EDUCATION COMMITTEE OF THE SOCIETY OF ACTUARIES SHORT-TERM ACTUARIA MATHEMATICS STUDY NOTE SUPPEMENT TO CHAPTER 3 OF INTRODUCTION TO RATEMAKING AND OSS RESERVING FOR PROPERTY AND CASUATY INSURANCE, FOURTH EDITION by Robert Brown, PhD, FSA, FCIA, ACAS and W Scott ennox, FSA, FCIA, FCAS Copyright 08 ACTE earning Posted by the Society o Actuaries with permission o the ACTE The Education Committee provides study notes to persons preparing or the examinations o the Society o Actuaries They are intended to acquaint candidates with some o the theoretical and practical considerations involved in the various subects While varying opinions are presented where appropriate, limits on the length o the material and other considerations sometimes prevent the inclusion o all possible opinions These study notes do not, however, represent any oicial opinion, interpretations or endorsement o the Society o Actuaries or its Education Committee The Society is grateul to the authors or their contributions in preparing the study notes STAM-4-8 Printed in USA
CHAPTER 3 SUPPEMENT (MARCH 5, 08 UPDATE) The methods presented in this chapter demonstrate how ultimate losses and reserves are calculated in practice rom historical data This section will ocus on the statistical methods that underlie the methods presented in section 36 We can restate Table 3 using the notation where i, denotes the cumulative loss payments or accident year i and development year Table 39 Cumulative oss Payments through Development Years Development Year () Accident Year (i) 0 3 4 5 6 7,0,,,3,4,5,6,7,0,,,3,4,5,6,7 3 3,0 3, 3, 3,3 3,4 3,5 3,6 3,7 4 4,0 4, 4, 4,3 4,4 4,5 4,6 4,7 5 5,0 5, 5, 5,3 5,4 5,5 5,6 5,7 6 6,0 6, 6, 6,3 6,4 6,5 6,6 6,7 7 7,0 7, 7, 7,3 7,4 7,5 7,6 7,7 8 8,0 8, 8, 8,3 8,4 8,5 8,6 8,7 The values o i, or i K are nown, where K is the highest development year, or 7 in this example The values o i, or i > K are unnown and represent the quantities that we want to estimate These values are the run-o triangle, or uture cumulative paid losses (the shaded values) The ultimate losses or each accident year are the values o i,k, or the last column o Table 39 Table 39 uses loss payments, so the loss reserve (unpaid losses) or each accident year is represented by the ollowing ormula: R i = i,k i,k-i (39) Formula 37a provided a ormula or estimating ultimate losses, as
Chapter 3 Supplement Estimated Ultimate osses = (osses Paid-to-Date) (37a) where is the loss-development actor rom a paid-loss-development triangle at duration (ie, rom development year to ) We can restate ormula 37a using the notation o this section, as: K i,k = ( i,k-i) (30) = K i The age-to-age development actors can be calculated by several dierent ormulas Using the mean, or volume-weighted average, one such ormula or is: K = K i, i, (3) This ormula can be obtained by maing certain assumptions and then employing a statistical estimation technique One such approach was developed by Mac (Mac, T, Measuring the Variability o Chain adder Reserve Estimates, Casualty Actuarial Society Forum, Spring 994, 0-8) Mac treats the values as random variables with the ollowing three properties: E (,,, ) = i, i,0 i, i, i, Var(,,, ) = α i, i,0 i, i, i, (,,, ) and (,0,,,, K, ) are independent or all i 3 i,0 i, ik, Here α is a parameter that relates the variance to previous values in the same way relates the mean to previous values Properties and have two consequences One is that the same actor applies regardless o the accident year (but does depend on the development year) The second is that a given value depends only on the previous development year and not on any
oss Reserving 3 prior years For example, i unusually low development is observed between 6, and 6,, the same development actor o 3 is used regardless, thereby potentially understating the ultimate losses and consequently the loss reserve or accident year 6 Mac does not mae an assumption regarding the distribution o the values Hence maximum lielihood estimation cannot be used Instead, he notes that given observations through development year, Property implies that i, / i, is an unbiased estimator o Then, or any set o weights, w,,, wk, with w, wk, =, K ˆ i, = wi, i, is also unbiased The variance o a weighted average o independent estimators (implied by Property 3) is minimized when the weights are inversely proportional to the variance o each term (See Example 35 at the end o this section or a proo when there are two estimators) Using Property, given observations through development year, wi, Var( /,, ) i, i, i,0 i, i, = Var(,, ) = = α α i, i,0 i, i, i, i, The sum o these weights is K i, α To ensure the weights add to one, divide by this sum to obtain w i, α i, i, = = K K i, i, α
4 Chapter 3 Supplement Finally, K ˆ K i, i, i, = = K K i, h, i, h= Mac shows that changing Property regarding the variance leads to alternative ormulas This stochastic orm o determining the values o i, or i > K, or uture cumulative paid losses in the run-o triangle, can be used to determine variances and thereore conidence intervals around the estimates o ultimate losses and consequently reserves We can also view the values a dierent way to help ormulate an alternative statistical representation o estimating ultimate losses From ormula 30, we can deine: K, ult = h, = 0,, K (3) h= Formula 3 gives us the age-to-ultimate development actors or each accident year We now rom the Bornhuetter Ferguson method that gives us the ratio o ultimate losses yet to be paid ater, ult development period to ultimate losses, thereore the ratio o ultimate losses paid by each development year to ultimate losses is represented by: r =, = 0,, K (33), ult We can then represent any value in the loss development triangle with the ollowing relationship: i, = r ik, ei,, = 0,, K (34) where e i, is an error term This ormulation has the advantage that uture
oss Reserving 5 values are not based solely on the most recent observed paid losses or each accident year In addition, an error term can be incorporated directly into the ormula Formula 34 can then be used to model the uture losses in the runo triangle and also determine variances and thereore conidence intervals around the estimates o ultimate losses and consequently reserves The example in this section uses historical cumulative paid losses to develop uture cumulative paid losses, including the ultimate losses The same approach can be used with incurred losses Example 35 et and Y be two independent random variables with variances σ and σ Y, respectively et Z = w ( w)y be a weighted average o the two variables Show that the variance o Z is minimized when w= c σ and = and determine the value o c w c/ σ Y Solution: Var( Z) = w σ ( w) σ Var( Z) = w w = w σ σ σ c w = = = σ σ σ σ σ σ σ ( ) σy 0 Y Y Y Y σ σ σ c w = = = σ σ σ σ σ σ σ σy where c = σ σ Y Y Y Y Y Y Y /