Economic Scenarios Generation for Insurance: ESG package and other tools (Pt. I : ESG) Rmetrics Workshop on R in Finance and Insurance 2014, Paris Thierry Moudiki, Frederic Planchet June 27, 2014 Thierry Moudiki, Frederic Planchet Economic Scenarios Generation for Insurance: June 27, 2014 1 / 22
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Economic Scenarios Generation for Insurance: ESGpackage and other tools (Pt. I : ESG)
Rmetrics Workshop on R in Finance and Insurance 2014, Paris
Thierry Moudiki, Frederic Planchet
June 27, 2014
Thierry Moudiki, Frederic Planchet Economic Scenarios Generation for Insurance: ESG package and other tools (Pt. I : ESG)June 27, 2014 1 / 22
ESG
R package ESG was designed to provide a minimal EconomicScenarios Generator (ESG) for valuation and capital requirementscalculations in Solvency II.Currently : projections of risk factors in a risk-neutral world.Available risk factors are : nominal rates, equity returns, propertyreturns, corporate bonds returns.
Thierry Moudiki, Frederic Planchet Economic Scenarios Generation for Insurance: ESG package and other tools (Pt. I : ESG)June 27, 2014 2 / 22
ESG
ESG current structure
Nominal rates
Equity prices Property returnsCorporate bonds
returns
Thierry Moudiki, Frederic Planchet Economic Scenarios Generation for Insurance: ESG package and other tools (Pt. I : ESG)June 27, 2014 3 / 22
ESG
Available risk factorsLet (Wt)t≥0 be a standard brownian motion.
Most simple and intuitive way : Euler scheme (1st order Ito-Taylordevelopment) :
rti+1 − rti = a(θ − rti )(ti+1 − ti) + σε√
ti+1 − ti
Another way : 2nd order development, Milstein scheme. More precise.But When σ is constant, not necessary. More complicated formula.Third way : exact simulation of the transition distribution betweenti+1 and ti :
rti+1 = e−a(ti+1−ti )rti + θ(1− e−a(ti+1−ti )) + σε
√1− e−a(ti+1−ti )
2aThierry Moudiki, Frederic Planchet Economic Scenarios Generation for Insurance: ESG package and other tools (Pt. I : ESG)June 27, 2014 6 / 22
ESG
0.012 0.013 0.014 0.015 0.016 0.017 0.018
010
020
030
040
050
0
Visualizing discretization bias (t = 2) on the example, through densities
N = 500 Bandwidth = 0.0002482
Den
sity
Euler simulationExact simulation
Thierry Moudiki, Frederic Planchet Economic Scenarios Generation for Insurance: ESG package and other tools (Pt. I : ESG)June 27, 2014 7 / 22
ESG
The package’s structureAn S4 object-oriented architecture, around 2 classes : ParamsScenariosand Scenarios, with associated getter and setter methods.
ParamsScenarios Scenarios
horizon A ParamsScenarios attribute
n ForwardRates slot
HW, BSHW, LMN parameters ZCRates slot
Equity/short rate correlation One slot for each model path
Thierry Moudiki, Frederic Planchet Economic Scenarios Generation for Insurance: ESG package and other tools (Pt. I : ESG)June 27, 2014 8 / 22
ESG
Using the package : 2 waysStep by step approach : Using successive getter and settermethods, to know exactly what is done at each step
Thierry Moudiki, Frederic Planchet Economic Scenarios Generation for Insurance: ESG package and other tools (Pt. I : ESG)June 27, 2014 9 / 22
Examples of use of ESG# loading ESGlibrary(ESG)# needed for yield curve interpolationlibrary(ycinterextra)# yield to maturitiestxZC <- c(0.01422,0.01309,0.01380,0.01549,0.01747,0.01940,
# maturitiesu <- 1:30# the yield curve must be interpolated on a monthly basisZC <- fitted(ycinter(yM = txZC, matsin = u,
matsout = seq(1, 30, by = 1/12),method = "SW"))
Thierry Moudiki, Frederic Planchet Economic Scenarios Generation for Insurance: ESG package and other tools (Pt. I : ESG)June 27, 2014 10 / 22
Examples of use of ESG : Step by step approach# object creationobjScenario <- new("Scenarios")# Setting the basic scenario's parametersobjScenario <- setParamsBaseScenarios(objScenario,
horizon = 5,nScenarios = 100)
# Parameters for BSHWobjScenario <- setRiskParamsScenariosS(objScenario,
# visualizing the results :par(mfrow=c(2, 2))matplot(t(y.step$shortRatePaths), type = 'l')matplot(t(y.step$stockPaths), type = 'l')matplot(t(y.interface$shortRatePaths), type = 'l')matplot(t(y.interface$stockPaths), type = 'l')
Thierry Moudiki, Frederic Planchet Economic Scenarios Generation for Insurance: ESG package and other tools (Pt. I : ESG)June 27, 2014 13 / 22
0 1 2 3 4 5
−0.
4−
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HW short rate (step by step)
time
valu
es
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020
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BSHW equity (step by step)
time
valu
es
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HW short rate (direct)
time
valu
es
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BSHW equity (direct)
time
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es
Thierry Moudiki, Frederic Planchet Economic Scenarios Generation for Insurance: ESG package and other tools (Pt. I : ESG)June 27, 2014 14 / 22
Example of Best Estimate Liability calculationWe consider an insurance company, offering a unit-linked contract
The insured party pays a premium equal to 1.The premium is invested in a stock : the unitMaturity : 10 yearsSystematic surrender rate : 2% (unavoidable)Economic surrender rate : 5% (depends on the economic situation).Added to systematic surrender rate whenever the unit-link falls belowthe initial value invested in it, which is 1The contract is entirely redeemed at maturity =⇒ surrender rateat maturity : 100%
In Solvency II, the Best Estimate liability related to the contract is equalto the average discounted value of its future cash-flows
Thierry Moudiki, Frederic Planchet Economic Scenarios Generation for Insurance: ESG package and other tools (Pt. I : ESG)June 27, 2014 15 / 22
Example of Best Estimate Liability calculation (cont’d)
r (s) : the systematic surrender rate (2%)r (e) : the economic surrender rate (5%)∀i = 1, . . . , 10
r (total)i :=
r (s) + r (e)11( SiSi−1
<1) 11(i≤9) + 100%× 11(i=10)
(rt)t≥0 : the instantaneous short rate (HW)(St)t≥0 : the value of the unit (BSHW)Resi = Resi−1 × Si
Si−1
(1− r (total)
i
); Res0 = 1 : the reserves
The Best Estimate liability associated to the contract is equal to :
BEL = E∗[ 10∑
i=1e−∫ i
0 ruduResi
]
Thierry Moudiki, Frederic Planchet Economic Scenarios Generation for Insurance: ESG package and other tools (Pt. I : ESG)June 27, 2014 16 / 22
Example of Best Estimate Liability calculation (cont’d)No close formula for the BEL. . .. . . Or difficult to derive =⇒ Monte Carlo simulation with ESGAn R function, calculFlux, is defined for the calculation of ALMcash-flows. calculFlux depends on projected short rates, projectedvalues of the unit, and the surrender rates, depending on the latterParameters for ALM projection :
k <- 0.12 # short rates' mean-reversion speedsTaux <- 0.05 # volatility of short ratessUC <- .16 # volatility of the unitrho_rS <- .5 # correlation unit vs short ratesH <- 10 # maturity of the contractnSimulations <- 1000 # number of simulationstauxRachatS <- .02 # systematic surrender ratestauxRachatC <- .05 # economic surrender rates
Thierry Moudiki, Frederic Planchet Economic Scenarios Generation for Insurance: ESG package and other tools (Pt. I : ESG)June 27, 2014 17 / 22
Example of Best Estimate Liability calculation (cont’d)
set.seed(10)# Simulation of the unit and of short rates with rStocktraj <- rStock(horizon=H, nScenarios=nSimulations, ZC=ZC,
# Short ratestrajectoiresTaux <- traj$shortRatePaths# Unit (a stock)trajectoiresUC <- traj$stockPaths# Future cash-flows and discount factorsFlux_futurs <- calculFlux(trajectoiresTaux,trajectoiresUC,
## [1] "Valeur de la moyenne des flux futurs actualisés par simulation : 0.991227449251919"
Thierry Moudiki, Frederic Planchet Economic Scenarios Generation for Insurance: ESG package and other tools (Pt. I : ESG)June 27, 2014 21 / 22
ESG
Future versionsMore flexibility on the interpolation of zero-rates (not onlymonthly frequency required)Projection is annual =⇒ impossible to obtain correct estimations ofdiscount factors =⇒ add an option for changing the samplingfrequencyAdding correlation/dependence between the risk factorsAdding real world modelsESGtoolkit, will be used in ESG
Thierry Moudiki, Frederic Planchet Economic Scenarios Generation for Insurance: ESG package and other tools (Pt. I : ESG)June 27, 2014 22 / 22