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					P/C RATEMAKING AND
LOSS RESERVING
by R.L. Brown & L. R. Gottlieb

     CHAPTER 3: RATEMAKING
         I. Introduction; II. Objectives;
  III. Frequency and Severity; IV. Data for
           Ratemaking; V. Premium Data;
    VI. Exposure Units; VII. The Expected
         Effective Period; VIII. Ingredients;
                 IX. Rate Changes
  Introduction & Objectives
I. Introduction: Methods are applicable outside of P &
    C (e.g. health)
II. Objectives:
A. Essential Objectives
   1. Cover expected losses and expenses
      a. include investment income
      b. no inter-cohort subsidies
      c. no subsidies across risk classes
      d. must cover to last dollar paid out (several years
         later)
      e. selling at a loss is O.K., but subsidy should come
         from I.C. owner’s equity                         2
  A. Essential Objectives
2. Produce rates that make adequate provision for
   contingencies

    a. The 100-year flood will happen!
    b. Tough to push rates up as industry is competitive
       and ph can often self-insure
    c. Inadequate rates endanger I.C. solvency




                                                           3
  A. Essential Objectives
3. Encourage loss control
   a. get ph to reduce claim freq., severity or both
   b. methods used
      1. good driver discounts
      2. discounts for sprinklers, burglar alarms
      3. discounts for accident prevention and rehab in
         W.C.
   c. leads to lower rates
   d. good for society
                                                          4
 A. Essential Objectives
4. Satisfy rate regulators
  a. rates must be adequate, not excessive and not
     unfairly discriminatory
  b. some methods that are actuarially acceptable
     may be refused (e.g., use of age, gender and
     marital status as rating variables)




                                                     5
  B. Non-Essential Objectives
B. Non-Essential but Desirable Objectives
  1. Produce rates that are reasonably stable
    a. rate changes must be prospectively justifiable

    b. reinsurance can help stability




                                                        6
  B. Non-Essential Objectives
2. Produce rates that are reasonably responsive
   to change

   a. real changes (vs. one-time aberration) should be
      reflected (e.g. change in speed limit)
   b. stability and responsive are in conflict (reflecting
      change only to the extent it is statistically
      credible helps)




                                                             7
  B. Non-Essential Objectives
3. Be simple and easy to understand

   a. ph must understand connection to loss control to
      make it happen
   b. agents and brokers must be capable of using rate
      manual
   c. computer systems are cheaper if less complex
      system
   d. must be able to “sell” system to management and
      regulators (who are often not actuaries)



                                                     8
  III. Frequency and Severity
A. Ratemaking Variables
1. Claim frequency distribution developed from recent
    experience data




(Aside: a loss may not result in a claim)



                                                  9
  III. Frequency and Severity
2. Loss distribution which models severity (S)




(Aside: total incurred claims = claims paid-to-date +
    unpaid loss reserve)




                                                        10
  III. Frequency and Severity
3. A rate of interest, i,or force of interest, 
4. Time to payment of claims (t)
5. Also define:




                                                   11
 IV. Data for Ratemaking

Data on claims, loss payments, and premiums
may be collected and tabulated in any of the
following three formats: accident year, policy
year, and calendar year. (A fourth basis, called
the reported year, which is used for claims-
made policies will not be discussed here.)




                                                   12
A. Accident Year

1.   The most common method for
                                      中大生校內捱車撞
     compiling actuarial data.
2.   All action of any loss             明報 2008-10-25
     occurring in accident year Z
     is held in the “accident year    沙田中文大學校園內昨午1 時許發生
     Z” file regardless of the date   交通意外。中大一年級女生楊 × 慧
     of action (e.g. payment).
                                      (19 歲) 在大學營修樓對開橫過馬
3.   You do not know the ultimate
     claim until the last dollar is   路時,遭一輛沿大學道西行的私家
     paid.                            車撞倒,楊女頭部及兩腳膝部受輕
4.   All immature data then           傷,救護員接報到場替她包紮,楊
     depend on reserve estimates.
                                      女由兩名同學陪同送院……

                                      33 歲私家車司機姚 × 志接受酒精測
                                      試後,證實並無飲酒,警方正調查
                                      意外原因。
                                                        13
B. Policy Year
1.   Any claim action arising on a policy that became
     effective in policy year Z is accounted for in the “policy
     year Z” file.
2.   If policy year Z goes from Jan. 1, Z to Dec. 31, Z, and if
     all policies are effective for one year, the exposure
     period for policy year Z goes from Jan. 1, Z to Dec. 31,
     Z+1 (i.e. 24 months).
3.   If all issue dates and accidents are uniformly distributed,
     the midpoint of PY(Z) is Dec. 31, Z or Jan. 1, Z+1.
4.   Thus, you do not have complete PY(Z) data until Dec. 31,
     Z+1 (and then it is still immature).
5.   However, pricing is done for policy years, so having policy
     year data is a consistent basis.
                                                              14
C. Calendar Year
1.   Any accounting action on a file in calendar year Z gets
     recorded in the “calendar year” Z file.
2.   Calendar year Z is complete and mature as at Dec. 31, Z.
3.   Incurred losses (Z) = Paid losses (Z) +  Reserve (Z)
     = Paid losses (Z) + Loss Reserve 12/31/Z - Loss Reserve
     12/31/Z-1.
4.   This system is not widely used as one cannot allocate
     “calendar year” action to any defined exposure period.

Note: Again, The REPORT YEAR, used for claims-made liability
    policies, is not discussed in the text.


                                                           15
V. The Premium Data
Written Premium vs. Earned Premium:
   Premium data usually come as “written”
   premiums (cash accounting) or “earned”
   premiums (accrual accounting).
  a. If $120 received Oct. 15, Z, then
     written premium in Z is $120, but
     earned premium is $25 for Z.
  b. The remaining $95 is unearned
     premium in Z, and will only be earned
     in year Z+1.
                                             16
  Question:

 The XYZ Insurance Company had an
  unearned premium reserve of $20
  million at the end of 2008. During 2009
  it wrote $25 million in annual
  premiums. At the end of 2009, its
  unearned premium reserve was $23
  million. What were earned premiums
  during 2009?


                                            17
 VI. The Exposure Unit
1. Premium = (rate) * (exposure units)
   (e.g. for W.C.  $100 of payroll)
2. A good exposure base should:
   a. be an accurate measure of quantitative exposure
      to risk
   b. be easy for I.C. to determine (at time of premium
      calculation)
   c. not to be subject to manipulation by ph
   d. be easy to record and admin


                                                      18
 VI. The Exposure Unit
2. A good exposure base should:
   e. be understood by ph
   f. not essential but desirable - automatically adjust
       with inflation (e.g. payroll)

3. You seldom get all criteria - must
   compromise (e.g. annual mileage is
   not used for car insurance)



                                                           19
 VII. The Expected Effective Period
1. There is a time lag from the most recent data to
   the expected effective period.
2. The effective period is a policy period, normally
   a policy year.
3. If rates take effect, Oct. 15, Z, and are to be
   left unchanged for exactly one year, on one-
   year policies, then the effective period starts
   Oct. 15, Z; ends Oct. 15, Z+2; and has a
   midpoint of Oct. 15, Z+1. (note: exposure starts
   at zero, peaks on Oct. 15, Z+l and ends at zero).
4. Claims data: actuary normally uses accident-
   year or policy-year data.

                                                       20
 VIII. Ingredients of Ratemaking
A. Loss-Development Factors
B. Trend Factors
C. Expenses
D. Loading for Profits and
   Contingencies
E. Credibility Factors
F. Investment Income
                               21
  Loss-Development Factors
1. These data come from the “Loss-Reserving”
   actuary. Your price must cover all claim costs for
   this cohort until the last $ is paid (many years out).
2. Early claims estimates may understate the ultimate
   value.
   a. reserve estimates are deficient or optimistic
   b. there exist claims incurred but not reported (IBNR)
   c. this leads to “+” loss-development (study Tables 3.1~3.3)
3. It is possible to have loss-development factors < 1
   a. reserve estimates prove to be conservative (even with
      IBNR)
   b. some lines allow for salvage and subrogation (i.e. claim
      payment recoveries)
                                                                  22
  Trend Factors
1. Must adjust past experience period data to mid-
   point of future exposure period.
2. Reasons for cost adjustment
   a.   economic inflation
   b.   judicial decision
   c.   change in mandated benefits
   d.   technical advances
   e.   legislative changes
   f.   change in U/W criteria or definitions
   g.   level of economic activity


                                                     23
 Trend Factors
3.   Note: not all of above can be modeled with
     trend factor (trend factor only projects past
     trends)
4.   If data accident year, mid-pt of A.Y.(Z) is
     June 30, Z or July 1, Z.
5.   If data policy year, mid-pt of P.Y.(Z) is Dec.
     31, Z or January 1, Z+1.
6.   If rates take effect at date t, Z for one year
     on one-year policies, mid-pt of future
     exposure period is t, Z+1.

                                                      24
Trend Factors
7.    Trend factor methods:
     a.   linear regression/projection from past period
     b.   log linear regression/projection (i.e. exponential trend)
     c.   fit frequency and severity separately vs. fitting loss cost
          (= f * S) data
     d. find slope of data trend line; then extrapolate only last
          one or two data points using this slope (say weighted
          70/30 for example)
8.    No overlap between loss-development and trend (overlap
      fallacy) since trend is from average past date of claim to
      future average claim date and loss development goes from
      future average claim date to final dollar paid.




                                                                        25
Example 3.1: Given the loss costs shown in Table 3.5,
   estimate the expected loss cost for rates that take
   effect September 1, 2008 on one-year policies. Assume
   that rates will be in effect for one year.

Table 3.5
Average Frequency, Average Severity, and Loss Cost
Accident   Average claim   Average Loss   Loss Cost per
 Year       Frequency        Severity     Unit Exposure
2003         .05141           2,323            119.39
2004         .05335           2,502            133.97
2005         .05311           2,445            129.89
2006         .05648           2,807            158.57
2007         .05765           3,274            188.72




                                                      26
     Expenses
1.    Loss adjustment expenses (LAE)
     a.   Allocated loss adjustment expenses (ALAE) (e.g. lawyers’
          fees, claim adjusters’ fees, court costs) go into claim file
          and are part of total incurred losses.
     b.   Unallocated loss adjustment expenses (ULAE) (e.g. head
          office claims dept., salaries) do not go into any claim file
          and are allocated en masse at year-end to line-of-business
          using an appropriate formula
2.    Commissions, premium taxes, licenses and fees
      and misc. (usually set as % of Gross Premium).


                                                                    27
 Expenses
3.   Expense ratio = all expenses (not LAE) as % of GP.
4.   Permissable loss ratio (PLR) = 1 - expense ratio
5.   .

6.   If expenses (not LAE) are partly fixed (F) per unit
     and partly variable (V) then:



7.   Book examples assume all non LAE expenses are
     variable unless indicated otherwise

                                                           28
Expense Loadings as a
Percentage of Premium




                        29
     Loading for Profit and Contingencies
1.    Need profit to compensate capital providers and
      need to cover “expected” deviations (insolvency is
      possible)
     a. implicit approach - adopt conservative
        parameters and live off the margins
     b. explicit approach - adopt best-estimate
        parameters and add an explicit margin for profit
        and cont.
     c. ignore investment income and call it profit and
        contingencies
     d. regulators prefer (b)
                                                      30
Concept Check
   Assume that the discounted expected
    claim costs and administrative costs for
    hurricane and fire insurance are the
    same. Which type of coverage will
    have higher fair premium? Why?




                                          31
 Credibility Factors
1. Sparse data will not be statistically credible.
2. Also, credibility strikes a balance between
   responsiveness and stability (only make a
   change to the extent it is credible).
3. You need more data for any set credibility
   for business that is more variable (e.g. auto
   liability vs. collision).



                                                32
     Credibility Factors
4.    Properties of credibility (Z)
     a. .
     b.        (more data, higher credibility)

     c.          (additional units of exposure provide
                     less than pro rata additional credibility)
5. Examples of credibility formulae:
   a.         ,            (E  measure of exposure)
                            (K  measure of variability)
   b.
              ,              (n  # of claims(e.g.)) (K 
      criteria for full credibility)
                                                              33
          Investment Income
1.       Should discount for time to payment
2.       Covered more later in loss reserving
        Example
          Assume no administrative costs
          one year policies, premium received at beginning
          certain claim costs = $100 paid according to table below
                                          Fair Premium




                                                                      34
    Example: Effect of
    Investment Income
   Assume
       no administrative costs
       one year policies, premium received at beginning
       certain claim costs = $100 paid according to table
        below
                                 Fair Premium




                                                             35
    Example: Effect of
    Investment Income
   Assume
       no administrative costs
       one year policies, premium received at beginning
       certain claim costs = $100 paid according to table
        below
                                 Fair Premium




                                                             36
    Example: Effect of
    Investment Income
   Assume
       no administrative costs
       one year policies, premium received at beginning
       certain claim costs = $100 paid according to table
        below
                                 Fair Premium




                                                             37
Effect of Investment Income Varies
 Across Lines of Business




                                38
P-C Industry Break-even
    Combined Ratio




                          39
IX. Rate Changes
1. Overall Average Rate Change
     Determine the average, or overall, rate
     change required.
2. Changing Risk Classification Differentials
     Decide on changes required in the
     differentials that apply to rate classification
     parameters.
3. Balancing back
     Adjust the results so that the overall change
     in premium income is actually desired.
                                                  40
A. Overall Average Rate Change
1.    Loss Cost Method
      .



        .



2.    Loss Ratio Method
      .



                                 41
A. Overall Average Rate Change
   LR Method
   where



   Denominator = today’s earned premium at current
    rates



   Or adjust account dept. records of “Earned
    Premiums” to “Earned Premiums @ Current Rate
    Level” using parallelogram method of Example 3.2


                                                       42
Example 3.2: Consider the following data:
    Calendar Year                      Earned Premium
       2004                                   3853
       2005                                   4600
       2006                                   5125

Average all policies have a one-year term, the policy issues
are uniformly distributed, and the following rate changes
have occurred:
       Date                               Rate Change
    July 1, 2002                               +12.5%
    November 15, 2004                          +10.0
    October 1, 2005                            +8.0
Rates are currently at the level set on October 1, 2005.
Calculate the earned premium at current rates for calendar
years 2004, 2005, and 2006.

                                                               43
Example 3.3: Calculate the new average gross rate given
   the following information:


    Expected Effective Incurred Losses
       (Trended and Developed)               30,000,000
    Earned Exposure Units                      1,000,000
    Earned Premium at Current Rates          45,000,000
    Present Average Manual Rate                       45
    Permissible Loss Ratio (1 – Expense Ratio)      .700




                                                           44
A. Overall Average Rate Change
3. If data are consistent, Loss Cost and Loss
   Ratio methods achieve same answer (see
   proof page 76/77 - this is important).
4. Credibility: if company data not fully
   credible (Z < 1) use:
   Cred-wtd Average Indicated Rate (or
   change) = Z (Company Indication) + (1-Z)
   Industry Indication


                                                45
B. Changing Risk Classification
Differentials
1.    Production of Rate Manual
     a. One cell is called the base cell (usually the cell with
         the largest credibility) and its rate is the Base
         Rate(BR)
     b. Then each risk class variable will have a vector of
         differentials (di) or relativities with Base Cell entry
         always set = 1.000.
     c. So for 3 risk class variables xi (i=1, l), yj (j=1, m) and
         zk (k=1, n), you can produce lmn rates




                                                                 46
 B. Changing Risk Classification
 Differentials
2.    Changing Differentials: Loss Ratio Method
     a. (lndicated Diff)i = (Existing Diff)I.


     Note: if experience period LRj = LRBASE then the existing
        differential need not be changed (understand?).
   i.e. the existing differential proved to be just right!
3. Changing Differentials: Loss Cost Method
   a. L.C. method takes you directly to the indicated diff.
        (one-step)
   b. .


                                                              47
Example 3.4: Determine new differentials for Class B
   and Class C, given the following information and
   assuming full credibility in all classes.


      Existing       Experience Period Loss
Class Differential    Ratio at Current Rates   Loss Cost

 A     1.00                 .65                   129
 B      .85                 .71                   120
 C     1.21                 .66                   157




                                                           48
Example 3.5: You are the consulting actuary for a
   property/casualty insurer that proposes to begin selling
   automobile insurance in State Z which has three
   classification territories. The insurer wants advice on
   what territorial differentials to adopt. You have the
   following information showing the present average
   territorial differentials used by the top five insurers.
    Territory                 Differential
        1                           1.00
        2                             .95
        3                           1.25
    The following industry statistics are also available for the latest
    calendar year.
                                    Earned Premium Incurred Losses
    Territory     Cars Insured      (in thousands)   (in thousands)
        1             65,354             12,046           7,215
        2             56,182             10,093           5,987
        3             24,858              5,840           3,580
What territorial differentials would you recommend, and what
    comments would you include in your report?
                                                                          49
B. Changing Risk Classification
Differentials
4.    Loss Ratio Diff  Loss Cost Diff if:
     a. populations across cells are heterogeneous
         (e.g. more young male drivers in Terr 1 than in Terr
         3).
     b. Since Loss Cost = $L/# exposure units, all exposure
         units are treated as equal.
     c. But they are not: e.g. if young male drivers pay
         twice the base rate then, in solving for the
         Territorial Diff young male drivers should count as 2
         units of exposure (see Example 3.6)



                                                             50
     B. Changing Risk Classification
     Differentials
     d.   The Loss Ratio method self adjusts for this problem
          to the extent that last year’s differentials were
          correct since



5.    With appropriate adjustments, the Loss Cost and Loss
      Ratio methods achieve same answer (see Appendix A
      now).
6.    Credibility: if cell credibility Z < 1, use (normally):
      New Diff = Z (Indicated Diff) + (1-Z)(Existing Dift)
      (that is, only change the differential to the extent that
      the need for a change is credible).
                                                                  51
C. Balancing Back
1.     Because dBR =1.000
     a. given an overall rate change indication of R, then
          equation BRNEW = BROLD (1 + R)
     b. And New Rate i,j,k = BRNEW . (new xi) . (new yj) .
          (new zk)
     c. But this combination may not produce a premium
          income increase of R as needed (see handout!)
     d. This is because the New Average Diff  Old Average
          Diff
     e. So: off-balance                     and
     f. balance-back factor is its reciprocal (see p.86)


                                                              52
C. Balancing Back
2.   Under the Lost Cost method, the balance back is
     achieved in one step:

     Base rate

3.   Review in detail example 3.7 (p.91-97)




                                                       53
Example 3.6: Given the following information, calculate the
   proposed Class 1A rate for Territory 2. Class differentials
   will not be changed, and the province wide rate change is
   +5%. The base cell is Territory 1, Class 1A.

            Earned     Existing              Average Existing
            Exposure   Average       Loss    Class Differentials
Territory    Units      Rate         Cost     within Territory

   1         2000         250         200        1.50
   2         1000         500         300        1.25




                                                                 54
Example 3.7: Given the following information, and
   assuming the revised rates take effect July 1, 2007 for
   one year on one-year policies, determine new rates
   for each of Class 1 and Class 2, for each of Territory 1
   and Territory 2. (Class ½ differentials will not change.)
   Use the loss ratio and loss cost methods, and base the
   overall average rate change on 2005 policy year data,
   assuming they are fully credible for that purpose. The
   permissible loss ratio is .600.

    Policy Year 2004 Losses
    As of March 31, 2006                     As of March, 2007
    Paid         Outstanding                 Paid       Outstanding
    400,000      100,000                     625,000           0

    Trend Factors
    July 1, 2006    July 1, 2006   January 1, 2006   January 1, 2006
    to               to              to              to
    July 1, 2007    July, 2008     July 1, 2007      July 1, 2008
    1.18             1.30           1.24             1.36
                                                                       55
                              Territory 1   Territory 2
Present Rates
     Class 1 (Differential)   100(1.00)     200(2.00)
     Class 2 (Differential)   300(3.00)     600(6.00)

Collected Earned Premium      1,000,000     1,000,000
Policy Year 2005 Incurred
Losses as of March 31, 2007     360,000       240,000

Earned Exposure Units
    Class 1                       5,000         2,000
    Class 2                       1,000           500

                                                          56

				
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