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

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2.1 Basics of Valuation 2.1.1 Valuation by Perfect Replication and Allocation Given a stochastic model for the future cash flows of a liability, the valuation of the liability at time t consists in assigning a monetary amount, the value, to the liability. According to SST principles, the valuation has to be marketconsistent. The valuation of (re)insurance liabilities is a special case of the more general valuation of contingent claims. Contingent claims are financial instruments for which the future claims amount is not fixed, but depends, in the meaning of probability theory, on the future state of the world. To highlight the overall context, we provide in the following some examples and remarks on the valuation of contingent claims, and its relation to the valuation of (re)insurance liabilities. The market-consistent value of a liability, in case the liability is traded in a liquid market, is simply its market price, i.e. the observed transfer price. Typically, there is no liquid market for the liabilities. In this case, the market-consistent value of such a non-traded financial instrument is taken to be the market value of a replicating portfolio, that is, a portfolio of liquidly and deeply traded financial instruments whose cash flows perfectly replicate the stochastic cash flows of the given instrument. In other words, the replicating portfolio replicates the (future) cash flows of the instrument exactly for any future state of the world. We call this approach valuation by perfect replication. Of course, market-consistent valuation in this sense is possible only if a replicating portfolio exists. From the point of view of probability theory, perfect replication means replication for any state of the world ω ∈ Ω. The actual probabilities assigned to each of these states of the world then become irrelevant. If perfect replication is not possible, a different form of replication could be considered, which we call law replication. Law replication means that we construct a portfolio which has the same probability distribution as the given financial instrument. Assume that valuation of instruments relies on a monetary utility function U, or alternatively on a convex risk measure ρ = −U, and that the functions U, ρ are law-invariant, i.e. they depend only on the law of a random variable, that is, on the distribution of the random variable. Then, with respect to such a monetary utility function, the initial instrument and the law-replicating portfolio have the same value. Clearly, perfect replication implies law replication but not vice versa. Regarding the latter statement, dependencies to other random variables are the same for the instrument and the portfolio in the case of perfect replication, but not for law replication. In particular, valuation of an instrument by a law-invariant monetary utility function or risk measure will disregard dependencies to other instruments. 10
We now return to the notion of perfect replication. We can distinguish two types of replicating portfolios: static portfolios, where the replicating portfolio is set up at the beginning and kept constant over time (although, of course, its value probably changes over time); and dynamic portfolios, where the initial portfolio is adjusted over time in a self-financing way (i.e. with no in- or out-flow of money, with the exception of the liability cash flows themselves) as information is revealed. Uniqueness of the value derived by perfect replication follows from a noarbitrage argument: If two portfolios of liquidly traded financial instruments perfectly replicate a given instrument, their values have to be equal, because otherwise an arbitrage opportunity would exist. To be more precise, because the market is liquid, the two portfolios can be bought and sold for their value (which is the market value), so the theoretical arbitrage opportunity can be realized. We mention two well-known examples of valuation by perfect replication. As a first example, consider the value at time t of a deterministic future payment at time t + d. This instrument is perfectly replicated by a defaultfree zero-coupon bond with maturity d. Hence, the value of the future payment is equal to the price of the bond at time t. The replicating portfolio is clearly static. As a second example, consider, at time t, a stock option with strike date t + d. The cash flow at t + d of the option is stochastic, and depends on the value of the stock at t + d. Nonetheless, option pricing theory shows that, under certain model assumptions, the option cash flow at t + d can be perfectly replicated by holding at time t a portfolio composed of certain quantities (long or short) of cash and the stock, and by dynamically adjusting this portfolio over time in a self-financing way. So this is a dynamic replicating portfolio. The value of the option at time t is again equal to the price of the replicating portfolio at t. In this example, the stochasticity of the cash flow becomes irrelevant, since perfect replication means that replication works for any state of the world. Assuming the option is part of a larger portfolio of financial instruments, its value is independent of the other instruments in the portfolio, on how their cash flows depend on each other, and on the volume of the portfolio. Observe that particularly the second example is not realistic in that it does not take into account transaction costs, which are inevitably part of the replication process. Crucially, when taking into account transaction costs, the costs of the replication are likely no longer independent of the overall portfolio. It might be, for instance, that certain transactions needed to dynamically adjust the replicating portfolio for different instruments cancel each other out, so that overall transaction costs are reduced. Therefore, the value of one instrument depends on the overall portfolio of instruments and is no longer unique - even though the replication is perfect and stochasticity 11
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 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 42 and 43: Hence, basis risk arises when the a
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
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Agribusiness (θ =0.2673) Marine (
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Basket Specific Models Basket speci
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is reasonable and, in particular, c
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Credit and Surety (Section 3.6, pp.
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An alternative approach is used, fo
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Historical losses are “as if” a
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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
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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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