From Principle-Based Risk Management to Solvency ... - Scor

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Hence, basis risk arises when the actual liability cash flows exceed the expected cash flows, i.e. when the discounted reserves are too small. The expression (2.6) demonstrating balance sheet solvency for year i in an economic balance sheet becomes Vi+1(ERPi+1) − Vi+1(ERPi) =E[Ri+1 |Fi+1] − E[Ri+1 |Fi], where Ri denotes the discounted outstanding reserves at time i, Ri := � pv (j+1/2→i)(Xj). (2.10) j≥i Assuming that we use a law-invariant risk measure ρ to capture the default option, the formula (2.9) for the amounts Ki available at the start of year i, determined at time t ≤ i, can then be written as Ki = ρ (pv (i+1/2→i)(Xi − E[Xi |Fi]) (2.11) + pv (i+1→i)(E[Ri+1 |Fi+1] − E[Ri+1 |Fi]) ), where the distributions of the involved random variables are conditional on the information available at time t ≤ i. Because Xi is known at time i +1, we have Xi = E[Xi |Fi+1], which implies that (2.11) for the amounts Ki can be written Ki = ρ(Mi), where Mi denotes the random variable of the mismatch Mi = E[Ri |Fi+1] − E[Ri |Fi], conditional on the information available at time t ≤ i. The mismatches Mi can be written as the change of the estimated ultimate amounts over a one-year period Mi = E[R0 |Fi+1] − E[R0 |Fi], (2.12) because the cash flows Xj for j =0...i− 1areknown at times t = i, i +1. Considering the cash flow of the corresponding one-year max cover, the expected value of the mismatch Mi conditional on the information at time t ≤ i is equal to zero. That is, the ERP is an unbiased optimal replicating portfolio. Consequently, the expected value of one-year max cover cash flow is equal to the coupon payment. In case we use a cost of capital margin approach, the expected return on the capital ˜ Ki for year i is then equal to the required return if the required return is provided by the coupon payment. Denoting the required return by η, the coupon face values ci are equal to ci = η ˜ Ki. 24
2.2 SST Valuation 2.2.1 Market Value Margin and Solvency Capital Requirement In SST terminology, the value of the basis risk is called the market value margin (MVM), and is equal to the difference between the market-consistent value of the liabilities and the value of the ORP. Denoting the portfolio of liabilities by L, the market-consistent value of the liabilities Vt(L) attimet is given by Vt(L) =V 0 t (L)+MVMt(L), where V 0 t (L) = Vt(ORPt) = value of the optimal replicating portfolio, MVMt(L) = market value margin = value of the basis risk. The SST Solvency Condition The basic idea of the SST to determine required regulatory capital is the following: A firm is considered solvent at time t = 0 if, at time t =1, with sufficiently high probability, the market(-consistent) value of the assets exceeds the market-consistent value of the liabilities, i.e. V1(A1) ≥ V1(L1). (2.13) Let us quickly comment on this solvency condition. Since we have expressed the value of the liabilities by means of a (super-)replicating portfolio, condition (2.13) guarantees that such a (super-)replicating portfolio can be purchased at time t = 1 by converting the assets to the (super-)replicating portfolio. This then ensures a regular run-off of the liabilities, which implies that the obligations towards the policyholders can be fulfilled 7 . It is assumed that the conversion of the assets can be achieved instantly. The SST approach is based on the cost of capital margin, and so the corresponding super-replicating portfolios have to be considered. These require capital, whose size depends on whether the first or the second strategy for the one-year max covers is selected (see Section 2.1.3) 8 . The SST is based on the second strategy, and thus the super-replicating portfolio for the liabilities at time t =1isgivenby � an unbiased optimal replicating portfolio ORP1 7 I.e. the policyholder’s claims can be paid, except when the default option is exercised. However, the default option is an integral part of the contract between policyholder and (re)insurance company. 8 Such capital would typically be provided by another (re)insurance company which takes over the run-off of the liabilities. 25
Page 1 and 2: From Principle-Based Risk Managemen
Page 3: Table of Contents Prefaces iv Ackno
Page 6 and 7: I shall not go into the different c
Page 8 and 9: cate the capital necessary for the
Page 10 and 11: and gave their support to this long
Page 12 and 13: NAME FUNCTION COMPANY Schindler Chr
Page 14 and 15: approach. The major risk factor cat
Page 16 and 17: force”), the volatility clusterin
Page 18 and 19: Part IX gives an outlook on the cur
Page 20 and 21: Table of Contents 1 Preliminaries 8
Page 22 and 23: 3.9.4 Dependency Structure and Aggr
Page 24 and 25: List of Tables 3.1 The portfolio de
Page 26 and 27: 1 Preliminaries Financial instrumen
Page 28 and 29: 2.1 Basics of Valuation 2.1.1 Valua
Page 30 and 31: has been removed. In the option exa
Page 32 and 33: Different types of “imperfect rep
Page 34 and 35: only depends on the law, i.e. the d
Page 36 and 37: For this reason, we define an unbia
Page 38 and 39: value of the assets has to exceed t
Page 40 and 41: educes the amounts Ki and thus the
Page 44 and 45: � the default option � sharehol
Page 46 and 47: Note that the set Ω⋆ 1 exclude t
Page 48 and 49: In view of the solvency condition (
Page 50 and 51: that year. The change in the estima
Page 52 and 53: The “stand-alone” shortfalls ES
Page 54 and 55: Further, we propose to calculate th
Page 56 and 57: We propose the following treatment
Page 58 and 59: In pricing, on the other hand, the
Page 60 and 61: We distinguish within the non-life
Page 62 and 63: To be more precise, a model is cons
Page 64 and 65: conditional on the information avai
Page 66 and 67: not for run-off business. The propo
Page 68 and 69: � Dependency between new, prior-y
Page 70 and 71: Run-Off Business By definition, run
Page 72 and 73: 3.2.4 New Business, Prior-Year Busi
Page 74 and 75: � Run-off business model for the
Page 76 and 77: � Basket specific models: In thes
Page 78 and 79: Agribusiness (θ =0.2673) Marine (
Page 80 and 81: Basket Specific Models Basket speci
Page 82 and 83: is reasonable and, in particular, c
Page 84 and 85: Credit and Surety (Section 3.6, pp.
Page 86 and 87: An alternative approach is used, fo
Page 88 and 89: Historical losses are “as if” a
Page 90 and 91: In general, the structure of a rein
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� Measures of profitability such
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affected airlines respectively manu
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Note that for the exposure loss mod
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3.7.4 Simulation Currently it is no
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with the modeling of the hazard, th
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3.8.2 Hazard Module Wind Windstorm
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ical source zones as fault, area or
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exposure retrieval and preparation,
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areas. Catrader performs analyzes o
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� paid external expenses (brokera
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� underwriting years k =0, 1 ...,
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� Volatility in the Mack model co
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the mean-squared error for the one-
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3.9.5 Data Requirements and Paramet
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where we might take ai =(1+θ) i fo
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4 Life and Health Methodology Our d
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4.2 GMDB Definition and Exposure Th
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4.3 Finance Re 4.3.1 Definition and
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are almost not affected by changes
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Fluctuation risk should be seen in
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Probability 10 x 10-4 Number of cla
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� Geography � Product types �
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� Determination of the frequency
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� Industry accident (contaminatio
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scenarios used, the same economic s
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4.7.1 Determining a Set of Replicat
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need to be evaluated is small compa
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Asset Strike Number of assets Europ
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Figure 4.4: Surface plot of GMDB ca
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Figure 4.7: GMDB with no maximum fo
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Standard deviation 400 000 350 000
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formula both including and excludin
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5 Dependency Structure of the Life
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6 Mitigation of Insurance Risk This
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We compute the two parameters which
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usiness written in the first year i
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For multi-location policies the qua
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8 Appendix 8.1 Portfolio Dependency
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Z1,1 Z1,1,1 Z1,1,2 Z1 Z1,2 Z1,3 Z Z
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To this probability, suitability (8
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insurance losses gives the insured
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Moreover, the copula C is given by
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continuous margins F1 and F2 and ex
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The third fallacy pertains to corre
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function C :[0, 1] d → [0, 1] has
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The d-dimensional Clayton copula is
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The two stochastic variables X and
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3. Use Equation (8.19) to compute t
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usiness and type of business, the e
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Further approaches to capital alloc
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(Xi)i=1,...,m, so for the net prese
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e a bounded, monotonously decreasin
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Definition 8.5 The composition C⋆
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in the portfolio. According to (8.5
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Now assume that H0 is an arbitrary
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Definition 8.13 For a copula C, the
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consequently, the total capital cos
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The time factor τX1+X2 of the sum
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where the probability measure GX,Z
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II Market Risk 192
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10.3.8 Interaction and Expectation
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List of Tables 12.1 Approximation f
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9 Introduction 9.1 Economic Risks i
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vations by means of statistical res
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the details needed in any concrete
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lation of future time intervals. Wh
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10.2 Bootstrapping - the Method and
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value 3 : xj = Ej−1[xj]+Ii (10.4)
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generation of the underlying econom
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this behavior conforms to theory an
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Calibrating the parameters of a GAR
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where u is a uniformly distributed
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� Do we use the same random varia
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example from 15 March 2007 to 15 Ju
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This is Equation (10.3) applied to
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The solid curve shows the mapping o
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as no uncertainty on the outcome re
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� Convert all ¯zj(T ) to the sim
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is to subtract the mean of all hist
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the basis of a study of the behavio
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10.3.11 Equity Indices Many applica
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Historical behaviors are reproduced
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this. The discrepancy Φi(x) − ˆ
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ate 15 against the USD, CPI and GDP
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10.5 Conclusion Refined bootstrappi
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we do not choose a time constant ex
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11 Interest Rate Hedges Under the d
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a year), then the coupon C (annuali
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11.5.2 Notes Inputs: � Notional (
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ment bonds. Portfolios of bonds are
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these are uniformly distributed ove
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� The initial market value of the
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FX rate, CTA acc init . In the Mell
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and the target MTM (= ¯ T ) which
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The coupon rate Cn of the new bond
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12.9 Cash Flow from or to the Portf
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In a multi-period simulation, the e
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The sum of all book values equals t
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calculations. The yield y ′ i of
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After the external cash flow and th
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investment. The two main categories
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gains and losses is U acc final = U
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of individual time steps as simple
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The book value in accounting curren
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13.7 Accounting Results in Local Cu
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Another useful variable is the tota
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study of that rate, we use an avera
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14.2 Timing: The Simulation Period
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2. The final asset component which
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15 Foreign Exchange Risk 15.1 FX Ri
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This is the factor we apply to esti
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III Credit Risk 296
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18.5 Modeling Credit Risks of a Rei
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17 Introduction 17.1 Credit Risk in
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Modeling credit spreads and default
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isks are the most important ones, s
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18.2 Variables to Describe Credit R
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slowly reacting variable subject to
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(18.3) by using some standard value
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The likely yield Y (T ) of a bond i
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of sector rotation. The market some
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Now the dynamic model for X/C can b
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and used by practitioners and which
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observed dependence between credit
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19 Limitations of the Credit Risk M
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Table of Contents 20 Operational Ri
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4. Incident Management, i.e. detect
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V The Economic Balance Sheet 328
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List of Tables 25.1 The classes of
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22 Aggregating Risk Inputs to Balan
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23 Legal Entities and Parental Guar
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2. In August of 2004, in order to r
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The asset-liability portfolio is co
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ealize the economic value of this e
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25 The Valuation of Assets The SST
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Assets Valuation Fixed maturities I
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with non-material value are taken a
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Liability Valuation Underwriting re
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VI Solvency Capital Requirements 35
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VII Process Landscape 352
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31.3 Roles and Responsibilities . .
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List of Figures 29.1 Embedment of t
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27 Summary The ultimate objective o
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28 Structure of the Process Landsca
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29 Overview 29.1 The Swiss Solvency
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Process step Proposal for asset man
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ALM Committee � Sign-off on adequ
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30 The ALM Process 30.1 General Ove
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Asset modeling: The asset portfolio
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Inflation index The inflation in a
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Simulate future time series Post pr
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30.3.2 Reserve Modeling The purpose
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Process Steps The following process
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30.3.3 Liability Modeling The main
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Retrocession Information on interna
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Calculate gross reinsurance loss in
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Unearned business modeler: The unea
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Sign-off procedures Overall, the li
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FX rates This data entity is taken
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Roles and Responsibilities The role
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Noninvested assets Balance sheet In
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Financial controller � Provide da
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31.2 Process Steps The following st
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Risk management team ECS Scenario d
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32 Appendix 32.1 Sign-off Matrix 32
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Steering committee: Project sponsor
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Function Body Responsible for Steer
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Table of Contents 33 Summary 412 34
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List of Tables 36.1 Economic variab
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33 Summary This Part of the documen
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and Bloomberg. One of the purposes
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35 Architecture of SST Relevant Sys
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35.1.2 Application Change Managemen
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35.3 Revision Control The key to ga
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IT SYSTEM DATA BASE DOCUMENT, DATA
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Entity Time series used Risk-free y
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Application FED User Bloomberg Othe
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For investment indices, there is al
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The initial data to the module are
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module� data�input liability�
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GCDP Aggregation of event losses MA
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36.4.6 Planning and Business Projec
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in Igloo and are used in conjunctio
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flows for this simulation step. The
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37 Architecture of New Developments
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changing dimensions: Codes such as
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each method has a corresponding set
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38 Appendix We list the IT systems
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IX Final Remarks 450
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Acronyms A AAD Annual aggregate ded
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L LE Legal entity. LGD Loss given d
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Index Accident and health model bas
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economic variables, 218-235 equity
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invested assets and credit risk, 30
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detrending, 211 economic variables,
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Life & Health business, 105 Foreign
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ing first development year, 63 occu
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etrocession, 332-341 retrocession a
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finance re, see Financial reinsuran
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terest rate, 200 inflation, seasona
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dependency structure with life and
Page 494 and 495:
Purchase guarantee, see liquidity a
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Shareholder equity one year max cov
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natural catastrophe modeling, 88 Vu
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Bürgi, R. (2007). ALM Information
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Gencyilmaz, C. (2007). Swiss Solven
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Müller, U. A. (2007b). Modeling cr
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