Proxy Techniques for Estimating Variable Annuity Greeks. Presenter(s): Aubrey Clayton, Aaron Guimaraes

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1 Sponsored by and Proxy Techniques for Estimating Variable Annuity Greeks Presenter(s): Aubrey Clayton, Aaron Guimaraes

2 Proxy Techniques for Estimating Variable Annuity Greeks Aubrey Clayton, Moody s Analytics Aaron Guimaraes, Thrivent Financial 17 November 2015 ( Hours) Equity-Based Insurance Guarantees Conference Nov , 2015, Chicago

3 Introduction 2

4 Hedge projection Typical hedge projection requirements» Calculate hedging gain/loss over projected run-off of liabilities for a Clearly Defined Hedging program» Nested stochastic: valuation at each time-step of each run-off path Typical current approach» Model points based on various criteria (in-/out-of-the-moneyness, etc.)» Seriatim calculated values of the Greeks for each model point» Run time for the hedge projections remains a huge issue» Brute force calculation requires massive computational resources 3

5 The argument for proxy functions Scenario requirements for a complex capital planning exercise: 6 6 planning 1,000 run-off 1,000 scenarios run-off scenarios * 1,000 * 10,000 9 scenarios quarters run-off * 480 * valuation 40 months years scenarios * 10,000 1,000 * scenarios * run-off 10,000 valuation * 480 scenarios valuation months * 10,000 * scenarios 480 months valuation * 10,000 * valuation 10,000 valuation scenarios scenarios * 3 sensitivities Is a full revaluation really needed? = billion scenarios» Tradeoff between scenario budget and number of liability model points» Proxy fitting calculations can be done in advance 4

6 Least Squares Monte Carlo 5

7 LSMC approach Define a large set of fitting scenarios Calculate many inaccurate liability values Fit a function through inaccurate values to build an accurate proxy function 6

8 Evolution and Adoption of LSMC PG v1 PG v2 PG v2.3 PG v3.0 PG v3.1 PG v American Option Pricing Longstaff Schwartz 2006 LSMC Application to Nested Simulation 2009 Solvency II and nested simulations Bauer et al 2009 LSMC applied to GMWB w/ ratchet Morrison Cathcart First UK insurer to adopt PG Banks adoption of LSMC for projection of counterparty exposure Risk magazine European Apr insurer implements Multi-year LSMC using PG projection of MC liabilities Tier 1 Asian July insurer adopts PG for economic Multi-year capital projection of CTE North American insurer Sept adopts PG for economic capital Multi-year and ORSA projection of 1-year VaR Nov Dynamic hedge projection for vanilla option Generali Jan implements Multi-period LSMC for internal modeling model of Greeks using LSMC VA provider uses LSMC Mar for assessing Survey of hedge Regression effectiveness Methods Dec Multi-period projection: Generali case study Japanese LSMC project for valuation and ORSA projection 7

9 Selection of Moody s Analytics proxy modeling research 1. Implementation of Least-Squares Monte Carlo (LSMC) in a Life Insurance Context a Case Study (Mar 2014) 2. One-year Projection of Run-off CTE Reserves (Mar 2013) 3. Multi-year Projection of Market-Consistent Liability Valuations (Apr 2013) 4. Multi-year Projection of Run-off CTE Reserves (Jul 2013) 5. Multi-year Projection of 1-year VaR Capital Requirements and Free Surplus (Sep 2013) 6. Proxy Function Fitting: Implementation Topics (Oct 2013) 7. A Survey of Regression Methods for Proxy Functions (Mar 2014) 8. Dynamic Hedge Projection and the Multi-period Modeling of Greeks (Nov 2013) 9. Multi-period Modeling of Greeks using LSMC: An Exotic Option Case Study (Jan 2014) 10. Quantifying the effect of Dynamic Hedging on 1-year VaR capital (Feb 2015) 11. Efficient Asset Allocation with LSMC (Jan 2014) 8

10 Four steps to derive the proxy function Identify risk and generate fitting points Calculate estimated value for each fitting point Run optimization to fit estimates Validate proxy function using accurate valuations 9

11 Risk factors to describe path dependency» For complex products, the value will depend on the path of economic risk factors up to that point» Summarise important features of the path in a small number of variables» Lookback option value depends on running maximum» VA guarantee value depends on moneyness of guarantee, etc. 10

12 From policy to portfolio Policy risk factors: issue age moneyness asset allocation Policy-level proxy function Market risk factors: yield curves equity returns credit spreads Portfoliolevel proxy function 11

13 Proxies using neural networks Neural networks are a functional form that can efficiently model non-linear behavior: Polynomial Regression p p (1) (2) y k = β 0 + β i xi + β ij xi x j + higher order terms i=1 i=1 j=1 i Neural Network and Activation Function y k = α k + i k w ik x i + j k w jk φ α j + i j w ij x i, φ x = exp( x) 12

14 Rho Vega Value Delta Example: Hedging a 10-year put option at time 9 Proxy functions capture the different behaviors of Greeks over the entire risk factor space, even close to option maturity Proxy Actual Proxy Actual Equity index Equity index Proxy Actual Proxy Actual Equity index Equity index 13

15 Delta Vega Value Equity Index Rho Hedge projection example scenario Liability Value (Actual) 0.5 Hedge Portfolio Value (Proxy) Hedge Portfolio (Actual) Equity Index Time (projection year) Actual Proxy Time (projection year) Actual Proxy Actual Proxy Time (projection year) Time (projection year) 14

16 CTE (90) Capital Requirement Example CTE(90) run-off capital requirements» 10-year at-the-money put option on S&P 500» CTE(90) run-off capital results for different re-balancing frequencies for the hedge portfolio 30% 25% 20% 15% 10% 5% Little benefit is obtained by moving from weekly to daily: instantaneous jumps cannot avoided by more frequent rebalancing 0% Cash Annual Quarterly Monthly Weekly Daily 15

17 Thrivent Case Study 16

18 Business needs» Ability to project hedge cash flows vital for a number of business applications» Regulatory requirements (AG VACARVM, C3-P2)» Pricing» Economic Capital 17

19 Hedge projection methods previously tried» Brute force calculation» Most accurate, but not scalable» Implicit method» Described in AG VACARVM practice note, crude» Tabular approach» Thrivent pre-computed Greeks for its guarantees varied by several risk dimensions (interest rates, moneyness, asset allocation, etc.) 18

20 Logistics Fast valuation models allow for many MC estimates 500,000 fitting points * 6 combinations of (Gender, Allocation) * 16 inner scenarios * 3 (Base value & Delta up/down) ~ 3 days of model run-time* *using 8-core desktop machine 19

21 Product risk factors Product:» Flexible premium deferred VA in waiting period» GLWB and annual ratchet, deferral bonuses» Three possible fund allocations: Conservative, Moderate, Aggressive Policy variables:» Gender» Asset Allocation» Issue Age» Contract Duration» Moneyness Separate functions Continuous Market risk variables:» Yield curve level» Yield curve slope» Yield curve curvature 20

22 Proxy liability value ($K) Proxy liability delta ($K) Results validation Good fit to value and delta across 2,450 chosen validation points True liability value ($K) -2 True liability delta ($K) 21

23 Liability Delta ($) Proxy function captures key properties Market exposure decreases with moneyness, even turning positive when out-of-the-money Moneyness Proxy Actual 22

24 Liability Value ($) Proxy function captures key properties Proxy shows the impact of annual resets as reset date approaches Contract Duration (Years) Proxy Actual 23

25 Conclusions Least Squares Monte Carlo offers an interesting and useful alternative for projecting VA hedges» Run-times hugely improved over brute-force methods» Can be more ambitious in the scope of the proxy functions as compared with the tabular approach To get the best results, understand the data you are trying to proxy» Liability Greeks may be discontinuous on some risk dimensions add risk factors to capture discontinuities in the proxy function 24

26 Questions? 25

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