P/C portfolios stochastic optimizationInvestment risk exposure
Case study and numerical evidencesConclusions
Optimal capital allocation and strategicportfolio selection for a large P/C insurer
Giorgio Consigli1
Vittorio Moriggia, Lorenzo Mercuri,Massimo di Tria2
1University of Bergamo2Allianz Investment Management
ICSP2013: MS Dynamic Stochastic Optimization andFinance 08-07-2013
G. Consigli P/C ALM via dynamic stochastic optimization
P/C portfolios stochastic optimizationInvestment risk exposure
Case study and numerical evidencesConclusions
Outline
1 P/C portfolios stochastic optimization
2 Investment risk exposure
3 Case study and numerical evidences
4 Conclusions
G. Consigli P/C ALM via dynamic stochastic optimization
P/C portfolios stochastic optimizationInvestment risk exposure
Case study and numerical evidencesConclusions
Introduction
The financial environment for global property and casualty (P/C) insurers haschanged dramatically in recent years mainly due to:
record P/C claims,
historically low fixed income and real estate returns,
ongoing introduction of regulatory, risk-based, capital requirements(Solvency II)
increasing market competition.
Jointly or individually those pressures have determined increasing liquiditydeficits in core P/C activity, stressed technical and financial scenarios andstimulated integration between insurance, investment and risk managementdivisions of insurance corporations.
G. Consigli P/C ALM via dynamic stochastic optimization
P/C portfolios stochastic optimizationInvestment risk exposure
Case study and numerical evidencesConclusions
Operational focus
We analyse the specific problem of an investment manager subject totechnical and maximum risk exposure constraints employing a dynamicportfolio strategy over a 10 year horizon. Key to the analysis:
the introduction of risk-adjusted performance measures as medium andlong-term investment targets,
the enlargement of the investment universe to include alternativeinvestments,
the control of risk capital exposure over time,
the sensitivity of short, medium and long term goals to stressedinsurance and financial scenarios.
G. Consigli P/C ALM via dynamic stochastic optimization
P/C portfolios stochastic optimizationInvestment risk exposure
Case study and numerical evidencesConclusions
References
Relevant previous contributions in the area include the seminal workby Cariño, Ziemba et al (1994): The Russel-Yasuda-Kasai model,Mulvey (2001): Multi-Period Stochastic Optimization Models forLong-term Investors, Gaivoronski A.A. et al (2001), de Lange P.E.,Fleten S-E. and Gaivoronski A. (2003), Mulvey and H. Erkan (2005)on Decentralized Risk Management for Global Property and CasualtyInsurance Companies, Mulvey (2007): Dynamic Financial Analysis forMultinational Insurance Companies.
G. Consigli P/C ALM via dynamic stochastic optimization
P/C portfolios stochastic optimizationInvestment risk exposure
Case study and numerical evidencesConclusions
Asset and liability scenarios
Uncertainty is modeled through scenario trees. For given tree structure,random price ρ and cash returns ξ for all asset classes are generatedthrough a set of econometric models and exogenously by a corporateeconomic scenario generator.The asset universe includes
fixed-income asset classes: Treasury 1-3, 3-5, 5-7, 7-10 and 10+ yearmaturity indices; 10 year Securitised, IG and SG Corporate Bondindices; 5 and 10 year inflation-linked fixed income indices;
equity assets: MSCI EMU (Public) Equity index and Private equity;
real estates: indirect GPR Europe real estate index and Direct realestate;
alternative investments: Infrastructure Cyclical and Defensive;Renewable energy and Commodities.
All insurance liabilities are inflation adjusted and generated exogenously bythe company insurance division.
G. Consigli P/C ALM via dynamic stochastic optimization
P/C portfolios stochastic optimizationInvestment risk exposure
Case study and numerical evidencesConclusions
Objective function
For t ∈ T , n ∈ Nt the following profit variables are considered in the objectivefunction:
V 1n = (Πf
n + Πtn) is the sum of technical and financial operating profits
before taxes – the operating profit
V 2n = (Πf
n − Φn)φ− κK fn defines the surplus investment value – the IVC
– generated by the portfolio manager and
From V 1 we denote with V 3n = V 1
n φ the company cumulative profit aftertaxes and for given RoRAC target z we denote with V 3 = zKn the targetcompany return.
Φn represents the cost of funding faced by the investment manager. Threetargets Vj for j = 1, 2, 3 are set by the management at the 1, 3 and 10 yearhorizon respectively.
G. Consigli P/C ALM via dynamic stochastic optimization
P/C portfolios stochastic optimizationInvestment risk exposure
Case study and numerical evidencesConclusions
Optimization problem
We solve the following stochastic program
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)
G. Consigli P/C ALM via dynamic stochastic optimization
P/C portfolios stochastic optimizationInvestment risk exposure
Case study and numerical evidencesConclusions
Key P/C assumptions and company risk capital
P/C premiums Rn are treated as exogenous and driven by marketcompetition. The investment manager does not have a reinsurance option.Furthermore:
Ln the insurance claims and Cn the associated operational costs areinflation adjusted.
Γn := Ln+CnRn
denotes a combined ratio monitored by the company toassess the insurance business efficiency
Λn are statutory inflation-adjusted reserves allocated at time tn
K tn denotes the actuarial risk capital to hedge against unexpected
losses from the core P/C business. We assume K tn = Λn × κt where κt
is a predetermined constant risk multiplier
K fn the investment risk capital estimated at node n. Kn = K f
n + K tn
defines the node n company global risk exposure
G. Consigli P/C ALM via dynamic stochastic optimization
P/C portfolios stochastic optimizationInvestment risk exposure
Case study and numerical evidencesConclusions
Investment risk capital
K fn = K f1
n + K f2n (8)
K f1n =
(∆A
n− −∆Λn−
)drn(tn − tn−) + K f1
n− (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 ∆Λ
n =∑
T>tnΛn,T × (t − tn) are the
asset and liability durations in node n. K f1 and K f2 define the interest rate andmarket risk exposure of the portfolio.K f
n := Xn − Λn − K tn represent a maximum risk exposure for the portfolio
manager whose aim is to maximise the portfolio expected return per unit riskcapital.
G. Consigli P/C ALM via dynamic stochastic optimization
P/C portfolios stochastic optimizationInvestment risk exposure
Case study and numerical evidencesConclusions
Correlations
In (8), kij = ki · kj · ρij , where ki is the i-th asset risk-charge and ρij is thecorrelation between assets i and j .
We consider three cases: ρ = I for independent risk factors, ρ := {ρij} aSolvency II compliant matrix and ρ := 1 a correlation matrix of all 1 for perfectpositive correlation.The following relationships hold ∀n ∈ Nt , t ∈ T and are taken into account inthe results’ validation:
K f2n (I) ≤ K f2
n (ρ) ≤ K f2n (1) (11)
The equation for K f2n with ρ = 1 becomes linear.
G. Consigli P/C ALM via dynamic stochastic optimization
P/C portfolios stochastic optimizationInvestment risk exposure
Case study and numerical evidencesConclusions
SLP, SQCP
The stochastic program (1) to (7) is solved under alternative specifications ofthe objectve function:
For K fn = K f
m +[K f1
n +∑
i X imki (tn − tm)
], the SLP deterministic
equivalent is solved with CPLEX dual simplex method
For K fn = K f
m +[K f1
n +∑
i
∑j X i
mX jmkij (tn − tm)
]we adopt the CPLEX
QCP solver. After solution we recover the risk capital original equationand analyse the risk capital dynamics and associated risk-adjustedreward measures.
G. Consigli P/C ALM via dynamic stochastic optimization
P/C portfolios stochastic optimizationInvestment risk exposure
Case study and numerical evidencesConclusions
Case study
We present a set of results focusing on:
the generation and solution times of a 768 scenario problems
the first stage implementable decisions under the SLP and SQCPformulations
the investment risk capital optimal absorption under different risk factorcorrelation assumptions along specific scenarios
the key variables sensitivity to increasing P/C loss ratios and risk capitalconstraints
the trade-off between portfolio liquidity and alternative investments
the targets attainability under the SLP and SQP approaches
The results have been collected on a HP workstation with Intel(R) Corei3-3220-CPU at 3.30Ghz processor and 5 Gb of RAM running Windows 8O.S. and 453 GB of hard disk.
G. Consigli P/C ALM via dynamic stochastic optimization
P/C portfolios stochastic optimizationInvestment risk exposure
Case study and numerical evidencesConclusions
Case study
We analyse a 768 scenario problem with 1, 3 and 10 years targets foroperating profit, IVC and RoRAC. The investment policy is constrainexd by aminimum 30% investment in Treasuries and maximum 50% investment inTIPS and corporates, 25% on equity, 20% on alternatives and commodities.A maximum 30% turnover is allowed from stage to stage. We present resultsfor α = 1 in the objective function.
The model has been instatiated in GAMS with scenarios and GUI and mainprogram in Matlab. We generate a deterministic equivalent in Gams andselect the solver for the large-scale MPS file from GAMS. As mentionedCPLEX solution algorithms have been adopted.
G. Consigli P/C ALM via dynamic stochastic optimization
P/C portfolios stochastic optimizationInvestment risk exposure
Case study and numerical evidencesConclusions
Results
Table : P/C problem dimension and solution time
SCENARIO TREE 768tree structure 6-4-4-2-2-2Scenario Gen CPU time (secs) 463DetEqv MPS file dimension SLP SQPSolver CPLEX LP CPLEX QCProws 364213 365683columns 511820 513290coeff non zero 4475177 4476647rows (after presolve) 75015 90348columns (after presolve) 104312 124277coefficients (after presolve) 1210776 1278791num quadr constraints – 4903MPS (CPU time, secs) 201 202.115Solution time (CPU time secs) 210 218.183
G. Consigli P/C ALM via dynamic stochastic optimization
P/C portfolios stochastic optimizationInvestment risk exposure
Case study and numerical evidencesConclusions
Optimal portfolio strategy
Figure : Optimal portfolios – linear and quadratic 768 scenario SPs
G. Consigli P/C ALM via dynamic stochastic optimization
P/C portfolios stochastic optimizationInvestment risk exposure
Case study and numerical evidencesConclusions
Default boundary and targets achievement
Figure : SLP Targets achievement likelihood and risk capital hedge
G. Consigli P/C ALM via dynamic stochastic optimization
P/C portfolios stochastic optimizationInvestment risk exposure
Case study and numerical evidencesConclusions
Default boundary and targets achievement
Figure : SQP Targets achievement likelihhod and risk capital hedge
G. Consigli P/C ALM via dynamic stochastic optimization
P/C portfolios stochastic optimizationInvestment risk exposure
Case study and numerical evidencesConclusions
Optimal risk capital allocation
Figure : Risk capital bounds – linear and quadratic SPsG. Consigli P/C ALM via dynamic stochastic optimization
P/C portfolios stochastic optimizationInvestment risk exposure
Case study and numerical evidencesConclusions
Sensitivity to loss ratios and risk capital limits
Figure : 768 scenario SLP optimal loss ratio-risk capital trade-offG. Consigli P/C ALM via dynamic stochastic optimization
P/C portfolios stochastic optimizationInvestment risk exposure
Case study and numerical evidencesConclusions
Portfolio liquidity and alternative investments
Figure : 768 scenario SLP optimal AI portfolios and liquidityG. Consigli P/C ALM via dynamic stochastic optimization
P/C portfolios stochastic optimizationInvestment risk exposure
Case study and numerical evidencesConclusions
The need by the Insurance company of a unified approach to coreinsurance business, investment management and risk management hasmotivated the present development
The mathematical model combines accounting and regulatoryequations wth the emerging financial and risk-management philosophyin the insurance business
The combination of capital allocation constraints and risk-adjustedperformance measurement supporting trade-off and stress-testinganalysis had a relevant impact on the company strategic planningpotential
Increasing emphasis goes into solution analysis
Scenario generation over a long term horizon is critical to the efficiencyof the suggested optimal policy
G. Consigli P/C ALM via dynamic stochastic optimization
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