Capital Allocation: Mathematics & Economics

Size: px

Start display at page:

Download "Capital Allocation: Mathematics & Economics"

Rosaline Underwood
5 years ago
Views:

Arizona Acuarial Club 2018 Spring Meeing Phoenix, AZ March 28, 2018 Capial Allocaion: Mahemaics & Economics Daniel Bauer Universiy of Alabama This presenaion is based on

1 Arizona Acuarial Club 2018 Spring Meeing Phoenix, AZ March 28, 2018 Capial Allocaion: Mahemaics & Economics Daniel Bauer Universiy of Alabama This presenaion is based on research projecs wih Qiheng Guo and George Zanjani. Financial suppor by he Casualy Acuarial Sociey (CAS) and ongoing suppor from Richard Derrig (deceased) is graefully acknowledged.

2 Page 2 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Capial Allocaion: I need o price o hi a 8% Reurn on Capial. How do I do i? Auo (Risk 1) I 1 Propery (Risk 2) I 2 Workers Comp (Risk 3) I 3

3 Page 2 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Capial Allocaion: I need o price o hi a 8% Reurn on Capial. How do I do i? Auo (Risk 1) I 1 Propery (Risk 2) I 2 Workers Comp (Risk 3) I 3 Capial

4 Page 2 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Capial Allocaion: I need o price o hi a 8% Reurn on Capial. How do I do i? Auo (Risk 1) I 1 Propery (Risk 2) I 2 Workers Comp (Risk 3) I 3?

5 Page 2 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Capial Allocaion: I need o price o hi a 8% Reurn on Capial. How do I do i? Auo (Risk 1) I 1 Propery (Risk 2) I 2 Workers Comp (Risk 3) I 3?

6 Page 3 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Gradien Approach o Capial Allocaion 1. Selec a risk measure 2. Calculae he gradien 3. Under cerain condiions, his leads you o an allocaion Auo (Risk 1) I 1 Propery (Risk 2) I 2 Workers Comp (Risk 3) I 3 ρ q (1) q (1) + ρ q (2) q (2) + ρ q (3) q (3) = A,... where A = ρ(i (1) + I (2) + I (3) ) and I (i) = q (i) L (i) gradien allocaion!

7 Page 4 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Moivaion and Overview Financial insiuions use risk measure gradiens o allocae capial o risks for purposes of pricing and performance measuremen Reurn on Risk-Adjused Capial (RORAC) RORAC for line k = (expeced reurn in line k) / (capial allocaed o line k) expeced reurn=u/w profi margin capial allocaed based on risk measure gradien Assess performance by comparing RORAC for line k o a arge ROE Criicism: Risk measure is arbirary and may no connec o he underlying economics of he business. Why should one-period underwriing reurn be correc meric for performance? Pricing while avoiding he "rigors of he pricing projec"? (Vener, 2010)

8 Page 5 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Moivaion and Overview We build economic models of he firm from he ground up and calculae marginal cos of risk. Yields "economic" capial allocaion.

9 Page 5 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Moivaion and Overview We build economic models of he firm from he ground up and calculae marginal cos of risk. Yields "economic" capial allocaion. Key Resul: RORAC emerges from economic consideraions, bu... If risk emerges from counerpary risk aversion, he correc risk measure akes an unconvenional form (generally no coheren or convex!): { } ρ(x) = exp E Q [X]. In muli-period conex, he componens of RORAC need o be redefined: Redefine reurn: Expeced reurn calculaions mus consider non-acuarial sources of coss (capial more or less cosly depending on loss realizaion) Redefine capial: "Capial" has o be conceived more broadly o include coningen sources of financing Redefine he benchmark: The cos of capial has o be adjused similarly a arge ROE on book equiy is no longer appropriae Differen developmen years "discouned" a differen raes, depending on company capializaion

10 Page 6 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Inroducion Profi Maximizaion and Marginal Cos of Risk Applicaion in he Conex of a CAT Reinsurer Applicaion o a Propery and Casualy Insurer Conclusion

11 Page 7 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Inroducion Reurn-on-Risk-Adjused-Capial (RORAC) RORAC i = [(Marginal) Reurn on Line i ] ρ [Cos-of-Capial] q i [(Marginal) Reurn on Line i ] [Cos-of-Capial] ρ q }{{} i Marginal Cos of Risk q i [(Marginal) Reurn on Line i ] [Cos-of-Capial] a i } {{ } To. Reurn Economic moivaion: Opimizaion of company s profis Deails RORAC i : [Premiums] E[Payoff] [CoC] a }{{}}{{} i p i i R i subjec o a risk measure consrain ρ(q 1 L q N L N ) a yields: ( ) pi R i q q i i { Acuarial Profi Allocaed Capial q i ρ q i q i = [CoC]

12 Page 8 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Inroducion Approach and Resuls We build (more) complex models of an insurer ha include: Risk-averse counerparies (policyholders) as he moivaion for holding capial Muliple periods, defaul is possible Various exernal financing opporuniies (inernal vs. exernal vs. emergency capial)

13 Page 8 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Inroducion Approach and Resuls We build (more) complex models of an insurer ha include: Risk-averse counerparies (policyholders) as he moivaion for holding capial Muliple periods, defaul is possible Various exernal financing opporuniies (inernal vs. exernal vs. emergency capial) Opimal RORAC (or, raher RARAC) calculaion: Adjus for company risk (weighing) All financial resources, coningen sources RAROC {}}{ Acuarial Profi Allocaed Capial? τ Endog. CoC : economizes on financing opp.

14 Page 8 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Inroducion Approach and Resuls We build (more) complex models of an insurer ha include: Risk-averse counerparies (policyholders) as he moivaion for holding capial Muliple periods, defaul is possible Various exernal financing opporuniies (inernal vs. exernal vs. emergency capial) Opimal RORAC (or, raher RARAC) calculaion: Adjus for company risk (weighing) All financial resources, coningen sources RAROC {}}{ Acuarial Profi Allocaed Capial? τ Endog. CoC : economizes on financing opp. We implemen / (numerically) solve he model and compare "opimal" and convenional RAROC calculaions While capial coss are sill mos imporan cos componen in non-exreme cases, ignoring addiional componens can lead o inefficien decisions In exreme cases, oher cos componens gain imporance

15 Page 9 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Profi Maximizaion and Marginal Cos of Risk Model Seup Loss L () i a ime in line i (non-negaive random variable, iid across )

16 Page 9 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Profi Maximizaion and Marginal Cos of Risk Model Seup Loss L () i a ime in line i (non-negaive random variable, iid across ) A he beginning of every (underwriing) period, firm chooses o underwrie cerain porion q (i) Resuling indemniy I (i) = I (i) (L (i) (proporional, generalizaions possible) of he risk a premium p (i), q (i) ) = L (i) q (i)

17 Page 9 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Profi Maximizaion and Marginal Cos of Risk Model Seup Loss L () i a ime in line i (non-negaive random variable, iid across ) A he beginning of every (underwriing) period, firm chooses o underwrie cerain porion q (i) Resuling indemniy I (i) = I (i) (L (i) (proporional, generalizaions possible) of he risk a premium p (i), q (i) ) = L (i) q (i) Capializaion: The company can raise or shed capial a he beginning of he period (R b ) a cos c 1, wih c 1 (x) = 0, x < 0 The company can also raise emergency funds a he end of he period (R e ) a (higher) cos c 2 Inernal cos of capial τ (< c1 (0+))

18 Page 9 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Profi Maximizaion and Marginal Cos of Risk Model Seup Loss L () i a ime in line i (non-negaive random variable, iid across ) A he beginning of every (underwriing) period, firm chooses o underwrie cerain porion q (i) Resuling indemniy I (i) = I (i) (L (i) (proporional, generalizaions possible) of he risk a premium p (i), q (i) ) = L (i) q (i) Capializaion: The company can raise or shed capial a he beginning of he period (R b ) a cos c 1, wih c 1 (x) = 0, x < 0 The company can also raise emergency funds a he end of he period (R e ) a (higher) cos c 2 Inernal cos of capial τ (< c1 (0+)) Law of moion: [ a = a 1 (1 τ) + R b c 1 (R b ) + i p i ] e r i I (i) + R e c 2 (R e )

19 Page 9 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Profi Maximizaion and Marginal Cos of Risk Model Seup Loss L () i a ime in line i (non-negaive random variable, iid across ) A he beginning of every (underwriing) period, firm chooses o underwrie cerain porion q (i) Resuling indemniy I (i) = I (i) (L (i) (proporional, generalizaions possible) of he risk a premium p (i), q (i) ) = L (i) q (i) Capializaion: The company can raise or shed capial a he beginning of he period (R b ) a cos c 1, wih c 1 (x) = 0, x < 0 The company can also raise emergency funds a he end of he period (R e ) a (higher) cos c 2 Inernal cos of capial τ (< c1 (0+)) Law of moion: [ a = a 1 (1 τ) + R b c 1 (R b ) + i p i ] e r i I (i) + R e c 2 (R e ) In case of defaul, remaining asses are paid o dollar a he same rae per dollar of coverage

20 Page 10 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Profi Maximizaion and Marginal Cos of Risk Some Implicaions of he Model Company s value funcion V depends on curren capial level V (a)

21 Page 10 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Profi Maximizaion and Marginal Cos of Risk Some Implicaions of he Model Company s value funcion V depends on curren capial level V (a) V saisfies Bellman equaion (sa. infinie-horizon dynamic problem)

22 Page 10 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Profi Maximizaion and Marginal Cos of Risk Some Implicaions of he Model Company s value funcion V depends on curren capial level V (a) V saisfies Bellman equaion (sa. infinie-horizon dynamic problem) R e {0, R }, e where R e "jus saves" company: If ([Fin. Resources] I), no need o raise (can raise cheaper a beginning of nex period) If ([Fin. Resources] < I), jus raise enough o survive (can raise cheaper a beginning of nex period) If V (0) < R e, opporune o le company defaul

23 Page 10 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Profi Maximizaion and Marginal Cos of Risk Some Implicaions of he Model Company s value funcion V depends on curren capial level V (a) V saisfies Bellman equaion (sa. infinie-horizon dynamic problem) R e {0, R }, e where R e "jus saves" company: If ([Fin. Resources] I), no need o raise (can raise cheaper a beginning of nex period) If ([Fin. Resources] < I), jus raise enough o survive (can raise cheaper a beginning of nex period) If V (0) < R e, opporune o le company defaul Bellman Equaion: (s.. several consrains) V (a) = [ ( ) E 1 {S I} j p(j) e r I τ a c 1 (R b ) + e r V (a new) ( [ 1 +1 {SD} (a + R b ) ) ] Three regions: I S no issues; S D defaul

24 Page 11 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Profi Maximizaion and Marginal Cos of Risk Connecing Risk and Reurn Premium: For empirical racabiliy, we assume policyholders assess company qualiy via he defaul probabiliy bu demand ges sauraed: p (i) = e r E[I (i) ] exp{α βp(i > D) γe[i ]} }{{} Mark-up over acuarial price

25 Page 11 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Profi Maximizaion and Marginal Cos of Risk Connecing Risk and Reurn Premium: For empirical racabiliy, we assume policyholders assess company qualiy via he defaul probabiliy bu demand ges sauraed: p (i) = e r E[I (i) ] exp{α βp(i > D) γe[i ]} }{{} Mark-up over acuarial price Policyholders assess risk via probabiliy of defaul ( raing) Margins decreasing in scale Generalizaions possible...

26 Page 11 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Profi Maximizaion and Marginal Cos of Risk Connecing Risk and Reurn Premium: For empirical racabiliy, we assume policyholders assess company qualiy via he defaul probabiliy bu demand ges sauraed: p (i) = e r E[I (i) ] exp{α βp(i > D) γe[i ]} }{{} Mark-up over acuarial price Policyholders assess risk via probabiliy of defaul ( raing) Margins decreasing in scale Generalizaions possible... Recall from he basic one-period model: The Marginal Cos of Risk [ ] [ ] ( ] ) I (i) I (i) Marg. Prem = E q 1 (i) {I A} + E I = A E [1 q (i) {I>A} + c 1(A)

27 Page 12 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Profi Maximizaion and Marginal Cos of Risk The Marginal Cos of Risk We have for he marginal cos for risk i {1, 2,..., N}: [ ] I (i) E exp {α βp(i D) γe[i]} (1 γe[i]) q (i) [ ] [ ] I (i) I (i) { = E q 1 (i) {I D} w(i) + E q (i) I = D E [ 1 {I>D} w(i) ]}, where he weighing funcion w is defined as: (1 c 1 (Rb )) [1 + V (S I)], I S w(i) = (1 c 1 (Rb )) [1 + ξ / 1 ξ ], S D) e r β p (j), I > D In paricular E[w(I)] = 1.

28 Page 12 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Profi Maximizaion and Marginal Cos of Risk The Marginal Cos of Risk We have for he marginal cos for risk i {1, 2,..., N}: [ ] I (i) E exp {α βp(i D) γe[i]} (1 γe[i]) q (i) [ ] [ ] I (i) I (i) { = E q 1 (i) {I D} w(i) + E q (i) I = D E [ 1 {I>D} w(i) ]}, where he weighing funcion w is defined as: (1 c 1 (Rb )) [1 + V (S I)], I S w(i) = (1 c 1 (Rb )) [1 + ξ / 1 ξ ], S D) e r β p (j), I > D In paricular E[w(I)] = 1. RAROC (RARAC?): [ ] [Marginal Revenue] E I (i) q (i) {I D} w(i) [ ] I (i) E q (i) I = D } {{ } / qi VaR α E [ 1 {I>D} w(i) ] }{{} CoC

29 Page 13 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion in he Conex of a CAT Reinsurer Inroducion Profi Maximizaion and Marginal Cos of Risk Applicaion in he Conex of a CAT Reinsurer Applicaion o a Propery and Casualy Insurer Conclusion

30 Page 14 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion in he Conex of a CAT Reinsurer Case Sudy: Daa from Ca Reinsurer Line Saisics Aggs Premiums Expeced Loss Sandard Deviaion Agg1 Agg2 Agg3 N American EQ Eas 6,824, ,175, ,321, N American EQ Wes 31,222, ,927, ,198, S American EQ 471, , , Ausralia EQ 1,861, ,712, ,637, Europe EQ 2,198, ,729, ,947, Israel EQ 642, , ,234, NZ EQ 2,901, ,111, ,860, Turkey EQ 214, , ,505, N Amer. Severe Sorm 16,988, ,879, ,742, US Hurricane 186,124, ,652, ,791, US Winersorm 2,144, ,967, ,611, Ausralia Sorm 124, , , Europe Flood 536, , ,092, ExTropical Cyclone 37,033, ,602, ,121, UK Flood 377, , ,221, US Brushfire 12,526, ,772, ,016, Ausralian Terror 2,945, ,729, ,829, CBNR Only 1,995, , ,453, Cer. Terrorism xcbnr 3,961, ,099, ,975, Domesic Macro TR 648, , ,316, Europe Terror 4,512, ,431, ,859, Non Cerified Terror 2,669, , ,138, Casualy 5,745, ,622, ,651, N American Crop 21,467, ,885, ,869,

31 (b) Resriced Mehods Page 15 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion in he Conex of a CAT Reinsurer Difference in Convenional Allocaions (a) All Mehods (w/o Exp3)

32 Page 16 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion in he Conex of a CAT Reinsurer Premium Funcion Specificaion: For company i in year where: log{p i } = α + α β d i γ E i + ε i di is he defaul rae according o he leer raing (fied based on AM Bes Raings) Ei is he expeced loss (based on average ne loss and loss adjusmen expense raio) Esimaed from NAIC daa for Reinsurance Companies according o Reinsurance Associaion of America s annual review Resuls: Variable Coefficien Sd. Error -value Inercep (α) Defaul rae (β) Expeced Loss (γ) 1.48 E E Year dummies are omied. Observaions: 288. Adj. R 2 = 26%

33 Page 17 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion in he Conex of a CAT Reinsurer Paramerizaions Parameer 1 ( base case") 2 ( profiable company") 3 ( empy company") τ 3.00% 5.00% 5.00% c (1) % 7.50% 7.50% c (2) E E E-010 ξ 50.00% 75.00% 20.00% r 3.00% 6.00% 3.00% α β γ 1.48E E E-010

34 Page 17 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion in he Conex of a CAT Reinsurer Paramerizaions Parameer 1 ( base case") 2 ( profiable company") 3 ( empy company") τ 3.00% 5.00% 5.00% c (1) % 7.50% 7.50% c (2) E E E-010 ξ 50.00% 75.00% 20.00% r 3.00% 6.00% 3.00% α β γ 1.48E E E-010

35 Page 18 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion in he Conex of a CAT Reinsurer Base Case Resuls (I) 12 q_1(a) q_2(a) q_3(a) q_4(a) 10 8 q e+09 2e+09 3e+09 4e+09 5e+09 6e+09 a Opimal porfolio for caasrophe reinsurer

36 Page 19 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion in he Conex of a CAT Reinsurer Base Case Resuls (II) 5e+08 R(a) 0-5e+08-1e e+09 R -2e e+09-3e e+09-4e e e+09 2e+09 3e+09 4e+09 5e+09 6e+09 7e+09 a Opimal raising decision for caasrophe reinsurer

37 Page 20 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion in he Conex of a CAT Reinsurer Base Case Resuls (III) 1.96e+09 V(a) 0.12 V (a) 1.94e e e V 1.88e+09 V e e e e e+09 2e+09 3e+09 4e+09 5e+09 6e+09 7e e+09 2e+09 3e+09 4e+09 5e+09 6e+09 a a Value funcion and derivaive caasrophe reinsurer

38 Page 21 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion in he Conex of a CAT Reinsurer Base Case Resuls (III) zero capial opimal capial high capial a 0 1,000,000,000 4,000,000,000 V (a) 1,885,787,820 1,954,359,481 1,880,954,936 R(a) 311,998, ,926,420,812 q 1 (a) q 2 (a) q 3 (a) q 4 (a) S 550,597,000 1,406,761,416 2,615,202,661 D 1,493,490,910 2,349,655,327 3,558,096,571 E[I] 199,297, ,561, ,841,815 p (i) /E[i] P(I > a) % 2.66% 0.002% P(I > S) 4.54% 0.45% 0.13% P(I > D) 0.002% 0.002% 0.002% c 1 (Rb ) 13.74% 4.65% 0.00% ξ 1 ξ P(S D} ] 2.90% 3.18% 2.54%

39 Page 21 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion in he Conex of a CAT Reinsurer Base Case Resuls (III) zero capial opimal capial high capial a 0 1,000,000,000 4,000,000,000 V (a) 1,885,787,820 1,954,359,481 1,880,954,936 R(a) 311,998, ,926,420,812 q 1 (a) q 2 (a) q 3 (a) q 4 (a) S 550,597,000 1,406,761,416 2,615,202,661 D 1,493,490,910 2,349,655,327 3,558,096,571 E[I] 199,297, ,561, ,841,815 p (i) /E[i] P(I > a) % 2.66% 0.002% P(I > S) 4.54% 0.45% 0.13% P(I > D) 0.002% 0.002% 0.002% c 1 (Rb ) 13.74% 4.65% 0.00% ξ 1 ξ P(S D} ] 2.90% 3.18% 2.54%

40 Page 22 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion in he Conex of a CAT Reinsurer Cos Allocaions in Base Case a = 0 a = 1bn a = 4bn Acuarial Value of Solven Paymens, (i) 199,259, ,502, ,752,070 (E[I 1 {I D} ]) 78.00% 80.73% 84.81% Company Valuaion of Solven Paymen, (ii) 12,917,945 38,621-5,274,818 (E[I (w(i) 1) 1 {I D} ]) 5.06% 0.01% -0.94% Capial cos, (iii) 43,298,096 74,781,276 90,335,366 (D [E[w(I) 1 {I>D} ]) 16.95% 19.26% 16.14% agg. marginal cos, (i)-(iii) 255,475, ,322, ,812, % % %

41 Page 22 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion in he Conex of a CAT Reinsurer Cos Allocaions in Base Case a = 0 a = 1bn a = 4bn Acuarial Value of Solven Paymens, (i) 199,259, ,502, ,752,070 (E[I 1 {I D} ]) 78.00% 80.73% 84.81% Company Valuaion of Solven Paymen, (ii) 12,917,945 38,621-5,274,818 (E[I (w(i) 1) 1 {I D} ]) 5.06% 0.01% -0.94% Capial cos, (iii) 43,298,096 74,781,276 90,335,366 (D [E[w(I) 1 {I>D} ]) 16.95% 19.26% 16.14% agg. marginal cos, (i)-(iii) 255,475, ,322, ,812, % % %

42 Page 22 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion in he Conex of a CAT Reinsurer Cos Allocaions in Base Case a = 0 a = 1bn a = 4bn Acuarial Value of Solven Paymens, (i) 199,259, ,502, ,752,070 (E[I 1 {I D} ]) 78.00% 80.73% 84.81% Company Valuaion of Solven Paymen, (ii) 12,917,945 38,621-5,274,818 (E[I (w(i) 1) 1 {I D} ]) 5.06% 0.01% -0.94% Capial cos, (iii) 43,298,096 74,781,276 90,335,366 (D [E[w(I) 1 {I>D} ]) 16.95% 19.26% 16.14% agg. marginal cos, (i)-(iii) 255,475, ,322, ,812, % % %

43 Page 23 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion in he Conex of a CAT Reinsurer RAROC calculaions, Base Case Allocaing Risk Adjusmen Line 1 Line 2 Line 3 Line 4 a = 0 VaR Allocaion D yes 2.90% 2.90% 2.90% 2.90% VaR Allocaion D no 3.44% 3.74% 3.61% 4.03% VaR Allocaion S no 8.52% 9.68% 10.74% 11.85% VaR Allocaion a no na na na na VaR Allocaion D red. form 2.87% 2.87% 2.88% 2.88% a = 1bn VaR Allocaion D yes 3.18% 3.18% 3.18% 3.18% VaR Allocaion D no 3.14% 3.20% 3.20% 3.16% VaR Allocaion S no 6.52% 5.27% 5.27% 5.16% VaR Allocaion a no 10.58% 8.67% 8.05% 5.51% VaR Allocaion D red. form 3.20% 3.20% 3.20% 3.21% a = 4bn VaR Allocaion D yes 2.54% 2.54% 2.54% 2.54% VaR Allocaion D no 2.37% 2.40% 2.41% 2.37% VaR Allocaion S no 10.80% 2.44% 2.89% 5.77% VaR Allocaion a no 2.13% 2.65% 2.03% 2.32% VaR Allocaion D red. form 2.60% 2.62% 2.61% 2.62%

44 Page 23 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion in he Conex of a CAT Reinsurer RAROC calculaions, Base Case Allocaing Risk Adjusmen Line 1 Line 2 Line 3 Line 4 a = 0 VaR Allocaion D yes 2.90% 2.90% 2.90% 2.90% VaR Allocaion D no 3.44% 3.74% 3.61% 4.03% VaR Allocaion S no 8.52% 9.68% 10.74% 11.85% VaR Allocaion a no na na na na VaR Allocaion D red. form 2.87% 2.87% 2.88% 2.88% a = 1bn VaR Allocaion D yes 3.18% 3.18% 3.18% 3.18% VaR Allocaion D no 3.14% 3.20% 3.20% 3.16% VaR Allocaion S no 6.52% 5.27% 5.27% 5.16% VaR Allocaion a no 10.58% 8.67% 8.05% 5.51% VaR Allocaion D red. form 3.20% 3.20% 3.20% 3.21% a = 4bn VaR Allocaion D yes 2.54% 2.54% 2.54% 2.54% VaR Allocaion D no 2.37% 2.40% 2.41% 2.37% VaR Allocaion S no 10.80% 2.44% 2.89% 5.77% VaR Allocaion a no 2.13% 2.65% 2.03% 2.32% VaR Allocaion D red. form 2.60% 2.62% 2.61% 2.62%

45 Page 23 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion in he Conex of a CAT Reinsurer RAROC calculaions, Base Case Allocaing Risk Adjusmen Line 1 Line 2 Line 3 Line 4 a = 0 VaR Allocaion D yes 2.90% 2.90% 2.90% 2.90% VaR Allocaion D no 3.44% 3.74% 3.61% 4.03% VaR Allocaion S no 8.52% 9.68% 10.74% 11.85% VaR Allocaion a no na na na na VaR Allocaion D red. form 2.87% 2.87% 2.88% 2.88% a = 1bn VaR Allocaion D yes 3.18% 3.18% 3.18% 3.18% VaR Allocaion D no 3.14% 3.20% 3.20% 3.16% VaR Allocaion S no 6.52% 5.27% 5.27% 5.16% VaR Allocaion a no 10.58% 8.67% 8.05% 5.51% VaR Allocaion D red. form 3.20% 3.20% 3.20% 3.21% a = 4bn VaR Allocaion D yes 2.54% 2.54% 2.54% 2.54% VaR Allocaion D no 2.37% 2.40% 2.41% 2.37% VaR Allocaion S no 10.80% 2.44% 2.89% 5.77% VaR Allocaion a no 2.13% 2.65% 2.03% 2.32% VaR Allocaion D red. form 2.60% 2.62% 2.61% 2.62%

46 Page 24 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion o a Propery and Casualy Insurer Inroducion Profi Maximizaion and Marginal Cos of Risk Applicaion in he Conex of a CAT Reinsurer Applicaion o a Propery and Casualy Insurer Conclusion

47 Page 25 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion o a Propery and Casualy Insurer 2L2DY Model Implemenaion Line 1: long-ailed line wih wo developmen years Line 2: shor-ailed line wih no developmen beyond he acciden year Line 1 AY \ DY L (1) 1 L (1) 2 2 L (1) 1 Line 2 AY \ DY 1 1 L (2) 1 2 L (2) 1 Table: Losses for A Company under 2L2DY

48 Page 25 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion o a Propery and Casualy Insurer 2L2DY Model Implemenaion Line 1: long-ailed line wih wo developmen years Line 2: shor-ailed line wih no developmen beyond he acciden year Line 1 AY \ DY L (1) 1 L (1) 2 2 L (1) 1 Line 2 AY \ DY 1 1 L (2) 1 2 L (2) 1 Table: Losses for A Company under 2L2DY I = q (1) L (1) 1 + q (2) L (2) 1 + q (1) L (1) 1

49 Page 26 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion o a Propery and Casualy Insurer 2L2DY Model Implemenaion Addiional assumpions of 2L2DY: Chain-Ladder (Mack, 1993): E(L (1) 2 L(1) 1 ) = (f 1)L(1) 1 V(L (1) 2 L(1) 1 ) = σ2 L (1) 1 f and σ 2 are he chain ladder facors and can be esimaed using riangle Condiional Normaliy: L (n) 1 L (n) 1 N (µ (n) 1, (σ(n) 1 )2 ) n = 1, 2 L (1) 2 L(1) 1 N ((f 1)L (1) 1, σ2 L (1) 1 ) Linear Correlaion Beween Lines corr(l (1) 1, L(2) 1 ) = ρ

50 Page 27 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion o a Propery and Casualy Insurer 2L2DY Model Implemenaion Calibraion: Loss riangles from NAIC Schedule P daa from he represenaive company. Line 1: Workers Compensaion Line 2: Commercial Auo Worker s Comp Comm Auo f N/A σ e5 N/A µ (n) e e8 (σ (n) 1 ) e e14 ρ Esimaed Parameer Values for Loss Triangles Cos and reurn parameers: τ = 0.03, c (1) 1 = 0.075, c (2) 1 = 1.0e 10, and reurn of asse invesmen is r = 0.03 (Par 1)

51 Page 28 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion o a Propery and Casualy Insurer 2L2DY Model Implemenaion Choice of premium funcion: People assess company qualiy via raing ha reflecs defaul probabiliy Increasing he scale of insurance business decreases profi margins [ ] { } p n = E e r q (n) L (n) 1 + e 2r q (n) L (n) 2 1 {n=1} L 1 exp α βp(i > D) γe(i)

52 Page 28 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion o a Propery and Casualy Insurer 2L2DY Model Implemenaion Choice of premium funcion: People assess company qualiy via raing ha reflecs defaul probabiliy Increasing he scale of insurance business decreases profi margins [ ] { } p n = E e r q (n) L (n) 1 + e 2r q (n) L (n) 2 1 {n=1} L 1 exp α βp(i > D) γe(i) Calibraed using NAIC premium, expense and paid loss daa from 2004 o 2013, and defaul probabiliy from Bes s Impairmen Rae and Raing Transiion Sudy Parameers Coefficien Sd. Error -value p-value α *** β *** γ 3.24e e ** Observaions: Adj. R 2 : Esimaed Premium Parameers

53 Page 28 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion o a Propery and Casualy Insurer 2L2DY Model Implemenaion Choice of premium funcion: People assess company qualiy via raing ha reflecs defaul probabiliy Increasing he scale of insurance business decreases profi margins [ ] { } p n = E e r q (n) L (n) 1 + e 2r q (n) L (n) 2 1 {n=1} L 1 exp α βp(i > D) γe(i) Calibraed using NAIC premium, expense and paid loss daa from 2004 o 2013, and defaul probabiliy from Bes s Impairmen Rae and Raing Transiion Sudy Parameers Coefficien Sd. Error -value p-value α *** β *** γ 3.24e e ** Observaions: Adj. R 2 : Esimaed Premium Parameers * Solve he Bellman equaion Value Ieraion mehod coded in Julia.

54 Page 29 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion o a Propery and Casualy Insurer Opimal Value, Capial and Insurance Porfolio

55 Page 30 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion o a Propery and Casualy Insurer RAROC Calculaions small pr. loss large pr. loss Allocaion Line 1 Line 2 Line 1 Line 2 a = 0 Correc Allocaion 29.96% 29.96% 12.53% 12.53% w/o Afer Shock 27.55% 29.96% 26.20% 12.53% Ac. only 32.84% 32.92% 13.65% 14.11% a = 1, 000, 000, 000 Correc Allocaion 3.43% 3.43% 3.53% 3.53% w/o Afer Shock 9.54% 3.43% 8.80% 3.53% Ac. only 3.44% 5.47% 3.35% 5.08%

56 Page 30 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Applicaion o a Propery and Casualy Insurer RAROC Calculaions small pr. loss large pr. loss Allocaion Line 1 Line 2 Line 1 Line 2 a = 0 Correc Allocaion 29.96% 29.96% 12.53% 12.53% w/o Afer Shock 27.55% 29.96% 26.20% 12.53% Ac. only 32.84% 32.92% 13.65% 14.11% a = 1, 000, 000, 000 Correc Allocaion 3.43% 3.43% 3.53% 3.53% w/o Afer Shock 9.54% 3.43% 8.80% 3.53% Ac. only 3.44% 5.47% 3.35% 5.08%

57 Page 31 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Conclusion Inroducion Profi Maximizaion and Marginal Cos of Risk Applicaion in he Conex of a CAT Reinsurer Applicaion o a Propery and Casualy Insurer Conclusion

58 Page 32 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Conclusion Conclusion Capial allocaion can be and should be grounded in an economic conex

59 Page 32 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Conclusion Conclusion Capial allocaion can be and should be grounded in an economic conex Marginal cos of risk is complex: Capial coss only one piece of marginal cos of risk, need o consider all (cos) aspecs Take he form of risk-adjusmen on company s valuaion ("effecively risk averse")

60 Page 32 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Conclusion Conclusion Capial allocaion can be and should be grounded in an economic conex Marginal cos of risk is complex: Capial coss only one piece of marginal cos of risk, need o consider all (cos) aspecs Take he form of risk-adjusmen on company s valuaion ("effecively risk averse")

61 Page 33 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Conclusion Conac Daniel Bauer Universiy of Alabama USA Thank you!

62 Page 34 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Conclusion Deailed One Period Model Insurer s problem: max A,{q (i) } p (j) E[I 1 {I A} ] A P(I > A) c 1 (A) p (i) = E[I (i) ] exp{α β E[I] γp(i > A)} I (i) = q (i) L (i), I = I (j), A asses, c 1 ( ) cos Premium funcion: scale and risk effec (could be generalized, of course)

63 Page 34 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Conclusion Deailed One Period Model Insurer s problem: max A,{q (i) } p (j) E[I 1 {I A} ] A P(I > A) c 1 (A) p (i) = E[I (i) ] exp{α β E[I] γp(i > A)} I (i) = q (i) L (i), I = I (j), A asses, c 1 ( ) cos Premium funcion: scale and risk effec (could be generalized, of course) Two levers, exposure and capializaion. Can rade off: [ ] [ ] [ ] I (i) p (i) I (i) I (i) E (1 γe[i]) = E q (i) E[I (i) ] q 1 (i) {I A} + E I = A ( P(I > A) + c q (i) 1 }{{ (A)) }}{{}}{{}}{{} cap. cos marginal premium ac. value cap. alloc

64 Page 34 Arizona Acuarial Club Spring Meeing March 28, 2018 Bauer Conclusion Deailed One Period Model Insurer s problem: max A,{q (i) } p (j) E[I 1 {I A} ] A P(I > A) c 1 (A) p (i) = E[I (i) ] exp{α β E[I] γp(i > A)} I (i) = q (i) L (i), I = I (j), A asses, c 1 ( ) cos Premium funcion: scale and risk effec (could be generalized, of course) Two levers, exposure and capializaion. Can rade off: [ ] [ ] [ ] I (i) p (i) I (i) I (i) E (1 γe[i]) = E q (i) E[I (i) ] q 1 (i) {I A} + E I = A ( P(I > A) + c q (i) 1 }{{ (A)) }}{{}}{{}}{{} cap. cos marginal premium ac. value cap. alloc If company is risk-averse, need o hink abou "uiliy" U: Go Back [ max A,{q (i) } E U( ] p (j) I 1 {I A} A 1 {I>A} c 1 (A)) p (i) = E[I (i) ] exp{α β E[I] γp(i > A)} [ I (i) Marg. Prem = E q 1 U ] [ ] I (i) ( (i) {I A} + E U I = A E [1 ] ) E[U] q (i) {I>A} + c 1 E[U] (A) }{{}}{{} w(i) w(i)

Data-Driven Demand Learning and Dynamic Pricing Strategies in Competitive Markets

Data-Driven Demand Learning and Dynamic Pricing Strategies in Competitive Markets Daa-Driven Demand Learning and Dynamic Pricing Sraegies in Compeiive Markes Pricing Sraegies & Dynamic Programming Rainer Schlosser, Marin Boissier, Mahias Uflacker Hasso Planer Insiue (EPIC) April 30,