Friedland14.Recoveries

Reading: Friedland, J.F., Estimating Unpaid Claims Using Basic Techniques, Casualty Actuarial Society, Third Version, July 2010. The Appendices are excluded.

Chapter 14: Recoveries: Salvage and Subrogation and Reinsurance

Pop Quiz

An insurer typically performs its reserve analysis using development on annual triangles. Due a marketing campaign however, it has experienced significant premium growth over the past 2 years. If Alice-the-Actuary did not make changes to her methods, would her estimates of ultimate claims be high, low or accurate? If not accurate, suggest an appropriate change to her analysis. Click for Answer 

Study Tips

VIDEO: F-14 (001) Recoveries → 5:00 Forum

Chapter 14 is not too hard. It should be a welcome relief after the Berquist-Sherman methods of Chapter 13. Poor Alice-the-Actuary has had some bad luck however. Maybe you can help her out! See further down. :-)

Estimated study time: 2-3 days (not including subsequent review time)

BattleTable

Based on past exams, the main things you need to know (in rough order of importance) are:

• (Reinsurance)

   - understand mechanics of quota-share (Q/S), excess-of-loss & stop-loss reinsurance

   - calculating ultimate claims net of reinsurance using net/gross/ceded loss triangles and appropriate tail factors

• (Salvage/Subrogation)

   - calculating ultimate salvage & subrogation using either the development method or the ratio method

   - selecting the best method, knowing the advantages & disadvantages of each

reference	part (a)	part (b)	part (c)	part (d)
E (2019.Fall #22)	(S/S) recoverable S/S: 1 - ratio method	(S/S) advantages: - ratio vs devlpt method
E (2019.Spring #21)	(Reins) identify scenario: - high reported claims	(Reins) ultimate: - discuss approaches
E (2019.Spring #23)	(Reins) retained IBNR: - rptd devlpt method
(2018.Spring #22)	BattleActs PowerPack
E (2017.Fall #16)	(Reins) CYEP: - calculate	(Reins) gross UEP: - calculate	(Reins) reported claims: - calculate AY value
E (2017.Fall #25)	(Reins) net unpaid claims: - reptd devlpt method
E (2016.Fall #24)	(S/S) ultimate S/S: - devlpt method	(S/S) ultimate S/S: - ratio method	(S/S) ultimate S/S: - any method
E (2015.Fall #23)	(Reins) net PY IBNR: - calculate	(Reins) net PY unpd clms: - calculate
E (2015.Spring #24)	(S/S) net ultimate clms: - ratio method
E (2014.Fall #21)	(Reins) reins program: - determine structure	(Reins) ceded IBNR: - calculate
E (2014.Spring #20)	(S/S) ultimate S/S: - ratio method
E (2013.Fall #22)	(Reins) net data: - assess reasonableness	(Reins) ultimate clms: - net/gross/ceded	(Reins) tail factors: - impact of Q/S & stop-loss
E (2013.Spring #24)	(S/S) ultimate S/S: - devlpt method	(S/S) ultimate S/S: - ratio method	(S/S) ultimate S/S: - best method

¹ Recoverable S/S is the same thing as unpaid S/S.

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In Plain English!

Intro

This chapter discusses 2 different types of recoveries:

• Salvage & Subrogation
• Reinsurance

Recoveries represent ways an insurer can get money back. Awesome!

Salvage is common in property insurance. A good example is last week when Alice-the-Actuary totalled her Lamborghini on the Ventura Freeway in L.A. (Aside from her ego, she wasn't hurt.) Her insurer paid her $150,000 for the loss. But her insurer then took possession of her wrecked car and recovered $20,000 in salvage by selling it to a scrap yard.

• the insurer's gross loss was $150,000
• the insurer's net loss was $150,000 - $20,000 = $130,000

When dealing with recoveries you have to understand whether the loss data is gross or net before flying off on your calculations. Now, salvage is pretty easy to understand but if I asked you to define subgrogation, could you do it?

definition: what is the definition of subrogation

• an insurer's right to recover claim payments to an insured from a 3rd party responsible for the damage

Subrogation is common in liability insurance. Let's say that Alice's accident on the Freeway was caused by an irritating little smart car that cut her off. Alice's insurer could subrogate against the insurer for the smart car and recover the net loss of $130,000. (The smart car wasn't damaged. Because, you know, it's smart.)

Reinsurance is the other type of recoverable discussed in this chapter of Friedland.

definition: what is the definition of reinsurance

• a transaction in which an insurer transfers a portion of their risk to another party (the reinsurer)

There are various reasons to purchase reinsurance, the most obvious being to reduce the likelihood of the insurer having to pay a large claim. Property insurers often purchase reinsurance to protect them from catastrophes.

• insurer losses gross of reinsurance are losses before reinsurance recoveries
• insurer losses net of reinsurance are losses after reinsurance recoveries
• ceded losses are reinsurer losses under the reinsurance treaty (refers to the portion of risk transferred from the primary insurer to the reinsurer)..

Formula:    (gross loss)   =   (net loss)   +   (ceded loss)

Different types of reinsurance are discussed further down.

Super-important advice: You have to be absolutely clear about whether your loss data is net, gross or ceded and whether those terms refer to salvage & subrogation or reinsurance (thx MPT!)

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Salvage/Subgrogation Recoveries

Alice-the-Actuary has decreed that we will henceforth use S/S as an abbreviation for Salvage/Subgrogation. (And I will henceforth ask our good friend Alice to stop using pretentious vocabulary such as "decreed" and "henceforth".)

Here is an old exam problem asking you for the ultimate S/S using both the development method and the ratio method. (The ratio method works like the development method except you do the calculations on a triangle of ratios of S/S to paid claims instead of directly to the paid claims. This is all laid out clearly in Alice's solution.)

E (2016.Fall #24)

The development method for S/S is exactly the same as the development method for any other triangle of loss data. I have included the result of the development method in my solution below but have omitted the intermediate steps. (Alice says you should know how to do that by now.) The solution shows the details for the ratio method:

Here is Alice's solution and 2 similar practice problems:

Solution: 2016.Fall #24

Practice: 2 problems like 2016.Fall #24

And here's something else you should probably know the answer to...

Question: what are 2 advantages of the ratio method versus the development method for calculating ultimate S/S

• LDFs in ratio method not as highly leveraged as in development method

(paid S/S data may be thin, especially at early maturities)

• ratio method recognizes the relationship between S/S and paid claims

(and the S/S-to-paid-claims ratios can be adjusted if they seem unreasonable)

Other than that, this section is pretty easy. Just make sure you practice the ratio method a few times.

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Reinsurance Recoveries

Common Types of Reinsurance

The source text (Friedland) has 3 very nice, simple examples of the different types of reinsurance you need to understand. (Links provided below.) The 3 types of reinsurance discussed in the text are:

1. quota-share
2. excess-of-loss
3. stop-loss

You're probably at least somewhat familiar with these different forms of reinsurance, but let's cover the key points in simple terms.

Quota-share reinsurance is a pro-rata contract where the insurer and reinsurer share premiums and losses according to a fixed percentage.

Suppose you have a gross loss of $1,000. If you also have a quota-share reinsurance treaty with a quota-share percentage of 70% then according to the Friedland source text:

• net loss (net of quota-share reinsurance) = $1,000 x 70% = $700
• ceded loss = $1,000 x 30% = $300

The allocation of gross premium works the same way. If the gross premium received by the insurer (thx CG!) is $200, then the net premium is $200 x 70% = $140, and the premium ceded to the reinsurer is $60. (thx MPT!)

On its own, this is very easy to understand. But remember: the quota-share percentage is applied to the gross loss to get the net loss. Confusion sometimes arises when the quota-share percentage is very low, like 25%. People sometimes think that means 25% of the gross loss is the ceded loss. That's wrong! (Multiply the gross loss by 25% to get the net loss.)

Here's a quick link to the Friedland's example of quota-share reinsurance:

Example: Quota-Share (source text)

Note 1: that if you have the gross and net loss triangles, it's very easy to determine whether or not you've got a quota-share contract.

Note 2: Other sources define the quota-share percentage the opposite way. In other words, a 70% quota-share agreement means that 70% of the premiums & losses are ceded. For this reason, it's a good idea to state your interpretation in any problem that uses quota-share reinsurance.

Excess-of-Loss reinsurance indemnifies the ceding company for losses that exceed a specified limit. (It is non pro-rata reinsurance.)

The contract would state whether the specified limit applies separately to individual losses or to aggregate losses.

Suppose a primary insurer has 2 claims for the amounts of $100 and $175.

Example 1: apply the individual limit of $150 to each individual loss

• loss 1 = $100→ no reinsurance recovery because loss ≤ $150
• loss 2 = $175→ reinsurance recovery = (loss 2) - limit = $175 - $150 = $25
• total recovery = $0 + $25 = $25

Example 2: apply the aggregate limit of $150 to aggregate losses

• aggregate loss = (loss 1) + (loss 2) = $100 + $175 = $275
• total recovery = (aggregate loss ) - limit = $275 - $150 = $125

Below is a link to the Friedland's individual excess-of-loss reinsurance example. The attachment point is $1m. That means all claims under $1m are covered by the primary insurer and the excess over $1m is covered by the reinsurer. The example doesn't provide details of individual claims but at least you can see how the triangles are different from the quota-share example. AY 2005 & 2007 had individual claims over $1m (as you can deduce from the ceded loss triangle) while AY 2006 & 2008 did not have any claims over $1m.

Example: Excess-of-Loss (source text)

Note: Friedland generally uses the term excess-of-loss reinsurance for individual excess-of-loss reinsurance, and stop-loss reinsurance for aggregate excess-of-loss. (See below.)

Stop-Loss reinsurance indemnifies all losses above a specified amount during the contract period.

This is really just the same as aggregate excess-of-loss, but we'll look at a more slightly more complicated example. Here are a few of the extra things you need to keep in mind:

• you can apply the stop-loss limit/attachment point on top of the net losses from an individual excess-of-loss policy

→ text example has individual claim attachment point = $500,000

• the contract can apply to AYs, PYs, CYs (Accident Years, Policy Years, Calendar Years)

→ text example applies to PYs

• the stop-loss limit/attachment point can be applied across multiple years and coverages

→ example combines 3 PYs: 2002/03, 2003/04, 2004/05

• there is often an upper limit on recoveries (beyond which the primary insurer again becomes fully responsible)

→ text example does not have an upper limit

Make sure you can follow the example below.

Example: Stop-Loss (source text)

Alice created a few more practice examples of stop-loss reinsurance similar to the example from Friedland.

Practice: 2 Stop-Loss problems

Combination Problem: The problem below combines excess-of-loss and stop-loss reinsurance.

E (2017.Fall #25)

Here is Alice's solution and 2 similar practice problems:

Solution: 2017.Fall #25

Practice: 2 problems like 2017.Fall #25

Tail Factors in Reinsurance

Recall that a tail factor for a loss triangle is the cumulative age-to-ultimate LDF. The "age" varies depending on how much credible data you have in your triangle. The age could be as low as 36 or 48 months if you only have a small triangle.

You already know that short-tail lines of business (Ex: property) often settle in a matter of months whereas long-tail lines (Ex: liability) can take many years. That means early estimates of reserves for short-tail lines should be more accurate than early estimates for long-tail lines. Ok, so what does this have to do with reinsurance. Consider the following question:

Question: For a primary insurer with a reinsurance treaty, how would development on gross reported data differ from development on net reported data

• It depends on whether the reinsurance treaty is quota-share or excess-of-loss.

quota-share reinsurance:

• no difference in development between gross and net triangles
• the net triangle is a multiple of the gross triangle (multiplier = quota-share percentage)

excess-of-loss reinsurance:

• development on gross experience may be more credible due to greater volume (includes all losses)
• development on net experience may be less volatile due to large claims being cut off (includes only data below reinsurance attachment point)

Now, if you want to estimate reserves for this primary insurer, there are 3 ways to proceed:

1. estimate reserves separately for gross and net triangles then ceded = gross – net
2. estimate reserves separately for gross and ceded triangles then net = gross – ceded
3. estimate reserves separately for net and ceded triangles then gross = net + ceded

To decide, you have to look at the data and the type of reinsurance treaty. If the ceded amount is small relative to the net amount, it probably wouldn't make sense to develop ceded triangles so option 1 might be best. Conversely if the ceded amount is large relative to the net (Ex: if a primary insurer wants to exit a line of business and they dumped the majority of the claims onto a reinsurer) then it might make more sense to use option 2. Anyway, while you're making this decision, there are a few facts about tail factors you may also want to consider:

Question: describe how tail factors between gross, net, and ceded triangle might differ for both quota-share and excess-of-loss reinsurance

quota-share reinsurance:

• no difference in tail factors between gross and net triangles
• the net triangle is a multiple of the gross triangle so ratios between columns (LDFs) are the same for gross and net

excess-of-loss reinsurance:

• Let TF_i = Tail Factor for (age i)-to-ultimate. Then the relationship between tail factors can be expressed as follows:

TFi (net)  ≤  TFi (gross)  ≤  TFi (ceded)

• TF_i (net) is the smallest because losses are capped once retention level or attachment point is hit → no more development
• TF_i (gross) is in the middle because the denominators in the LDF ratios are bigger than for the ceded triangle
• TF_i (ceded) is the biggest because all development is in the tail after the retention level is hit (up to the reinsurance limit)

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More Exam Problems

Here are a few more older exam problems. They are worth fewer BRQ points since they aren't as recent.

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POP QUIZ ANSWERS

For a growing book of business, her estimates would tend to be low:

• growth shifts the average accident date to a later point in each development period – this makes the period less mature

→ a less mature development period needs a higher CDF

→ historical CDFs based on losses before the shift in average accident date will be too low

→ final estimates will be too low

A solution would be to use accident quarters instead of accident years. There would still be a tendency for estimates to be too low (due to shifts in average accident date just as with annual triangles) but the distortion would be less pronounced because quarters are shorter than years. A large company with significant data resources may even decide to switch to accident months.

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