Optimal long-term management of a P/C insurance portfolio ... · Stochastic optimization model...

P/C economicsStochastic optimization model

StochasticsDSP interfaces and implementation

Case study and numerical evidencesInsurance portfolio stress-testing

Conclusions

Optimal long-term management of a P/Cinsurance portfolio with endogenous risk

control

Giorgio Consigli1

Vittorio Moriggia, Lorenzo Mercuri, Elena BenincasaMichele Gaffo 2, Massimo di Tria, Davide Musitelli, Luca Valagussa

1University of Bergamo2Allianz Investment Management

PhD AEB Seminar 20-05-2014G. Consigli P/C ALM via dynamic stochastic optimization




Conclusions

Outline

1 P/C economics

2 Stochastic optimization model

3 Stochastics

4 DSP interfaces and implementation

5 Case study and numerical evidences

6 Insurance portfolio stress-testing

7 Conclusions

G. Consigli P/C ALM via dynamic stochastic optimization




Conclusions

Premiums, reserves and P/C liabilities

Figure : Stylized annual P/C business drivers





Conclusions

Investment management

We focus on a P/C portfolio management problem formulated as adynamic stochastic program under an extended set of financial,technical and regulatory (Solvency II compliance) constraints.

The investment manager faces a 10 year decision horizon and seeksan optimal trade-off between short, medium and long-term goals. Anextended asset universe is considered including liquid and illiquidassets.

The insurance reserves are treated as exogenous liabilities by theportfolio manager. A partial asset-liability duration matching ispursued.





Conclusions

Risk assessment

The technical and investment divisions must comply with risk capitalrequirements determined by an independent Risk managementdivision.The company total risk capital is defined as the sum of the actuarialand the investment risk capital.A maximum investment risk exposure in the form of ascenario-dependent default boundary must be satisfied by theportfolio manager.A multistage stochastic programming approach is adopted to capturethe key elements of the investment problem in presence of technicalas well as regulatory constraints.





Conclusions

Risk exposure

Figure : Overall risk exposure





Conclusions

References

Cariño D.R., Kent T., Myers D.H., Stacy C., Sylvanus M., Turner A.L.,Watanabe K. and Ziemba W.T. (1994): The Russel-Yasuda-Kasaimodel: An asset-liability model for a Japanese insurance companyusing multistage stochastic programming Interfaces 24(1)

John M Mulvey (2001): Multi-Period Stochastic Optimization Models forLong-term Investors, Quantitative Analysis in Financial Markets (vol 3),(M. Avellaneda, ed.), 2001 World Scientific Publishing Co., Singapore

Gaivoronski A.A., Hoyland K. and de Lange P.E. (2001): Statutoryregulation of casualty insurance companies: an example from Norwaywith stochastic programming analysis. In Stochastic optimization:algorithms and applications (Gainesville, FL, 2000), volume 54 of Appl.Optim., Kluwer Acad. Publ





Conclusions

References

de Lange P.E., Fleten S-E. and Gaivoronski A. (2003): Modelingfinancial reinsurance in the casualty insurance business via stochasticprogramming. PhD thesis, Department of Industrial Economics andTechnology Management, N.T.N.U., Trondheim (NO)

John M Mulvey and H. Erkan (2005): Decentralized Risk Managementfor Global Property and Casualty Insurance Companies, in Applicationsof Stochastic Programming, Siam Publication

John M Mulvey and W Kim (2007): The Role of Alternative Assets forOptimal Portfolio Construction, in Encyclopedia of Risk Assessment,John Wiley and Sons

John M Mulvey (2007): Dynamic Financial Analysis for MultinationalInsurance Companies in Handbook of Asset and Liability Management,Vol. 2, (with W. Pauling, S. Britt, F. Morin) S.A. Zenios and W.T. ZiembaEds





Conclusions

References

G.Consigli, M. di Tria, M.Gaffo, G.Iaquinta, V.Moriggia and A.Uristani(2011): Dynamic portfolio management for property and casualtyinsurance. Handbook on Stochastic Optimization Methods in Financeand Energy, Bertocchi, Consigli and Dempster Eds. Fred HillierInternational Series in Operations Research and Management Science,Springer (US)

G.Consigli and M. di Tria (2012): Optimal long-term Property andCasualty ALM with risk capital control, Asset-Liability Management forFinancial Institutions, Swarup B. Ed., QFinance Series Bloomsbury Pbl(London)





Conclusions

Notation and model variables

For t ∈ T , a discrete and finite time space, n ∈ Nt are nodallabels with children n+ and parent node n−. We denote by a(n)the sequence of all ancestors for any final or intermediate nodeand with c(n), the subtree originating from n. tn is the time ofnode n

Rn are P/C premiums, Ln the insurance claims and Cn theassociated operational costs in nodes n. Γn := Ln+Cn

Rndenotes

the combined ratio

Λn are insurance reserves allocated at time tn for all future claims

Πfn is the cumulative investment profit at node n, while Πt

n is thecumulative profit generated by the technical division





Conclusions

Notation and model variables

K tn is the technical risk capital and K f

n the investment risk capitalestimated at node n. Kn = K f

n + K tn defines the node n company

risk exposure

We assume K tn = Λn × κt where κt is a predetermined constant

risk multiplier

xi,m,n are portfolio holdings of asset i , node n, bought inm ∈ a(n). X−n =

∑i∑

m x−i,m,n and X +n =

∑i x+

i,n are sellings andbuyings. Xi,n =

∑m∈a(n) xi,m,n and Xn =

∑i Xi,n. We denote with

A the asset universe.





Conclusions

Short, medium and long-term risk-adjusted returns

For t ∈ T , n ∈ Nt

V 1n = (Πf

n + Πtn) is the sum of technical and financial profit before

taxes

V 2n = (Πf

n − Φn)φ− κK fn defines the surplus investment value

generated by the portfolio manager and

From V 1 we denote with V 3n = V 1

n φ the company cumulativeprofit after taxes and for given RoRAC target z we denote withV 3 = zKn the target company return.

Φn is a theoretical cost of funding faced by the investment manager:Φn =

∑m∈n− Λmζ

−n (tn − tn−) + Φn−. The three targets are assumed to

be set at the 1, 3 and 10 year horizon respectively.





Conclusions

In extreme summary the optimization problem is:

maxx∈X

(1− α)∑

j

λjE [V jn|Σn]− α

∑j

λjE [V j − V jn|Σn]

(1)

s.t for all n ∈ Nt , t ∈ T

Xn = Xn−(1 + ρn) + X +n − X−n + zn (2)

zn = X−n − X +n + Rn − Ln − Cn + Xn−ξn + zn−(1 + rn) (3)

Πfn = Xn−ξn +

∑m∈a(n)

X−m,nγm,n + Πfn− (4)

X +n + X−n = ϑXn−(1 + ρn) (5)

K fn ≤ Xn − Λn − K t

n (6)∑i

X inli ≤ l (Xn) (7)





Conclusions

Investment risk capital

K fn = K f1

n + K f2n (8)

K f1n =

(∆A

n− −∆Λn−)

drn(tn − tn−) + K f1n− (9)

K f2n =

√∑i∈A

∑j∈A

Xi,n−Xj,n−kij (tn − tn−) + K f2n− (10)

where ∆An =

∑i∑

m∈a(n) xi,m,n∆i,n and ∆Λn =

∑T>tn Λn,T × (t − tn) are

the asset and liability durations in node n. K f1 and K f2 define theinterest rate and market risk exposure of the portfolio.





Conclusions

Correlations and investment risk exposure

In (8), kij = ki · kj · ρij , where ki is the i-th asset risk-charge and ρij isthe correlation factor between assets i and j .

We consider three cases: ρ = I for independent risk factors, ρ := ρija Solvency II compliant matrix and ρ := 1 a correlation matrix of all 1for perfect positive correlation.The following relationships hold ∀n ∈ Nt , t ∈ T and are taken intoaccount in the results’ validation:

K f2n (I) ≤ K f2

n (ρ) ≤ K f2n (1) (11)





Conclusions

ρ = 1

For ρ = 1, kij is symmetric with diagonal elements k2i and

off-diagonals kikj . Thus:

K f2n =

√∑i∈A

∑j∈A

Xi,n−Xj,n−kij (tn − tn−) (12)

=

√∑i∈A

(Xi,n−ki )2(tn − tn−) (13)

=∑i∈A

Xi,n−ki (tn − tn−) (14)

The expression for K f2n with ρ = 1 becomes linear.





Conclusions

ρ = I

From equation (14) we also have:∑

i∈A (Xi,n−ki )2>∑

i∈A

(X 2

i,n−k2i

),

then K f2n (1) > K f2

n (I). Finally for 0 < ρij < 1, ∀i , j we have∑i∈A

(X 2

i,n−k2i)<∑i∈A

∑j∈A

Xi,n−Xj,n−kij <∑i∈A

(Xi,n−ki )2

.





Conclusions

SLP, SQCP, SOCPThe stochastic program (1) to (7)is solved under alternativespecifications of the objectve function:

For K fn = K f

m +[K f1

n +∑

i X imki (tn − tm)

], the multistage

stochastic linear program is solved with CPLEX dual simplexmethod

For K fn = K f

m +[K f1

n +∑

i∑

j X imX j

mkij (tn − tm)]

we adopt theCPLEX QCP solver

For K fn = K f

m +

[K f1

n +

√∑i

(∑j X j

maij (tn − tm))2]

we use

CONOPT solver. Here aij is the generic ij-th element of theCholeski factorisation of the risk matrix

Under any of the above after the solution we recover the originalinvestment risk capital formulation and analyse the solution output.G. Consigli P/C ALM via dynamic stochastic optimization




Conclusions

Scenario generation

Scenario generation is required to determine the tree process for allthe random coefficients in equations (1) to (7).The liability model includes the inflation-adjusted claims (for givennominal premiums) and the reserves.The asset model deals with asset price and cash returns. For the firsttime in this application we have interfaced a proprietary CorporateEconomic scenario generator, adopted by the company asforecasting tool, with a dedicated scenario generator constructedprimarily to deal with investment returns on alternatives, real estateand commodity.





Conclusions

Liability scenariosP/C liabilities are inflation adjusted and their time evolution dependson the P/C policy renewals. We distinguish an ongoing from a run-offbusiness assumption. In the latter premiums are assumed not to berenewn after say 1 year and the reserves will accordingly decay overtime.

Figure : Run off liability profile





Conclusions

Asset return model

The asset universe includes a set of investment opportunities, whoseprice ρ and cash returns ξ must be specified and input to theoptimization problem:

Barcap Treasury 1-3, 3-5, 5-7, 7-10 and 10+ year maturity

10 year Securitised, IG and SG Corporate Bond indices

5 and 10 year inflation-linked fixed income indices

MSCI EMU (Public) Equity index and Private equity index

Indirect GPR Europe real estate index and Direct real estate

Infrastructure Cyclical and Defensive investments (100 years)

Renewable energy and Commodities





Conclusions

Asset returns

Figure : Scenario module





Conclusions

Asset returns

Figure : infrastructures and commodity model





Conclusions

Asset returns

Figure : Real estate model





Conclusions

Asset returns

Figure : Private equity





Conclusions

Generic implementation framework

Figure : Implementation frameworkG. Consigli P/C ALM via dynamic stochastic optimization




Conclusions

Case study

We present a set of results focusing on:

a comparison between the static 1 year and the dynamic model

the generation and solution time for stochastic programs of increasingdimension under the different solution approaches

the investment risk capital evolution under different risk factorcorrelation assumptions

the associated first stage and scenario dependent optimal allocationstrategies

the targets attainability results

optimal risk-adjusted performance surfaces as a function of the weights

stress testing results on the performance measures





Conclusions

AIM view1

Figure : former AIM approachG. Consigli P/C ALM via dynamic stochastic optimization




Conclusions

AIM view2

Figure : current approachG. Consigli P/C ALM via dynamic stochastic optimization




Conclusions

Case study

Figure : Case study Dynamic P/C portfolio optimization





Conclusions

Hardware: HP proprocessor: AMD A4-3420 APU 2.8 GHzCpu: 1 physical processor - 2 cores - 2 logical processorsRam: 4 GBO/S: Windows 8 Pro 64bHdisk: 450 GB





Conclusions

Dynamic versus static, Ongoing versus Run-off P/Canalysis

Figure : Static vs dynamic 1y return surplus





Conclusions

Results

Table : P/C problem dimension and solution time

SCENARIO TREE 64 256tree structure 2-2-2-2-2 4-4-2-2-2-2Scenario Gen CPU time (secs) 79.578 204DetEqv MPS file dimension LP SQP NLP LP SQP NLPSolver CPLEX LP CPLEX QCP CONOPT CPLEX LP CPLEX QCP CONOPTrows 31125 31251 32763 123399 123899 129899columns 43484 43610 45122 172812 173312 179312coeff non zero 226219 226345 229873 1064102 1064602 1078602rows (after presolve) 7031 8399 27760 33144columns (after presolve) 9378 11133 37075 43973coefficients (after presolve) 102571 108115 406437 428199numbinaries na na na nanum quadr constraints na 418 na 1603MPS (CPU time, secs) 1.326 1.809 2.418 11.092 12.215 20.591Solution time (CPU time secs) 1.467 1.965 2.824 11.95 13.089 22.448





Conclusions

Table : P/C problem dimension and solution time

SCENARIO TREE 512 768tree structure 4-4-4-2-2-2 6-4-4-2-2-2Scenario Gen CPU time (secs) 308 463DetEqv MPS file dimension LP SQP NLP LP SQP NLPSolver CPLEX LP CPLEX QCP CONOPT CPLEX LP CPLEX QCP CONOPTrows 242855 243835 255595 364213 365683 383323columns 341260 342240 354000 511820 513290 530930coeff non zero 2591364 2592344 2618803 4475177 4476647 4517807rows (after presolve) 54874 65108 75015 90348columns (after presolve) 72957 86097 104312 124277coefficients (after presolve) 790322 835597 1210776 1278791numbinaries na na na nanum quadr constraints na 3091 na 4903MPS (CPU time, secs) 47.627 52.837 98.031 201 202.115 293.906Solution time (CPU time secs) 50.622 55.786 96.862 210 218.183 332.985





Conclusions

Risk-adjusted performance

Figure : Risk control and risk-adjusted performanceG. Consigli P/C ALM via dynamic stochastic optimization




Conclusions

SLP – Optimal strategy

Figure : SLP 768 0502030 HN and optimal strategy along the meanscenario





Conclusions

SQP – Optimal strategy

Figure : SQP 768 0502030 HN and optimal strategy along the meanscenario





Conclusions

SOCP – Optimal strategy

Figure : SOCP 512 0502030 HN and optimal strategy along themean scenario





Conclusions

Default boundary and targets achievement

Figure : SLP Targets achievement likelihhod and risk capital hedge





Conclusions


Figure : SQP Targets achievement likelihhod and risk capital hedge





Conclusions


Figure : SOCP Targets achievement likelihhod and risk capital hedge





Conclusions

Short-medium-long term trade-offs

Figure : Trade off analysis and IVC





Conclusions

Stress-testing analysis

The following open questions are addressed: to which extent a condition ofincreasing trouble in the core insurance business needs to be compensatedby a relaxation of the risk capital constraints? Furthermore: in presence of atrade-off between short, medium and long-term targets in the decision model,how would the targets attainability be affected by deteriorating technicalratios?

Consider the following definition of the investment risk capital upper bound.As χ increases above 1 the risk constraint is relaxed:

K fn ≤ χ · [Xn − Λn − K t

n]. (15)





Conclusions

Stress-testing analysis

Let Υ(ls, χs, 1,Ω) denote the stochastic program solution for given lossratio ls, risk capital coefficient χs, α = 1 in (1) and fixed number offinancial and insurance scenarios Ω.

We analyse Υ(ls, χs, 1,Ω) for different input values ls = 0.52, 0.56,0.59, 0.63, 0.66, 0.7 and χs = 0.6, 0.72, 0.84, 0.96, 1.08, 1.2 on asequence of problem instances with fixed goals V 2 and Z on the IVCand RoRAC respectively.





Conclusions

Optimal risk capital allocation

Figure : Investment risk capital requirements for different correlationmatrices ζ





Conclusions

Stress-testing analysis 1

Figure : IVC and loss ratio-risk capital trade-offG. Consigli P/C ALM via dynamic stochastic optimization




Conclusions


Consider V 2 first: the IVC will deteriorate if, for given income taxes andexogenous cost of capital, the portfolio manager will be unable to pushportfolio net returns up without increasing the portfolio cost of capital.

From Figure 22 as the loss coefficient increases the IVC at the threeyear horizon decreases: the derivative is negative and decreasingrapidly as the loss ratio reaches 0.65.

On the other hand V 2 appears relatively independent from the allocatedrisk capital, suggesting that to achieve the specified target, as the lossratio deteriorates, the portfolio manager won’t find convenient to moveinto riskier strategies, which would require incremental investment riskcapital.





Conclusions


Figure : RoRAC and loss ratio-risk capital trade-offG. Consigli P/C ALM via dynamic stochastic optimization




Conclusions


We can again evaluate the dependence of the RoRAC on the loss ratio andthe risk tolerance coefficient through the Matlab interactive interface: the 3Dsurface can be fitted in this case with a hyperplane with equationZ = 0.864K f − 0.55l , with estimated first order partial derivatives∂Z∂l = −0.55 and ∂Z

∂K f = 0.864.Contrary to the previous case, the relaxation of the investment risk capitalconstraint is in this case exploited to compensate the stress generated byincreasing losses.





Conclusions

The need by the Insurance company of a unified approach tocore insurance business, investment management and riskmanagement has motivated the present development

The dynamic stochastic programming approach allows also thecombination of short medium and long-term goals, making1-year static optimization no longer practical

The mathematical model combines accounting and regulatoryequations wth the emerging financial and risk-managementphilosophy in the insurance business

The extension of the planning horizon makes possible throughappropriate cash-flow models to incorporate alternativeinvestments in the asset universe





Conclusions

The combination of capital allocation constraints andrisk-adjusted performance measurement supporting trade-offand stress-testing analysis had a relevant impact on thecompany strategic planning potential

Increasing emphasis goes into solution analysis

Scenario generation over a long term horizon is critical to theefficiency of the suggested optimal policy


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