Milliman Why Can’t The Accountants Get Reserves Right? by Roger M. Hayne, FCAS, MAAA 2005 CAS Spring Meeting

Milliman What is A Reserve? Accountants say it is to be a “reasonable estimate” of the unpaid claim costs Accountants say it is to be a “reasonable estimate” of the unpaid claim costs CAS says that “an actuarially sound loss reserve … is a provision, based on estimates derived from reasonable assumptions …” CAS says that “an actuarially sound loss reserve … is a provision, based on estimates derived from reasonable assumptions …”

Milliman First Question – An Estimate of What? There seems to be agreement on this – an “estimate” of amount unpaid. There seems to be agreement on this – an “estimate” of amount unpaid. Is it an estimate of the average amount to be paid? No Is it an estimate of the average amount to be paid? No It is an estimate of the most likely amount to be paid? No It is an estimate of the most likely amount to be paid? No It is an estimate of the amount to be paid It is an estimate of the amount to be paid

Milliman Some Accounting Pronouncements “Earnings fluctuations are inherent in risk retention, and they should be reported as they occur” “Earnings fluctuations are inherent in risk retention, and they should be reported as they occur” “The sole result of accrual, for financial accounting and reporting purposes, is allocation of costs among accounting periods” “The sole result of accrual, for financial accounting and reporting purposes, is allocation of costs among accounting periods” Accounting goal to provide a accurate picture of the financial condition of a company Accounting goal to provide a accurate picture of the financial condition of a company

Milliman Simple Example One year insurance contract One year insurance contract During the year a claim has occurred During the year a claim has occurred The day after the contract expires the claim will be settled by the toss of a fair coin The day after the contract expires the claim will be settled by the toss of a fair coin – Costs $1,000,000 if heads – Costs $0 if tails Assume (for simplicity) Assume (for simplicity) – No interest income – No other expenses – Market will pay a risk charge of 20% of premium Total premium = $625 thousand Total premium = $625 thousand

Milliman Accounting Results End Year 0 End Year 0 – Earned Premium $625,000 – Reserves $500,000 – Profit $125,000 End Year 1 – Earned Premium $0 – Reserve Decrease $500,000 – Payments $1,000,000 (50%) $0 (50%) – Profit ($500,000) (50%) $500,000 (50%)

Milliman Fair Picture? The accounting snap-shot at the end of year 0 is a healthy company with $125,000 profit The accounting snap-shot at the end of year 0 is a healthy company with $125,000 profit The same company, with no changes at all at the end of year 1 shows either $500,000 additional profit or $500,000 loss in that year – for doing NOTHING The same company, with no changes at all at the end of year 1 shows either $500,000 additional profit or $500,000 loss in that year – for doing NOTHING Does this really make sense? Does this really make sense?

Milliman Oh, I Forgot to Tell You Surplus at start of year 0 = $200,000 Surplus at start of year 0 = $200,000 At end of year 0, $325,000 surplus – can write more business At end of year 0, $325,000 surplus – can write more business At end of year 1 At end of year 1 – $825,000 surplus (50%) – Insolvent (50%) Fair picture? Fair picture?

Milliman It Works on the Average The argument is that on the average there is no income in year 1 The argument is that on the average there is no income in year 1 Will be solvent based on the average Will be solvent based on the average Two actuaries hunting rabbits – one shoots 2 feet in front, the other 2 feet behind and celebrate that on the average they “got it” Two actuaries hunting rabbits – one shoots 2 feet in front, the other 2 feet behind and celebrate that on the average they “got it” Ask the rabbit Ask the rabbit

Milliman Wait a Minute Accountants say Accountants say “The Board cannot sanction the use of an accounting procedure to create the illusion of protection from risk when, in fact, protection does not exist” Doesn’t the rabbit prove that booking the mean really violates this precept? Doesn’t the rabbit prove that booking the mean really violates this precept?

Milliman Another Minute Accountants say Accountants say “to report activity when there has been none would obscure a fundamental difference in circumstances between enterprises that transfer risks and others that do not” Doesn’t booking the mean do precisely that in this example? Doesn’t booking the mean do precisely that in this example?

Milliman The Reality Given current knowledge there is a distribution of possible future outcomes – the reserve properly is a DISTRIBUTION Given current knowledge there is a distribution of possible future outcomes – the reserve properly is a DISTRIBUTION The “estimate of future payments” definition says the reserve is a point on this distribution The “estimate of future payments” definition says the reserve is a point on this distribution It does not say which one It does not say which one We are measured by how close the “estimate” is to what actually happens We are measured by how close the “estimate” is to what actually happens

Milliman Margin, What Margin? The rabbit will have a problem with any number booked The rabbit will have a problem with any number booked Why fixate on the mean? Why fixate on the mean? “Margin” implies something extra “Margin” implies something extra I assert – “Margin? We don’t need no steenkin’ margin.” I assert – “Margin? We don’t need no steenkin’ margin.”

Milliman Heresy? CAS says CAS says “A reserve estimate should take into account the degree of uncertainty inherent in its projections …Further, an explicit provision for uncertainty may be warranted when the indicated reserve value is subject to a high degree of variability”

Milliman Not Really – Look at Pricing Classic Risk Theory result: Classic Risk Theory result: No matter how much surplus an insurer starts with, the long-term probability of it going broke is 100% if it prices at the expected cost Economists say a risk-taker should be rewarded for taking risk Economists say a risk-taker should be rewarded for taking risk Both imply that the only economically viable insurance premium is above expected costs Both imply that the only economically viable insurance premium is above expected costs

Milliman There is No Margin Notice that economic theory does not say that the price is the expected cost plus something Notice that economic theory does not say that the price is the expected cost plus something The cost is the cost The cost is the cost The fact is, the cost is above the mean The fact is, the cost is above the mean Thus, “mean” and “margin” are artificial constructs because the cost is above the mean Thus, “mean” and “margin” are artificial constructs because the cost is above the mean

Milliman Margin is an Illusion Just as what the market will pay is indivisible the economic value of the agreement to pay future claims is indivisible Just as what the market will pay is indivisible the economic value of the agreement to pay future claims is indivisible The fact that risk takers should be compensated for taking risk means that the economic value is higher than the mean The fact that risk takers should be compensated for taking risk means that the economic value is higher than the mean

Milliman One Thought Amortize the pricing “risk load” over the life of the reserves Amortize the pricing “risk load” over the life of the reserves See Philbrick’s discussion from several years ago See Philbrick’s discussion from several years ago Still maintains the dichotomy between “mean” and “margin” Still maintains the dichotomy between “mean” and “margin”

Milliman Times Change Price reflects risk at time policy is issued Price reflects risk at time policy is issued Reserve should reflect risk at the time the financial statement is prepared Reserve should reflect risk at the time the financial statement is prepared Events unanticipated at time policy is issued and not contemplated in the price may have a significant influence on reserves Events unanticipated at time policy is issued and not contemplated in the price may have a significant influence on reserves – Asbestos exposure in a 1965 GL policy

Milliman Context New CAS Statement of Principles talks in terms of context New CAS Statement of Principles talks in terms of context Reserves are not set in a vacuum Reserves are not set in a vacuum Should consider the company issuing the statement Should consider the company issuing the statement Why not explicitly recognize company position when setting reserves? Why not explicitly recognize company position when setting reserves?

Milliman Minimum Penalty Since any number will be “wrong” let me submit a reasonable estimate of reserves (complements of Rodney Kreps) Since any number will be “wrong” let me submit a reasonable estimate of reserves (complements of Rodney Kreps) Suppose Suppose – (a really BIG suppose) we know the probability density function of future claim payments and expenses is f(x) – For simplicity one year time horizon – g(x,μ) denotes the decrease in shareholder (policyholder) value of the company if reserves are booked at μ but payments are actually x.

Milliman Minimum Penalty (Cont.) A rational reserve (i.e. “estimate of future payments”) is that value of μ that minimizes A rational reserve (i.e. “estimate of future payments”) is that value of μ that minimizes i.e. the expected penalty for setting reserves at μ over all reserve outcomes i.e. the expected penalty for setting reserves at μ over all reserve outcomes

Milliman Some Observations In order to yield the mean, the function g(x,μ) must be In order to yield the mean, the function g(x,μ) must be It is symmetric (same penalty for being y units high as for being low) It is symmetric (same penalty for being y units high as for being low) It is quadratic (being 2 units off is 4 times as bad as being 1 unit off) It is quadratic (being 2 units off is 4 times as bad as being 1 unit off) Reasonable? Reasonable?

Milliman Some Observations Taking g(x,μ) as Taking g(x,μ) as Gives the median as the “least painful” Gives the median as the “least painful” The mode is obtained with g being a “point mass” when x = μ The mode is obtained with g being a “point mass” when x = μ A given probability level can be obtained by using straight lines with different slopes A given probability level can be obtained by using straight lines with different slopes

Milliman A Reasonable g Not likely symmetric Not likely symmetric Likely flat in a region “near” μ Likely flat in a region “near” μ Increases faster when x is above μ than when x is below Increases faster when x is above μ than when x is below Likely increases at an increasing rate when x is above μ Likely increases at an increasing rate when x is above μ Such a function generally gives an estimate above the mean Such a function generally gives an estimate above the mean

Milliman Conclusion Biggest accounting myth – the reserve reflects actual future payments Biggest accounting myth – the reserve reflects actual future payments The true economic value of a reserve is likely greater than the expected present value of future payments The true economic value of a reserve is likely greater than the expected present value of future payments The chance of any point on the distribution is virtually zero The chance of any point on the distribution is virtually zero So why focus on a “mean” and “margin”? So why focus on a “mean” and “margin”? Take a holistic, economically rational approach Take a holistic, economically rational approach