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

Portfolio Replication – Variable Annuity Case Study

Curt Burmeister Senior Director Algorithmics

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.)”

Why Replicating Portfolios?

Asset Portfolio Liability Replicating S tra_te gic _Op tim iza tio n ag ain st _R ep lic atin g P ort_fo lio (E_C , S olv en cy II, LD I, e tc) Stra tegi c O ptim izat ion agai nst L iabi lity Por tfolio (E C, S olve ncy II, LDI, etc)

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

Uses of Replicating Portfolios

Current applications

1. Economic capital

2. Sensitivity analysis and hedging

3. Benchmarks for investment managers

Future applications

Steps in the Portfolio Replication Process

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)

Output

Replicating portfolio = set of optimized weights (holdings) for the asset universe

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?)

Minimize Cash Flow Deviations

( )

( )

s

t a a L

t Times s Scenarios a Assets

min

w

p

x cf

t,s

cf

t,s

∈ ∈ ∈

⎛

₋

⎞

⎜

⎟

⎝

⎠

∑

∑

∑

( , ) ( , ) t s a a L

w weight on the errors at time t probability of scenario s p

x position in asset a

cf t s is the cash flow for asset a on scenario s and time t

Variable Annuity Portfolio – # Policies by GMDB type

~15000 Total Policies 15398 4394 3521 1501 5982 Total 4218 5071 6109 Total 2147 682 0 1389 2005-2007 1583 879 444 2165 2002-2005 664 1960 1057 2428 < 2002 Combo Ratchet Roll-up ROP Policy Date

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

4394 446 3587 361 Combo 6192 984 455 4391 1 1501 627 419 Roll-up 15398 3521 5982 Total 1567 296 195 3 7645 2241 1396 2 Total Ratchet ROP ITM Band

Variable Annuity Cash Flows

Scenario dependent cash flows include

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

Market Indices

1. US Interest Rate Curve

2. Russell 1000

3. S&P 500

4. Nasdaq 100

5. MSCI EAFE Index

6. MSCI Emerging Market Free

7. MSCI REIT Index

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)

Cash Flow Comparison

PV Liabilities/PV Replicating Portfolio across scenarios

Market Risk Sensitivities

-9428.05 -1.16 884.76 0 -.05 507 EQ Vega (1%) EQ Gamma ($1) EQ Delta ($1) IR Vega (1%) IR Gamma (1 bp) IR Delta (1bp) IR Deltas (1 bp) 0 50 100 150 200

Stress Test Scenarios (not used in optimization)

RN liability value and RP values are also computed under 8 instantaneous shocks

Equity shocks ƒ -10% ƒ +10% ƒ -30% ƒ +30% IR shocks ƒ -100bp ƒ +100bp ƒ -300bp ƒ +300bp

Relative Changes for Shock Scenarios

+30% EQ +10% EQ -10% EQ -30% EQ +300bps +100bps -100bps -300bps 186 80 -154 -539 128 47 -60 -211 VA Portfolio 452 163 -180 -608 125 50 -64 -236 Replicating Portfolio -20% 0% 20% 40% 60% 80% 100% 120% 140% 160% -300 bps -100 bps +100 bps +300 bps -30% EQ -10% EQ +10% EQ +30% EQ % E rro r

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