View Full Version : Expected Reinsurer Deficit

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.

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.

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.

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.

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!

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