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triplea
10-16-2007, 04:35 PM
1. (2 points) Let X and Y be two random variables denoting reported and ultimate losses, respectively. Explain what each of the following means, and state the loss development method under which the condition holds, as explained in Brosius:

i. E[Var[Y|X]] = 0

ii. Cov(X, Y) = 0

iii. Cov(X, Y-X) = 0

iv. Var[E[Y|X]] = 0

incandescense
10-16-2007, 05:26 PM
1. (2 points) Let X and Y be two random variables denoting reported and ultimate losses, respectively. Explain what each of the following means, and state the loss development method under which the condition holds, as explained in Brosius:

i. E[Var[Y|X]] = 0

EPV = 0 means Z = 1, or the Link-Ratio method gives the correct ultimate loss.

ii. Cov(X, Y) = 0

Ultimate loss has no correlation with reported loss, so this is equivalent to the budgeted loss method.

iii. Cov(X, Y-X) = 0

The reserve has no correlation with reported losses, so this is equivalent to the BF Method.

iv. Var[E[Y|X]] = 0

VHM = 0 means that Z = 0, or full credibility to Budgeted Loss Method.

triplea
10-16-2007, 05:28 PM
I like the answers to ii and iii, but I think more should be said regarding what E[Var[Y|X]] really means (same thing for iv).

triplea
10-16-2007, 05:32 PM
So, I would say E[Var[Y|X]] is the variance of the loss reporting pattern, thus if it is equal to 0, then the loss reporting pattern does not change (or has no variability), therefore the link ratio method is implied here.

Var[E[Y|X]] is the variance of the loss occurring process, thus if this is equal to 0, there is no change in the loss occurring process (or has no variability), which calls for the fixed prior, or the budgeted loss, method.

Something like that...

regression
10-16-2007, 10:55 PM
Var[E[Y|X]] or Var[E[X|Y]]?

So, I would say E[Var[Y|X]] is the variance of the loss reporting pattern, thus if it is equal to 0, then the loss reporting pattern does not change (or has no variability), therefore the link ratio method is implied here.

Var[E[Y|X]] is the variance of the loss occurring process, thus if this is equal to 0, there is no change in the loss occurring process (or has no variability), which calls for the fixed prior, or the budgeted loss, method.

Something like that...

2M
10-16-2007, 10:59 PM
The variance formulas they have above are backwards it is Var[X|Y]

triplea
10-16-2007, 11:09 PM
So it should be E[Var[X|Y]] and Var[E[X|Y]]??

triplea
10-16-2007, 11:09 PM
Why is that?

2M
10-16-2007, 11:22 PM
The way they are used in the problems are all in terms of reported to ultimate (X to Y)

When you say E[X|Y] = 30%, you are saying 30% of your ultimte losses have been reported.