Measuring Loss Reserve Uncertainty

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• Measuring Loss Reserve Uncertainty
• William H. Panning
• EVP, Willis Re
• Casualty Actuarial Society
• Annual Meeting, November 2006
• 2006 Hachemeister Award Presentation

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Agenda

• What is Loss Reserve Uncertainty (LRU) and why is
  it important?
• How does this new method for measuring LRU differ
  from existing methods?
• How does this new method work and how do we know
  it is accurate?
• What are the practical advantages and limitations
  of this method?

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• What is Loss Reserve Uncertainty (LRU)
• and why is it important?

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1.1 What is LRU and why is it important?

• LRU is a measure of potential loss reserve
  development --the degree to which actual future
  loss payments may ultimately deviate favorably
  or unfavorably from the currently forecast
  amounts that constitute loss reserves
• LRU differs from pricing uncertainty -- the
  potential deviation between forecast loss (when a
  policy is written) and paid losses plus estimated
  reserve (at the end of an accident year).

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1.2 Why does measuring LRU matter?

• ERM LRU is a key component of Enterprise Risk
  Management, which requires measuring and managing
  all of a firms significant risks
• Capital needs knowing LRU assists a firm in
  determining the appropriate amount of capital to
  hold and whether some risk reduction action may
  be appropriate
• Interpreting calendar year deviations LRU can
  tell us whether their magnitude is significant
  and worthy of attention
• Comparisons with other firms are we better or
  worse?

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1.3 To whom should LRU matter?

• Issues
• Estimating surplus adequacy
• Capital allocation and pricing
• Managerial feedback is this deviation
  significant?
• Reinsurance

• Audiences
• Management
• Rating agencies and regulators
• Analysts and investors
• Management actuary, CFO, CEO
• Management
• Rating agencies
• Management

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• 2. How does this new method for measuring LRU
  differ from existing methods?

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2.1 How does this new method differ from
existing ones?

• Existing Methods
• Most are ad hoc (algorithmic democracy), lack
  criteria for fit
• Others require costly, opaque software,
  specialized expertise
• None have been validated
• Chain ladder focus creates bias
• Susceptible to statistical pitfalls

• New Method
• Based on a standard, minimum squared error linear
  regression
• Simple, transparent, implemented in Excel
  spreadsheet
• Validated using simulated data
• Use of regression minimizes bias
• Avoids these pitfalls

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2.2 Additional objectives features of the new
method

• It should be based on widely-available public
  data
• Schedule P Part 3 Paid Loss triangles
• The results should enable comparisons
• Across different lines of business within a firm
• For the same line of business across different
  firms
• Between forecast and actual calendar year
  payments
• The results should be scalable
• Unaffected by irrelevant differences in the size
  of reserves
• Applicable to reserves estimated by other methods

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• 3. How does this new method work and
• how do we know it is accurate?

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3.1 The starting point Paid Loss Triangles
Note that development years start with DY0
The cumulative numbers in each row converge to
ultimate values.
Reserve sum of ultimates minus boxed diagonal
values
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3.2 Essential steps in estimating LRU

• Step 1 Use the numbers already available to find
  a common underlying pattern for estimating future
  loss payments
• The chain ladder method does this by calculating
  link ratios the average ratio of numbers in a DY
  to the corresponding numbers in the preceding DY
• Step 2 Measure the variability of the available
  numbers around this underlying pattern
• Step 3 Use this measure to estimate the
  variability and correlation of forecast future
  payments and total reserve
• NOTE Steps 2 and 3 depend crucially on doing
  Step 1 correctly

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3.3 Three statistical pitfalls in Step 1, and
their solutions

• The chain ladder has no objective criterion for
  measuring and maximizing goodness of fit to
  existing data
• Solution use linear regression, to minimize
  total squared error
• The use of cumulative data creates serial
  correlation
• Solution use incremental data
• 
  Development year
• 0
  1
  2 . . .
• Acc Yr (Aea) (Aea)(Beb)
  (Aea)(Beb)(Cec) . . .
• 3. Heteroskedasticity (non-constant SD) s(ea)
  ? s(eb) ? s(ec)
• Solution analyze each development year separately

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3.4 Estimating reserves from incremental paid
loss data
Use these numbers (X)
To fit these numbers (Y)
Then use these numbers
To forecast these numbers

• Use linear regression to fit paid losses in
  future DYs (up to DY7), with DY0 as the
  independent variable in all cases
• Use estimated regression coefficients to forecast
  future loss payments

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3.5 Steps in calculating the standard deviation
of reserves

• Use the Salkever (textbook) method to calculate
  the standard deviation of each forecast future
  paid loss
• Use the linear regression results to calculate
  the variance-covariance matrix of forecast errors
  for each future DY
• Finally, aggregate these results to obtain LRU by
  Development Year, Calendar Year, and Total Reserve

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Calculating the SD of Forecast Paid Losses for DY2
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3.6 How do we know that the method is accurate?

• We created 10,000 simulated paid loss triangles
  where we knew the true underlying values
  (ultimate paid losses as well as the actual paid
  losses at any given point in the process)
• We used the new method to estimate ultimate paid
  losses
• We compared the known true values to the
  estimates obtained from the simulated triangles
• These estimates were, on average, identical to
  the true values underlying the simulation
• For estimated reserves
• For loss reserve uncertainty, which includes
  parameter risk
• NOTE No other method for estimating LRU has been
  validated (to the best of my knowledge)

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3.7 Validation statistics
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• 4. What are the practical advantages and
  limitations of this method?

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4.1 Avantage The results of this method are
scalable

• This method measures LRU in dollars. A better
  measure is the coefficient of variation, or CV,
  which is LRU as a of the estimated reserve
• CV is unaffected by reserve size, and so can be
  compared
• across different business lines for the same firm
  
• or across the same line for different firms
• I believe that the CV can be legitimately applied
  to reserve estimates obtained in other ways
  (e.g., using claims data)

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4.2 Advantage Comparisons across lines of
business
S Coefs Ratio of remaining payments to dollars
paid in initial development year (measures length
of payout) E(Res) Estimated Reserve SD/E(Res)
Coefficient of Variation, or standard deviation
of reserve divided by estimated
reserve SD/E(CY) Coefficient of Variation of
forecast payments in the next Calendar
Year NOTE All results are based on 2003
Schedule P Part 3
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4.3a Advantage Comparisons across firms PPA
S Coefs Ratio of remaining payments to dollars
paid in initial development year (measures length
of payout) E(Res) Estimated Reserve SD/E(Res)
Coefficient of Variation, or standard deviation
of reserve divided by estimated
reserve SD/E(CY) Coefficient of Variation of
forecast payments in the next Calendar
Year NOTE All results are based on 2003
Schedule P Part 3
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4.3c Advantage Comparisons across firms WC
S Coefs Ratio of remaining payments to dollars
paid in initial development year (measures length
of payout) E(Res) Estimated Reserve SD/E(Res)
Coefficient of Variation, or standard deviation
of reserve divided by estimated
reserve SD/E(CY) Coefficient of Variation of
forecast payments in the next Calendar
Year NOTE All results are based on 2003
Schedule P Part 3
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4.4a Comparisons of forecast versus actual
paids WC
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4.4b Comparisons of forecast versus actual
paids CMP
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4.5a Limitations of this method data
peculiarities
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4.5b Limitations of this method data
peculiarities
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4.5c Limitations of this method data
peculiarities
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4.6 What are the limitations of this method?

• Some versions of this method permit a gradual
  speedup or slowdown in payment patterns over
  time, but otherwise it assumes a stable past and
  future environment with regard to
• Underwriting criteria
• Exposure types
• Reinsurance parameters
• Legal and Regulatory environments
• Estimating the tail is difficult with this method
  as with others
• It necessarily relies on public data, and so does
  not reflect claims-level data available only to
  the firm

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• 5. Conclusions

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5.1 Conclusions

• This method produces validated results
• It is the only method that has been validated
• It is reasonably simple
• It avoids serious statistical pitfalls
• It is based on standard textbook methods
• It can be implemented in a spreadsheet
• It can be explained to colleagues and superiors
• It enables intra-firm comparisons of different
  lines of business
• It enables comparisons of a line of business
  across firms
• It enables detection of emerging problems that
  need attention

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5.2 How can I learn more about this method?

• Send me an email at
• Bill.Panning_at_Willis.com