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Why Can’t The Accountants Why Can’t The Accountants Get Reserves Right?Get Reserves Right?

byby

Roger M. Hayne, FCAS, MAAARoger M. Hayne, FCAS, MAAA

2005 CAS Spring Meeting2005 CAS Spring Meeting

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What is A Reserve?What is A Reserve?

Accountants say it is to be a Accountants say it is to be a “reasonable estimate” of the “reasonable estimate” of the unpaid claim costsunpaid claim costs

CAS says that “an actuarially CAS says that “an actuarially sound loss reserve … is a sound loss reserve … is a provision, based on estimates provision, based on estimates derived from reasonable derived from reasonable assumptions …”assumptions …”

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First Question – An First Question – An Estimate of What?Estimate of What?

There seems to be agreement on this There seems to be agreement on this – an “estimate” of amount unpaid.– an “estimate” of amount unpaid.

Is it an estimate of the average Is it an estimate of the average amount to be paid? Noamount to be paid? No

It is an estimate of the most likely It is an estimate of the most likely amount to be paid? Noamount to be paid? No

It is an estimate of the amount to be It is an estimate of the amount to be paidpaid

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Some Accounting Some Accounting PronouncementsPronouncements

““Earnings fluctuations are inherent in Earnings fluctuations are inherent in risk retention, and they should be risk retention, and they should be reported as they occur”reported as they occur”

““The sole result of accrual, for The sole result of accrual, for financial accounting and reporting financial accounting and reporting purposes, is allocation of costs among purposes, is allocation of costs among accounting periods”accounting periods”

Accounting goal to provide a accurate Accounting goal to provide a accurate picture of the financial condition of a picture of the financial condition of a companycompany

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Simple ExampleSimple Example

One year insurance contractOne year insurance contract During the year a claim has occurredDuring the year a claim has occurred The day after the contract expires the claim The day after the contract expires the claim

will be settled by the toss of a fair coinwill be settled by the toss of a fair coin– Costs $1,000,000 if heads Costs $1,000,000 if heads – Costs $0 if tailsCosts $0 if tails

Assume (for simplicity) Assume (for simplicity) – No interest incomeNo interest income– No other expensesNo other expenses– Market will pay a risk charge of 20% of premiumMarket will pay a risk charge of 20% of premium

Total premium = $625 thousandTotal premium = $625 thousand

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Accounting ResultsAccounting Results

End Year 0End Year 0– Earned Premium Earned Premium

$625,000$625,000– Reserves $500,000Reserves $500,000– Profit $125,000Profit $125,000

End Year 1End Year 1– Earned Premium $0Earned Premium $0– Reserve Decrease Reserve Decrease

$500,000$500,000– PaymentsPayments

$1,000,000 (50%)$1,000,000 (50%) $0 (50%)$0 (50%)

– ProfitProfit ($500,000) (50%)($500,000) (50%) $500,000 (50%)$500,000 (50%)

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Fair Picture?Fair Picture?

The accounting snap-shot at the end The accounting snap-shot at the end of year 0 is a healthy company with of year 0 is a healthy company with $125,000 profit$125,000 profit

The same company, The same company, with no changes with no changes at allat all at the end of year 1 shows either at the end of year 1 shows either $500,000 additional profit or $500,000 $500,000 additional profit or $500,000 loss in that year – for doing NOTHINGloss in that year – for doing NOTHING

Does this really make sense?Does this really make sense?

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Oh, I Forgot to Tell YouOh, I Forgot to Tell You

Surplus at start of year 0 = $200,000Surplus at start of year 0 = $200,000 At end of year 0, $325,000 surplus – At end of year 0, $325,000 surplus –

can write more businesscan write more business At end of year 1At end of year 1

– $825,000 surplus (50%)$825,000 surplus (50%)– Insolvent (50%)Insolvent (50%)

Fair picture?Fair picture?

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It Works on the AverageIt Works on the Average

The argument is that The argument is that on the averageon the average there is no income in year 1there is no income in year 1

Will be solvent based on the averageWill be solvent based on the average Two actuaries hunting rabbits – one Two actuaries hunting rabbits – one

shoots 2 feet in front, the other 2 feet shoots 2 feet in front, the other 2 feet behind and celebrate that on the behind and celebrate that on the average they “got it”average they “got it”

Ask the rabbitAsk the rabbit

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Wait a MinuteWait a Minute

Accountants sayAccountants say““The Board cannot sanction the use The Board cannot sanction the use

of an accounting procedure to of an accounting procedure to create the illusion of protection from create the illusion of protection from risk when, in fact, protection does risk when, in fact, protection does not exist”not exist”

Doesn’t the rabbit prove that Doesn’t the rabbit prove that booking the mean really violates booking the mean really violates this precept?this precept?

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Another MinuteAnother Minute

Accountants sayAccountants say““to report activity when there has to report activity when there has

been none would obscure a been none would obscure a fundamental difference in fundamental difference in circumstances between enterprises circumstances between enterprises that transfer risks and others that do that transfer risks and others that do not”not”

Doesn’t booking the mean do Doesn’t booking the mean do precisely that in this example?precisely that in this example?

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The RealityThe Reality

Given current knowledge there is a Given current knowledge there is a distribution of possible future distribution of possible future outcomes – the reserve properly is a outcomes – the reserve properly is a DISTRIBUTIONDISTRIBUTION

The “estimate of future payments” The “estimate of future payments” definition says the reserve is a point definition says the reserve is a point on this distributionon this distribution

It does not say which oneIt does not say which one We are measured by how close the We are measured by how close the

“estimate” is to what actually happens“estimate” is to what actually happens

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Margin, What Margin?Margin, What Margin?

The rabbit will have a problem The rabbit will have a problem with any number bookedwith any number booked

Why fixate on the mean?Why fixate on the mean?““Margin” implies something extraMargin” implies something extra I assert – “Margin? We don’t I assert – “Margin? We don’t

need no steenkin’ margin.”need no steenkin’ margin.”

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Heresy?Heresy?

CAS says CAS says ““A reserve estimate should take into A reserve estimate should take into

account the degree of uncertainty account the degree of uncertainty inherent in its projections …Further, inherent in its projections …Further, an an explicitexplicit provision for uncertainty provision for uncertainty may be warranted when the may be warranted when the indicated reserve value is subject to indicated reserve value is subject to a high degree of variability”a high degree of variability”

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Not Really – Look at PricingNot Really – Look at Pricing

Classic Risk Theory result:Classic Risk Theory result:No matter how much surplus an insurer No matter how much surplus an insurer

starts with, the long-term probability of it starts with, the long-term probability of it going broke is 100% if it prices at the going broke is 100% if it prices at the expected costexpected cost

Economists say a risk-taker should be Economists say a risk-taker should be rewarded for taking riskrewarded for taking risk

Both imply that the only economically Both imply that the only economically viable insurance premium is above viable insurance premium is above expected costsexpected costs

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There is No MarginThere is No Margin

Notice that economic theory does not Notice that economic theory does not say that the price is the expected cost say that the price is the expected cost plus somethingplus something

The cost is the costThe cost is the cost The fact is, the cost is above the meanThe fact is, the cost is above the mean Thus, “mean” and “margin” are Thus, “mean” and “margin” are

artificial constructs because the cost is artificial constructs because the cost is above the meanabove the mean

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Margin is an IllusionMargin is an Illusion

Just as what the market will pay Just as what the market will pay is indivisible the economic value is indivisible the economic value of the agreement to pay future of the agreement to pay future claims is indivisibleclaims is indivisible

The fact that risk takers should be The fact that risk takers should be compensated for taking risk compensated for taking risk means that the economic value is means that the economic value is higher than the meanhigher than the mean

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One ThoughtOne Thought

Amortize the pricing “risk load” Amortize the pricing “risk load” over the life of the reservesover the life of the reserves

See Philbrick’s discussion from See Philbrick’s discussion from several years agoseveral years ago

Still maintains the dichotomy Still maintains the dichotomy between “mean” and “margin”between “mean” and “margin”

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Times ChangeTimes Change

Price reflects risk at time policy is Price reflects risk at time policy is issuedissued

Reserve should reflect risk at the time Reserve should reflect risk at the time the financial statement is preparedthe financial statement is prepared

Events unanticipated at time policy is Events unanticipated at time policy is issued and not contemplated in the issued and not contemplated in the price may have a significant influence price may have a significant influence on reserveson reserves– Asbestos exposure in a 1965 GL policyAsbestos exposure in a 1965 GL policy

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ContextContext

New CAS Statement of Principles New CAS Statement of Principles talks in terms of contexttalks in terms of context

Reserves are not set in a vacuumReserves are not set in a vacuumShould consider the company Should consider the company

issuing the statementissuing the statementWhy not explicitly recognize Why not explicitly recognize

company position when setting company position when setting reserves?reserves?

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Minimum PenaltyMinimum Penalty

Since any number will be “wrong” let me Since any number will be “wrong” let me submit a reasonable estimate of reserves submit a reasonable estimate of reserves (complements of Rodney Kreps)(complements of Rodney Kreps)

Suppose Suppose – (a really BIG suppose) we know the probability (a really BIG suppose) we know the probability

density function of future claim payments and density function of future claim payments and expenses is f(expenses is f(xx) )

– For simplicity one year time horizonFor simplicity one year time horizon– g(g(xx,,μμ) denotes the decrease in shareholder ) denotes the decrease in shareholder

(policyholder) value of the company if reserves (policyholder) value of the company if reserves are booked at are booked at μμ but payments are actually but payments are actually xx..

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Minimum Penalty (Cont.)Minimum Penalty (Cont.)

A rational reserve (i.e. “estimate of A rational reserve (i.e. “estimate of future payments”) is that value of future payments”) is that value of μμ that minimizesthat minimizes

i.e. the expected penalty for setting i.e. the expected penalty for setting reserves at reserves at μμ over all reserve over all reserve outcomesoutcomes

0

P g fx, x dx

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Some ObservationsSome Observations

In order to yield the mean, the function In order to yield the mean, the function g(g(xx,,μμ) must be) must be

2( , )g x x

It is symmetric (same penalty for being It is symmetric (same penalty for being y units high as for being low)y units high as for being low)

It is quadratic (being 2 units off is 4 It is quadratic (being 2 units off is 4 times as bad as being 1 unit off)times as bad as being 1 unit off)

Reasonable?Reasonable?

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Some ObservationsSome Observations

Taking g(Taking g(xx,,μμ) as) as

Gives the median as the “least painful”Gives the median as the “least painful” The mode is obtained with g being a The mode is obtained with g being a

“point mass” when “point mass” when xx = = μμ A given probability level can be obtained A given probability level can be obtained

by using straight lines with different by using straight lines with different slopesslopes

g ,x x

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A Reasonable gA Reasonable g

Not likely symmetricNot likely symmetricLikely flat in a region “near” Likely flat in a region “near” μμ Increases faster when Increases faster when xx is above is above

μμ than when than when xx is below is belowLikely increases at an increasing Likely increases at an increasing

rate when rate when xx is above is above μμSuch a function generally gives Such a function generally gives

an estimate above the meanan estimate above the mean

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ConclusionConclusion

Biggest accounting myth – the reserve Biggest accounting myth – the reserve reflects actual future paymentsreflects actual future payments

The true economic value of a reserve is likely The true economic value of a reserve is likely greater than the expected present value of greater than the expected present value of future paymentsfuture payments

The chance of any point on the distribution is The chance of any point on the distribution is virtually zerovirtually zero

So why focus on a “mean” and “margin”?So why focus on a “mean” and “margin”? Take a holistic, economically rational Take a holistic, economically rational

approachapproach