An Application of Optimisation & Simulation for Solvency II
By Vojo Bubevski (vojo.bubevski@landg.com)
12 April, 2011
Abstract
The Solvency II regulations are fundamentally redesigning the capital adequacy regime for European
(re)insurers and will be effective from 31 October 2012. The Solvency II objectives are to establish a
set of EU-wide capital requirements and risk management standards replacing the current Solvency I
requirements.
Solvency II establishes two levels of capital requirements: i) Minimal Capital Requirement (MCR), i.e. the
threshold below which the authorization of the (re)insurer shall be withdrawn; and ii) Solvency Capital
Requirement (SCR), i.e. the threshold below which the (re)insurer will be subject to a much higher
supervision. The SCR should deliver a level of capital that enables the (re)insurer to absorb significant
unforeseen losses over a specified time horizon. It should cover, at a minimum, insurance, market,
credit and operational risks, corresponding to the Value-at-Risk (VAR) of the (re)insurer’s own basic
funds, subject to a confidence level of 99.95% over a one-year period.
Solvency II offers two options for calculating SCR, i.e. by applying either: i) a standard model, which will
be provided by the regulator; or ii) an internal model, which will be developed by (re)insurers. A
standard model cannot consider the company’s specific factors, thus the SCR will be higher. In
contrast, an internal model results in a lower SCR as all the (re)insurer’s specific factors are
considered. Therefore, Solvency II offers capital-reduction incentives to insurers that invest in
developing best practices in risk management and control.
Basic principles of an internal model are presented from practical aspect. The model uses Palisade’s
@RISK® and RISKOptimizer®, i.e. Optimization and Monte Carlo simulation, to calculate the VAR
(SCR). It is applied on real market data. It can help (re)insures to reduce their SCR (VAR) providing
higher underwriting capabilities and increasing their competitive position, which is their ultimate
objective.
12 April, 2011
1
Agenda
•
•
•
•
•
•
•
•
•
•
Introduction
Solvency II
Risk
Risk Models & Risk Metrics
Solvency II Standard Model – An Example
Solvency II Internal Model – Examples using Palisade @RISK® &
RISKOptimizer®
Models’ Results Comparison
Conclusion
References
Questions & Answers
12 April, 2011
2
Introduction:
• Insurance Companies: Economy & competition put pressure to apply more risky
strategies to gain higher returns;
• Supervisory Authorities: Demand for guaranties for the policy holders and solvency of
the re(insurers);
• Solvency I (January 2004): Solvency capital is derived deterministically from the
premiums and losses not considering the companies’ individual business profiles and their
specific asset portfolios;
• Credit Crunch (2007-2008): Failure of institutions to appreciate and manage the interconnection between the risks inherent in their business;
• Observation: Current risk & decision management methods are insufficient;
• Need for Change: New risk & decision management methods are required;
• Value-Based Management: Strategic & tactical decisions based on rate of return and
objective risk assessment considering the correlation between the risks and identifying the
sources of risks;
• Solvency II (November 2012): New level of supervisory requirements challenging every
aspect of the insurance businesses and promoting Value-Based-Management;
• Major Solvency II Activities: Asset Management, Actuarial Practice, Underwriting Policy,
Reinsurance Policy, Controlling, Internal Revision, Planning and IT;
12 April, 2011
3
Solvency II: An Overview
• Objectives:
–
–
–
–
Establish new supervisory system combining quantitative & qualitative methods (Three Pillars
Approach)
Provide for assessment of overall solvency of insurance companies/groups based on risksensitive approach;
Give additional incentive for proper risk management;
Ensure consistency between financial sectors;
• Pillar 1 – Establishes Quantitative Requirements for insurance companies:
–
–
Minimal Capital Requirement (MCR): The threshold below which the authorization of the
(re)insurer shall be withdrawn;
Solvency Capital Requirement (SCR): The threshold below which the (re)insurer will be subject
to a much higher supervision. The SCR should deliver a level of capital that enables the
(re)insurer to absorb significant unforeseen losses over a specified time horizon. It should
cover, at a minimum, insurance, market, credit and operational risks, corresponding to the
Value-at-Risk (VAR) of the (re)insurer’s own basic funds, subject to a confidence level of
99.95% over a one-year period.
• Pillar 2 – Institutes Qualitative Control of insurance companies;
• Pillar 3 – Found Rules on Supervisory & Public disclosure (reporting).
12 April, 2011
4
Solvency II: An Overview…
• Purposes of Solvency Capital Requirement:
–
–
–
–
–
–
Provides a guarantee fund to cover possible losses;
Motivates companies to avoid undesirable level of risk;
Promotes a risk measurement and risk management culture within a company;
Provides a supervisors’ tool to assume control over failing or failed companies;
Alerts supervisors to emerging market trends;
Ensures that the insurance portfolio of a failed insurer can be transferred to another provider
with a higher certainty.
• Fulfillment of Solvency Capital Requirement:
–
The sum of company’s available assets must exceed the sum of technical liabilities . Not all the
assets of the company are allowed to be considered in this calculation;
• Standard Model – Uniform model for EU to calculate SCR using generic methods:
–
i) Provided by the regulator; ii) Applies generic assumptions; iii) Conservative as specific
company’s spectrum of relevant risks is not considered; iv) Results in a higher SCR.
• Internal Model – Created by the company’s Risk Management Department:
–
i) Accounts for the specific company’s spectrum of relevant risks and their correlations; ii)
Applies specific company’s assumptions; iii) Offers better and more reliable information on the
company’s solvency situation; iv) Reduces SCR; v) Must not be less prudent than the standard
model; vi) Must be validated and approved by the competent authorities.
12 April, 2011
5
Risk
• A Definition:
–
Risk is the chance of something happening that will have an impact on objectives. It is measured in terms of
consequences and likelihood.
• Risk – The core of Insurance Business:
–
–
Insurance Contract is a classic example of Risk Transfer;
Individuals, companies and governments protect themselves from various risks such as death, disability,
property damage due to a natural disaster, illegal actions, accident, unexpected expenses as result of business
interruption, legal proceedings, bad debt, etc.
• Additional Insurance Business Common Risks:
–
–
–
Company specific risks, which arise from individual company activities;
Systematic risks, which are shared whole (re-)insurance industry;
Systemic risks, which arise from changes in the general economic conditions and concern all industries;
• Solvency II Risk Classification:
–
–
–
–
–
Underwriting Risk: Specific insurance risk arising from the underwriting of insurance contracts;
Credit Risk: Risk of default and change in the credit quality of the issuers of securities, counterparties and
intermediaries;
Market Risk: Risk arising from the level or volatility of the market prices of financial instruments;
Operational Risk: Risk of loss resulting from inadequate or failed internal processes, people, systems or from
external events;
Liquidity Risk: Exposure to loss in the event that insufficient liquid assets will be available.
• Impact of four quantifiable risks on the economic capital of insurance companies:
–
i) Investment ALM Risk (Market Risk): 64%; ii) Operating Risk: 27%; iii) Credit Risk: 5%; iv) Insurance Risk: 4%.
12 April, 2011
6
Risk Models and Risk Metrics (VAR & Tail VAR)
• Risk Models Classification:
–
Scenario Base Models: Imply measuring the impact of specific scenarios on the total Profit &
Loss Distribution. A scenario is as a complete alternate state of the world (e.g. major financial
crisis, earthquakes, windstorms, etc.);
– Static Factor Models: The risk capital calculation is based on a linear combination of static
factors (i.e. risk weights) multiplied with company specific size measures;
– Covariance Models: The calculation of resulting portfolio value (also SCR) is based on the
sensitivity of Risk-Based Capital to changes in determined risk factors;
– Stochastic Models: Advanced risk models including economical scenario generators, which
produce a pseudorandom realization of the risk factors. The risk capital is determined by
applying a risk measure to the total Profit & Loss Distribution.
• Value-at-Risk (VAR): The most used metrics (Harry Markowitz, “Portfolio Selection”, 1952);
– VAR is the predicted worst-case loss at a specific confidence level over a certain period of time
based on probability distribution. It is characterized by two parameters, i.e. time horizon and a
percentile. α-percentile is the probability of realization which is below α. I.e. α is reverse to the
Confidence Level (α = 100% - Confidence Level). For Solvency II, the Confidence Level is
99.5%, thus α = 0.5%.
• Tail Value-at-Risk (Tail VAR), also referred to as Expected Shortfall:
– An alternative risk metrics which is defined as the mean of the outcomes that exceed VAR level.
Tail VAR returns a greater loss figure than the VAR and in the context of Solvency II will require
respectively larger solvency capital. It provides a higher Confidence Level.
12 April, 2011
7
Standard Model: An Example of Market Risk Model (ALM)
• Swiss Solvency Test (SST) Model (i.e. SST Asset Model):
–
–
–
• SST
–
–
–
A risk-factor based Covariance Model, which considers the effects caused by changes in the
market factors on assets and liabilities;
Calculations are based on the market value of the assets and liabilities, normally distributed risk
factors, linear correlation between risk factors and Tail VAR;
Includes 23 risk factors: Discrete term interest rate structure (9 factors), Implied volatility of
interest rates, Exchange rates of CHF (four factors incl. EUR, GBP, USD & JPY), Implied
volatility of FX rates, Share price index, Private equity, Hedge funds, Participations, Other
equity, Implied volatility of share price index, Property and Credit spread;
Asset Model Process:
Inference (proscribed by regulator / modeled by companies): i) Modeling of individual risk
factors; ii) Derivation of a joint distribution of the risk factors;
Mapping (performed by companies): i) Derivation of the sensitivities of Assets & Liabilities to
individual risk factors; ii) Derivation of the total sensitivity of RBC (i.e. SCR) to individual risk
factors;
Transformation (proscribed by regulator): i) Derivation of joint distribution of RBC deltas; ii)
Application of the Tail VAR.
• Major Limitations of Swiss Solvency Test (SST) Model (i.e. SST Asset Model):
–
–
–
–
No differentiated approach to investments (e.g. rating based, previous performance, etc);
Linear dependency between the risk factors and the RBC is assumed;
Volatilities of the risk factors are proscribed deterministically;
Approximated Tail VAR is applied at the end on the results of the deterministic modelling.
12 April, 2011
8
Internal Market Risk Model (ALM): Fund 1 Return Distribution
Fit Comparison for Ln(1 + Fund 1 Return)
RiskBetaGeneral(0.33128,0.44435,-0.00017875,0.0036737)
-0.163
7000
3.576
5.0%
10.8%
90.0%
79.3%
5.0%
9.8%
6000
Input
5000
Minimum -0.000179
Maximum
0.00367
Mean
0.00152
Std Dev
0.00162
Values
47
4000
3000
BetaGeneral
2000
Minimum -0.000179
Maximum
0.00367
Mean
0.00147
Std Dev
0.00143
1000
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
0
Values in Thousandths
12 April, 2011
9
Internal Market Risk Model (ALM): Fund 2 Return Distribution
Fit Comparison for Ln(1 + Fund 2 Return)
RiskBetaGeneral(0.44456,0.52559,-0.00068717,0.0035115)
-0.659
2500
3.340
5.0%
7.3%
90.0%
81.9%
5.0%
10.8%
2000
Input
Minimum -0.000687
Maximum
0.00351
Mean
0.00127
Std Dev
0.00162
Values
47
1500
BetaGeneral
1000
Minimum -0.000687
Maximum
0.00351
Mean
0.00124
Std Dev
0.00149
500
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
0
Values in Thousandths
12 April, 2011
10
Internal Market Risk Model (ALM): Fund 3 Return Distribution
Fit Comparison for Ln(1 + Fund 3 Return)
RiskLogistic(0.0086857,0.029917)
-0.0925
5.0%
3.3%
10
0.0798
90.0%
88.2%
5.0%
8.5%
9
8
Input
7
Minimum -0.1506
Maximum
0.0836
Mean
0.00426
Std Dev
0.0537
Values
47
6
5
Logistic
4
Minimum
−∞
Maximum
+∞
Mean
0.00869
Std Dev
0.0543
3
2
1
0.15
0.10
0.05
0.00
-0.05
-0.10
-0.15
-0.20
0
12 April, 2011
11
Internal Market Risk Model (ALM): Fund 4 Return Distribution
Fit Comparison for Ln(1 + Fund 4 Retrn)
RiskLogistic(0.0074862,0.029965)
-0.0941
5.0%
3.3%
9
0.0786
90.0%
88.2%
5.0%
8.5%
8
7
Input
Minimum -0.1518
Maximum
0.0827
Mean
0.00305
Std Dev
0.0538
Values
47
6
5
4
Logistic
Minimum
−∞
Maximum
+∞
Mean
0.00749
Std Dev
0.0544
3
2
1
0.15
0.10
0.05
0.00
-0.05
-0.10
-0.15
-0.20
0
12 April, 2011
12
Internal Market Risk Model (ALM): Fund 5 Return Distribution
Fit Comparison for Ln(1 + Fund 5 Return)
RiskLogistic(0.0062831,0.030012)
-0.0956
5.0%
3.2%
10
0.0775
90.0%
88.2%
5.0%
8.5%
9
8
Input
7
Minimum -0.1530
Maximum
0.0818
Mean
0.00184
Std Dev
0.0538
Values
47
6
5
Logistic
4
Minimum
−∞
Maximum
+∞
Mean
0.00628
Std Dev
0.0544
3
2
1
0.15
0.10
0.05
0.00
-0.05
-0.10
-0.15
-0.20
0
12 April, 2011
13
Internal Market Risk Model (ALM): Minimal SCR – Maximal VAR
Maximal VAR Portfolio ‐ Minimal Solvency Capital Requirement
Fund 1
Fund 2
Monthly Return
Fund 3
0.001466673
Fund 4
Fund 5
Total
0.001236822
0.0086857
0.0074862
0.0062831
Monthly Sigma
0.001450572
0.001513639
0.053405725
0.053379087
0.053265154
Yearly Return
0.017600079
0.014841863
0.1042284
0.0898344
0.0753972
Yearly Sigma
0.005024929
0.0052434
0.185002858
0.184910583
0.184515904
Asset Allocation
0.113424523
0.129607724
0.62518024
0.037091935
0.094695577
1
Correlation
Ln(1 + F1)
Ln(1 + F1)
Ln(1 + F2)
1
Ln(1 + F3)
Ln(1 + F4)
Ln(1 + F5)
0.992694444
‐0.243339016
‐0.24329726
‐0.243263357
Ln(1 + F2)
0.992694444
1
‐0.223112875
‐0.222717225
‐0.222332623
Ln(1 + F3)
‐0.243339016
‐0.223112875
1
0.999995559
0.999982439
Ln(1 + F4)
‐0.24329726
‐0.222717225
0.999995559
1
0.999995632
Ln(1 + F5)
‐0.243263357
‐0.222332623
0.999982439
0.999995632
1
Portfolio Return 0.07955335
Portfolio Return Mean
0.106196776
Portfolio Variance
0.234397662
Portfolio Sigma
0.484146323
Portfolio VAR (0.5%)
>=
0.08
8%
‐0.170381837
Percentile (X for given P) (Best Simulation) = -0.1704
12 April, 2011
14
Internal Market Risk Model (ALM): Minimal SCR – Maximal VAR
RISKOptimizer: Optimization Summary
Performed By: Vojo
Date: 06 March 2011 23:36:42
Model: Max VAR Portfolio.xlsx
Goal
Cell to Optimize
Statistic to Optimize
Statistic Parameter
Type of Goal
'The Model'!$D$18
Percentile (X for given P)
0.005
Maximum
12 April, 2011
15
Internal Market Risk Model (ALM): Minimal SCR – Maximal VAR
12 April, 2011
16
Internal Market Risk Model (ALM): Minimal SCR – Maximal VAR
12 April, 2011
17
Internal Market Risk Model (ALM): Minimal SCR – Maximal VAR
12 April, 2011
18
Internal Market Risk Model (ALM): Maximal Mean Return
Maximal Mean Return Portfolio
Fund 1
Monthly Return
Monthly Sigma
Yearly Return
Yearly Sigma
Asset Allocation
SigmaY*AsstAllocn
Fund 2
Fund 3
Fund 4
Fund 5
Total
0.001466673
0.001236822
0.0086857
0.0074862
0.0062831
0.001426431
0.001487721
0.054472553
0.054794804
0.054685974
0.017600079
0.014841863
0.1042284
0.0898344
0.0753972
0.004941303
0.005153617
0.188698458
0.189814768
0.18943777
0.003254566
0.016745434
0.9
0.04
0.04
1.60818E‐05
8.62995E‐05
0.169828612
0.007592591
0.007577511
1
Correlation
Ln(1 + F1)
Ln(1 + F1)
Ln(1 + F2)
Ln(1 + F3)
Ln(1 + F4)
Ln(1 + F5)
Ln(1 + F2)
Ln(1 + F3)
Ln(1 + F4)
Ln(1 + F5)
1
0.992694444
‐0.243339016
‐0.24329726
‐0.243263357
0.992694444
1
‐0.223112875
‐0.222717225
‐0.222332623
‐0.243339016
‐0.223112875
1
0.999995559
0.999982439
‐0.24329726
‐0.222717225
0.999995559
1
0.999995632
‐0.243263357
‐0.222332623
0.999982439
0.999995632
1
Portfolio Return Portfolio Return Mean
Portfolio Variance
Portfolio Sigma
Portfolio VAR (0.5%)
Risk Free Rate (assumed 5%)
Portfolio Sharpe Ratio
0.100720638
Mean (Best Simulation) = 0.1229
0.122856705
0.410571771
0.640758746
‐0.763815244
0.05
0.1137038
12 April, 2011
19
Internal Market Risk Model (ALM): Maximal Sharpe Ratio
Maximal Sharpe Ratio Portfolio
Fund 1
Fund 2
Fund 3
Fund 4
Fund 5
Total
0.001466673
0.001236822
0.0086857
0.0074862
0.0062831
0.001447794
0.001511091
0.053255056
0.053184966
0.053230668
0.017600079
0.014841863
0.1042284
0.0898344
0.0753972
0.005015305
0.005234574
0.184480924
0.184238127
0.184396443
0.228655293
0.018298648
0.692479157
0.058632855
0.001934047
0.001146776
9.57856E‐05
0.127749195
0.010802407
0.000356631
Monthly Return
Monthly Sigma
Yearly Return
Yearly Sigma
Asset Allocation
SigmaY*AsstAllocn
1
Correlation
Ln(1 + F1)
Ln(1 + F1)
Ln(1 + F2)
Ln(1 + F3)
Ln(1 + F4)
Ln(1 + F5)
1
0.992694444
‐0.243339016
‐0.24329726
‐0.243263357
Portfolio Return Portfolio Return Mean
Portfolio Variance
Portfolio Sigma
Portfolio VAR (0.5%)
Risk Free Rate (assumed 5%)
Portfolio Sharpe Ratio
Ln(1 + F2)
Ln(1 + F3)
Ln(1 + F4)
Ln(1 + F5)
0.992694444
‐0.243339016
‐0.24329726
‐0.243263357
1
‐0.223112875
‐0.222717225
‐0.222332623
‐0.223112875
1
0.999995559
0.999982439
‐0.222717225
0.999995559
1
0.999995632
‐0.222332623
0.999982439
0.999995632
1
0.081885001
0.107600683
0.230716614
0.480329693
Result (Best Simulation) = 0.1199
‐0.201416194
0.05
0.119919055
12 April, 2011
20
Internal Market Risk Model (ALM): Minimal Downside Risk
Minimal Downside Risk Portfolio (Min Target(0.08))
Fund 1
Fund 2
Fund 3
Fund 4
Fund 5
Total
Monthly Return
0.001466673
0.001236822
0.0086857
0.0074862
0.0062831
Monthly Sigma
0.001426431
0.001487721
0.054472553
0.054794804
0.054685974
Yearly Return
0.017600079
0.014841863
0.1042284
0.0898344
0.0753972
Yearly Sigma
0.004941303
0.005153617
0.188698458
0.189814768
0.18943777
Asset Allocation
0.003254566
0.016745434
0.9
0.04
0.04
SigmaY*AsstAllocn
1.60818E‐05
8.62995E‐05
0.169828612
0.007592591
0.007577511
1
Correlation
Ln(1 + F1)
Ln(1 + F2)
Ln(1 + F3)
Ln(1 + F4)
Ln(1 + F5)
Ln(1 + F1)
1
0.992694444
‐0.243339016
‐0.24329726
‐0.243263357
Ln(1 + F2)
0.992694444
1
‐0.223112875
‐0.222717225
‐0.222332623
Ln(1 + F3)
‐0.243339016
‐0.223112875
1
0.999995559
0.999982439
Ln(1 + F4)
‐0.24329726
‐0.222717225
0.999995559
1
0.999995632
Ln(1 + F5)
‐0.243263357
‐0.222332623
0.999982439
0.999995632
1
Portfolio Return 0.100720638
Portfolio Return Mean
0.122856705
Portfolio Variance
0.034215907
Portfolio Sigma
0.184975423
Portfolio VAR (0.5%)
‐0.763815244
Target (P for given X) (Best Simulation) = 0.4727
12 April, 2011
21
Internal Market Risk Model (ALM): Minimal Probability of Loss
Minimal Probability of Loss Portfolio (Minimal Target(0))
Fund 1
Fund 2
Monthly Return
0.001466673
Monthly Sigma
0.001444208
Yearly Return
0.017600079
Yearly Sigma
0.005002884
Asset Allocation
0.01
SigmaY*AsstAllocn
5.00288E‐05
Fund 3
0.001236822
0.001490878
0.014841863
0.005164553
0.002770611
1.4309E‐05
Fund 4
0.0086857
0.05525772
0.1042284
0.191418357
0.9
0.172276521
Fund 5
0.0074862
0.055364661
0.0898344
0.191788811
0.04
0.007671552
Total
0.0062831
0.055570209
0.0753972
0.192500851
0.047229389
0.009091698
1
Correlation
Ln(1 + F1)
Ln(1 + F1)
Ln(1 + F2)
Ln(1 + F3)
Ln(1 + F4)
Ln(1 + F5)
1
0.992694444
‐0.243339016
‐0.24329726
‐0.243263357
Ln(1 + F2)
Ln(1 + F3)
Ln(1 + F4)
Ln(1 + F5)
0.992694444
‐0.243339016
‐0.24329726
‐0.243263357
1
‐0.223112875
‐0.222717225
‐0.222332623
‐0.223112875
1
0.999995559
0.999982439
‐0.222717225
0.999995559
1
0.999995632
‐0.222332623
0.999982439
0.999995632
1
Portfolio Return Portfolio Return Mean
Portfolio Variance
Portfolio Sigma
Portfolio VAR (0.5%)
0.101177022
0.104563916
0.035730163
0.189024238
‐0.99488519
Target (P for given X) (Best Simulation) = 0.4091
12 April, 2011
22
Internal Market Risk Model (ALM): Minimal Risk – St. Deviation
Minimal Risk Portfolio ‐ Minimal Standard Deviation
Fund 1
Fund 2
Fund 3
Fund 4
Fund 5
Total
Monthly Return
0.001466673
0.001236822
0.0086857
0.0074862
0.0062831
Monthly Sigma
0.001429882
0.001491334
0.053664559
0.053587252
0.053744215
Yearly Return
0.017600079
0.014841863
0.1042284
0.0898344
0.0753972
Yearly Sigma
0.004953258
0.005166133
0.185899487
0.185631687
0.186175421
0.13429812
0.057712464
0.485270702
0.02901306
0.293705654
0.000665213
0.00029815
0.090211575
0.005385743
0.054680774
Asset Allocation
SigmaY*AsstAllocn
1
Correlation
Ln(1 + F1)
Ln(1 + F1)
Ln(1 + F2)
1
Ln(1 + F3)
Ln(1 + F4)
Ln(1 + F5)
0.992694444
‐0.243339016
‐0.24329726
‐0.243263357
Ln(1 + F2)
0.992694444
1
‐0.223112875
‐0.222717225
‐0.222332623
Ln(1 + F3)
‐0.243339016
‐0.223112875
1
0.999995559
0.999982439
Ln(1 + F4)
‐0.24329726
‐0.222717225
0.999995559
1
0.999995632
Ln(1 + F5)
‐0.243263357
‐0.222332623
0.999982439
0.999995632
1
Portfolio Return 0.078550162
Portfolio Return Mean
0.088692491
Portfolio Variance
0.022515637
Portfolio Sigma
Portfolio VAR (0.5%)
0.150052115
‐0.386396248
Standard Deviation (Best Simulation) = 0.1501
12 April, 2011
23
Internal Market Risk Model (ALM): Minimal Variance
Minimal Variance Portfolio
Fund 1
Monthly Return
Monthly Sigma
Yearly Return
Yearly Sigma
Asset Allocation
SigmaY*AsstAllocn
Fund 2
0.001466673
0.00142988
0.017600079
0.004953251
0.267078683
0.001322908
Fund 3
0.001236822
0.001489523
0.014841863
0.005159859
0.026966994
0.000139146
Fund 4
0.0086857
0.053368073
0.1042284
0.184872426
0.650903743
0.120334154
Fund 5
Total
0.0074862
0.0062831
0.053448075
0.053506146
0.0898344
0.0753972
0.185149562
0.185350728
0.017714614
0.037335967
0.003279853
0.006920249
1
Correlation
Ln(1 + F1)
Ln(1 + F2)
Ln(1 + F3)
Ln(1 + F4)
Ln(1 + F5)
1
0.992694444
‐0.243339016
‐0.24329726
‐0.243263357
0.992694444
1
‐0.223112875
‐0.222717225
‐0.222332623
‐0.243339016
‐0.223112875
1
0.999995559
0.999982439
‐0.24329726
‐0.222717225
0.999995559
1
0.999995632
‐0.243263357
‐0.222332623
0.999982439
0.999995632
1
Ln(1 + F1)
Ln(1 + F2)
Ln(1 + F3)
Ln(1 + F4)
Ln(1 + F5)
Portfolio Return Portfolio Return Mean
Portfolio Variance
Portfolio Sigma
Portfolio VAR (0.5%)
0.077349911
0.083904635
0.016949151
0.130188904
‐0.196435206
Variance (Best Simulation) = 0.0169
12 April, 2011
24
Comparison of Portfolios Vs Maximal VAR Portfolio
Portfolio
Mean Return
Mean Return Delta
VAR
VAR Delta
Maximal VAR
0.106196776
0
-0.170381837
0
Maximal Mean Return
0.122856705
0.016659929
-0.763815244
-0.593433407
Maximal Sharpe Ratio
0.107600683
0.001403907
-0.201416194
-0.031034357
Minimal Downside Risk
0.122856705
0.016659929
-0.763815244
-0.593433407
Minimal Probability of Loss
0.104563916
-0.00163286
-0.99488519
-0.824503353
Minimal Risk
0.088692491
-0.017504285
-0.386396248
-0.216014411
Minimal Variance
0.083904635
-0.022292141
-0.196435206
-0.026053369
12 April, 2011
25
Conclusion: Major Points to Emphasise
• Solvency II Standard Model:
– Provided by the regulator;
– Uniform across EU to calculate SCR using generic methods;
– The risk models are deterministic, not stochastic;
– Applies generic assumptions;
– Specific company’s spectrum of relevant risks is not considered;
– It is conservative and results in a higher SCR;
• Solvency II Internal Model:
– Created by the company’s Risk Management Department;
– Applies specific company’s assumptions and accounts for the specific company’s
spectrum of relevant risks and their correlations;
– Offers better and more reliable information on the company’s solvency situation
reducing SCR;
– Must not be less prudent than the standard model;
– Must be validated and approved by the competent authorities.
• Microsoft™ Excel® and Palisade™ @RISK® & RISKOptimizer® were used;
• Presented RISKOptimizer® Models are advanced stochastic models using optimization
and simulation;
• Presented models can help (re)insures to reduce their SCR (VAR) providing higher
underwriting capabilities and increasing their competitive position, which is their ultimate
objective.
• .
12 April, 2011
26
References
1.Cruz, Marcelo, The Solvency II Handbook, Risk Books, London, 2009.
2.Winston, Wayne, Decision Making Under Uncertainty, Palisade
Corporation, Ithaca NY, 2010.
3.Rusalovskiy, Artem, Challenges of Solvency II Implementation, VDM
Verlag Dr. Müller, Germany, 2008.
12 April, 2011
27
Questions & Answers
12 April, 2011
28
Thank You
12 April, 2011
12 April, 2011
29