You are given the following:

• The current rate per exposure unit equals $750.
• Without inflation, projected pure premium equals $550 over the next twelve months.
• Without inflation, fixed underwriting expenses equal $50 per exposure unit.
• Aggregate exposure over the next twelve months is uniformly distributed through time.
• The annual loss inflation rate is 4%.
• Fixed underwriting expenses grow at an annual rate of 2%.
• Loss adjustment expenses are negligible.

If the insurer leaves rates unchanged, determine the upper bound on the variable expense % which ensures that the underwriting profit on policies sold in the next 12 months will exceed 15% of written premium.

• 1.25%
• 1.53%
• 1.66%
• 2%
• 2.25%

We recall the fundamental insurance equation equals

[[math]] P = L + E_V + E_F + Q\cdot P. [[/math]]

Dividing by the exposure, we obtain

[[math]] 750 = \overline{L} + 750V + 50 + 750Q . [[/math]]

The projected pure premium, [math]\overline{L}[/math], equals $550 multiplied by the trend factor. The forecast period is the next two calendar years, so the trend factor equals 1.04 and the projected pure premium equals $572. Similarly, the projected fixed expense per exposure unit equals $50 multiplied by the trend factor 1.02 or $51. According to the equation above, the condition [math]Q\gt 0.15 [/math] implies that

[[math]] 750Q = 125 - 750V \gt 112.5 [[/math]]

or [math]V [/math], the variable expense %, must be less than 1.66%.