Equity-Based Insurance Guarantees Conference November 1-2, New York, NY. Operational Risks

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1 Equity-Based Insurance Guarantees Conference November -, 00 New York, NY Operational Risks Peter Phillips

2 Operational Risk Associated with Running a VA Hedging Program Annuity Solutions Group Aon Benfield Securities, Inc hours November, 00 EBIG Conference November, 00

3 Legal Disclaimer This was prepared for informational purposes only and is intended only for the designated recipient. It is neither intended, nor should be considered, as () an offer to sell, () a solicitation or basis for any contract for purchase of any security, loan or other financial product, (3) an official confirmation, or (4) a statement of Aon Benfield Securities, Inc. or any of their affiliates. With respect to indicative values, no representation is made that any transaction can be effected at the values provided and the values provided are not necessarily the value carried on Aon Benfield Securities, Inc. books and records. The recipient of this document is advised to undertake an independent review of the legal, tax, regulatory, actuarial and accounting implications of any transaction described herein Aon Benfield Securities, Inc. does not provide legal, tax, regulatory, actuarial or accounting opinions. Any offer will be made only through definitive agreements and such other offering materials as provided by Aon Benfield Securities, Inc. or their appropriately licensed affiliate(s) prior to closing which contain important information regarding, among other things, certain risks associated with any transaction described in this document and should be read carefully before determining to enter into such a transaction. November, 00

4 Agenda Section Introduction Section Model Implementation Risks Section 3 Quantifying Greeks Estimation Error Section 4 Intraday Greeks Approximation Risks Section 5 Summary November, 00 3

5 Section : Introduction 4

6 Introduction Running a Variable Annuities hedging program is extremely challenging and prone to serious risks, such as: market risk, strategy risk and operational risk The performance of a hedging program, capital requirements and reserves are highly dependent on the choice of stochastic models that are used to model the Liability We believe that Rho is the largest risk factor by far many VA products, however, some direct writers that hedge their Rho risk are not using a stochastic interest rate model (such as Hull-White F/F model) due to runtime constraints, and as more direct writers use stochastic interest rate models, care must taken to ensure they are properly implemented There are multiple approaches to calculate a point-in-time Greeks on a book of business Some companies are not aware of the estimation risk associated with different approaches which in turn can have a potentially important impact on hedge program performance Most companies are not able to calculate book level seriatim Greeks on an intraday basis, and therefore use approximation techniques, such as grid interpolation or Taylor Series expansion ( by using st, nd and some 3 rd order Greeks) These techniques work great when the market goes sideways What happens when the market moves sharply? This presentation will concentrate on some of the operational risks associated with: Implementation risk associated with stochastic interest rate scenario generators Point-in-time Greeks estimation risk Intraday Greeks estimation risk for trade decision support November,

7 Section : Model Implementation Risks 6

8 Model Implementation Risks Design Inventory Design choices to make when implementing a Liability model Economic Scenario Generator How do you model bond fund risk? How do you model interest rate risk? How do you model volatility risk? Which calibration technique should you use? Liability Cash Flow Model Time step discretization logic Daily, Monthly or Yearly? How to properly implement stub logic? Mortality model selection Lapse model selection Survivorship model selection Joint Life modeling Cohorting logic and rationale Mixed, single or double precision Stand alone implementation testing of Models Integration testing of the Liability Cash Flow Models and Scenario Generation Models November,

9 Model Implementation Risks Model Description Hull-White Two-Factor Model Description The short interest rate is defined as: r( t) ( t) X ( t) Y ( t) With the two factors X and Y defined by the linear SDE s dx axdt dw ( t) dy bydt dw ( t) W and W define a two-dimensional wiener process with correlation: The deterministic function φ is given by: ( T ) dw dw dt at bt at bt e e e e aa bb M f (0, T ) Where f(0,t) is the initial instantaneous forward rate term structure ab November,

10 9 Model Implementation Risks Scenario Generator The scenario generator is constructed by discretizing the solution to the linear SDE s that define X and Y Hull-White Two-Factor Scenario Generation The scenario generator is constructed by discretizing the solution to the linear SDE s that define X and Y Where Δt is the time step and Z and Z are independent standard normal random variables (0,) (0,) ) ( ) ( (0,) ) ( ) ( i i t b i t b i i t a i t a i Z Z e b t Y e t Y Z e a t X e t X Due to runtime constraints an Euler scheme ( st order approximation) is often used. This leads to the simplification: (0,) (0,) ) ( ) ( ) ( (0,) ) ( ) ( ) ( i i i i i i i Z Z t t Y t b t Y Z t t X t a t X November, 00 8

11 Model Implementation Risks Testing of Proper Implementation Probability of Negative Rates The short rate has a normal distribution with mean and variance given by: r ( t) ( t) r ( t) a at bt ( ab) t e e e b This allows for the possibility of negative rates and also serves as a test of the implementation. The probability of negative rates is given by: a b ( t) r r ( t) 0 r ( t) where Φ denotes the standard normal cumulative distribution function. P Aside from the theory one needs to consider in practice what do with negative short rates Use them Set them to a very small number November,

12 Model Implementation Risks Testing of Proper Implementation Plot of Probability of Negative Rates Using the Euler approximation to the SDE s results in a higher probability of negative rates November, 00 0

13 Model Implementation Risks Testing of Proper Implementation Probability Distribution Check The short rate in the HWF model is normally distributed The mean and variance of the short rate can be checked against the theoretical results of the model Discretization errors introduced in simulating the short rate can lead to an incorrect distribution November, 00

14 Model Implementation Risks Discount Factors Discretization Error And The Affect on Discount Factors Often times one is interested in calculating discount factors from the short rate. This involves computing: DF exp t r( u) du 0 The discretization errors introduced by the scenario generator for the short rate are propagated through to the discount factors. Furthermore, additional errors can be introduced by numerically approximating the integral in the expression above. A common approach is to use: Y ( t ) t 0 r ( u ) du i j r ( t ) j t j t j But this can be avoided by simulating paths of the pair {r(t),y(t)} ( )} without discretization error given that, under the Hull-White model, the pair is jointly Gaussian November, 00 3

15 Model Implementation Risks Testing of Proper Implementation Interest Rate Option Monte Carlo Valuation versus Hull-White Closed Form Use Interest Rate Scenario Generator to price European Interest Rate options with known closed-form solution HWF parameters Risk-free rate: 5% Volatility: % Mean-Reversion Strength:.0 European Bond call option Bond notional: $00 Bond maturity: 0 years Strike price: $77 Option Expiry: 5 years Monte Carlo runs 00 paths each November,

16 Model Implementation Risks Conclusion It is not simple as it looks to implement and thoroughly test a stochastic interest rate model Testing for proper model implementation is not the same thing as running model calibration tests When implementing a stochastic interest rate model it is important to validate that the scenario generator leads to results that are consistent with the model, otherwise the pricing of instruments that rely on the output of such scenario engines may be grossly inaccurate The Hull-White model is a simple and well understood model, but there are several ways to implement the model. For example, one can use bond prices, short rates or forward rates in the simulation process There are many parts to a term structure model (interpolation, discretization) and each in turn can have an impact on the validity of the outputs of the model so implementation testing is a must If you are going to use a third party model or use an internal IRSG you should exhaustively check to make sure it has been implemented properly November,

17 Section 3: Quantifying Greeks Estimation Error 6

18 Quantifying Greeks Estimation Error Overview Monte Carlo Simulation Error Greeks drive the trading activity and the risk management strategy of a dynamic hedging program There are no simple closed-form solutions for VA Greeks and hence the reliance on Monte Carlo simulation How accurate are these important numbers? What are the different methods to calculate the Greeks for a portfolio of policy holders? How many stochastic paths are required to get acceptable level of convergence? What is the error associated with the estimation for first, second and even third order and cross Greeks? Confidence intervals for Liability models are easy to calculate if you are using the same paths for every policyholder, like where companies load a flat file to feed the scenario generation process in a projection system that is use to calculate the Greeks for a hedge program However many companies use a different seed for each policyholder, because they have stub logic or because they seek rapid convergence for expected values, and finding a confidence interval here becomes a more difficult and complicated process We believe a hedge program manager should know the sampling error for any important Greek in the hedge program The experiments on the following slides are based on: $30Bn of Guaranteed Withdrawal For Life product with million policy holders November,

19 Quantifying Greeks Estimation Error Standard Confidence Interval Book Level Standard Confidence Interval When every policyholder is valued using a common set of scenarios the 95% confidence interval for the FMV is simple to calculate The formula for a confidence interval for a mean is Where Z = Z a/. For example, the value of Z in a 95% confidence interval is.96 because P(Z >.96) = For a 90% confidence interval, Z =.645. Co ount $ Delta in millons for the book using 00 scenarios and same seed Mean 5.39 Std Total $ Delta in Millons November,

20 Quantifying Greeks Estimation Error Using Different Seed for Every Policy Confidence Interval When Using Different Seed for Every Policy If a different seed is used for each policyholder how do you calculate a CI? There are several techniques but we will talk about two techniques today Resampling Well suited technique when the theoretical distribution of a statistic is complicated or unknown It is distribution-independent and is indirect way to asses the properties of the distribution underlying the sample and the statistics of interest Provides us with a way of understanding the consequences of sampling variability for drawing inferences about the population based on our data Care must be taken to do this properly Re-run the valuation process again and again but with different starting seeds for each policyholder A brute force way to understand the variability of your statistics of interest Requires very fast and flexible simulation environment November,

21 Quantifying Greeks Estimation Error Resampled Confidence Interval The Bootstrapping Method Here we resample with replacement from each drawing Now we can obtain fast convergence for an expected result Resampled $ Delta in millons for the book using diffrent seeds Mean 4.94 Std.005 Count $ Delta in Millions November,

22 Quantifying Greeks Estimation Error Brute Force Confidence Interval Re-simulating the Greeks 00 times In this case we calculate the expected value of over MM individual policyholders Here we re-run the valuation process 00 times, calculating base, up and down values, or calculating 300 MM expected policyholder values in the process Next we approximate where 95 of the observations are found to estimate the confidence interval Resampled $ Delta in millons for the book using 00 scenarios with diffrent se 300 Count Mean 4.94 Std $ Delta in Millions November, 00 0

23 Quantifying Greeks Estimation Error Summary of Results This table compares the 95% confidence interval for $ Delta Method Mean 95% Confidence Interval Same Seed 00 paths 5,387,378,88,404 Different Seed 00 path Resample 4,943,870 0,64 Different Seed Brute Force 00 samples 4,94,68,36 Delta--the amount the FMV will change for a % move in the market However you are generating your scenarios, and whatever Greeks you are calculating, make sure you think about sampling error If a number is used in a hedging operation chances are it is important, and if it is important, then a confidence interval should be provided with the number It is important to remember that some Greeks have huge sampling errors but are still very important, such as Gamma and other second order Greeks You should measure the accuracy of each one of your Greeks on a regular basis because ) the higher order Greeks are difficult to estimate in practice, ) they can have a large impact intra day trading decisions, and 3) they can change rapidly as markets move November, 00

24 Section 4: Intraday Greeks Approximation Risks 3

25 Intraday Greeks Approximation Risks Overview Most Direct Writers make today s trading decisions using information on the Liability from last night Trading grids are generated via overnight runs using closing values from the day before Hedge Portfolio asset positions and Greeks are updated as the markets change throughout the day, but the Liability value and Greeks, are estimated on a heuristically basis and hence so are net exposures for the hedge program Overnight runs create trading grids, and once combined with some type of interpolation or extrapolation process, are used to re-estimate the Liability Greeks during normal market hours The curse of dimensionality and of runtimes limit the number of book level valuations that can be completed in the overnight runs to derive this information or grid The curse of Liability dimensionality is caused by following: Equity sub accounts ( 5-8 ) + volatility (3-5) + term structure buckets ( 5-8) >> 0 dimensions If you pick points for each dimension, the meshgrid would have over million points Chances are you want to calculate Greeks too, so there would many calculation to perform at each one of these million points What is done in practice? Some companies rely on FMV surface and use multi-dimensional interpolation processes Other companies rely on Taylor Series Expansion A few calculate real-time Greeks November,

26 Intraday Greeks Approximation Risks Test Design for Taylor Series Expansion (TSE) Here we take a book of about million policyholders and evaluated the book across a mesh grid, with 5 different account value levels and 3 different interest rate levels This creates 5*3 or 345 different points (shocks) where the book value and the Greeks are calculated We calculate Delta and Rho using 00 paths with different seeds at each of these points and then compare the results to a second order estimate for Delta and Rho using TSE with cross Greeks In the naïve approach the calculated base case and Greeks results will feed the intraday estimate of Delta and Rho using a TSE We have simplified the Liability estimation problem because we have boiled it down to only two state variables, changes in total account value and parallel changes in the yield curve, versus the real problem direct writers face in practice November,

27 Intraday Greeks Approximation Risks Result Summary The % errors are large considering a starting dollar Delta of 5 MM and can easily exceed MM. Note the critical role second order Greeks play in TSE for large market movements November,

28 Intraday Greeks Approximation Risks Test Design for Fair Market Value (FMV) surface Here we take a book of about M policyholders and calculate the FMV and the Greeks for the book across a mesh grid, with 5 different account value levels and 3 different interest rate levels However we select only a portion of the FMV points to feed a two dimensional spline We then calculate Delta and Rho using the spline and compare to actual results This test will allow us to get a better sense of how well a FMV surface created from yesterday s close will help us estimate the intraday Liability Greeks in practice Remember we have boiled it down to only two state variables, changes in total account value and parallel changes in the yield curve, versus most direct writers who face the curse of dimensionality and are basically limited to under 00 runs to create the trading grid November,

29 Intraday Greeks Approximation Risks Interpolated FMV overnight run spline training set Multidimensional splines need to be provided appropriate training set and even then can fail to provide reasonable outputs, and here we use the yellow FMV points in our two dimensional spline November,

30 Intraday Greeks Approximation Risks Interpolated FMV Delta spline error Here we can see that using a FMV surface from the day before is subject to spline issues. It is almost like you are not sure what size of error you will get and often it can be very large November,

31 Intraday Greeks Approximation Risks Interpolated FMV DVO spline error Here we can see using a FMV surface from the day before is subject to spline issues for interest rate sensitivities too November,

32 Intraday Greeks Approximation Risks Delta error expressed in S&P500 emini contracts to reblance Trading extra contracts means adding noise to your hedge program, day after day, and when there is a large market movement the noise problem can increase exponentially November,

33 Intraday Greeks Approximation Risks Conclusion Making trading decisions today based on information from yesterday may result in adding a lot of noise to your hedge program results Taylor Series Expansion interpolation results The size of the noise or error grows with the size and nature of the market change using a TSE as intraday interpolator Second order and cross Greeks drive the performance of intraday Greeks estimation when markets move a lot but the problem is these Greeks are notoriously unstable On the plus side a TSE interpolator is simple to set up and use FMV surface interpolation results Even in two dimensions a FMV interpolation process to estimate the Liability Greeks and value is subject to large errors at times, and the pattern of the errors is hard to understand Overall Summary Trading today is based on grids from yesterday using FMV spline is far from a perfect approach The best alternative is to trade with timely and accurate information VA hedge program managers should monitor this risk in their hedge program and try to measure and assess the impact of trading with stale information on their hedge breakage numbers November,