P/C RATEMAKING AND LOSS RESERVING by Robert Brown and Leon Gottlieb CHAPTER 3: LOSS RESERVING The financial condition of an insurance company can not be adequately assessed without sound loss reserve estimates. -- CAS Chapter 5 HCPI106-7/2003

I. Introduction Ratemaking and Loss Reserving are two most important P&C actuarial jobs Qualified actuary must attest to adequacy of year-end reserves (by law) to protect policyholders to allow distribution of reported profits to indicate level of solvency to potential investors (may be conservative) to provide ultimate claims estimates to pricing actuary

II. How Outstanding Claims Payments Arise There can be several delays in the claims-payment process between accident incurral and reporting between reporting and recording in head office between claim file opened and final dollar paid (i.e., growth or decline on known claims) Large claims tend to settle last Try to model future claim payment patterns based on past patterns

A Hypothetical Claim 10/15 Claim Event 2013 2014 2015 2016 2017 Loss Reserve Carried 12/31 $0

A Hypothetical Claim MM/DD/YY 12/20/14 Claim Reported to Agent, but Not Yet Recorded by Insurer $0 10/15 Claim Event 2013 2014 2015 2016 2017 Loss Reserve Carried 12/31 $0

A Hypothetical Claim MM/DD/YY 09/09/15 Pay $3,000 Medical Expenses; Offer $15,000 Settlement $0 12/20/14 Claim Reported to Agent, but Not Yet Recorded by Insurer $0 10/15 Claim Event 01/07/15 Claim Recorded 2013 2014 2015 2016 2017 Loss Reserve Carried 12/31 $20,000

A Hypothetical Claim MM/DD/YY 09/09/15 Pay $3,000 Medical Expenses; Offer $15,000 Settlement $0 12/20/14 Claim Reported to Agent, but Not Yet Recorded by Insurer $0 10/15 Claim Event 01/07/15 Claim Recorded 02/02/16 Offer Refused; Go to Court 2013 2014 2015 2016 2017 Loss Reserve Carried 12/31 $20,000 $40,000

A Hypothetical Claim MM/DD/YY 09/09/15 Pay $3,000 Medical Expenses; Offer $15,000 Settlement $0 12/20/14 Claim Reported to Agent, but Not Yet Recorded by Insurer $0 10/15 Claim Event 01/07/15 Claim Recorded 02/02/16 Offer Refused; Go to Court 10/06/17 Court Decision; Pay $32,000 2013 2014 2015 2016 2017 Loss Reserve Carried 12/31 $20,000 $40,000 $0

A Hypothetical Claim MM/DD/YY 12/20/14 Claim Reported to Agent, but Not Recorded by Insurer 09/09/16 Pay $3,000 Medical Expenses; Offer $15,000 Settlement 10/15/13 Claim Event 01/07/15 Claim Recorded 02/02/16 Offer Refused; Go to Court 10/06/17 Court Decision; Pay $32,000 2013 2014 2015 2016 2017 Loss Reserve Carried 12/31 $0 $0 $20,000 $40,000 $0

A Hypothetical Claim $43,000 12/31 $35,000 10/06 $40 $23,000 12/31 $30 $20 $3,000 09/09 $10 2014 2015 2016 2017 $0 2013 History of the Incurred Loss Development for the hypothetical claim

A Real HK case

III. Definition of Terms A. Basic Elements Claim frequency distribution developed from recent experience data (Aside: a loss may not result in a claim)

III. Frequency and Severity Loss distribution which models severity (S) (Aside: total incurred claims = claims paid-to-date + unpaid loss reserve)

III. Frequency and Severity A rate of interest, i, or force of interest,  Time to payment of claims (t): Payout Pattern Also define:

A. Individual Claim File Estimate Field adjuster establishing claim file upon notice of (potential) claim File info Date of occurrence (accident) Date of claim Assigned lawyer Examining physician Payments-to-date

Individual Claim File Estimate Field adjuster to estimate E[future payments] based on Severity of potential loss Likely time to settle (ALAE will go up) Inflation between now and settlement Any recent impacts of courts or legislation Other pertinent info Case reserves =  All individual claim file estimates (split by line of business and accident year)

B. Gross IBNR (Reserves) Gross IBNR reserve (or bulk reserve) is total of: future development (+, -) on known claims files that are closed but may be re-open (e.g., W.C.) claims incurred but not reported (pure IBNR) claims reported (to field adjuster) but not recorded (in head office), RBNR

C. Paid Loss Development Paid age-to-age loss development:  cumulative payments on a defined set of claims between successive valuations (e.g., annual/quarterly) Age-to-ultimate loss development:  from defined date (or age) evaluation to ultimate payment amount Paid-loss development factor (L.D.F.) = NOTE: there are no estimates in these numbers; they are 100% known and 100% objective

D. Incurred Loss Development Same definition as above except you are using incurred data ( = pd-to-date + reserve O/S estimate) future Incurred L.D.F. can be < 1 if reserve estimates are conservative because of salvage or subrogation

E. Salvage and Subrogation Salvage: e.g., if a policyholder has a car accident and the car is “written off,” the policyholder will receive a check for the value of the car at the time of claim. From that point on, the car is owned by the insurer. The right to any salvage value belongs to the insurer, and will appear in the accounts as a recovery of costs. Subrogation: e.g., if a policyholder has a house fire, the insurer pays for the house to be repaired or rebuilt within the limits of policy. If it can be shown that the fire was caused by the negligence of a 3rd party, the insurer has the right under subrogation to pursue recovery in a lawsuit against the 3rd party. These can lead to a less-than-one final loss-development factor.

F. Loss Adjustment Expenses Must reserve for future ALAE and ULAE ALAE (e.g., lawyers’ fees) are usually in claim file paid and incurred so are covered ULAE (e.g., head office) are allocated at year-end (to line of business) by formula

G. Fast Track Reserves Used for high freq, low severity, fast closing lines (e.g., auto collision) Head office actuary provides an average claim size based on past experience All files use this number initially Final number is entered once known (e.g., claim settled) If claim file open too long, individual estimates take over

IV. Professional Considerations Setting reserves is not a black box computer calculation Must have intimate knowledge of company all definitions (e.g. what is a claim) management attitude toward settling claims (i.e. fast/slow)  company’s retention limit Must be cognizant of external factors economic inflation recent court decisions & recent legislation Must set up data collection criteria compromise of homogeneity and statistical credibility Review accuracy of data Calculate reserves using more than one method Reconciliation of different answers will narrow the range of most likely estimate

V. Checking the Data Methods are based on there being patterns in claims settlement Review with care any past data that do not fit pattern Calendar year action will impact the “diagonals” of the loss-development triangles (as seen later) Document all modifications/adjustments (and keep on file)

VI. Loss Reserving Methods (There are as many loss reserving methods as there are actuaries!) Case Reserve Estimates Plus To Case file estimates add Gross IBNR or future development on known claims files that have closed but may re-open IBNR = total of pure IBNR and RBNR Method is very subjective may be used to hide profit and avoid tax or make the year profitable Could result in insolvency

B. The Expected Loss Ratio Method Estimate ultimate expected loss ratio = E [LR] Then E [LR]   earned premium for fiscal period = E [$L, ultimate] Finally: E [Loss Reserve] = E [$L, ultimate] – [$L, pd-to-date] Again, too subjective May lead to illogical answers (e.g. if you have just taken a rate decrease for purely competitive reasons) At times, may be the only method available: new line of business gov’t regs require minimum loss ratio targets for certain volatile lines of business (e.g. Employee Fidelity Insurance) can be used in conjunction with other methods (see Bornhuetter-Ferguson)

C. The Chain-ladder or Loss-Development Triangle Method Historical data present as a triangle of paid or incurred development Create a cumulative paid or incurred triangle (Note: activity in row labeled Accident Year Z and column labeled development year t, took place in Calendar year Z+t) Calculate age-to-age loss development factors, LDF, (or link ratios) (see tables 3.2 and 3.3) Look for patterns in these LDF Use these patterns to create the missing half of the triangle Advantages of using paid loss purely objective, contains no reserve estimates Advantages of incurred loss data reserve estimate contains valuable information

C. The Chain-ladder or Loss-Development Triangle Method Do both paid and incurred and reconcile the differences (ultimately they must be the same as all incurrals become paid) Part of work is data review (remember calendar-year activity appears along diagonals of the triangle--e.g. legislative change or management style shift) Reserve = E[ultimate paid or incurred (same)] - [Pd-to-date] Note: Mean LDF (as defined in text) is a volume-weighted average (i.e. years of greater volume are give greater “weight”) If there is consistent growth or decline in a column of LDF may project the trend into future values (i.e. not use a constant) A LD triangle with n rows and n-l columns, creates a statistical model with 2n-l parameters (n-l LDF applied to n “jump-off” points -not good modeling!) Not surprising, stability is not a characteristic of the LDF method

Stability is not a characteristic of the LDF!! Example 1: Increase by 10% the cumulative payments for accident year 2006, development year 1, from $22,253 to $24,478.  The total loss reserve using 5-year average modeling assumption would rise 4.1% Example 2: The Claims Department changed its claims settlement philosophy for 2007 so that all losses were paid more promptly. In fact, all of the payments along the calendar year 2007 diagonal increased by 10% (except for AY 2000, development year 7 which has been assumed to be fully mature).  Logically it should lead to smaller reserve, but that resulted in a 9.2% larger reserve using the Chain-ladder method.

D. The Bornhuetter-Ferguson Method (BF) A combination of the E[LR] and LDF-triangle methods Adds stability to LDF-triangle method and Adjusts E[LR] as time provides more information From p.73-4 it can be shown that from the LDF method: E [Loss Reserve] = E [$L, ultimate] ( ), where (cumulative LDF from now to )

D. The Bornhuetter-Ferguson Method (BF) The BF method For each Acc Year row estimate E[LR] based on most recent info Then: E[$L, ultimate] = E[LR]  (Earned Premiums) Then: E[Loss Reserve] = E[$L, ultimate] ( ) Hence a combination of E[LR] and LDF methods BF answer will (must) lie between E[LR] answer and LDF answer

Example 3.1: You have chosen the following paid loss-development factors to model the lower half of a claims paid rectangle: 1/0 2/1 3/2 4/3 /4 1.41 1.22 1.16 1.08 1.04 You are setting reserves for the annual report as of December 31, 2006. For accident year 2005 you have claims paid-to-date lf $420,000 at report duration 1, which is December 31, 2006. The earned premium calculated for accident year 2005 is $1,000,000 and the expected loss ratio is .600. For the 2005 accident year, determine the estimated loss reserve using each of the expected loss ratio method, the chain-ladder method, and the Bornhuetter-Ferguson method.

Example 3.2: Given Find the Total Actuarial Reserve using: Earned Premium = $800,000 E[LR] = .680 Sum Individual Claim File Estimates ($Paid+Reserves) = $500,000 $Pd-to-date = $300,000 Find the Total Actuarial Reserve using: (a) The Loss Ratio Method; (b) The Chain-Ladder Method; (c) The Bornhuetter Ferguson Method.

E. Estimates Split into Frequency & Severity Requires for each Acc Year, E[# of claims] this data is known very rapidly (i.e. not as much gross IBNR on # of claims as $ Loss) Given a triangle of cumulative loss payments, and a vector of E[# claims] for each Acc Yr, you can create a triangle called Cumulative Loss Payments per Claim Incurred (a severity triangle) usually done with “$ paid” but can be “incurreds” helpful if payments adjusted for inflation and represent constant $ values (more so in the late 1980’s) Really good if you can create a triangle of Cumulative Closed Claim Files (see table 3.9) This can then be transformed into %-of-claims-closed triangle to show the “speed of finalization” (see table 3.10)

E. Estimates Split into Frequency & Severity You can thus see if the company is speeding up or slowing down settling claims You can then “adjust” paid losses to a consistent %-of-claims-closed basis (see Q3.11 and do it) This is important - e.g. the company decides to settle faster, but does not tell the actuary reserves should go down the claims-paid data suddenly show larger LDF in the last diagonal with larger model LDF, reserves go up - Wrong!

F. Summary Do more than one method and reconcile differences May build some level of conservatism in, depending on the use of the results (e.g. solvency vs. pricing)

VII. Discounting Loss Reserves For the life side, discounting for time value is a given Historically, most P & C reserves were carried undiscounted (hence some implicit conservatism) Explicitly calculated provision for adverse deviation is probably actuarially preferable Reserves for tax purposes must now be discounted future development on known claims

VII. Discounting Loss Reserves To discount create lower half of triangle = future cash flow assume payments are made at mid-pt of development year choose an interest rate and re-value at present value note: any given discount factor, vt, will be applied across a diagonal (Tables 3.17 and 3.18) Now need an explicit provision for adverse deviations

VII. Discounting Loss Reserves Issues is the mid-pt assumption valid especially for the first year and last year? what interest rate should I use? (answer: the net rate being earned by the assets backing these liabilities) asset-liability matching has not been the norm for P & C co’s it should be net of expenses how does one include unrealized capital gains? what are the tax implications?

VII. Discounting Loss Reserves Discounted loss reserves are the correct ones to use in pricing (pricing actuary should discount back to average date of premium receipt) Discounting is becoming much more common (often forced by tax or pricing agencies) The best way to learn this stuff is to do lots of exercises. Books are always better than movies (SAP vs. GAPP) - as well as power point slides!!