Actuarial Outpost Variance Payment Per Payment
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 Short-Term Actuarial Math Old Exam C Forum

#1
06-18-2018, 09:30 PM
 underactuary212 CAS SOA Join Date: Dec 2017 Posts: 10
Variance Payment Per Payment

Lets say youre given a 2 parameter Pareto (alpha = 2 theta = 300) with deductible 500

Is there a quick way to calculate variance of payment per payment variable without grinding the integrals?

Similarily lets say youre given exponential with theta = 300 and deductible 500? is there a way to calculate variance of payment per paymet
#2
06-18-2018, 10:34 PM
 pershing Member SOA Join Date: Jul 2012 College: Starfleet Academy Posts: 50

If X~Pareto(alpha = a, theta = 300) and d=500, then X-500|X>500 ~Pareto(alpha = a, theta' = 300+500=800).

V(X-500|X>500)=(800/(a-1))^2 * (a/(a-2)). Note when a=2, variance of payment per payment isn't defined.

If Y~Exponential(theta = 300) and d=500, then Y-500|Y>500 ~Exponential(theta = 300).

V(Y-500|Y>500)=theta^2=300^2.

These are the two nicest distros when it comes to variance of payment per payment variables!

Uniform(0, theta) is pretty nice, too:

If Z~Uniform(0, theta) and d=500 (theta>500), then Z-500|Z>500 ~Uniform(0, theta-500).

V(Z-500|Z>500)= (theta-500)^2/12.
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#3
06-18-2018, 10:37 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,049

Pareto per payment random variable is Pareto with theta increased by the deductible. Variance is mean squared x (alpha)/(alpha - 2). So the the variance would not exist with alpha = 2.

Exponential for the per payment random variable is the original exponential.
#4
06-19-2018, 07:54 AM
 underactuary212 CAS SOA Join Date: Dec 2017 Posts: 10

Oh ok that makes sense thanks. I was familiar with the distributions of e(d) but I didn't realize you could use that same distribution to calculate first and second moments and then the variance.