Exam 5: Insurance to Value - Possible Error

[Izzy Mandelbaum]

On page 26 of Anderson's paper, the author shows a table with characteritics of the size of loss distribution and the corresponding behavior of the pure premium rate as a function of the face amount of insurance.

The table implies that, when small losses outnumber large ones (and hence the second derivative is negative), the rate decreases at a decreasing rate. In other words, as the face amount increases, the successive differences between rates get smaller and smaller. This is supported by Example 9 on the following page.

I believe that this conclusion is wrong, and that the exact opposite is true. When the loss distribution is such that small losses are more probable than large losses, it can be shown that the second derivative of R with respect to F (symbols as defined in paper) is in fact negative, as stated by the author. However, this implies that the rate will decrease at an increasing rate, which contradicts the author's statement. Drawing a graph should help clarify the situation.

Does anyone else see a problem with the reasoning in the paper? This question was actually tested in the May 1997 exam (question number eight). At that time, the answer which I claim is incorrect was accepted as correct.

[Renshi G]

On page 26 of Anderson's paper, the author shows a table with characteritics of the size of loss distribution and the corresponding behavior of the pure premium rate as a function of the face amount of insurance.

The table implies that, when small losses outnumber large ones (and hence the second derivative is negative), the rate decreases at a decreasing rate. In other words, as the face amount increases, the successive differences between rates get smaller and smaller. This is supported by Example 9 on the following page.

I believe that this conclusion is wrong, and that the exact opposite is true. When the loss distribution is such that small losses are more probable than large losses, it can be shown that the second derivative of R with respect to F (symbols as defined in paper) is in fact negative, as stated by the author. However, this implies that the rate will decrease at an increasing rate, which contradicts the author's statement. Drawing a graph should help clarify the situation.

Does anyone else see a problem with the reasoning in the paper? This question was actually tested in the May 1997 exam (question number eight). At that time, the answer which I claim is incorrect was accepted as correct.

I think you may want to try some concrete numerical examples to see what happens. :)

[Plant Food]

The problem is with the expression "small/large losses outnumber large/small losses". The author doesn't quantify this. To understand what's going on look at the following.

Case 1: s(L)=-2L+2 (0<L<1) Using Equation 23 we get dR/dF=-f(1-(2/3)F) and d^2R/dF^2=(2/3)f>0

Case 2: s(L)=1 (0<L<1) Using Equation 23 we get dR/dF=-f/2 and d^2R/dF^2=0

Case 3: s(L)=2L (0<L<1) Using Equation 23 we get dR/dF=-f(2/3)F and d^2R/dF^2=(-2/3)f<0

If you draw the graphs of the 3 cases, you'll see what she's trying to say. I suspect that the change from positive to negative has more to do with the skewness or the relationship of the mean to the median than the fuzzy expression "small/large losses outnumber large/small losses". In general, d^2R/dF^2=(-f/F^2)s(F)-2(f/F^3)int_0^F{Ls(L)dL}

(Gratuitous shots at actuaries) It amazes me that, for a group of people that pride themselves on their mathematical ability, they can be so imprecise.