  Actuarial Outpost TIA Exam 2 #5 Variance with a deductible
 Register Blogs Wiki FAQ Calendar Search Today's Posts Mark Forums Read
 FlashChat Actuarial Discussion Preliminary Exams CAS/SOA Exams Cyberchat Around the World Suggestions

 Short-Term Actuarial Math Old Exam C Forum

#1
 ericp Member Join Date: Aug 2007 Posts: 283 TIA Exam 2 #5 Variance with a deductible

I don't understand why the solution to this problem used a method other than using the formulas given for the exponential but no mention is made why they were not.

Given an exponential with a deductible and we are asked to find the variance of the payment, why can't the formula (Ex^2 - E[x^d]^2) - (same for Ex)^2 be used?

The solution to this problem uses a form of the double expectation/variance formula for the variance. Well, that formula never made any intuitive sense to me so I am wondering why that was chosen for the solution to this problem rather than what would seem to be the simpler approach using the formulas given and then just plug in values.

Anyone know?
If not, does anyone understand the method that was used in the solution?

Thanks.
#2
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,579 Quote:
 Originally Posted by ericp I don't understand why the solution to this problem used a method other than using the formulas given for the exponential but no mention is made why they were not. Given an exponential with a deductible and we are asked to find the variance of the payment, why can't the formula (Ex^2 - E[x^d]^2) - (same for Ex)^2 be used? The solution to this problem uses a form of the double expectation/variance formula for the variance. Well, that formula never made any intuitive sense to me so I am wondering why that was chosen for the solution to this problem rather than what would seem to be the simpler approach using the formulas given and then just plug in values. Anyone know? If not, does anyone understand the method that was used in the solution? Thanks.
I don't know the exact problem but I assume it is find the variance of YL not YP given X is exponential. This problem can be done in less than a minute. If X is less than d, then the mean and variance of YL are both 0. If X is greater than d, then the mean is theta and the variance is theta^2. The probabilities are F(d) and 1 - F(d) from the exponential.

Calculate the variance as the mean of the variances plus the variance of the means (which can be done using the bernoulli shortcut). The formula simplifies and might be worth memorizing.
#3 Jim Daniel Member SOA Join Date: Jan 2002 Location: Davis, CA College: Wabash College B.A. 1962, Stanford Ph.D. 1965 Posts: 2,713 Quote:
 Originally Posted by Academic Actuary I don't know the exact problem but I assume it is find the variance of YL not YP given X is exponential. This problem can be done in less than a minute. If X is less than d, then the mean and variance of YL are both 0. If X is greater than d, then the mean is theta and the variance is theta^2. The probabilities are F(d) and 1 - F(d) from the exponential. Calculate the variance as the mean of the variances plus the variance of the means (which can be done using the bernoulli shortcut). The formula simplifies and might be worth memorizing.
Another way to think of this calculation is through mixture distributions. That leads to the fact that E[YL^k] = E[YP^k] x Pr[ there is a positive payment] . If X is Exponential with mean theta and there is just an ordinary deductible, then of course YP is Exponential with the same mean theta.
__________________
Jim Daniel
Jim Daniel's Actuarial Seminars
www.actuarialseminars.com
jimdaniel@actuarialseminars.com

 Thread Tools Search this Thread Show Printable Version Email this Page Search this Thread: Advanced Search Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is Off

All times are GMT -4. The time now is 12:05 AM.

 -- Default Style - Fluid Width ---- Default Style - Fixed Width ---- Old Default Style ---- Easy on the eyes ---- Smooth Darkness ---- Chestnut ---- Apple-ish Style ---- If Apples were blue ---- If Apples were green ---- If Apples were purple ---- Halloween 2007 ---- B&W ---- Halloween ---- AO Christmas Theme ---- Turkey Day Theme ---- AO 2007 beta ---- 4th Of July Contact Us - Actuarial Outpost - Archive - Privacy Statement - Top