Portfolio Replication Variable Annuity Case Study. Curt Burmeister Senior Director Algorithmics

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1 Portfolio Replication Variable Annuity Case Study Curt Burmeister Senior Director Algorithmics

2 What is Portfolio Replication? To find a portfolio of assets whose value is equal to the value of a liability portfolio under today s market conditions and future market conditions (scenarios.) 2007 Algorithmics Incorporated. All rights reserved. 2

3 Why Replicating Portfolios? Asset Portfolio Strategic Optimization against Liability Portfolio (EC, Solvency II, LDI, etc) Strategic Optimization against Replicating Portfolio (EC, Solvency II, LDI, etc) Liability Portfolio Portfolio Replication Optimization Replicating Portfolio 2007 Algorithmics Incorporated. All rights reserved. 3

4 Why Replicating Portfolios? 1. Performance - It can be difficult for existing actuarial projection systems run the 10,000 50,000 scenarios required for an economic capital calculation but it is typically much easier for asset systems to run 50,000 scenarios for an asset portfolio. 2. Leverages existing liability models - Most firms have invested years customizing their liability models. The portfolio replication approach uses the cash flows generated from the existing actuarial models. 3. A common API - Many firms use multiple actuarial projection systems throughout the organization and the portfolio replication process defines a de facto interface to all the actuarial projection systems 2007 Algorithmics Incorporated. All rights reserved. 4

5 Uses of Replicating Portfolios Current applications 1. Economic capital 2. Sensitivity analysis and hedging 3. Benchmarks for investment managers Future applications 1. Regulatory capital & Solvency II 2007 Algorithmics Incorporated. All rights reserved. 5

6 Steps in the Portfolio Replication Process Output Replicating portfolio = set of optimized weights (holdings) for the asset universe Optimization Setup Choice of objective function and attribute to be matched Weight and trading cost constraints Other constraints (duration, max # holdings, etc) Liability Portfolio Simulate liability portfolio across scenarios and through time calculating the relevant attributes (e.g. cash flow, THEO/Value.) Scenarios Can be real world, risk neutral, stress tests, or combination Must evolve all risk factors (e.g. IR, EQ,& FX) through time Asset Universe Simulate asset universe across scenarios and through time calculating relevant attributes (e.g. cash flow, THEO/Value, etc) 2007 Algorithmics Incorporated. All rights reserved. 6

7 Algo Risk Replicating Portfolio Architecture 2007 Algorithmics Incorporated. All rights reserved. 7

8 Art versus Science 1. What type of scenarios and how many (i.e. risk neutral or real world?) 2. Which assets are included in the tradable universe (type, maturity, strike, underlying, etc?) 3. What is the objective function? 4. What are the time steps (i.e. bucketing?) 5. What are the constraints? 2007 Algorithmics Incorporated. All rights reserved. 8

9 Minimize Cash Flow Deviations s min w t p xacfa t,s cfl t,s t Times s Scenarios a Assets ( ) ( ) w p x t a cf cf s a L weight on theerrors at time t probability of scenario s position in asset a (, t s) isthecash flow for asset a on scenario s and time t (, t s) isthe cash flow for the liability portfolioon scenario s and time t 2007 Algorithmics Incorporated. All rights reserved. 9

10 Variable Annuity Case Study 2007 Algorithmics Incorporated. All rights reserved. 10

11 Variable Annuity Portfolio # Policies by GMDB type ~15000 Total Policies Policy Date ROP Roll-up Ratchet Combo Total < Total Algorithmics Incorporated. All rights reserved. 11

12 Variable Annuity Portfolio # Policies by ITM Band 1: MGDB Guarantee / Account Value < 0.9 2: 0.9 MGDB Guarantee / Account Value 1.1 3: 1.1 MGDB Guarantee / Account Value ITM Band ROP Roll-up Ratchet Combo Total Total Algorithmics Incorporated. All rights reserved. 12

13 Variable Annuity Cash Flows Scenario dependent cash flows include Market Indices 1. guaranteed minimum death benefit 2. general account release 3. commissions 4. expenses 5. mortality/expense charge 6. revenue sharing 7. surrender charges 8. per policy fees 1. US Interest Rate Curve 2. Russell S&P Nasdaq MSCI EAFE Index 6. MSCI Emerging Market Free 7. MSCI REIT Index 8. Lehman US Aggregate 2007 Algorithmics Incorporated. All rights reserved. 13

14 Annual Variable Annuity Cash Flows 2007 Algorithmics Incorporated. All rights reserved. 14

15 Replicating Universe and Optimization Setup Replicating Instruments Zero Coupon Bonds Swaptions (physical settlement), Equity Forwards (on each index) European Equity Options (on each index) Optimization Setting Choices Bucketing: Low/High (match each cash flow vs. match bucketed cash flow) Value Constraint Yes/No (match replicating portfolio value to RN liability value) Trade Penalties Yes/No 2007 Algorithmics Incorporated. All rights reserved. 15

16 Bucketed Variable Annuity Cash Flows 2007 Algorithmics Incorporated. All rights reserved. 16

17 Cash Flow Comparison Variable Annuity Cash Flows Replicating Portfolio Cash Flows 2007 Algorithmics Incorporated. All rights reserved. 17

18 Replicating Portfolio 2007 Algorithmics Incorporated. All rights reserved. 18

19 PV Liabilities/PV Replicating Portfolio across scenarios R 2 = 97.1% 2007 Algorithmics Incorporated. All rights reserved. 19

20 Market Risk Sensitivities IR Deltas (1 bp) y 2y 3y 4y 5y 7y 10y 15y 20y 30y IR Delta (1bp) 507 IR Gamma (1 bp) -.05 IR Vega (1%) 0 EQ Delta ($1) EQ Gamma ($1) EQ Vega (1%) Algorithmics Incorporated. All rights reserved. 20

21 Stress Test Scenarios (not used in optimization) RN liability value and RP values are also computed under 8 instantaneous shocks Equity shocks -10% +10% -30% +30% IR shocks -100bp +100bp -300bp +300bp 2007 Algorithmics Incorporated. All rights reserved. 21

22 Relative Changes for Shock Scenarios 160% 140% 120% 100% % Error 80% 60% 40% 20% 0% -20% -300 bps -100 bps +100 bps +300 bps -30% EQ -10% EQ +10% EQ +30% EQ -300bps -100bps +100bps +300bps -30% EQ -10% EQ +10% EQ +30% EQ Replicating Portfolio VA Portfolio Difference Algorithmics Incorporated. All rights reserved. 22

23 Ideas for improving the replicating portfolio 1. Adding new instrument types to the asset universe (i.e. composite indices) 2. Reduce the number of market indices 3. Increase the number of scenarios 4. Adjust the strikes to better match the guarantees 2007 Algorithmics Incorporated. All rights reserved. 23

24 Q&A 2007 Algorithmics Incorporated. All rights reserved. 24

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