allianz lebensversicherungs-ag · 19. mai 2003 das risikokapitalmodell der allianz...
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Lebensversicherung
Das Risikokapitalmodell derAllianz Lebensversicherungs-AG
Ulm19. Mai 2003
Dr. Max HappacherAllianz Lebensversicherungs-AG
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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LebensversicherungTable of contents
1. Introduction: Motivation, Group-wide Framework
2. Internal Risk Model: Basics, Life Approach
3. Standalone Risks and Liability Response
4. Simulation of Assets & Fair Value of Liabilities
5. Liability Response: Dynamic Reseves/Benefits
6. Liability Response: Valuation of Liabilities
7. Risk Aggregation, Diversification and Mitigation
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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LebensversicherungTable of contents
1. Introduction: Motivation, Group-wide Framework
2. Internal Risk Model: Basics, Life Approach
3. Standalone Risks and Liability Response
4. Simulation of Assets & Fair Value of Liabilities
5. Liability Response: Dynamic Reseves/Benefits
6. Liability Response: Valuation of Liabilities
7. Risk Aggregation, Diversification and Mitigation
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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Lebensversicherung
Main goals for Allianz internal risk analysisSupporting value management
• Systematic risk assessment based on internal data
• Improving risk adjusted profitability measurement
• Portfolio management / asset mix
• Capital allocation
• Compare outcome against rating agencies’ models
Meeting legal requirements
• Risk reporting → More detailed quantitative requirements by DRSC or SEC
• Solvency → New solvency requirements on the way; new Basel II-like requirements for insurance companies (Solvency II) and financial conglomerates allow for internal risk analysis models
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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LebensversicherungRisk analysis project covering AZ Group flagshipsPC: SGD, AZRE, Cornhill, AZ Elementar, AZ Suisse, RAS, Lloyd Adriatico, AZ
Seguros, AGF, FFIC, AIC, AZ Australia, Hermes/Euler
Life: AZ Leben, AZ Elementar, AZ Suisse, RAS, Lloyd Adriatico, AZ Seguros, AGF, AZ Life, AZ First Life
Banking: Dreba Group
Consistent framework of risk categories for all segments
Respect material differences between & within segments by specific risk models
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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Lebensversicherung
Consistent framework for risk analysis, risk capital & capital allocation
Risks of AZ Leben
Insurance Risks
ProspectiveClaims
Pricing
Retrospective Claims
PC Risk
PremiumRisk
ReserveRisk
Non-Cat
Cat
Link to S&P standard categoriesC3 C4 C5 C2 C1 C6
Operating Risk
LifeRisk
Market Risk
Counter-party Risk
ReinsurerCredit
Worthiness(Security)
Market RisksCredit RisksOther Risks
Business Risk- Lapse- Cost
Operational Event Risk- IT failure- Litigation- Fraud
Biometric Risks- Mortality- Longevity- Disability- Calamity
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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LebensversicherungTable of contents
1. Introduction: Motivation, Group-wide Framework
2. Internal Risk Model: Basics, Life Approach
3. Standalone Risks and Liability Response
4. Simulation of Assets & Fair Value of Liabilities
5. Liability Response: Dynamic Reseves/Benefits
6. Liability Response: Valuation of Liabilities
7. Risk Aggregation, Diversification and Mitigation
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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Lebensversicherung
Capital requirement depends on point of view
Internal risk approach measures the required risk capital on an economic point of view
ASSETS
LIABILITIES
Required Capital Net Asset Value =
Economic Capital
Excess Capital
Regulatory Capital
Rating Agency Capital
(Economic) Risk Capital• Market Value
• Best Estimates
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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Lebensversicherung
Economic Capital is volatile due to unexpected developments affecting assets and liabilities
ASSETS• Market Value
LIABILITIES• Fair Value
ECONOMICCAPITAL
(NAV)
Market Crisis
Large Claims
All expected profits and losses are included in the best estimate NAV, unexpected developments induce volatility or risk into the business
INSOLVENCY
Probability 0
Economic Capital
BE NAV
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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LebensversicherungRisk Capital: Potential loss of expected NAV for the next 12 months (VaR approach)
NAV0
Probability Density
Required Risk CapitalExpected NAV
SOLVENCY STANDARD
AAA0.01%
AA0.03%
A0.07%
Worst Case
Risk capital is the capital required to run the business tied toa given solvency / worst case level
Key parameters• Confidence interval
/ solvency level• Risk distribution• Risk correlation• Risk concentration
Diversification
Worst case
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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LebensversicherungModelling risks - conceptional overview
Equity
Interest rate
Real estate
Standalone risks
Liability response via simulation
CV Credit
CV Life
CV Business
CV Op. Event
TCV
TCV -- Total C
hange in Total C
hange in ValueValue
Risks shareable with policyholder(correlated)
Non-shareable risks
BE
WC IR
WC EQ/RE
WC Shar.
CV IR
CV EQ/RE
CV Shar.
Aggregation
Diversification
CV
CV
IRIRC
VC
VEQEQ
Busin
Busin ..
Op.
Op.
LifeLife
Mitigation
ShareholderShareholder
PolicyholderPolicyholder
Credit
Life
Business
Operationalevent
FX
CV FX
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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LebensversicherungRisk sharing with policyholder
... completely via profit sharing (´´liability response´´)• Equities (EQ)• Real estate (RE)• Interest rate (IR)
... partially via profit sharing:• Credit (Default)• FX• Life (calamity, volatility, trend)• Business (lapse, new business, cost inflation)
... not sharable via profit sharing:• Operational event
} via asset returns
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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LebensversicherungTable of contents
1. Introduction: Motivation, Group-wide Framework
2. Internal Risk Model: Basics, Life Approach
3. Standalone Risks and Liability Response
4. Simulation of Assets and Fair Value of Liabilities
5. Liability Response: Dynamic Reseves/Benefits
6. Liability Response: Valuation of Liabilities
7. Risk Aggregation, Diversification and Mitigation
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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Lebensversicherung
Change in Value (CV) as difference between asset value impact and liability response
Value Best Estimate Asset Value
Worst Case
Asset Value Impact
BestEstimateLiabilityValue
LiabilityValueImpact
AssetValueImpact Liability
ValueImpact
InvestmentChange in Value (CV)
WorstCase
Modelling life risks: worst casechange in value
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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Lebensversicherung
Standalone risks for life insuranceWorst case shock scenarios
• According to 0.07%-quantile of corresponding distribution
• Impact of shock in the first year• Propagation for next n years necessary for
valuation of liabilitiesThree shocks are considered:
• Equity: 1st year loss of roundabout 40 %• Interest rates: down shift of yield curve by 1,5 %• Biometrical, operational and business risks lead to
an impairment of gross profits
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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Lebensversicherung
Equity risk... before liability response
• Market value view (after hedging)• Risk driver: volatility of stock returns• Equity portfolio modeled by 4 indices
(DAX100, MSCI: EMU, EUROPE, WORLD)• Lognormal distribution• Best estimate (BE) return: 8,5%• Total volatility:
» historical volatilities of indices» covariance matrix for correlation effects
• Issues: time frame, ex post / ex ante, private equity, derivatives
Worst Case
BestEstimate
Risk Capital
Probability
PortfolioValue
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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Lebensversicherung
IR down risk: Risky present value due to volatile discount rateDown-side Risk
• Down shift of interest rate curve in year 1• Shifted IR curve also used in years 2 to n• Lower discount rate leads to higher values of (fixed income)
assets and liabilities
• Increase of liabilities higher due to higher duration• Impact of lower asset returns on the liabilities lessened by
guaranteed rate for P/H
Yield (%)
Time
Yield Curve
Time
PresentValue
t=0 t=1 t=2 t=3 t=4
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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Lebensversicherung
Other sharable risks... before liability response
Riskcategory
Subrisks Method Value[Mio. €]
RatioPH:SH
Issues
Investment(cont’d.)
Credit
FX
Distribution/ SimulationAnalytic
> 1000
small
50:50 Hugeportfolio
Hedges
Life CalamityVolatilityTrend
GdV-ModelGdV-ModelGdV-Model
< 10*> 10*> 100*
100:080:2080:20
Availabilityof internaldata
Business LapseNew businessCost inflation
GdV-ModelGdV-ModelProxy
> 10> 100> 10
80:2080:2080:20
Availabilityof internaldata
Total aggregated sharable risks: ~ 1 Mrd. €
* Values after reinsurance
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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LebensversicherungTable of contents
1. Introduction: Motivation, Group-wide Framework
2. Internal Risk Model: Basics, Life Approach
3. Standalone Risks and Liability Response
4. Simulation of Assets & Fair Value of Liabilities
5. Liability Response: Dynamic Reseves/Benefits
6. Liability Response: Valuation of Liabilities
7. Risk Aggregation, Diversification and Mitigation
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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LebensversicherungFair value of liabilities...
... is the present value of future cash flows
... is influenced by asset movements:• Liability cash flows (usually) depend on future
portfolio (asset) returns• Profit sharing creates a tie between asset return
and liability value• Management rules to describe the
interdependence... depends on interest rates:
• Present value of future liability cash flows changes with moving interest rates
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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LebensversicherungFair value of liabilities: Models
Deterministic world:• Future asset returns are assumed to be given• Future cash flows on basis of a known profit
sharing scheme• Options of insurer and PH are negligible
Real world:• Returns are volatile• Profit sharing according to actual return• Options of insurer and PH have to be considered
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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Lebensversicherung
Guaranteed rates have negative financial value for the shareholder
Investment return paths may be below guaranteed rate
Investment Return
Guaranteed rate
Time
Best Estimate Return
The risk of investment return below guaranteed rate is pure shareholder risk
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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Lebensversicherung
The cash flows of profit sharing products are only dependent on the investment return above the guaranteed rateInvestment
Return
Guaranteed rate
Time
LIABILITY CASH FLOWS(Schematic)
Thus financially the liability cash flow structure of this product is similar to a floor on the investment return
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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Lebensversicherung
Pricing this “floor” can easiest be done by simulation
1. Simulate different scenarios of future interest rates and inv. returns
2.Calculate in each scenario the corresponding future liability cash flows
Year 1 Year 2 Year 3 Year 4 Year 5
Investment Return 7.0% 6,5% 5,5% 2,5% 3,0%Interest Rates 5.5% 4,5% 3,0% 2,0% 2,8%
3.Discount these cash flows at the simulated interest rates
Year 1 Year 2 Year 3 Year 4 Year 5
Cash Flow 70 65 55 30 30
Scenario i
Scenario i
4.Fair Value: Average value over all scenarios
∑ = +⋅= 5
1 ))(_1(1)(___
k kkRateInterestkFlowCashiScenarioValue
( ) niScenarioValueValueFair n
i/___
1∑ ==
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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Lebensversicherung
Portfolio return: Combined return of asset classes
Return
Time
Return
Time
Return
Time
Return
Time
FIXED INCOME RET. EQUITY RETURN REAL ESTATE RETURN
PORTFOLIO RETURNS
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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LebensversicherungSimulation of equity / real estate
Input:• Expected asset performance• Volatility of the portfolio (cont. compounded)
Output: 1-year performance P
Model: ;
therefore:
with i the cont. compounded return expectation
2σ
µ
2/1)(2+=+= σρµ ePEσρ,
+= LogNValue
ValuePt
t ~1
2−=
2−)+=
22 σσµρ i1log(
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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LebensversicherungSimulation of FI-returns
Input:• Portfolio statement comprising the cash flows of
the FI portfolio (CFs for each investment)• Model for simulation of future interest rates
Assumptions:• Reinvestment in risk free bonds with same
duration as initial portfolio• Bonds are held to maturity
Simulation model for interest rates:• Crucial: IR model has to be arbitrage free
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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LebensversicherungTable of contents
1. Introduction: Motivation, Group-wide Framework
2. Internal Risk Model: Basics, Life Approach
3. Standalone Risks and Liability Response
4. Simulation of Assets & Fair Value of Liabilities
5. Liability Response: Dynamic Reseves/Benefits
6. Liability Response: Valuation of Liabilities
7. Risk Aggregation, Diversification and Mitigation
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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LebensversicherungDynamic reserves and benefits
Problem: For each simulation run a new liability projection necessary
Retrospective Approach:• Two projections are used: declared profit sharing
(BE) and minimum guaranteed (MG)• Basis is the minimum guaranteed projection run• For each simulation run dynamic reserves are
generated retrospectively via interpolation• Dynamic benefits are generated via scaling
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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Lebensversicherung
From simulated returns todynamic reserves and benefits
• PH credited rate crt as moving n-year average ofsimulated returns (after shareholder profit)
• Initial reserves R0 and initial cash flows Rim1identical for BE and MG
• Dynamic reserves calculated retrospectively:
• Dynamic benefits according reserves at b.o.y.:
• Valuation reserves are used as buffer
) Rim – Rim (R – R
R – R Rim Rim tMGtBEtMGtBE
tMGt tMGt 1111 ++++ +=
tttt Rim )) * R(mg, cr ( R ++= −1max1
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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LebensversicherungTable of contents
1. Introduction: Motivation, Group-wide Framework
2. Internal Risk Model: Basics, Life Approach
3. Standalone Risks and Liability Response
4. Simulation of Assets & Fair Value of Liabilities
5. Liability Response: Dynamic Reseves/Benefits
6. Liability Response: Valuation of Liabilities
7. Risk Aggregation, Diversification and Mitigation
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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LebensversicherungValuation of cash flows
Central problem:
Appropriate discount factors for cash flows
Approach:• Simulate asset returns using actual return
expectations ( i. e. 5 % on RE, 8,5 % on EQ,...)• Discount cash flows at risk free rate
Problem:• Return expectations are risk adjusted• Discounting at risk free rates overstates PV
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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LebensversicherungRisk neutral valuation
2 alternatives:a) Adjust discount factors to intrinsic risk of cash flowsb) Simulate asset returns using risk free rates and
discount liability cash flows at risk free rates
No applicable way to obtain option adjusted discount factors for liability cash flows in a).
Alternative b):
Concept of risk neutral valuation
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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LebensversicherungSummary:Liability CF simulation and valuation
(200)(150)(100)
(50)-50
Year
CF's
3Determine liability
cash flows
1 Generate risk free rates
0%4%8%
12%16%
0 20 40 60Year
Rate
2 Generate risk free asset returns
-30%
20%
0 20 40 60Year
Return
4 Discount @ risk freerate
liability valuei = X Mio. €
5
liability value =
Y Mio. €
Average liability value
Liability value: Done 1000 times in each simulation
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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Lebensversicherung
Simulation used to calculate the liability value in a given shock scenario
Portfolio ofAssets
Portfolio ofLife Policies
Simulation of investment returns
after next year
DynamicLiabilities
PV of Assets
PV of
Liabilities
PV of the Net
Position•BE Case•WC Shocks
Shock Scenarios
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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LebensversicherungTable of contents
1. Introduction: Motivation, Group-wide Framework
2. Internal Risk Model: Basics, Life Approach
3. Standalone Risks and Liability Response
4. Simulation of Assets & Fair Value of Liabilities
5. Liability Response: Dynamic Reseves/Benefits
6. Liability Response: Valuation of Liabilities
7. Risk Aggregation, Diversification and Mitigation
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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Lebensversicherung
Boundary conditions for WC scenarios:
Scaling of assets and liabilities:• A: 100% = capital investments + cash• L: 100% = insurance reserves - free RfB
Asset mix: constant
Liability response & ALM model
WC scenario EQ/RE FI IR curveBE BE BE BE
EQ/RE WC BE BE
IR down BE WC WC (down)
Sharable BE –WC Shar.
BE BE
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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Lebensversicherung
Standalone risks ...before and after liability reponse
ScenarioAsset Value
Impact
Liability Value Impact
Change in
ValueEQ -100 +33 -66RE -100 +33 -66
IR down +100 -150 -50Shareable -100 +10 -90
Liability response is a 1st order effect
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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LebensversicherungAggregation to total risk: TCVStandalone change in value after liability response: CViare aggregated to total change in value (TCV) viaparametric aggregation:
jji
iji CVCCV TCV ∑∑∑∑ ⋅⋅⋅⋅⋅⋅⋅⋅====,
with covariance matrix Cij expressing interdependence between standalone risks.
Disaggregation for risk analysis: ii
divi CVCVTCV CV ⋅⋅⋅⋅∂∂∂∂∂∂∂∂====,
Diversification benefits: ∑∑∑∑∑∑∑∑ ≤≤≤≤====i
ii
divi CVCVTCV ,
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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Lebensversicherung
Risk mitigation:Hierarchy of buffers
S/H Risk Capital
Withoutrisk mitigation
TCV
With risk mitigation
TVR fr. RFB
1/2 LTD
1/2 LTD
SE
Σ1 Σ2 Σ3Shareholder part qSH = 0,1 * TVR/ ΣΣΣΣ1
Shareholder risk capital: RC = qSH • TCV if TCV < ΣΣΣΣ1
TVR: Total valuation reservesfr.RfB: free RfBLTD: Liability for terminal dividendsSE: Shareholder Equity
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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Lebensversicherung
Review: standalone risks... before and after ...E.g. credit risk: 100 €
• After 50:50 risk sharing with P/H: 50,0 + 50,0
• After liability response: 50,0 + 45,0 = 95,0
• After diversification: 65
• After risk mitigation: 5
19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG
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LebensversicherungBackup: Interest rate model
Y0(t): current, continuously comp. zero coupon yield => Discount factor:=> Continuously compounded forward rate:
For tenor structure T define forward rate processes as
with
Choose volatility to meet 5-year swap volatility, (5 = average duration of FI portfolio)
2
1 0 1( , ) ( , ) , where ,2
tX it i i i i t if T T f T T e X N σ σ+ +
= × −
�
[ ] [ ] ( )221 0 1 1 0 1( , ) ( , ), ( , ) ( , ) e -1 t i i i i t i i i iE f T T f T T Var f T T f T T σ
+ + + += =
~
)/()()((),( 12110220210 ttttYttYttf −−=
))(exp( 0 ttY ⋅−