Introduction & Background

Transcription

1 Taking the lid of Least Squares Monte Carlo urak Yelkovan 08 Novemer 03 Introduction & ackground

2 Introduction Proxy models are simplified functions that Represent liailities and/or assets Can very quickly e revalued under revised risk driver values Are calirated to results from detailed actuarial models Proxy models are used for rapid updates of the alance sheet for: Regular solvency monitoring Capital calculation l with Monte Carlo simulations Various forms of proxy models are in use with the most popular approach in the UK eing the use of polynomials. Example with two risks, x =x, x ), up to quadratic terms: f x c a x a x a x x a x a x, x ) Linear relationship Interaction etween risks Non-linear relationship 3 Introduction The proxy modelling caliration inputs consist of two items: Stressed scenarios outer scenarios ) alance sheet in stressed scenarios Inner scenarios: For deterministically valued usiness, there is one inner scenario. Outer scenarios ase case Inner scenarios Stress impacts Outer scenarios: These can e prepared in different ways Manually selected, often according to a rule Randomly generated A comination of manual and random Manual Random For stochastically valued usiness, there could e thousands of inner scenarios. 4

3 Proxy modelling errors Measuring the error There are different measures for errors. The most RMSE: commonly Root mean square used error are: Maximum asolute error Commonly used. The RMSE is minimised using regression techniques i.e. minimising least squares). No firm relation etween RMSE and SCR: RMSE is an average error. The error in the SCR estimate may e larger ecause not all points have an equal impact on the SCR. Under certain conditions, the error function using least squares is equivalent to a Legendre polynomial: Minimising the maximum asolute error minimax ) is difficult to compute. Firm relationship etween maximum error and SCR: The error on the SCR cannot exceed the maximum error of the proxy model. Hence, minimax is theoretically a etter ojective to optimise than the RMSE. Under certain conditions, the error function using minimax is a equivalent to a Cheyshev polynomial: Maximum error Maximum error Regression is often preferred to a minimax fit as it is easier to compute ut it is difficult to translate a RMSE into error ounds for the SCR. 5 The proxy modelling process Risk factors and caliration simulations Which risk factors / alance sheet states should e used? How to draw from a risk neutral return distriution starting at t=? How many RN sims per point? Type of risk factors and caliration distriution of caliration points Validation of caliration simulations Specification of Regression Regression method e.g. least squares with asis, non-parametric methods,...) Transformation of risk factors for regression z.. ln or Nd)) as enlargement of asis function space Criterion for selection of regression terms from a numer of asis functions Control variates Testing procedures and quality criteria Statistical assessment, analysis of residuals Plausiility of results Choice of Out-of-sample test points Quantitative quality criteria Choice of software Licensed software vs. in-house development Integration in the overall SCR calculation / more general process Definition of interfaces Automation of the process LSMC and Proxy Models 6 0 &W Deloitte GmH 3

4 Proxy modelling errors A taxonomy There are three types of errors with proxy models. These are summarised elow including considerations how these can e minimised and reduced. When using specific methods, it is important to e aware of the nature of the errors in order to improve the fit. Description Considerations Sampling error inner scenarios) Error from randomness in fitting to stress results ased on Monte Carlo scenarios i.e. options and guarantees) Oserved for all stochastically valued usiness Can e reduced y running more simulations LSMC can correct for this error Can e measured using regression techniques Fitting error Error from using a fitting technique that does not produce the optimal fit i.e. does not achieve minimax) for the choice of asis functions Present in all proxy models that are not fitted to minimise the maximum error Can e reduced y using more appropriate stress tests for caliration and/or minimax fit Cannot e measured using regression techniques Spanning error Error in the est fit to a given choice of asis functions i.e. fit cannot e improved y altering the fitting technique or outer scenarios) Present in all proxy models Can e reduced y increasing the polynomial order or use of terms that capture the liaility ehaviour more closely e.g. replicating portfolios and implied parameter lack Scholes) Cannot e measured using regression techniques 7 Proxy modelling LSMC 4

5 LSMC introduction Run time of detailed actuarial models is a challenge, particularly for stochastically modelled usiness. LSMC makes etter use of outer scenarios which results in lower run time of detailed actuarial models for caliration than the traditional curve fitting approach to stress tests. However, er LSMC requires more onerous validation as none of the caliration stress test results help in assessing the goodness-of-fit see illustration elow). For given functions, the fits can e equally good for stress tests with least squares and LSMC. Caliration of general proxy model manually selected outer scenarios) Caliration effort = # outer scenarios * # inner scenarios LSMC Increase numer of outer scenarios and decrease numer of inner scenarios LSMC is ased on polynomials and suffers the same challenges as the traditional polynomial curve fit to stress tests. 9 LSMC in more detail LSMC uses polynomials uses randomly generated outer scenarios is applied to stochastically valued usiness i.e. for cost of guarantee) Outer scenarios Inner scenarios Inner scenarios must e generated using a market consistent ESG reflecting the market conditions implied y the outer scenarios. ase case Stress impacts Outer scenarios must e expressed in terms of the risk drivers X, X,.. used in the proxy model. 0 5

6 LSMC in more detail - Process Define framework Determine approach and parameters e.g. risk drivers and distriutions numer of inner scenarios validation scenarios Produce outer scenarios Sample from multivariate distriution Comined stresses for market and nonmarket risks Run ESG Calirate and run ESG for each market risk outer scenario Calirate and run ESG for each validation scenario Run asset and liaility models Set up and run full asset and liaility models with ESG files for each outer scenario Run full asset and liaility model for each validation scenario Fit formulae Use simple least squares regression to fit polynomial formula using results of each outer scenario Validate results against validation runs Test fit LSMC in more detail The strength of polynomials is that they are a generic family of curves: Can e used for lots of classes of usiness. Can e used when not all the factors impacting the usiness are transparent. Increasing the order of polynomials will ultimately result in convergence to true function see elow) UT a very high order with complexity and run time implications) may e required for the fit to e good enough. However, there are limitations in practice: Option ehaviour is poorly descried y polynomials. This can e seen on the second derivative of the option value: The second derivative of an option price is a ell shaped curve. The second derivative of a polynomial is still a polynomial. 6

7 Refining proxy models - Illustration A simple step for annuities is to model cashflows directly rather than fitting a polynomial to the EL this can improve results consideraly. The following example looks at fitted and actual results under five different yield curve stresses YC to YC5). Challenge: Replication of historical yield curves with limited numer of yield risk factors. Spot rate Term Present value vs. cash flows 3 Refining proxy models - Illustration Fitting to the implied parameters of a lack-scholes formula can e used as an approach to using polynomial asis functions. This only applies to options and guarantees GAO example e.g. With-Profits usiness). Thechart elow shows a comparison of polynomialand and implied parameter lack Scholes fitting. The approach has the advantage of 5 scenarios at the in 00 net asset stress were tested matching the ehaviour of the liaility value for oth approaches. The implied parameter lack Scholes errors are aout whilst eing ale to use low order 50% smaller on average. polynomials. It makes validation easier as it is possile to check inputs as well as the formula for reasonaleness. To use the full enefit of this approach, it should e applied at a granular level e.g. grouped model points). Option pricing formulae such as lack-scholes) can allow for option-type ehaviour and reduce spanning error. 4 7

8 LSMC The caliration prolem The ESG needs to e recalirated for each outer scenario. There are two options: Use of latent parameters: The outer scenarios consist of simulated ESG parameters. No recaliration required as all required parameter can e read off the outer scenarios. However, the outer scenarios must e calirated to ESG parameters: What is a -in-00 move of the mean reversion rate? Use of oservale market variales: The outer scenarios consist of yield curves, volatilities, etc., and hence a recaliration is required. This approach is more intuitive for plausiility checks and etter communication. Latent Parameters Analytical in simple cases or Monte Carlo Simulation Caliration May require Monte Carlo goal seek Oservale Risk Drivers Outer scenarios: A popular choice for generating the outer scenarios is the uniform distriution. However, our research showed that outer scenarios need to e generated in a specific way to give a etter fit: The wider the range of the outer scenarios, the etter the fitting is. Having more outer scenarios at the edges of the interested range gives a etter fit, i.e. the use of the uniform distriution is not optimal. The quantity to fit drives the choice of the range of the outer scenarios. The selection of outer scenario distriutions for LSMC is still often a matter for sujectivity. 5 LSMC in more detail LSMC uses linear regression to fit a formula. Linear regression is a well researched area and a numer of statistical techniques are availale to test a regression fit. However, statistical techniques applied to proxy model fits should e handled with care. Statistical goodness-of-fit measures can e misleading, understating the fitting error. Regression techniques assume that residuals are independent and identically distriuted. ut they are not: Spanning error, which occurs ecause the underlying function to fit is not a polynomial. The first picture shows random errors, the second spanning error. Volatility stress tests lead y construction to residuals that are more volatile than stress results without volatility stress. It is tempting to look at confidence intervals when assessing the error of a LSMC fit. Example: Put option exposed to interest rate level, equity level and equity volatility risk The following chart shows the 0 scenarios around the fitted 99.5 th percentile. For each scenario the following quantities are shown: Fitted result dark lue line) True result green and red diamonds) Confidence intervals light lue lines) Oservation: The use of confidence intervals failed in this example as all ar three validation scenarios lie outside the confidence intervals. Statistical methods to assess the goodness-of-fit fail as the assumptions to apply these tests are not met. Users potentially get false comfort from good fitting measures. 6 8

9 9 n p Y Y Y Y ; ; Using Linear Regression to Fit Proxy Functions T T T T n np n n Y Y p n Y I N Y Y ˆ ) ˆ) ˆ) ˆ ) ˆ) ) ˆ ) 0, Var n = # fitting scenarios p = # asis functions Y = OF in each fitting scenario = asis functions in each fitting scenario Want to estimate OF) p n T T T T p n T p n ~ student ) ˆ ˆ ) ˆ) ~ ) Var Risk factor Selection A fundamental prolem given the large amount of risk drivers availale to select as a part of the proxy modelling process and not unique to LSMC) is selecting the underlying polynomial for use in the intended algorithm. There are many ways in which the underlying terms are selected or de-selected, the most common are listed elow: Use of Akaike Information Criteria AIC): The AIC is the estimate of a constant plus the relative distance etween unknown true likelihood function of the data and the fitted likelihood function of the model. AIC = k - lnl) Use of ayesian Information Criteria IC): The IC is an estimate of a function of the posterior proaility of a model eing true under a certain ayesian setup. IC penalises model complexity more heavily. 8 IC = klnn)-lnl) Where k is the numer of free parameters, L is the Maximum Likelihood Function of the model. A lower value of AIC and IC is preferred.

10 Optimisation We can optimise our LSMC estimate in a numer of ways, with the most common eing listed elow: Add increasing numer of risk factors, cross terms, and increase the power and order of the polynomial. Whilst this does provide increasing effectiveness at diminishing rates of information, there may e certain scenarios for which the LSMC does not provide a very good fit nonetheless. Attempt to minimise the maximum error within a tolerale level versus the least squares, although there may e increased error around the iting/critical scenario as a result. Use Control Variates to segregate known simulation errors from the total monte carlo simulation error. Using Control Variates reduces the overall variance of the polynomial estimate proxy function and hence leads to more stale results with reduced confidence intervals. An example of using Control variates and how to calculate them: complex liaility standard features exotic feature Model an exact closed form value. Eg. ZC, vanilla swaptions etc. Remaining Simulation variaility 9 LSMC in more detail ID the formula shape There are two main approaches for identifying the polynomial terms: Manually identified y looking at selected stress impacts. Automatically identified using statistical techniques. Manual Automatic Specific outer scenarios are defined in order to identify the The most common approaches are step-wise optimisations: shape y risk and risk pairs: Polynomial shapes are identified in isolation for each risk driver y looking at single risk stresses. Transformation of risk drivers are applied if they improve the fit to lower order polynomials. Then cross terms are fitted. Cannot e used with randomly generated outer scenarios. At each step in these algorithms, possile additional formula terms are tested and the term that most improves the fit is selected. The algorithm can e forwards, ackwards or oth ways selecting. 0 0

11 LSMC in more detail Step-wise regression techniques can e used to automate the selection of formula terms ut these techniques should e applied with care: Assumptions underlying the algorithms do not apply, so the fits may not e optimal. Over-fitting can e a prolem. Confidence intervals generated y these tools can e invalid, particularly for LSMC fits, due to assumptions not eing met. Significant out-of-sample testing must still e done to judge the success of the fit. The techniques do not consider transforming risk drivers to improve the fit. Example of over-fitting: The following two oservations indicate over-fitting: Residuals on the validation runs are large while residuals on caliration runs are close to zero. Residuals on the validation runs increase as more terms are added. We elieve that optimisation techniques can e useful in automating and speeding up fitting processes, ut they should e used with care and comined with manual investigations. Comparison of methods Curve fitting to stress tests and LSMC are oth using polynomial asis functions and are therefore suject to spanning error. The two approaches can get the same answer when the outer scenarios are unched in a small numer of fitting scenarios. LSMC can use higher order polynomials without significantly increasing the run time for creating the fitting inputs. With curve fitting to stress tests, the fitted curve can e compared against alance sheet impacts of outer scenarios to provide additional visual comparison of the fit while alance sheet impacts from LSMC processes have too ig a sampling error to e used. Curve fits to stress tests can e derived using existing models without coding changes while the use of LSMC requires changes to liaility models to use the scenarios generated for the fitting. The generation of fitting scenarios for LSMC outer scenarios followed y inner scenarios) is complex and requires a tool.

12 LSMC An example Example Maturity Guarantee Unit Value ma aturity moneyness = guaranteed value minus unit value Time 4

13 achelier s 900) Option Pricing Model Random Walk Model Assume future moneyness = current moneyness +N0s N0,s ) Option price is the conditional )expected future moneyness Function of current moneyness and of the volatility s Contemporary option pricing models are more complicated than this 5 Can we Avoid Nested Scenarios? Guarantee cost at Maturity Scenarios Moneyness in one year 6 3

14 Linear fit compared to Nested Monte Carlo Maturity Guarantee cost at Scenarios Linear ased on scens Nested MC Moneyness in one year 7 Higher Order Polynomials Guarantee cost at Maturity Scenarios Linear Quadratic Cuic Quartic Quintic Sextic Nested MC Moneyness in one year 8 4

15 Errors do not look like Random Noise! 6% 5% 4% 3% % % 0% % % 3% 4% 5% Error oneyness % max m Quadratic Cuic Quartic Quintic Sextic 9 D Spanning Error: Accuracy Potential Max error as % maximum money yness 5% Quadratic 4% 3% % Quartic % Sextic Octic 0% Maximum moneyness as multiple of volatility 30 5

16 What Aout Varying Volatility? Max error as % moneyness 0% 9% 8% 7% 6% 5% 4% 3% % % 0% Maximum vol as multiple of moneyness Quadratic Cuic Quartic Quintic Sextic Heptic Octic 3 LSMC Conclusion 6

17 The final word isn t written yet Least squares with polynomial asis functions common today): Advantages Polynomials are easy to understand and simple to implement Extremely fast calculation solution of a linear system of equations) Can get the same answer as curve fitting to stress tests when the outer scenarios are unched in a small numer of fitting scenarios Can user higher order polynomials due to the richness of the outer scenarios Useful toolkit to have ut not a panacea Error estimates comine inner scenario sampling error and estimation of asis function loadings Disadvantages Requires a tool that automates the recaliration of the ESG to each outer scenario Requires changes to liaility models to use the simulations generated for the fitting More extensive validation Model error! True liaility function may not e well-approximated y polynomials) Liaility estimate at every caliration point influences estimate at every other, no matter how distant Difficult to apply inner scenario validation tools aritrage-free and market-consistent tests, caliration tests) for each outer scenario Methodology improvements in this area are likely, so it is desirale to have the option to upgrade the asis functions without changing the rest of the process. 33 Possile alternatives in the future Fully non-parametric regression methods Advantages Very successful for many applications in other fields Appears to work well when used for proxy models for a few risk factors Possile to extract OF-properties such as smoothness of the OF function Aility to validate the noise in the model Disadvantages High dimensions and scarcity of caliration points an even greater challenge Mathematics ecomes consideraly more complex No clear method to identify the optimal nonparametric method 34 7

18 Questions Comments Expressions of individual views y memers of the Institute and Faculty of Actuaries and its staff are encouraged. The views expressed in this presentation are those of the presenter. 08 Novemer