Hedging VA GuaranteesFrom Defining Objectives to Optimizing a Program

C. Dan Cazacu and Alan YangRGA Re

Motivation

• Atilio Meucci’s 2011 “Prayer”

– Methodology for portfolio and risk allocation

• EBIG Conference in 2011

– Many participants mentioned reinsurance as a potential solution for VA Guarantees

– Availability of reinsurance

Table of contents

1. Introduction to VA Guarantees

2. Risk Identification

3. Hedge Objectives

4. Case study: Optimization

5. Model risk and Black Swans

6. Operational Risk and Governance

7. Concluding Remarks

Introduction to Variable Annuities• What are VA’s?

• Contract: policyholder pays initial premium, receives promise of future payments starting at annuitization date

• Accumulation and payout phases

• During accumulation phase premium invested in mutual funds (equity market exposure and risk)

• Used in retirement planning

• VA’s selling points are:

– Tax advantages

– Equity based guarantees: Living benefits and Death benefits

4

Introduction to DBs• Death benefit: paid to beneficiary upon death of

policyholder• Amount paid may be the higher of account value and :

– Return of premium– Ratchet– Roll-up– Combination of Ratchet and Roll-up

• Cost is percentage of account value, over the life of the policy

• Protects beneficiary from a down equity market at time of death of policyholder

5

Introduction to DBs (2)Modeling of DBs

– Annual or quarterly models

– Series of put options

– Probability of exercise: tied to survival/mortality

– Projections for 20, 30 or more years

– Path dependent

– Monte Carlo simulation of multiple assets at policy-level (computation intensive)

– Policyholder behavior (lapse, partial withdrawal)

– Impact of living benefits (and associated policyholder behavior) on DBs

6

Relation of DB to Options

• The difference between DB and AV represents the payout of the option sold to the client

• Mortality and survival gives– the probability of exercise of the option– the expected time of exercise

• Time of Death is the actual time of exercise• Some options are path dependent!

7

In an up Scenario: DB is not worth much

Up scenario

-

50,000

100,000

150,000

200,000

1 2 3 4 5 6 7 8 9 10 11

AV

Standard

Ratchet

Roll-up 5.5

Max

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Value of guarantees shown in down scenario

Down Scenario

-

20,000

40,000

60,000

80,000

100,000

120,000

140,000

160,000

180,000

200,000

1 3 5 7 9 11

AV

Standard

Ratchet

Roll-up 5.5

Max

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Ratchet valuable in an up and down scenario

Up and down

-

20,000

40,000

60,000

80,000

100,000

120,000

140,000

160,000

180,000

200,000

1 2 3 4 5 6 7 8 9 10 11

AV

Standard

Ratchet

Roll-up 5.5

Max

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Introduction to LBs

• Accumulation Benefit– Increases benefit base to initial premium if market down

– Exercisable ten years from issue

• Income Benefit– Rolls-up benefit base by x%

– May be combined with a ratchet feature

– Exercisable after ten or more years

• Withdrawal benefit– Guarantees minimum annual withdrawals as x% of initial

premium for y years or for life of annuitant

– Exercisable immediately

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Introduction to Living Benefits(2)

• Paid for with fees over the life of the policies

• Modeling issues:

– Similar to DB modeling issues

– Probability of exercise tied to survival

– Policyholder behavior: if benefit in the money, policyholder may restrain making withdrawals which he would normally make in order to exercise option

– Impact of LBs on DBs: additional fees taken out of account value and different persistency

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Living Benefits: AB10 and AB20AB10 and AB 20 Payouts

60,000

80,000

100,000

120,000

140,000

160,000

180,000

200,000

0 2 4 6 8 10 12 14 16 18 20

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Living Benefits: IB IB Payout

60,000

110,000

160,000

210,000

260,000

0 2 4 6 8 10 12 14 16 18 20

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The Lifetime withdrawal benefit

• Guarantee a minimum level of annual withdrawals for the lifetime of the annuitant ( even if withdrawals reduce contract value to zero)

• Computation of the Benefit Base:

Initial Premium

Quarterly Ratchet

7% Compounding Step-up• MAW – maximum annual withdrawal percentages (for example:

– 5% for age 59.5 to 69– 6% for age 70 to 79– 7% for age 80+

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• Essentially, guarantees are similar to put options sold on a basket of assets

• Policyholder Behavior is an essential additional factor

• Additional features bring strong path dependency

Death Benefits and Living Benefitssummarized as options

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• Main risk to the company (direct writer/ reinsurer): financial loss due to insufficient fees/ large claims due to market fluctuations.

• Divide and conquer:– Equity level risk Delta

– Interest rate risk Rho

– Volatility risk Vega

– Other market risks ( Gamma, Cross-Greeks)

– Model risk

– Operational risk

Risk Identification

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The Greeks

• Delta Δ = ∂ V / ∂S , sensitivity of the option with respect to the change in the underlying

• Gamma Γ = ∂ Δ /∂S, sensitivity of Delta with respect to the change in the underlying

• Rho ρ = ∂ V /∂r, sensitivity of the option with respect to the change in the interest rate r

• Theta Θ = ∂ V /∂t, sensitivity of the option with respect to the change in time t

• Vega = ∂ V /∂σ, sensitivity of the option with respect to the change in volatility σ

V here is the value of the option (PV of cost), S is the underlying (stock, AV for us)

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Put value (BS) and DeltaValue of European Put, K=100

-5

0

5

10

15

20

80 84 88 92 96 100 104 108 112 116 120

Underlying S

Price

at T

ime T

- t=0

.5

Value of european put Delta hedge

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Delta of put as a function of SDelta of BS European Put, K=100

(0.90)

(0.80)

(0.70)

(0.60)

(0.50)

(0.40)

(0.30)

(0.20)

(0.10)

-

80 84 88 92 96 100 104 108 112 116 120

Underlying S

Delta

at T

ime T

- t=1

Delta of put

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Gamma of Put as function of S

Gamma of BS European Put, K=100

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

80 84 88 92 96 100 104 108 112 116 120

Underlying S

Gamm

a at T

ime T

- t=.5

Gamma of put

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Graph of Rho ( r is constant?)

• Note in BS, Risk free rate is assumed constant, so rho is zero.

• The sensitivities would mark a transition form one BS world to another,

with a different risk free rate.Rho of BS European Put, K=100

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

80 84 88 92 96 100 104 108 112 116 120

Underlying S

Rho a

t Tim

e T- t=

.5

Rho of put

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Vega as function of S(σ is constant ?)

Vega of BS European Put, K=100

0

5

10

15

20

25

30

80 84 88 92 96 100 104 108 112 116 120

Underlying S

Vega

at T

ime T

- t=.5

vega of put

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Black- Scholes Framework

• Key Assumptions– Underlying follows lognormal

random walk (can be relaxed)

– Risk free interest rate is constant (can be a known function of time)

– No dividends on the underlying

(may have continuous dividends)

– Delta hedging done continuously

– No transaction fees

– No arbitrage

• Stochastic process:

dS =μS dt + σS dX

• Leads to PDE:

Θ + ½ σ2 S2 Γ + rSΔ –rV = 0

• Which has as solution the Price of Call Options:

C = S N(d1) – K exp (-r (T-t))N(d2)

where N is the normal distribution,

d1=(log(S/K) + (r + ½ σ2 )(T-t))/ σ√(T-t)

d2=(log(S/K) + (r - ½ σ2 )(T-t))/ σ√(T-t)

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Hedging VA Guarantees

• Take positions to neutralize the change in liabilities due to movements of:

– Equity (delta hedging)

– Interest (rho hedging)

– Volatility (vega hedging)

• Common Hedging Instruments:

– Index Futures, Interest rate swaps, Options

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Delta Hedging: Concept• Risk is that underlying funds

(proxy: S&P 500 index) decline, values of guarantees increase (red line put)

• Delta: slope of tangent

• To hedge, we short S futures (blue line): match slopes

• If S declines, Delta becomes more negative (yellow dotted line), so we have to short more, i.e. “sell low”

• “Sell low, buy high” to mitigate risk

• The trading losses are also mitigated (up to a certain level of volatility) by the time decay of the option

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-10

-5

0

5

10

15

20

80 84 88 92 96 100 104 108 112 116 120

Pri

ce

at

Tim

e T

-t=

1

Underlying S

Value of European Put, Strike 100

Value of european put

Delta hedge

New delta

Rho Concept• Risk is that as interest rates

decrease the value of the option will increase

• Rho is sensitivity of value of the option with respect to interest change

• When equities are down due to slowing economy, flight to treasuries, interest rates come down

• Hedge Rho, by entering into swaps (long term receive fixed)

• As interest rates come down, more Rho, more swaps are needed

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0

5

10

15

20

25

80 84 88 92 96 100 104 108 112 116 120

Pri

ce

at

Tim

e T

-t=

1

Underlying S

Value of European Put at 3% and 0.25% interest rate

Value of european putValue at low interest

Vega Concept

• Risk is that volatility will go up and increase the value of the guarantees

• Vega is rate of increase of option with Volatility

• In a High Vol environment, we will experience more trades to rebalance the Delta position, and losses on “Buy high, sell low”

• Mitigate by hedging Vega

28

0

5

10

15

20

25

80 84 88 92 96 100 104 108 112 116 120Pri

ce a

t T

ime T

-t=

1Underlying S

Value of European Put, at 20% vol and 30% vol

Value of european put

Hedging Objectives

• Need to be defined:

– Economic, Accounting, Tail?

– The answer may vary by company

• Once this defined

– Define thresholds for Greek exposures

– Define derivatives usage limits

– Define liquidity and capital constraints

Case Study: Model Specification

• Using a state space model, We model the joint distribution of the yield curve with index return and implied volatility, which are the key risk factors for VA product.

• Some Key Assumptions:

Interest rate: Mean-reverting process,

Affine Term Structure,

Equity Index: Stochastic Volatility,

Index Return Jump,

Flight to Quality Effect

Implied Volatility: Jones Equation, Leverage Effect

Case study: Model Estimation

• The model is estimated utilizing swap term structure, equity index return and VIX index level back to May of 1994.

• The model is estimated using a Markov Chain Monte Carlo (MCMC) Bayesian Approach.

“Learn” about the unobserved latent factors using the observed interest rate data and S&P 500 Index Returns.

Allows flexible amount of prior information to be updated with the large panel of observed data.

Integrate out the parameters in order to reflect parameter uncertainty in the forecasted scenarios.

In Sample Fit For Index Return and Volatility

Jun. 05, 07 Jan. 09, 08 Aug. 14, 08 Mar. 23, 09 Oct. 26, 09 Jun. 03, 10 Jan. 06, 11 Aug. 12, 110

0.1

0.2

0.3

0.4

0.5

0.6

0.7Volatilities at Each Point

Nov. 20, 08

Dec. 01, 08

Mar. 02, 09Mar. 09, 09

Mar. 30, 09

Apr. 08, 09May. 20, 10

Jun. 07, 10

Jun. 30, 10

Aug. 08, 11

Aug. 10, 11

Aug. 11, 11

Means For Sim Data: Real Value Starting Point

VIX-In-Sample Fit

Jun. 05, 07 Jan. 09, 08 Aug. 14, 08 Mar. 23, 09 Oct. 26, 09 Jun. 03, 10 Jan. 06, 11 Aug. 12, 11-0.1

-0.05

0

0.05

0.1

0.15In-Sample Forecast of Index Return

Dates

Mo

nth

ly R

etu

rn

Dec. 01, 08

Feb. 04, 10

Aug. 08, 11

Aug. 10, 11

Mar. 23, 09

Nov. 24, 08

Oct. 28, 08

Oct. 13, 08

Upper 95% Lower 5% Observed Mean Forecast

Index Return In-Sample Fit

Make sure the model capture the dynamic of historical market behavior

Out of Sample Forecasting: Swap Term Structure

2012 2017 2022 2027 2032 2037 0%

2%

4%

6%

8%

10%

12%

Year

Ra

te

RF Ten Year Maturity Scenarios

min 1% 5% 25% 50% 75% 95% 99% max

10Y Swap Rate Evolution

Forecasting Interest Rate Surface

Scenarios Generation • Use Monte Carlo method to generates 10,000 of possible

outcomes from the model, to describe the distribution of market in the futures.

• Evaluate both Hedging Instruments (Asset) and Liability under each time point and each scenario.

• On each “run”, implement the hedging strategy, calculate the amount of derivatives we need, tracking market value, Greeks, liquidity requirement and any other important measure for the hedging position.

Hedging Strategy Testing

• Monte Carlo Simulation helps us to address not only average scenarios of the out come, but also tail risk of the portfolio.

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

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8

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10x 10

6

T

Perc

entile

Future Margin Requirement2

-400

-300

-200

-100

0

100

200

300

400

1 11 21 31 41 51 61 71

Earnings Volatility Before Hedging

(Orange Region)Earnings Volatility After

Hedging Strategy Testing

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9x 10

6

T

Perc

entile

Liability Vega

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9x 10

6

T

Perc

entile

Asset Vega

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5x 10

6

T

Perc

entile

Vega Exposure

Hedging Strategy: Innovation and Optimization

To standardize our optimization procedure, we have to define:A. Target Function: f(x) (The economic output we want to optimize)

Total P&L, Total Liquidity requirement…

B. Constraint: g(x) (Define the resource we allocate to this project)

Derivatives Position should be less than…

5% Value at Risk should not exceed…

C. Variable: x (The hedging strategy we define)

Threshold and Hedge Effectiveness,

Different Hedging instruments,

Specific Strategy Given certain market condition.

Hedging Strategy: Innovation and OptimizationAn Example

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

8

T

Perc

entile

Future Margin Requirement1

Basic Strategy: 90% Hedge Effectiveness for Delta

Improved Strategy: 90% Hedge Effectiveness for Delta +Unwind Out of Money Option

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12x 10

7

T

Perc

entile

Future Margin Requirement1

Hedging Strategy: Innovation and OptimizationAn Example

• Find out Economic Interpretation for Improved strategy

Model Risk

• Taleb’s Black Swan Definition: A highly improbable event, with great impact, explainable after the fact

• Name explanation: before discovery of Australia all swans known (to Europeans) were white

40

Example of Black Swans

• Positives

– Discovery of Americas by Columbus

– Invention of the Internet

• Negatives

– Pandemics

– Losses due to a catastrophic earthquake and/or tsunami

– Financial crisis of 2008

41

The Thanksgiving paradox

• Turkeys are fed everyday by humans

• They would expect, based on past performance, to continue to be fed

• Something else happens just before Thanksgiving

42

Modeling Black Swan issue

• (One) Key Idea: The (log)normal distribution does not accurately describe the evolution of stocks: we are using the wrong probability distribution

• Recent example: May 2010 “Flash-Crash”

43

What could happen to hedging? (a few scenarios)

• Markets could freeze (or be halted)

– Can not trade, experience a gap

– Continuous rebalancing assumption is key

• Dealer that holds your derivatives may go bankrupt

– Example: Lehman Brothers in 2008

• Dealer may change collateral requirements or spreads in crisis

44

What to do about it? (a few thoughts)

• Use more than one model to value risk (think ranges, not values)

• Use different hedging instruments

• Measure counterparty risk

• Diversify counterparty risk

• Measure and manage collateral

• Monitor liquidity of markets you are in

45

Other Risks to Consider

• Lapse and longevity assumptions

• Basis risk: performance of the underlying mutual funds relative to indices they are mapped to

• Correlations and Higher Order Greeks

Operational Risks

• Risks related to trading operations

– Person

– System

• Risks related to periodic valuation of liabilities and Greeks (whether in-house or outsourced)

Governance

• Oversight needed

– For definition and execution of strategy

– For maintaining the program within the limits agreed on

• Multi-departmental effort: Need to include not just Risk Management, but also Investments, Valuation, Pricing, Treasury

• DUP and ISDAs

Concluding remarks

• Measuring, Managing and Mitigating risks associated with VA guarantees:

– Is not easy

– Is intellectually challenging and interesting

– Can help the company write a profitable product

• The views and methodologies presented herein reflect the authors opinions and do not necessarily reflect those of RGA

Disclaimer

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