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General Apathy
03-22-2012, 11:27 AM
Help me settle a bet.

What is the ERD at the 99th percentile given the following data:

Percentile Loss
99.0% 4,799,769
99.1% 4,877,292
99.2% 4,982,982
99.3% 5,134,494
99.4% 5,401,793
99.5% 5,626,858
99.6% 5,803,433
99.7% 6,088,999
99.8% 6,476,636
99.9% 7,021,987

Colymbosathon ecplecticos
03-22-2012, 11:38 AM
Odds are that you're wondering what to do with the first loss amount. The first term should be ((4,877,292 - 4,799,769)/2 + 4,799,769) *0.001, the trapezoid method. Other numerical methods could also be used, such as Simpson's Rule.

You will always have a problem with what to do at the other end. You might fit a curve in various ways, but I suspect that the bet is about whether to use the first point or not.

PS How did the world-wide roller-coaster tour work out?

tommie frazier
03-22-2012, 12:04 PM
the loss there is the reinsurer's loss?

idk, like 56k?

General Apathy
03-22-2012, 12:14 PM
Deficit Average Defecit
99.0% 4,799,769 912,950
99.1% 4,877,292 77,523 Probability
99.2% 4,982,982 183,213 1%
99.3% 5,134,494 334,725 Expected Deficit
99.4% 5,401,793 602,024 9,130
99.5% 5,626,858 827,089 Percent of Premium
99.6% 5,803,433 1,003,664 0.190%
99.7% 6,088,999 1,289,230
99.8% 6,476,636 1,676,867
99.9% 7,021,987 2,222,218

This is how I set it up. ERD of .19%.

is this way off? My brain doesn't work today.

MountainHawk
03-22-2012, 12:19 PM
That's if the premium = 99%ile.

I didn't think ERD had a percentile. It's just integral (0,infinity) of Pr(Loss) max(Loss-Premium,0) dLoss

Your calculation is an estimate of it if the Premium is 4,799,769.

General Apathy
03-22-2012, 12:21 PM
That's if the premium = 99%ile.

I didn't think ERD had a percentile. It's just integral (0,infinity) of Pr(Loss) max(Loss-Premium,0) dLoss

Your calculation is an estimate of it if the Premium is 4,799,769.

Yes sorry. I meant if the premium was 4,799,769...

Klaymen
03-22-2012, 01:51 PM
Hope you aren't writing Exam 9 questions :)

Doctor Who
03-22-2012, 03:06 PM
Nah, he's going to write questions for the SOA.

M^3
03-22-2012, 03:08 PM
As MH mentioned, ERD doesn't have a percentile. Also, remember to put everything on a PV basis.

MountainHawk
03-22-2012, 03:12 PM
As MH mentioned, ERD doesn't have a percentile. Also, remember to put everything on a PV basis.
PV? Interest rates are 0, done. ;-)

General Apathy
03-22-2012, 06:43 PM
As MH mentioned, ERD doesn't have a percentile. Also, remember to put everything on a PV basis.

I've never seen ERD expressed as anything but a percentile

See

www.casact.org/affiliates/maf/0906/downs.ppt

www.casact.org/education/ratesem/2008/handouts/ruhm.ppt

http://www.variancejournal.org/issues/01-01/009.pdf

Agree PV but its immaterial these days.

M^3
03-22-2012, 06:51 PM
I've never seen ERD expressed as anything but a percentile

See

www.casact.org/affiliates/maf/0906/downs.ppt

www.casact.org/education/ratesem/2008/handouts/ruhm.ppt

http://www.variancejournal.org/issues/01-01/009.pdf

Agree PV but its immaterial these days.

Can you show me in your links where it says ERD is defined based on a fixed percentile? (Note: It can't possibly be based on a fixed percentile due to the definition of it... you can back into a percentile if you want from it though.)

A quote from one of Goldfarb's papers, where he discussed how CTE compares to EPD (which is basically the ERD from a policyholder perspective):

The Expected Policyholder Deficit (EPD) is closely related to the CTE risk measure. However, the
CTE is conditional on the losses exceeding an arbitrarily selected percentile while the EPD is
somewhat less arbitrary. The EPD is driven by the average value of the shortfall between the assets
and liabilities. All liability scenarios are included in this calculation, in contrast to the CTE risk
measure that uses only those scenarios for which the liabilities exceed a selected percentile. But in
the EPD calculation, scenarios for which there is no “shortfall” are assigned a value of zero.

http://www.casact.org/library/studynotes/goldfarb8.2.pdf

Happy Spiaggia
03-22-2012, 06:56 PM
Can you show me in your links where it says ERD is defined based on a fixed percentile? (Note: It can't possibly be based on a fixed percentile due to the definition of it... you can back into a percentile if you want from it though.)

A quote from one of Goldfarb's papers, where he discussed how CTE compares to EPD (which is basically the ERD from a policyholder perspective):

http://www.casact.org/library/studynotes/goldfarb8.2.pdf

ERD in all those links are expressed as a percentile and this is how it is used in practice.

M^3
03-22-2012, 06:59 PM
ERD in all those links are expressed as a percentile and this is how it is used in practice.

As defined in the third link, ERD = (probability of net income loss * average severity of income loss ) / net premium.

How is this a percentile?

MountainHawk
03-22-2012, 07:38 PM
I've never seen ERD expressed as anything but a percentile

See

www.casact.org/affiliates/maf/0906/downs.ppt

www.casact.org/education/ratesem/2008/handouts/ruhm.ppt

http://www.variancejournal.org/issues/01-01/009.pdf

Agree PV but its immaterial these days.
It's expressed as a percent of premium.

It is not based on a percentile of the loss distribution. You have to consider all loss positions.

Happy Spiaggia
03-22-2012, 08:41 PM
Yes percentage not percentile.

It has the % thingy after it. that's what i mean.

Avi
03-26-2012, 01:12 PM
I agree with CE that it certainly depends on what is beyond the 99.9 %-ile.

The way I've always understood it, the ERD is a number in the same units as loss is measured, which is usually dollars. It may be converted to a percentage of premium; it can also be converted to an expected number of bananas, given a Bananas to dollar conversion factor :)

More precisely:

\textrm{ERD} = \int_0^\infty \max(\textrm{Resinsurer Loss}, 0) \;f(x)\;dx

Knowing that the reinsurer starts losing money at a ground-up severity of x (and not counting tax and other issues) simplifies the equation to:

\textrm{ERD} = \int_x^\infty (t-x) \;f(t)\;dt

Which is just another way of stating Overall Mean - LAS(x).

In your case, we are faced with the issue CE mentioned above, which is that we don't have any knowledge about what happens after the 99.9%-ile. Assuming that is the max, and using the trapezoid approximation for integrals, I pretty much agree with your estimate.

CDF Loss Exp. Deficit
99.00% 4,799,769 -
99.10% 4,877,292 38.76
99.20% 4,982,982 130.37
99.30% 5,134,494 258.97
99.40% 5,401,793 468.37
99.50% 5,626,858 714.56
99.60% 5,803,433 915.38
99.70% 6,088,999 1,146.45
99.80% 6,476,636 1,483.05
100.00% 7,021,987 3,899.09
ERD 9,054.99
%Prem 0.19%
Just for kicks, fitting a generalized Pareto (not the KPW one, the EVT one - and I used Kreps's and not McNeil's parametrization) by minimizing the squared difference between the empirical and fitted distribution points (In a sense, finding the parameters that minimize the Cramér-von Mises criterion) using Excel (which is the pits, I agree, but I was too lazy to write a fitting routine in R right now) and then using those parameters to calculate E(X) - LAS(x) where x is the \$ value that starts the loss, I get:

a 48.41772323
q 989001.625
Squared Error 5.88921E-07
99.00% 4,799,769 0.99019674
99.10% 4,877,292 0.990870386
99.20% 4,982,982 0.991713451
99.30% 5,134,494 0.992785634
99.40% 5,401,793 0.994344721
99.50% 5,626,858 0.995388647
99.60% 5,803,433 0.996068507
99.70% 6,088,999 0.996959046
99.80% 6,476,636 0.997849537
99.90% 7,021,987 0.998673714

Mean 1,009,859
LAS(4799769) 998,967
ERD @99% 10,892
ERD 0.23%
Which stands to reason, as the curve extends beyond the 99.9 %-ile.

As an aside, what was the bet?

:)

General Apathy
03-28-2012, 05:05 PM
Thanks Avi!