GIRO Convenion 23-26 Sepember 2008 Hilon Sorreno Palace A Pracical Sudy of Economic Scenario Generaors For General Insurers Gareh Haslip Benfield Group Agenda Inroducion o economic scenario generaors Building blocks of an ESG Time horizons Ineres rae modelling Real world versus risk neural Calibraion echniques Equiy reurn modelling Wha is an ESG? A sochasic model of key economic variables, e.g. equiy reurn, inflaion, ineres raes, FOREX ec Two key purposes: risk managemen or pricing Tha is: eiher real world or risk neural For general insurance we are primarily concerned wih real world ESG s (risk neural is used in life insurance o price coningen liabiliies)
Wha do general insurers need in an ESG? Primarily ineresed in risk managemen Need o projec how he asses and liabiliies of an insurer will evolve ime Asse value driven by equiy reurn, ineres raes, inflaion Liabiliies are impaced by claims inflaion and ineres raes (discouning) Building Blocks of an ESG Can model inflaion, nominal ineres raes, equiy reurns as disinc random processes and inroduce correlaions beween hem o reflec he co-dependencies Or build he model up in erms of key economic building blocks: inflaion, real ineres, excess equiy reurn Nominal Ineres Equiy Growh Building Blocks of an ESG The second approach is normally favoured and can also be applied o model claims inflaion Claims inflaion = Price inflaion + XS Inflaion Excess claims inflaion can be modelled using variey of processes (e.g. Jump processes for cour claims inflaion) Moor BI Employer PI Medical
Some ESG s in he Public Domain The Wilkie model A Sochasic Asse Model & Calibraion For Long-Term Financial Planning Purposes by John Hibber, Philip Mowbray & Craig Turnbull (June 2001) Modeling of Economic Series Coordinaed wih Ineres Rae Scenarios Sponsored by he CAS and he SOA (July 2004) Undersanding ESG s Many companies purchase ESG s from exernal providers How does his fi ino regulaory models? Aricle 119 Saisical Qualiy Tes - Should be able o jusify he assumpions underlying he inernal model o he supervisory auhoriies UK ICA The firm remains responsible for he reliabiliy of he underlying assumpions - his responsibiliy canno be passed on o a hird pary I is imporan ha acuaries undersand he models and calibraion process No sufficien o delegae his process enirely o he provider Time Horizons How he ESG is calibraed depends on he use 1 year model Shor erm focus Calibraion should reflec curren marke condiions, no long erm averages How can he marke value of asses change over he nex year? This is significan issue for he equiy model Need a forward looking measure of volailiy Less significan for ineres raes and inflaion
Modelling Ineres Raes (and inflaion) Shor rae models Popular choice for ESG s, idea is o develop a model for he real world evoluion of he insananeous ineres rae...... and use his o compue he full erm srucure of ineres raes under he risk-neural measure Correlaion wih oher economic variables is provided hrough Brownian moion shocks A Selecion of Shor Rae Models One facor models have he following form: dr ( r ) d σ ( r ) dw = μ + Vasicek: ( r ) α( μ r ), σ ( r ) σ μ = = Cox-Ingersoll-Ross: Black-Karasinski: ( r ) = α( μ r ), σ ( r ) σ r μ = ( r ) α( ln( μ) ln( r )), σ ( r ) σ μ = = Using a Shor Rae Model Recall ha we are ineresed in modelling he real world evoluion of ineres raes...... bu we also need he risk neural model for deriving he yield curve This is a common poin of confusion in heir use Moving from real world o he risk neural world is achieved via he marke price of risk This represens a erm premium ha invesors demand for holding money for longer periods
The Marke Price of Risk Example: Vasicek Real world model: dr ( μ r ) d σdw = α + If we assume marke price of risk is: λ () = λ Then risk neural model is: (via Girsanov) ~ dr = α( μ r ) d + σ ( λ( ) d + dw ) λσ ~ dr = α μ + r d + σdw ( ) α Vasicek Real World vs. Risk Neural Real world: Risk neural: dr dr ( μ r ) d σdw = α + = α ~ λσ ( μ + r ) d + σdw α Model differs only by he long erm mean reversion rae How do we esimae he parameers? Calibraion Mehods 1) Proxy Approach - Mehod The shor rae canno be direcly observed in he marke, bu shor erm ineres raes should be similar e.g. 1 monh ineres rae The real world model for he shor rae is fied o he hisorical proxy ime series daa Use maximum likelihood esimaion or mehod of momens
Calibraion Mehods 1) Proxy Approach - Difficulies Hisorical daa for very shor daed reasury sock limied Usually have o look a 3 monh rae as proxy We do no make use of hisorical informaion abou he yield curve a oher duraions In realiy he erm srucure provides informaion abou he expeced pah of he shor rae Only provides real world parameers The marke price of risk is no recoverable Calibraion Mehods 2) Cross-Secional Approach Mehod Hisorical daa is rich full yield curve is available a every dae Cross-secional mehod uses all daa I assumes n spo raes are observed wihou error (o back-ou he unobservable shor rae) and remaining spo raes are observed wih error Parameers found using maximum likelihood esimaion along all hisorical daes and across all spo raes Calibraion Mehods 2) Cross-Secional Approach Properies Complicaed o implemen and runs slowly Maximum likelihood problem ill-posed Many local maxima This is due o he model implied shor rae changing for each combinaion of parameers Provides boh real world parameers and average marke price of risk Opimal in he sense ha i incorporaes all available informaion
Calibraion Mehods 3) Swapion Implied - Mehod Raher han use hisorical daa, we can use observed marke prices o calibrae risk neural model Parameers are chosen o minimise he sum of square difference beween modelled and observed swapion prices Provides calibraion suiable for pricing liabiliies coningen on ineres raes e.g. guaranees Calibraion Mehods 3) Swapion Implied - Difficulies The mehod is only sricly necessary if you need o price ineres rae derivaives For risk managemen we need real world model I s possible o move back o real world using an assumpion for he marke price of risk......bu volailiy assumpion is unlikely o be valid Swapion implied volailiy will no be consisen wih realised volailiy Alhough arguably a forward looking measure of ineres rae volailiy is more appropriae for 1 year models......can consider sochasic volailiy approaches (complicaed for ineres rae modelling) Example of Proxy Mehod Daa se: Daily US Treasury Bond Yields Proxy: 3 monh spo rae Model: 1 Facor Black-Karasinski Real world: d ln r = α ln μ ln r d + Risk neural: d ln r α( ~ ln μ ln r ) d + ~ σλ μ = μ exp ( ) σdw ( ) ε ~ σdw =
Example of Proxy Mehod Hisorical MLE esimae provides real world model parameers: ˆ α = 0.14115, ˆ μ = 0.026338, ˆ σ = 0.35224 How o find λ? The risk neural version of he model should be consisen wih he curren yield curve Therefore we should use λ o fi he model o he curren yield curve Example of Proxy Mehod We herefore need o calculae spo raes under he Black-Karasinski model for parameers ˆ, α ˆ, μ ˆ σ and some value for λ There is no closed form for spo raes under BK They are compued by numerical mehods Trees, finie differences, Mone Carlo We apply a rinomial ree (see Brigo & Mercurio) λ is seleced o minimise he sum of square error beween acual and modelled spo raes Example of Proxy Mehod Varying λ changes he shape of he yield curve 5.00% 4.50% 4.00% BK Implied Yield Curve vs. Acual Spo Rae 3.50% 3.00% 2.50% 2.00% 1.50% 0 5 10 15 20 25 30 Term Acual BK Spo Opimal λ = -0.09749 provides a good fi Noe ha hrough ime varying parameers an exac fi is also possible
Example of Proxy Mehod Oupu of a single scenario of he real world evoluion of he shor rae: 0.12 Evoluion of he Shor Rae 0.1 0.08 r 0.06 0.04 0.02 0 0 2 4 6 8 10 12 14 Time (Years) Comparison of Ineres Rae Models Probabiliy densiy for he 3 models calibraed o he daa se (5 year spo rae) 100 90 80 70 Comparison of Ineres Rae Models Probabiliy Densiy 60 50 40 30 20 10 CIR Vasicek BK 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Ineres Rae Comparison of Ineres Rae Models Black-Karasinski has heavy righ ail (due o log-normally disribued shor rae) No represenaive of hisorical ineres rae Cox-Ingersoll-Ross is underweigh in he ails Vasicek provides closes mach o hisorical disribuion of ineres raes Issue wih negaive ineres raes can be ignored for real ineres rae models...... or simply runcaed a zero if modelling nominal ineres rae
Comparison of Ineres Rae Models Useful o look a 0.5 h and 99.5 h perceniles for % change in spo raes over 1 year Ineres Rae Model 0.5 h Percenile of 5 Year Spo Rae Cox-Ingersoll-Ross -31% 49% Vasicek -55% 61% Black-Karasinski -42% 81% Solvency II -40% 56% 99.5 h Percenile of 5 Year Spo Rae Again i can be seen ha BK oversaes righ hand ail and CIR underweigh in ails: Vasicek bes choice Shor Rae Models - How Many Facors? Common criicism of one facor models is hey induce perfec correlaion beween spo raes of differen duraions Below is a able of realised correlaions on spo rae from US Treasury bonds Correlaion 1 year 2 year 3 year 5 year 7 year 10 year 20 year 30 year 1 year 1.00 0.99 0.97 0.91 0.87 0.81 0.68 0.64 2 year 0.99 1.00 0.99 0.96 0.93 0.88 0.78 0.77 3 year 0.97 0.99 1.00 0.99 0.96 0.92 0.83 0.84 5 year 0.91 0.96 0.99 1.00 0.99 0.97 0.91 0.92 7 year 0.87 0.93 0.96 0.99 1.00 0.99 0.95 0.94 10 year 0.81 0.88 0.92 0.97 0.99 1.00 0.98 0.98 20 year 0.68 0.78 0.83 0.91 0.95 0.98 1.00 0.99 30 year 0.64 0.77 0.84 0.92 0.94 0.98 0.99 1.00 Shor Rae Models - How Many Facors? Clear ha spo raes beyond 5 year duraion are all perfecly correlaed Slighly less correlaion beween < 5 year and longer erm raes No necessarily a problem for non-life insurers who Do no have complicaed fixed income porfolios Are no valuing embedded ineres rae opions (life insurers need more facors o mach he swapion implied volailiy surface) If desirable o de-correlae spo raes move o muli-facor models N.B. calibraion becomes more involved
Excess Equiy Reurn Models Wha is required from he equiy model? Only ineresed in real world evoluion of excess equiy reurn Unless equiy derivaives held in asse porfolio I should capure he heavy ails observed in hisorical equiy reurn I mus be suiable for shor erm projecions i.e. volailiy should reflec curren marke condiions Excess Equiy Reurn Models There are many differen models described in he lieraure for modelling equiy reurn Exponenial Brownian moion Regime swiching models (e.g. Hardy) Exponenial jump diffusion (e.g. Meron, Kou) Sochasic volailiy (e.g. Heson) Which one is appropriae? Exponenial Brownian Moion This is he model underlying he Black- Scholes equaion ds = S ( μ d + σdw ) Easy o parameerise o hisorical daa, bu his provides hisorical average volailiy Wha ime period is appropriae for measuring hisorical volailiy? Difficul o achieve appropriae calibraion ha capures shor erm volailiy
Regime Swiching Model The economy is assumed o have wo saes: Sable sae wih low variance and seady reurns Volaile sae wih reurns highly posiive or negaive Equiy risk premium is assumed o follow a log-normal disribuion in boh saes (bu wih differen mean and variance) Swiching beween saes follows a coninuous wo sae Markov process Easy o calibrae and provides good fi o hisorical daa Alhough calibraed parameers change if discreisaion sep is alered Exponenial Jump Diffusion Exension o exponenial Brownian moion o include jumps ( μd + dw + ( e ) dn ) ds = S σ 1 J dn() indicaes if a jump has occurred in [, +d] J is he size of he jump Meron Normal disribuion Kou Asymmeric double exponenial disribuion Works very well for risk neural modelling (Exoics) Poor fi for real world jump frequency is found o be approx. 40 per annum using MLE mehods Heson Model (Sochasic Volailiy) Exension o exponenial Brownian moion o allow for ime varying volailiy ds = S dy = κ (1) ( μd + Y dw ) (2) ( γ Y ) d + σ Y dw Y() represens he variance of sock prices Model provides fi o curren marke condiions Good choice for Solvency II models capuring 1 year equiy risk
Excess Equiy Models Fied o S&P 500 Regime Swiching and Exp BM similar in disribuion They boh have higher upside semi-variance ha hisorically observed 1 Year Excess Equiy Reurn Models 3 2.5 Probabiliy 2 1.5 1 Heson Regime Swich Exp BM 0.5 Heson model capures curren marke condiions and provides beer fi o hisorical daa 0 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Excess Equiy Reurn Excess Equiy Models Fied o S&P 500 Useful o compare 0.5 h percenile for excess equiy reurn under each calibraed model Equiy Model 0.5 h Percenile Over 1 Year Exponenial Brownian -32% Moion Regime Swiching -38% Heson Model -42% Solvency II (Global) -32% Significan difference in 1 year change in marke value when using curren marke condiions Are Solvency II assumpions appropriae? Calibraion Issues Expeced excess equiy reurn is hisorically much more sable han nominal equiy reurn bu sill difficul o choose appropriae fuure value wha s an appropriae ime window? Annual Reurn in Excess of Cash 0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 Rolling 5 Year Average Reurn in Excess of Cash For S&P 500 03/01/67 03/01/77 03/01/87 03/01/97 03/01/07 Dae
Calibraion Issues Similar issue for choosing volailiy assumpion Wha s an appropriae hisorical ime horizon o measure realised volailiy? Can resolve his using Heson model by using forward looking volailiy measure E.g. VIX index for S&P 500 Implied volailiy for oher markes Quesion of appropriae adjusmen implied volailiy normally over-esimaes realised volailiy Conclusion Many differen models for ineres raes and equiy reurn Each have respecive pros/cons, bu some clear winners Complicaed underlying financial heory Vial ha acuaries undersand he implicaion of model choice and calibraion mehodology ESG s for one year models should reflec curren marke condiions, no hisorical condiions