.

Previously, the company’s workers’ compensation insurance premium was 12966. By how much will the premium decrease if the company decides to purchase blue sofas? (Everything else remains the same.)

**Solution S5-11-2.** The manual premium remains the same at 12871 (See Solution S5-11-1), but now it is multiplied a factor of (1 – 0.02) = 0.98.

Total Premium = Higher of

i=1NΣ(Class rate * $Payroll for class /100) (12871)

*(1.0 – Blue Sofa Credit) (1 – 0.02 = 0.98)

*(1.0 + Green Sofa Debit) (1 + 0 = 1)

*(1.0 – Rounded Corners Credit) (1 – 0 = 1)

*(1.0 + Spikes Debit) (1 + 0 = 1)

+ Expense Constant (95)

and

Minimum Premium specified in the rating manual ($1000 in this

example).

12871*0.98 + 95 = 12708.58 > 1000.

The premium will decrease by 12966 – 12708.58 = **$257.42**.

**Problem S5-11-3.** A company currently employs 2 writers of actuarial study guides who are paid $40000 each per year. The company has both green and blue sofas and has rounded corners sticking out of walls. At most how many more writers of actuarial study guides can the company hire while keeping its workers’ compensation insurance premium at 1000?

**Solution S5-11-3.** We first find the current manual premium. There is only one class of workers, so Manual Premium = Class rate * $Payroll for class /100 = 0.45*2*40000/100 = 360.

We multiply the manual premium by the following factors:

(1.0 – Blue Sofa Credit) = (1-0.02) = 0.98;

(1.0 + Green Sofa Debit) = (1+0.05) = 1.05;

(1.0 – Rounded Corners Credit) = (1-0.10) = 0.90;

Prior to the application of the expense constant, we thus have a value of 360*0.98*1.05*0.90 = 333.396. Each employee would contribute 333.396/2 = 166.698 to the manual premium, adjusted by the factors above.

To 333.396 we add the expense constant of 95 and get 428.396. This is how much the company would have paid in premium if it were not for the 1000 minimum. The company can add 1000 – 428.396 = 571.604 to what its premium would have been without the minimum before reaching 1000.

Thus, hiring an additional 571.604/166.698 = 3.4289179352 writers of actuarial study guides would not contribute to an increase in the premium. Since writers of actuarial study guides only come in discrete units, the company can hire **at most 3 more writers of actuarial study guides**.

**Problem S5-11-4.** A company has the following characteristics:

There are 7 asteroid miner employees, 5 of whom get paid $46,000 per year, and 2 of whom get paid $150,000 per year.

There are also 2 “other” employees, each of whom gets paid $90,000 per year.

The company’s office walls have spikes sticking out them. The company does not have any sofas or rounded corners.

Find the total premium that the company will have to pay to Srekrow Insurance Company to insure all of its employees.

**Solution S5-11-4.** We first find i=1NΣ(Class rate * $Payroll for class /100). Here, N = 2.

For asteroid miners, Class rate * $Payroll for class /100 = 3.56(5*46000 + 2*150000)/100 = 18868.

For the “other” employees, Class rate * $Payroll for class /100 = 1.00(2*90000)/100 = 1800.

Thus, the manual premium is 18868 + 1800 = 20668.

Total Premium = Higher of

i=1NΣ(Class rate * $Payroll for class /100) (20668)

*(1.0 – Blue Sofa Credit) (1 – 0 = 1)

*(1.0 + Green Sofa Debit) (1 + 0 = 1)

*(1.0 – Rounded Corners Credit) (1 – 0 = 1)

*(1.0 + Spikes Debit) (1 + 0.50 = 1.50)

+ Expense Constant (95)

and

Minimum Premium specified in the rating manual ($1000 in this example).

20668*1.50 + 95 = 31097 > 1000, so the premium for this company is **$31,097.**

**Problem S5-11-5.** A company has the following characteristics:

There are 7 asteroid miner employees, 5 of whom get paid $46,000 per year, and 2 of whom get paid $150,000 per year.

There are also 2 “other” employees, each of whom gets paid $90,000 per year.

The company’s office walls have spikes sticking out them. The company does not have any sofas or rounded corners.

The company decides to diversify into hippopotamus training. However, it does not wish to see an increase in workers’ compensation premiums as a result of doing so. To compensate for the increased payroll, the company decides to eliminate the spikes sticking out of its office walls, to purchase blue sofas, and to install rounded corners on all the walls.

Each hippopotamus trainer gets paid $50000 per year. By at most how many hippopotamus trainers could the company hire without increasing its workers’ compensation premium after instituting the safety improvements above?

**Solution S5-11-5.** From Solution S5-11-4, the current premium is 31097, and the current manual premium is 20668.

With the new credits received from the safety improvements, the manual premium is multiplied by the following factors:

(1.0 – Blue Sofa Credit) (1 – 0.02 = 0.98);

(1.0 – Rounded Corners Credit) (1 – 0.10 = 0.90).

20668*0.98*0.90 + 95 = 18324.176 = the new total premium before any new workers are hired.

The company can afford to increase its premium by 31097 – 18324.176 = 12772.824 through hiring new hippopotamus trainers.

How much will each hippopotamus trainer cost in extra premium?

The manual premium *per trainer* is 1.35*50000/100 = 675.

Multiplying this value by the two credit factors, we get 675*0.98*0.90 = 595.35. This is the added cost in premium of hiring one trainer.

Thus, the company can hire at most 12772.824/595.35 = 21.45431091 additional trainers.

Since hippopotamus trainers only come in discrete units, the company can hire **at most 21 more hippopotamus trainers.**

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