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
 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.
 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 amount to be
paid
Milliman
Some Accounting
Pronouncements
“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”
 Accounting goal to provide a accurate
picture of the financial condition of a
company

Milliman
Simple Example



One year insurance contract
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
– Costs $1,000,000 if heads
– Costs $0 if tails

Assume (for simplicity)
– No interest income
– No other expenses
– Market will pay a risk charge of 20% of premium

Total premium = $625 thousand
Milliman
Accounting Results

End Year 0

End Year 1
– Earned Premium
– Earned Premium $0
$625,000
– Reserves $500,000
– Profit $125,000
– 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 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?
Milliman
Oh, I Forgot to Tell You

Surplus at start of year 0 = $200,000
 At end of year 0, $325,000 surplus –
can write more business
 At end of year 1
– $825,000 surplus (50%)
– Insolvent (50%)

Fair picture?
Milliman
It Works on the Average

The argument is that on the average
there is no income in year 1
 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”
 Ask the rabbit
Milliman
Wait a Minute

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?
Milliman
Another Minute

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?
Milliman
The Reality

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
 It does not say which one
 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
 Why fixate on the mean?
 “Margin” implies something extra
 I assert – “Margin? We don’t
need no steenkin’ margin.”

Milliman
Heresy?

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:
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
 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
 The cost is the cost
 The fact is, 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
 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
 See Philbrick’s discussion from
several years ago
 Still maintains the dichotomy
between “mean” and “margin”

Milliman
Times Change

Price reflects risk at time policy is
issued
 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
– Asbestos exposure in a 1965 GL policy
Milliman
Context
New CAS Statement of Principles
talks in terms of context
 Reserves are not set in a vacuum
 Should consider the company
issuing the statement
 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)
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

P      g  x,   f  x dx
0

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
g( x,  )   x   
2

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)
 Reasonable?
Milliman
Some Observations

Taking g(x,μ) as
g  x,    x  
Gives the median as the “least painful”
 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

Milliman
A Reasonable g
Not likely symmetric
 Likely flat in a region “near” μ
 Increases faster when x is above
μ than when x is below
 Likely increases at an increasing
rate when x is above μ
 Such a function generally gives
an estimate above the mean

Milliman
Conclusion
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 chance of any point on the distribution is
virtually zero
 So why focus on a “mean” and “margin”?
 Take a holistic, economically rational
approach

Milliman