Individual Claim Development An Application Bas Lodder 9 March 2015

Chain Ladder Based Methods Limitations Individual Claim Development - Bas Lodder, 09/03/2015 Classical chain ladder (CL) based claim reserving methods are standard practice for attritional claims large volumes historically homogeneous risks claims with expected development based on AY / DY only (no calendar year effect) sufficient historical claims information Counterexample: motor liability includes large claims claim inflation effects property damage vs. bodily injury  general non-homogeneity changes in legal environment (whiplash, „Via sicura“)  historical non-homogeneity lump-sum and annuity payments  shocks, complex tail behaviour We need to find an alternative reserving method for motor liability claims 2

Alternative Claim Reserving Methods Individual Claim Development Individual Claim Development - Bas Lodder, 09/03/2015 Classical CL methods aggregate historical claim payments per risk in an AY – DY triangle Can we improve our estimates if we skip the aggregation step? Option 1: deterministic individual claim development triangulation: similar to classical CL methods best estimate by nearest neighbour approach Option 2: stochastic individual claim development triangulation: development pattern based on number of payments per claim stochastic simulation of future payments (frequency and severity) literature: Antonio et al. (2012), Pigeon et al. (2013), Pigeon et al. (2014) We need a second opinion for our CL best estimate 3 Chosen method: deterministic individual claim development

Deterministic Individual Claims Development (ICD) Methodology Individual Claim Development - Bas Lodder, 09/03/2015 Given historical individual incremental claim payments C i,k, we need to estimate future claim payments Ĉ i,k for each claim i and each DY k ≤ k max = max k (C i,k ) Ĉ i,k = α i,k * Σ j: AY(j) + k ≤ CY ((D i,j ) β * C j,k ), with D i,j = distance measure based on historical claim development difference α i,k = 1 / (Σ j: AY(j) + k ≤ CY ((D i,j ) β ) (scale factor) β (-∞, 0) (shape factor) Options for calculating D i,j claims basis: paid or incurred method: additive or multiplicative differences: absolute or squared 4 C i,k Ĉ i,k AY, i k (DY) CY

Deterministic Individual Claims Development (ICD) Example Individual Claim Development - Bas Lodder, 09/03/2015 First we calculate the distances D i,j : D 3,1 = │C 3,0 - C 1,0 │ + │C 3,1 - C 1,1 │ = │ │ + │ │ = 70 Similarly, we find D 3,2 = 30, D 4,1 = 20, D 4,2 = 20 and D 4,3 = 10 Next, we calculate the scaling factors α i,k, with, say, β = -1 and β = -2 β = -1: α 3,2 = 1 / (1/D 3,1 + 1/D 3,2 ) = 1 / (1/70 + 1/30) = 21 β = -2: α 3,2 = 1 / (1/(D 3,1 ) 2 + 1/(D 3,2 ) 2 ) = 1 / (1/ /900) = / 58 ≈ 760 Similarly, we find α 4,1 = 5 resp and α 4,2 = 10 resp. 200 Now, we derive the expected claim payments Ĉ i,k (for sake of simplicity, we take β = -1) Ĉ 3,2 = α 3,2 * (C 1,2 /D 3,1 + C 2,2 /D 3,2 ) = 21 * (0/ /30) = 28 Ĉ 4,1 = α 4,1 * (C 1,1 /D 4,1 + C 2,1 /D 4,2 + C 3,1 /D 4,3 ) = 5 * (80/ / /10) = 55 Ĉ 4,2 = α 4,2 * (C 1,2 /D 4,1 + C 2,2 /D 4,2 ) = 10 * (0/ /20) = 20 5 D i,j is constant in k, α i,k is not!

ICD: An Application Process, Data, Assumptions Individual Claim Development - Bas Lodder, 09/03/2015 Model implementation with Frank Cuypers and Simone Dalessi, Prime Re Services methodology VBA-based Excel template testing loops Data and assumptions in this presentation input data: large (0.2 – 1.0 MCHF) and mid-size (0.1 – 0.2 MCHF) Swiss Mobiliar motor liability claims claims basis: paid method: additive differences: absolute β = -2, i.e. Ĉ i,k = α i,k * Σ j: AY(j) + k ≤ CY ((D i,j ) -2 * C j,k ) 6

Mid-size claims – Actual vs. Expected, CY claims Individual Claim Development - Bas Lodder, 09/03/2015 Although claim amounts vary by AY, both CL and ICD estimates seem to be quite accurate 7

Mid-size claims – Estimation Error, CY claims Individual Claim Development - Bas Lodder, 09/03/2015 Large historical single claim payments cause overestimation in AYs ICD outperforms CL in these AYs since it puts negligible weight on the claim for which these payments were made Other claim developments cause similar estimation errors to both methods 8

Mid-size claims – Actual vs. Expected, CY claims Individual Claim Development - Bas Lodder, 09/03/2015 We randomly picked 4 claims per AY to reduce the number of claims The smaller number of claims causes shocks in claim payments The shock in AY 2004 was known by CY 2011, the one in AY 2008 came later 9

Mid-size claims – Estimation Error, CY claims Individual Claim Development - Bas Lodder, 09/03/2015 Due to the smaller number of claims, deviations from actual paid amounts are larger In particular, the payment in DY 5 of AY 2008 came unexpected Contrary to our overall expectation of smaller data sets, ICD does not significantly outperform CL in this example, except for AYs 2009 and

Large claims – Actual vs. Expected, CY ‘600 claims Individual Claim Development - Bas Lodder, 09/03/2015 The decline paid claim amounts is caused by Decline in whiplash claims Difference in claim maturity 11

Large claims – Estimation Error, CY ‘600 claims Individual Claim Development - Bas Lodder, 09/03/2015 The decline in whiplash claims causes large estimation errors in both methods Otherwise, ICD performs slightly better here 12

Large claims – Actual vs. Expected, CY claims Individual Claim Development - Bas Lodder, 09/03/2015 The decline in claims payments over time is mostly caused by frequency and therefore the reduced data set (4 claims per AY) is not affected 13

Large claims – Estimation Error, CY claims Individual Claim Development - Bas Lodder, 09/03/2015 As expected, the reduction of claims causes larger estimation errors ICD still performs slightly better than CL 14

ICD: An Application Conclusions Individual Claim Development - Bas Lodder, 09/03/2015 For the examples shown, ICD seems to be at least as good a method as CL ICD outperforms CL if large claims with unusual patterns are included in claims history Performance does not seem to depend on volume No outperformance if claims history contains significant calendar year effects (changes in legal environment, inflation) or changes in claims handling speed 15

Micro-Level Reserving Why (not)? Individual Claim Development - Bas Lodder, 09/03/2015 Deterministic ICD can perform well if claims data do not contain calendar effects Challenges IBNYR claims need to be estimated separately – comparison with CL only possible after removing IBNYR claims a large amount of individual claims data needs to be processed  IT / actuarial tools the model presented provides a best estimate, error estimates can be derived as well Like any claims reserving method, ICD requires actuarial judgement! ensuring homogeneity in claims history parameter choice / model options sensitivity testing – robustness! understanding differences to CL and other models Outlook: stochastic ICD more suitable in case of changing claims handling speed can provide a distribution of ultimate claim amounts 16

ICD: An Application Further questions Individual Claim Development - Bas Lodder, 09/03/2015 ? 17

Dessert: Personal Liability (0.1 MCHF – 5 MCHF) Incurred data (300 claims) Individual Claim Development - Bas Lodder, 09/03/2015 This data set contains some large all-or-nothing claims, mostly in earlier AYs In CL, such claims affect age-to-age factors, causing low estimates In ICD, such claims will obtain negligible weights The overall negative deviation is due to a combination of conservative claim reserves faster settlement of claims over time 18