1. 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

2. 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

3. 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

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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. A Real HK case

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

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

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

15. 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

16. 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)

17. 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

18. 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

19. 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

20. 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.

21. 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

22. 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

23. 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

24. 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)

25. 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

26. 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)

27. 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

28. 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

29. 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.

30. 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 )

31. 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

32. Example 3.1: You have chosen the following paid loss-development factors to model the lower half of a claims paid rectangle:
1/ /1 3/ /3 /4
You are setting reserves for the annual report as of December 31, For accident year 2005 you have claims paid-to-date lf $420,000 at report duration 1, which is December 31, The earned premium calculated for accident year 2005 is $1,000,000 and the expected loss ratio is 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.

33. 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.

34. 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)

35. 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!

36. 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)

37. 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

38. 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

39. 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?

40. 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!!