2014 Fall #18
Looking at the first solution I do not understand how the area of the shaded region equals the equation they came up with. I do not understand why (rG-rH)*(1-rG/2) is equal to the rectangular section or the shaded region. (rG-rH) being the height makes sense to me but why is the width of the rectangle 1-rG/2. I have the same concern for the triangle section width being (rG-rH)/2.
Comments
We're told the correct distribution for losses is uniform on [$0, $90,000] so E is $45,000.
Since we're told to keep the same minimum and maximum premium we need to find r_H and r_G, the entry ratios associated with the minimum and maximum premiums respectively.
Since the loss distribution is uniform the largest entry ratio possible is 90,000 / 45,000 = 2.
Further, the entry ratios increase linearly as you range through all possible losses that come from a uniform loss distribution. So the Lee diagram looks like a line from (0,0) to (1,2). We then overlay r_G and r_H; the shaded region in between is the ratable loss.
The area of the shaded region is determined by the first balance equation - we calculate it to be 0.633. The area of a triangle is 1/2 * base * height. We view the shaded region as the difference between the triangle with base determined by r_H and the one with base determined by r_G.
Since the hypotenuse of these triangles is determined by the line from (0,0) to (1,2) we can write the equation of this line as y = 2x. Plugging in r_H and r_G in turn for y means the base of the triangles are 1-r_H/2 and 1-r_G/2 respectively.
So the shaded area is 0.5*(1-r_H/2)*(2-r_H) - 0.5*(1-r_G/2)*(2-r_G) = (1-r_H/2)*(1-r_H) - (1-r_G/2)*(1-r_G).
The first sample solution takes a slightly different approach by looking at the shaded area as a trapezoid and breaking it into the sum of a triangle and rectangle. The rectangle has height r_G - r_H and width 1- r_G/2. The triangle has height r_G - r_H and base (1-r_H/2) - (1-r_G/2) = (r_G - r_H)/2
wondering why we cannot directly calculate rG and rH since we have min and max premium, and E? (i.e. rG = max prem / E and rH = min prem / E)
The entry ratio rG is the actual loss associated with the maximum premium divided by the expected loss. We do not directly know the numerator so cannot calculate as you suggest.