Computation of the Aggregate Claim Amount Distribution Using R and actuar. Vincent Goulet, Ph.D.

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1 Computation of the Aggregate Claim Amount Distribution Using R and actuar Vincent Goulet, Ph.D.

2 Actuarial Risk Modeling Process 1 Model costs at the individual level Modeling of loss distributions 2 Aggregate risks at the collective level Risk theory 3 Determine revenue streams Ratemaking 4 Evaluate solvability of insurance portfolio Ruin theory

risks at the collective level Risk theory 3 Determine revenue

3 What actuar Is A package providing additional Actuarial Science functionality to the R statistical system Distributed through the Comprehensive R Archive Network (CRAN) Currently provides: 17 additional probability distributions loss modeling facilities aggregate claim amount calculation fitting of credibility models ruin probabilities and related quantities simulation of compound hierarchical models

additional probability distributions loss modeling facilities aggregate claim amount calculation

4 Yes But... Why R? Compare x <- matrix(2, 3, 10:15) vs x _ Multi-platform Interactive State-of-the-art statistical procedures, random number generators and graphics

5 Collective Risk Model Let We have the random sum We want to find S : aggregate claim amount N : number of claims (frequency) C j : amount of claim j (severity) F S (x) = Pr[S x] S = C C N = Pr[S x N = n] Pr[N = n] n=0 = F n C (x) Pr[N = n] n=0

(frequency) C j : amount of claim j (severity) F S (x) = Pr[S

6 Collective Risk Model Let We have the random sum We want to find S : aggregate claim amount N : number of claims (frequency) C j : amount of claim j (severity) F S (x) = Pr[S x] S = C C N = Pr[S x N = n] Pr[N = n] n=0 = F n C (x) Pr[N = n] n=0

7 Collective Risk Model Let We have the random sum We want to find S : aggregate claim amount N : number of claims (frequency) C j : amount of claim j (severity) F S (x) = Pr[S x] S = C C N = Pr[S x N = n] Pr[N = n] n=0 = F n C (x) Pr[N = n] n=0

8 What actuar Tries Not To Be Magic! (Clueless user)

9 What actuar Tries Not To Be Magic! (Clueless user)

10 What actuar Tries Not To Be Magic! (Clueless user)

11 What actuar Tries Not To Be Magic! (Clueless user)

12 What actuar Tries Not To Be Magic! (Clueless user)

13 What actuar Tries Not To Be Magic! (Clueless user)

14 What We re Presenting Here Today (Insightful user) aggregatedist() F S (x)

15 How We Can Tackle the Problem 1 Carry out the convolutions FC k (x) for k = 0, 1, 2,... 2 Use a Normal approximation ( ) x µs F S (x) Φ 3 Use the Normal Power II approximation ( ) F S (x) Φ x µ S γ S γ S σ S γ 2 S σ S 4 Use simulation: F S (x) F n (x) = 1 n n j=1 I{x j x}

16 How We Can Tackle the Problem 1 Carry out the convolutions FC k (x) for k = 0, 1, 2,...! 2 Use a Normal approximation ( ) x µs F S (x) Φ 3 Use the Normal Power II approximation ( ) F S (x) Φ x µ S γ S γ S σ S γ 2 S σ S 4 Use simulation: F S (x) F n (x) = 1 n n j=1 I{x j x}

17 How We Can Tackle the Problem 1 Carry out the convolutions FC k (x) for k = 0, 1, 2,...! 2 Use a Normal approximation! ( ) x µs F S (x) Φ 3 Use the Normal Power II approximation ( ) F S (x) Φ x µ S γ S γ S σ S γ 2 S σ S 4 Use simulation: F S (x) F n (x) = 1 n n j=1 I{x j x}

( ) x µs F S (x) Φ 3 Use the Normal Power II approximation ( ) F S (x)

18 How We Can Tackle the Problem 1 Carry out the convolutions FC k (x) for k = 0, 1, 2,...! 2 Use a Normal approximation! ( ) x µs F S (x) Φ 3 Use the Normal Power II approximation! ( ) F S (x) Φ x µ S γ S γ S σ S γ 2 S σ S 4 Use simulation: F S (x) F n (x) = 1 n n j=1 I{x j x}

( ) x µs F S (x) Φ 3 Use the Normal Power II approximation!

19 How We Can Tackle the Problem 1 Carry out the convolutions FC k (x) for k = 0, 1, 2,...! 2 Use a Normal approximation! ( ) x µs F S (x) Φ 3 Use the Normal Power II approximation! ( ) F S (x) Φ x µ S γ S γ S σ S 4 Use simulation: γ 2 S F S (x) F n (x) = 1 n σ S n j=1 I{x j x}!

20 Most Commonly Used Method 5 Recursive method (Panjer algorithm): [ 1 f S (x) = (p 1 (a + b)p 0 )f C (x) 1 af C (0) min(x,m) ] + (a + by/x)f C (y)f S (x y) with y=1 f S (0) = P N (f C (0))

21 Most Commonly Used Method 5 Recursive method (Panjer algorithm):! [ 1 f S (x) = (p 1 (a + b)p 0 )f C (x) 1 af C (0) min(x,m) ] + (a + by/x)f C (y)f S (x y) with y=1 f S (0) = P N (f C (0))

22 Discretization of Continuous Distributions > discretize(pgamma(x, 2, 1), from = 0, to = 5, method = "upper") pgamma(x, 2, 1) x

23 Discretization of Continuous Distributions > discretize(pgamma(x, 2, 1), from = 0, to = 5, method = "lower") pgamma(x, 2, 1) x

24 Discretization of Continuous Distributions > discretize(pgamma(x, 2, 1), from = 0, to = 5, method = "rounding") pgamma(x, 2, 1) x

25 Discretization of Continuous Distributions > discretize(pgamma(x, 2, 1), from = 0, to = 5, method = "unbiased", lev = levgamma(x, 2, 1)) pgamma(x, 2, 1) x

26 Computing the Aggregate Claim Amount Distribution aggregatedist() is the unified interface to all 5 supported methods Computer intensive calculations are done in C Output is a function to compute F S (x) in any x R methods to plot and compute related quantities

27 Example Assume N Poisson(10) C Gamma(2, 1) > fx <- discretize(pgamma(x, 2, 1), from = 0, to = 22, step = 2, method = "unbiased", lev = levgamma(x, 2, 1)) > Fs <- aggregatedist("recursive", model.freq = "poisson", model.sev = fx, lambda = 10, x.scale = 2)

28 Example (continued) > plot(fs) Aggregate Claim Amount Distribution Recursive method approximation F S (x) x

29 Example (continued) > summary(fs) Aggregate Claim Amount Empirical CDF: Min. 1st Qu. Median Mean 3rd Qu. Max > knots(fs) [1] [18] [35] > Fs(c(10, 15, 20, 70)) [1]

30 Example (continued) > mean(fs) [1] > VaR(Fs) 90% 95% 99% > TVaR(Fs) 90% 95% 99%

31 One more thing...

32 What If Recursions Do Not Start? For example, in the Compound Poisson case f S (0) = e λ(1 f C(0)) If λ is large, f S (0) = 0 numerically One solution: 1 divide λ by 2 n 2 convolve resulting distribution n times with itself

33 Example > Fsc <- aggregatedist("recursive", model.freq = "poisson", model.sev = fx, lambda = 5, convolve = 1, x.scale = 2) > summary(fsc) Aggregate Claim Amount Empirical CDF: Min. 1st Qu. Median Mean 3rd Qu Max

34 Concluding Remarks See cran.r-project.org/package=actuar for the package Package vignettes provide complete documentation Please cite the software in publications: > citation(package = "actuar") To cite actuar in publications use: C. Dutang, V. Goulet and M. Pigeon (2008). actuar: An R Package for Actuarial Science. Journal of Statistical Software, vol. 25, no. 7, URL [...] Contribute!

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