An insurer uses classification ratemaking to set rates and uses a single rating variable to classify policies with three different levels: level I, level II and level III. The pure premium for each level is assumed to be gamma distributed with unknown parameters [math]\theta [/math] and [math]\alpha [/math] that change across levels. More precisely, we have the following:

Level	[math]\alpha [/math]	[math]\theta [/math]	Exposure weight
I	2	500	50%
II	3	450	25%
III	3	475	25%

Marketing wants level II rates to be 10% higher than level 1 rates and level III rates 15% higher than level I rates. If the insurer targets an expected profit percentage of 15% and there are no fixed or variable expenses, determine the rate per exposure unit for level II policies.

• 1,379.92
• 1,450.56
• 1,457.25
• 1,516.49
• 1,578.22

We define the following variables:

• [math]R_{i,I} [/math] equals the expected loss per exposure unit for level [math]i [/math] divided by the expected loss per exposure unit for level I.
• [math]R_{i,T} [/math] equals the targeted ratio of the rate for level [math]i[/math] over the rate for level I.
• [math]\mu [/math] equals the expected loss per exposure unit for level I.
• [math]B[/math] equals the rate per exposure unit for level I.
• [math]w_i [/math] denotes the exposure weight for level [math]i[/math].

Given the definitions above, the exposure weighted expected pure premium equals

[[math]] \mu \sum_{i}w_i R_{i,I} [[/math]]

, the exposure weighted rate per exposure unit equals

[[math]] B \sum_{i}w_i R_{i,T} [[/math]]

and the expected loss ratio equals

[[math]] \frac{\mu}{B} \frac{ \sum_{i}w_i R_{i,I} }{\sum_{i}w_i R_{i,T} }. [[/math]]

Using the values given in the question, we have

Level i	[math]R_{i,I} [/math]
I	1
II	1.35
III	1.425

, [math]\sum_{i}w_i R_{i,I} = 1.19375[/math] and [math] \sum_{i}w_i R_{i,T} = 1.065[/math]. In particular, the expected loss ratio equals 1,120.89 divided by [math]B[/math]. Since the target expected loss ratio is 0.85, the rates per exposure unit are as follows: $1,318.69 for base level I, $1,450.56 for level II and $1,516.49 for level III.