Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring

Transcription

1 Non-life insurance mathematics Nils. Haavardsson, University of Oslo and DNB Skadeforsikring

2 Introduction to reserving Introduction hain ladder The naive loss ratio Loss ratio prediction Non-life insurance from a financial perspective: for a premium an insurance company commits itself to pay a sum if an event has occured Policy holder signs up for an insurance Policy holder pays premium. ontract period, in which premium is earned clas might occur Incurred clas will be reported and settled Issues that need to be solved: How much premium is earned? How much premium is unearned? How do we measure the number and size of unknown clas? How do we know if the reserves on known clas are sufficient?

3 The premium reserve is split in two parts: Provision for unearned premiums Provisions for unexpired risks Premium reserves Introduction hain ladder The naive loss ratio Loss ratio prediction Earned and unearned premium: Written premium is earned evenly/uniformly over the cover period The share of the premium that has been earned is the past te s proportion of the total period If a larger premium has been received the difference is the unearned premium Example: An insurance policy starts on September 1 01 and is valid until August The premium for the entire period is 400. At 31 December we have received two quarterly premiums or 100. We have then earned (4/1)*400 = 800. Unearned is =400 1/ / /8-013 Unexpired risk reserve Regard entire period covered by the insurance rom a point in te, say 31/1-01, we look forward to all the clas and expenses that could occur after this point. all them 311 If 311>uture premiums yet not due (P)+unearned premium reserve (UP) the difference is accounted as unexpired risk reserve In example assume 311 = 1800>P+UP= =1600, so unexpired risk reserve is 00 3

4 las reserves Introduction hain ladder The naive loss ratio Accident date Reporting date las payments las close las reopening las payments las close Loss ratio prediction las reserving issues: How do we measure the number and size of unknown clas? (IBNR reserve, i.e., Incurred Bot Not Reported) How do we know if the reserves on known clas are sufficient? (RBNS reserve, i.e., Reserved But Not Settled) las occurence year las losses settled la payments plus clas handling expenses 4

5 las occurence year Introduction The development of clas losses settled las losses settled for each clas occurence year are often not paid on one date but rather over a number of years hain ladder The naive loss ratio Loss ratio prediction Incremental clas loss settlement data presented as a run-off triangle Incremental clas loss Development year settlements omments: The development year for a clas settlement amount reflects how long after the clas occurence year the amount was settled. An amount settled during the clas occurence year was settled in development year 0 In the example the largest development year for any clas occurence years is 7 The data shown represents the incremental clas losses settled in the development year or any cell in the table, the value shown represents the incremental clas loss amount that was settled in calendar year Each diagonal set of data represents the amounts settled in a single calendar year Green cells represent observed data all red represent te periods in the future for which we wish to estate the expected clas settlements amounts

6 Introduction Assumptions underlying the LM hain ladder The naive loss ratio Loss ratio prediction Patterns of clas loss settlement observed in the past will continue in the future The development of clas loss settlement over the development years follows an identical pattern for every clas occurence year But the observed clas loss settlement patterns may change over te: hanges in product design and conditions hanges in the clas reporting, assessment and settlement processes (example: different owners) hange in the legal environment Abnormally large or small cla settlement amounts hanges in portfolio so that the history is not representative for predicting the future (example: strong growth)

7 las occurence year Introduction hain ladder LM in practice The naive loss ratio Loss ratio prediction Determining the LM estator for the cumulative clas loss settlement factor umulative clas loss Development year settlements /18300=1,1151 LM estator for clas loss settlement factor = =0407 1,9989 1,3140 1,4 1,1151 1,0491 1,0118 1,0035 omments: These LM estators for the cumulative clas loss settlement factors are used to estate the cumulative clas loss settlement amount in the future or each clas occurence year the last historical observation is used together with the appropriate LM estator for the development factor to estate the cumulative settlement amount in the next development year This value is, in turn, multiplied by the estator for the development factor for the next development year and so on.

8 las occurence year Introduction hain ladder LM in practice The naive loss ratio Loss ratio prediction Determining the estated cumulative clas loss settlements in future periods umulative clas loss Development year settlements *1,0491 *1,0118 *1, = 6715 = 6794 = LM estator for clas 1,8508 1,3140 1,4 1,1151 1,0491 1,0118 1,0035 omments: The values shown in the red cells are the estators for future cumulative clas settled These estates are always based on the latest available cumulative clas settlement amounts for the relevant clas occurence year, i.e., the estated future cumulative clas settlements are always based on the last green diagonal of data It is now sple to derive the estated incremental clas settlement amounts for the future periods An incremental settlement amount is the difference between tow consecutive cumulative settlement amounts

9 las occurence year Introduction hain ladder LM in practice The naive loss ratio Loss ratio prediction Determining the estated incremental settlement amounts from the estated cumulative amounts Incremental clas loss Development year settlements = 314 = 79 =

10 las occurence year Introduction hain ladder LM in practice Determining the estated incremental settlement amounts from the estated cumulative amounts The naive loss ratio Loss ratio prediction Incremental clas loss Development year settlements Estated clas loss alendar year settlement amounts = omments: Group the estated incremental clas loss settlement amounts by the year in which they will be settled These cash flows can then be discounted to determine the technical provisions Norwegian State Treasury Bonds (Statsobligasjoner in Norwegian) may be used as discount factor Example: a cash flow due in 017 is discounted with a 4 year old Norwegian State Treasury Bond etc. Why do we hope that the development year does not exceed 10??

11 It can be useful to be able to predict loss ratio requency Severity Loss ratio ore ,0 % Large 13,1 5,1 16,7 % 71,7 % 100 % 90 % Log Normal with conditional mean 5.1 M 80 % 70 % 60 % loss ratio 5 Poisson with mean % adjusted loss ratio core loss ratio 0 40 % adjusted core loss ratio 15 frequency 30 % 10 adjusted frequency 5 0 % % 0 % October 013

12 Overview Important issues Models treated urriculum Duration (in lectures) What is driving the result of a nonlife insurance company? insurance economics models Lecture notes 0,5 Poisson, ompound Poisson How is cla frequency modelled? and Poisson regression Section 8.-4 EB 1,5 How can clas reserving be modelled? hain ladder, Bernhuetter erguson, ape od, Note by Patrick Dahl How can cla size be modelled? Gamma distribution, lognormal distribution hapter 9 EB How are insurance policies priced? Generalized Linear models, estation, testing and modelling. RM models. hapter 10 EB redibility theory Buhlmann Straub hapter 10 EB 1 Reinsurance hapter 10 EB 1 Solvency hapter 10 EB 1 Repetition 1 1

13 Overview of this session The Bornhuetter erguson model Stochastic clas reserving in non-life insurance 13

14 Bornhuetter erguson The Bornhuetter-erguson method Stochastic clas reserving The Bornhuetter-erguson method is more sophisticated than the Naive loss method It looks on where in te clas will be reported or paid It is very silar to an ordinary budgeting model used by businesses You budget for future clas by period The sum of these future budgeted clas is the IBNR reserve 14

15 Bornhuetter erguson The Bornhuetter-erguson method Stochastic clas reserving More formally, the following principles apply: ( B1) predictor of ( B ) or ( B 3) we know Expected clas Unemerged clas is independent of The final are known, in the meaning E[ E[ clas ] ] are considered identical known, i.e. to the mean are independent of that emerged we have a clas, in (B3) is the factor that would develop losses from development period j to the end for accident year i. could have been determined by the hain Ladder technique N E[ ] 15

16 The Bornhuetter-erguson method Bornhuetter erguson Stochastic clas reserving The unbiased Bonrhuetter-erguson predictor is given by B This predictor takes emerged clas into account as it swaps past expected emergegence with real emergence (i.e. it is better than the Naive Loss Ratio) (1) can be re-written B W 1 (1 W We make a further assumption hain ladder type estate 1 ( 1 )* (1 ) 1 N )* N with W «Known» expected clas ( B 4) var[ * ] *var[ ] 1 (1) () 16

17 Bornhuetter erguson The Bornhuetter-erguson method Stochastic clas reserving ( Theorem) combination of (in the meaning Proof: we ultately want to weight between chain ladder and the naive loss ratio method. Introduce the two random variables How should L and NL be weighted to minize total error? Note that L NL The weights plicitly defined in () produces the best the two predictors L minizing quadratic loss) N * Error from hain Ladder estate Error from naive loss method and N 17

18 The Bornhuetter-erguson Total W W W ( B W L method (1 W ) (1 W ) NL )( N W N ) W N Bornhuetter erguson Stochastic clas reserving (3) We want to find the best combination of the two predictors (L + NL) so that the error in (3) is minized Thus, we need to solve the problem min W { E{ Total } } min W { E( B ) }} (4) 18

19 The Bornhuetter-erguson method In general for an estator W of a parameter theta: Mean square error of E( W EW ) varw ( EW ) ( EW E( W EW ( EW ) ) estator W is E( W W ) EW varw EW ) EW rom (4) and (5) we see that we want to minize the variance of The strategy is now to use Lemma 6. on the two variables We can see that L and NL have the same mean, 0. We also need to prove that and are uncorrelated L NL To show the uncorrelatedness of the components we need the auxillary result, ov[ ov[ ov[ var[,,,( ] ] ov[ ] ) 0 if, var[ ] ] ov[ ] :,( var[ B and L NL ]/ )] Bornhuetter erguson Stochastic clas reserving (5) 19

20 The Bornhuetter-erguson method Using the calculation rules for covariances and remembering that covariances between a random variable and a constant vanishes, we get that ov[ L ov[, NL, ] ov[ (6) proves uncorrelatedness. ] ov[, We also need to calculate the variance of N var[ ] var[ ], var[ var[ NL L ] ] var[ var[, Using Lemma 6. we find that the optal weights are W / 1 ( 1) 1 1 ( 1) ] ( ] ov[ 1 1), N ] ] L and NL ] 0 Bornhuetter erguson Stochastic clas reserving (6) 0

21 Bornhuetter erguson Bornhuetter-erguson example Stochastic clas reserving Tidsperioden Tidsperioden Periode + 0 Periode + 1 Periode + Periode + 3 Periode + 4 Skadereserver (Tidsperioden) methode Dato + 0 Dato + 1 Dato + Dato + 3 Tidsperioden Tidsperioden Skadeprosent Standard triangel 1,9543 1,176 1,0351 1,009.. År-vektet triangel,105 1,185 1,0344 1,009.. Sisteverdivektet triangel 1,8031 1,173 1,0367 1,009.. Gjennomsnittsv ektet triangel,4015 1,1884 1,0340 1,009.. K last-w eighted hain Ladder 1,971 1,1884 1,0340 1,009.. hain Ladder uten de ekstreme utviklings,0553 1,173 1,0340 1,009.. Oppdater,0484 1,1800 1,0347 1,

22 Bornhuetter-erguson example Bornhuetter erguson Stochastic clas reserving

23 Bornhuetter erguson Bornhuetter-erguson example Stochastic clas reserving Te period Year Period + 0 Period + 1 Period + Period + 3 Period + 4 las reserves (The te period) earned premium periode+4 calculated clas reserve calculated clas reserves chain ladder (left axis) clas reserves Bornhuetter erguson (left axis) predicted loss ratio (right axis) , , , , ,00 3

24 omparison Bornhuetter-erguson and chain-ladder Bornhuetter erguson Stochastic clas reserving , , ,00 60,00 40,00 clas reserves chain ladder (left axis) clas reserves Bornhuetter erguson (left axis) predicted loss ratio (right axis) , ,00 4

25 Bornhuetter erguson How good is our model? Stochastic clas reserving Up to now we have (only) given an estate for the mean/expected ultate cla We would also like to know how good this estate predicts the outcome of random variables How accurate are our reserves estates? Imagine a non-life insurance company with a total clas reserve of NOK and a profit-loss statement given below The earnings statement is slightly positive ( ) If the clas reserves are reduced by 1% this doubles the income before taxes Only a slight change of the clas reserves may have an enormous pact on the earning statement Therefore it is very portant to know the uncertainties in the estates Earning statement at 31 December a) Premium earned b) las incurred current accident year c) Loss experience prior years d) Underwriting and other expens e) Investment income Income before taxes

26 The mean square error of prediction measures the quality of the estated clas reserves Bornhuetter erguson Stochastic clas reserving Assume that we have a random variable and that W estates Then the mean square error of W for is given by. We also have that E( W EW ) varw ( EW ) ( EW E( W EW ( EW ) ) W ) EW varw EW EW Normally we measure the quality of the estators and the predictors for the ultate clas by means of second moments such as the mean square error of prediction defined above We really want to derive the whole predictive distribution of stochastic clas reserving. Most often it is not feasible to calculate this distribution analytically Therefore we have to rely on numerical algorithms such as Bootstrapping methods and Monte arlo Sulation methods to produce a sulated predictive distribution for the clas reserves if E( W ) 6

27 Bornhuetter erguson The previous example revisited Stochastic clas reserving Year Period + 0 Period + 1 Period + Period + 3 Period + 4 las reserve (te period) las reserve/ Payments(%) Standard error Relative standard error (%) , , , , , , , , ,9 clas reserves chain ladder clas reserves Bornhuetter erguson clas reserve stochastic method w ith Bootstrap

28 Imagine you want to build a reserve risk model There are three effects that influence the best estat and the uncertainty: Payment pattern RBNS movements Reporting pattern Up to recently the industry has based model on payment triangles: Year Period + 0 Period + 1 Period + Period + 3 Period What will the future payments amount to?? Bornhuetter erguson Stochastic clas reserving 8

29 Mill NOK Mill NOK Payment pattern, reporting pattern and RBNS movement Bornhuetter erguson Stochastic clas reserving Payment pattern Reporting pattern 1, 1, 1 1 0,8 0,6 Product A Product B 0,8 0,6 Product A Product B 0,4 Product 0,4 Product 0, 0, 0 Year 1 Year Year 3 Year 4 Year 5 0 Year 1 Year Year 3 Year 4 Year 5 RBNS movement year 1 RBNS movement year Jan eb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Product A Product B Product Jan eb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Product A Product B Product 9

30 Bornhuetter erguson Reserve risk distribution Stochastic clas reserving Year Period + 0 Period + 1 Period + Period + 3 Period + 4 Relative standard error (%) , , , ,9 One possible stochastic models for the chain-ladder technique: the logarithm of the cumulative clas amounts Y=log () and the log-normal class of models Y m with as independent normal random errors 30

31 Bornhuetter erguson Worst case for reserve risk Stochastic clas reserving 1600 Development result product A % = -74 M 31