Replicating Portfolios Complex modelling made simple1c6dd1f6... · Presentation1 0 50 100 150 200 250 300 0 50 100 150 200 250 300 USD million USD million Replicating portfolio cash
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© 2010 Towers Watson. All rights reserved.
Replicating PortfoliosComplex modelling made simpleSAV Versammlung
by Jolanta Tubis10 September 2010
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 2Presentation1
Agenda
Smart modelling
The replicating portfolioApproachCase studies
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 3Presentation1
Why smart modelling?
Life insurance is about hedging exposures
For savings products life insurance is about individualised guaranteesEach policyholder has potentially different strike prices (=guaranteed benefits), terms and benefits typesThe resulting overall exposure for the insurance company is— complex
— non-linear
— difficult to manage
But this gives life insurance a unique selling propositionIn fact a life insurance portfolio is a portfolio of optionsBut how does this portfolio look like?
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 4Presentation1
Why smart modelling?
Smoothed investment return
Ben
efit
s Shareholder
Interest rate guarantee
EEV gives us some insight from time to time, but with great effort.Can we use the EEV-efforts to get more? Can we reduce effort and improve accuracy?
In order to manage the liabilitieswe need:- the whole picture- daily
Basecase
SensitivitySensitivity
And we have issues with accuracy and run-time
We have some information, but not enough and not often enough
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There are many promising approaches to solve some of our technical problems
Weighted Monte CarloImproves accuracy and fit to calibrationStill requires stochastic calculations for each piece of information
Change of measure – importance samplingRelevant, if not necessary, for stochastic determination of economic capital Can be very successfully combined with the replication portfolio approach
Control variatesImproves accuracy and fit to calibrationStill requires stochastic calculations for each piece of information
Moment matchingImproves accuracyStill requires stochastic calculations for each piece of information
Replicating portfoliosImproves accuracy and fit to calibrationCan be used as control variateEasy to understand and applySome relevant information can be determined without stochastic runs Enables timely and relevant management information
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 6Presentation1
Agenda
Smart modelling
The replicating portfolioApproachCase studies
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 7Presentation1
What is a replicating portfolio?
A replicating portfolio is a portfolio (of assets) that agrees in value with your liabilities under a range of economic conditions (=scenarios)
Portfolio of tradable options –
e.g. swaptions
Portfolio of functions of the scenarios – e.g. annuity functions
Portfolio of non-traded options –e.g. asset share
options
Replicating portfolio
Stochastic cash-flows
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 8Presentation1
The key problem is the determination of the „candidateassets“…
The „candidate assets“ should be able to reflect all relevant features of the contingentcash-flows, like
Dependency on core asset classesDependency on interest ratesPath-dependent features like e.g.— smoothing of returns— look-back-features
Typically following candidate assets are sufficient:The underlying core asset classes (in contract currency)Zero bondsSwaptionsPlain vanilla call and put options
For a relevant range of strike prices and termsIn some circumstances path dependent options are required-> Actuarial judgement is important
To avoid overfitting
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 9Presentation1
After determining the candidate assets we can determine a portfolio as linear combination that is highly correlated with the liabilities
Asset 6Asset 5Asset 4Asset 3Asset 2Asset 1Cash flow at time t
Scenario
A6,6A6,5A6,4A6,3A6,2A6,1L66
A5,6A5,5A5,4A5,3A5,2A5,1L55
A4,6A4,5A4,4A4,3A4,2A4,1L44
A3,6A3,5A3,4A3,3A3,2A3,1L33
A2,6A2,5A2,4A2,3A2,2A2,1L22
A1,6A1,5A1,4A1,3A1,2A1,1L11
Value of asset in scenario
L1 = w1*A1,1 + w2*A1,2 + w3*A1,3+…
L2 = w1*A2,1 + w2*A2,2 + w3*A2,3+…
L3 = w1*A3,1 + w2*A3,2 + w3*A3,3+…
…………Subject to constraints…
“replicating portfolio”“replicating portfolio” used as the basis of the estimation of the sensitivity
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 10Presentation1
Replicating portfolios can be derived from standard liability model runs
Standard Liability Model Runs
Cash Flow Outputs
Liability Models
Replicating Portfolio Tool
Replicating Portfolio
Inputs/ Scenario
Files
Scenarios
Cash Flows
Potential Replicating Portfolios
Optimization Engine
We still need a base run
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 11Presentation1
Replicating portfolios can simplify your life substantially
The approach enables you toRecalculate results for changed market parameters (asset prices, interest rates, volatility etc.)Calculate sensitivities (greeks like delta, vega, rho etc.)Improve accuracy and reduce the number of necessary runsProject asset-dependent variables, e.g. required capital, in stochastic runs
…Without the need to re-project the liabilitiesWhich is usually the onerous part of the simulation
But the most important advantage is the fact that a replicating portfolio simplifies communication dramatically
A replication portfolio is a description of your liabilities in terms of assets
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A typical replicating portfolio
2.523Floating strike lookback option on DAX – 20 years – strike 1234
312Put Option on DAX – 10 years – Strike 500
0.20.25Swaption – EUR – 1 year term – 5 years tenor – Strike 2%
…
0.50.57Swaption – EUR – 10 years term – 10 years tenor – Strike 4%
123
1
3.5
2
2
4
Current value
3.55Zero Bond EUR – 30 years
210Put Option on DAX – 10 years – Strike 1234
…
22Zero Bond EUR – 1 year
120.5SMI
Value in 1 moth under stress test
XYZ
Notional in bn EUR
Candidate Asset
45Total
…
21DAX
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 13Presentation1
Replicating portfolios are in fact control-variates
replicating portfolio
stochastic cashflows”residuals”
=replicating portfolio (closed form solution)
+”residuals” (low volatility = high accuracy)
A replicating portfolio of the liabilities forms an ideal control variate
nσ
≈Estimation error
=
+
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 14Presentation1
Goodness of fit – typical analysis
124.66Value of the replicating portfolio
128.33Value, market-consistent scenarios
124.62Exact value, closed form solution
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 15Presentation1
Specific issues – replicating portfolio for requiredeconomic capital
In general the calculation of required capital requires full stochasticapproach (nested stochastic simulations)
The replicating portfolio approach allows to avoid the stochastic valuations and therefore to reduce the number of necessary calculations substantially
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 16Presentation1
Specific issues – replicating portfolio for requiredeconomic capital
The approximation must be good in the quantile considered – not onlyaround the median
The optimisation approach typically enforces a good fit around the medianThis is where most scenarios areNot such a good approach for required economic capital purposes…
Large market shocks should be replicated adequately
It is important to ensure that the asymptotic behaviour of the replicationportfolio cash-flows are reasonable
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 17Presentation1
Agenda
Smart modelling
The replicating portfolioApproachCase studies
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 18Presentation1
Applications of the replicating portfolio approach
VA portfolios- hedging
SST target capital for a block of GMxBs – European reinsurer
Typical German with profits business – the book value effect
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 19Presentation1
Replicating portfolio approach for VA dynamic hedging
Numerical problemsThe Greeks calculation usually requires repeated stochastic simulation for a large number of scenarios over a huge portfolio of contracts— time-consuming— numerical errors
Hedge effectiveness testing usually requires nested stochastic simulations
Our research proves The Replicating portfolio approach can be successfully used to fit VA business with— Complex path dependent policyholder behaviour— Complex guarantee and asset mix structure
The replicating assets are plain vanilla assets that will allow for — Faster and more accurate valuation— Hedging and market risk estimation, meaning derivation of Greeks— Hedge effectiveness testing without time consuming nested stochastic simulations
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 20Presentation1
Replicating assets
Replicating assetsPlain vanilla put optionsBasket options (including the forward starting versions) on actual underlying— Simulating the asset mix of the underlying asset portfolio through combinations of
70-80% equity and 30-20% bonds
Knock-out basket options— When index level exceeds a certain level the option is knocked out, if the index
never exceeds the knock-out level the option is in-force
Closed forms or numerical approximations available
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 21Presentation1
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Replicating portfolio cash flows
MoSes cash flows
Results of central projectionScatter plot: Central scenarios
R2 Measure: 0.96VRM: 5.16
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 22Presentation1
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Replicating portfolio cash flows
MoSes cash flows
Results of central projectionScatter plot: Equity stress scenarios
R2 Measure: 0.96VRM: 5.24
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 23Presentation1
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Replicating portfolio cash flows
MoSes cash flows
Results of central projectionScatter plot: Interest stress scenarios
R2 Measure: 0.97VRM: 5.47
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 24Presentation1
Applications of the replicating portfolio approach
VA portfolios- hedging
SST target capital for a block of GMxBs – European reinsurer
Typical German with profits business – the book value effect
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 25Presentation1
SST target capital for a block of GMxBs – European reinsurer
Case study from 2005 (!)
Includes policyholder behaviour (lapsation)
Liabilities not straightforward: ratchets includedThus the replication portfolio included floating strike discrete lookback options— Good approximation formula available
Used for SST purposesValid approach as asymptotic behaviour is clear!
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Plain vanilla contracts, no policyholder behaviour
-1.200.000
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AV(t)
CF(
t)
CF
Premium
Claims
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 27Presentation1
Plain vanilla contracts, with policyholder behaviour
Cash flows by asset value after 10 years
-1200000
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0 50000000 100000000 150000000 200000000 250000000 300000000 350000000
AV(t)
CF(
t)
CF
Premium
Claims
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 28Presentation1
The replication portfolio included floating strike discretelookback options
Number of assets purchased in each periodQuarter
Assets Strike 1 2 3 4 5 6 7 8 9 10European put 1 0.5 0 0 0 0 0 0 0 0 0 0European put 2 0.8 0 0 0 0 0 0 0 0 -4'012'000 -4'012'000European put 3 1.111 -653'000 -653'000 -653'000 -653'000 -2'825'000 -2'825'000 -2'825'000 -2'825'000 -2'351'000 -2'351'000European put 4 1.667 -527'200 -527'200 -527'200 -527'200 -333'700 -333'700 -333'700 -333'700 -315'000 -315'000European put 5 2.222 -472'900 -472'900 -472'900 -472'900 -301'500 -301'500 -301'500 -301'500 -285'100 -285'100European put 6 2.778 -418'600 -418'600 -418'600 -418'600 -269'300 -269'300 -269'300 -269'300 -255'300 -255'300European put 7 3.333 -364'300 -364'300 -364'300 -364'300 -237'100 -237'100 -237'100 -237'100 -225'400 -225'400European put 8 3.889 -310'000 -310'000 -310'000 -310'000 -204'900 -204'900 -204'900 -204'900 -195'500 -195'500European put 9 4.444 -255'700 -255'700 -255'700 -255'700 -172'600 -172'600 -172'600 -172'600 -165'700 -165'700European put 10 10 287'300 287'300 287'300 287'300 149'500 149'500 149'500 149'500 133'000 133'000Equity N/A 690'100 690'100 690'100 690'100 430'400 430'400 430'400 430'400 404'600 404'600Bond N/A 97'740 97'740 97'740 97'740 57'990 57'990 57'990 57'990 53'760 53'760Lookback put 1 0.5 -1'145'000 -1'145'000 -1'145'000 -1'145'000 -181'500 -181'500 -181'500 -181'500 -131'700 -131'700Lookback put 2 0.8 -1'145'000 -1'145'000 -1'145'000 -1'145'000 -181'500 -181'500 -181'500 -181'500 -131'700 -131'700Lookback put 3 1.111 -653'000 -653'000 -653'000 -653'000 -1'253'000 -1'253'000 -1'253'000 -1'253'000 -1'264'000 -1'264'000Lookback put 4 1.667 -527'200 -527'200 -527'200 -527'200 -333'700 -333'700 -333'700 -333'700 -315'000 -315'000Lookback put 5 2.222 -472'900 -472'900 -472'900 -472'900 -301'500 -301'500 -301'500 -301'500 -285'100 -285'100Lookback put 6 2.778 -418'600 -418'600 -418'600 -418'600 -269'300 -269'300 -269'300 -269'300 -255'300 -255'300Lookback put 7 3.333 -364'300 -364'300 -364'300 -364'300 -237'100 -237'100 -237'100 -237'100 -225'400 -225'400Lookback put 8 3.889 -310'000 -310'000 -310'000 -310'000 -204'900 -204'900 -204'900 -204'900 -195'500 -195'500Lookback put 9 4.444 -255'700 -255'700 -255'700 -255'700 -172'600 -172'600 -172'600 -172'600 -165'700 -165'700Lookback put 10 10 287'300 287'300 287'300 287'300 149'500 149'500 149'500 149'500 133'000 133'000
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 29Presentation1
Goodness of fit
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 30Presentation1
Applications of the replicating portfolio approach
VA portfolios- hedging
SST target capital for a block of GMxBs – European reinsurer
Typical German with profits business – the book value effect
© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 31Presentation1
Fallstudie: RP für das deutsche gewinnberechtigte Geschäft
Gesellschaft ABC schreibt hauptsächlich das typische deutsche gewinnberechtigte Geschäft: Kapital und Renten
Für die Fallstudie wurden die Zahlen „anonymisiert“
Approximation des VIFs unter MCEV
y = xR2 = 0.9525
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-5 0 5 10 15 20 25 30
VIF per Szenario
RP
R2 von 95% zeigt eine gute Anpassung des
RP zu dem Dividenden-Cash
Flow VIF
aus
RP
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Repricing: Um die Qualität zu verifizieren prüfen wir, ob wir mittels RP die ökonomischen Sensitivitäten replizieren können
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ORG RP
15%12%
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-25%
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-15%
-10%
-5%
0%
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15%
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Basis Zins+1% Zins-1% Aktien/Immo -10%
ORG RP
Die Sensitivität Zins+1% ist beim RP unterschätzt
Das geschätzte RP repliziert sehr gut die
Sensitivitäten: Aktien -10% und Zinsen -1%
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Veränderung der Aktienpreise
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Geschätztes RP erlaubt mühelos das ganze Spektrum der Sensitivitäten zu berechnen
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© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.towerswatson.com 34Presentation1
RP spiegelt das Risiko des deutschen gewinnberechtigten Geschäftes wider
MCEV kann man mit „long“-Positionen in Aktien/Immobilien/Bonds und „short”-Positionen in Derivate replizieren
Relativ niedriger Anteil der Swaptions liegt an „Moneyness“ der Swaptions (deep out-of-the-money)
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Equityindex
Equity putoptions
RE index RE putoptions
Swaptions Zerobonds
Bondindex
Replicating Portfolio Illustrativ
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Contact
Jolanta TubisSeefeldstrasse 214Postfach8034 ZürichTel.: 043 488 4486Fax: 043 488 4444jolanta.tubis@towerswatson.com
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