September 21, 2006

2006 Seminar for the Appointed Actuary

Colloque pour l’actuaire désigné 2006

2006 Seminar for the Appointed Actuary

Colloque pour l’actuaire désigné 2006

Canadian Institute

of Actuaries

Canadian Institute

of Actuaries

L’Institut canadien desactuaires

L’Institut canadien desactuaires

September 21, 2006

Stochastic Equity Modeling

Dr. Julia Lynn Wirch-Viinikka

AVP Investment Products Pricing

AEGON Canada

AEGON Canada 3

2006

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6

2006

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September 22, 2006

Agenda

• Equity Risk – where is it?

• Stochastic Modeling – what is it?

• What options do we have for modeling equity risk?

• How do we start?

• How do we improve our model?

• How do we illustrate our results?

• How complicated can it get?

AEGON Canada 4

2006

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6

2006

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September 22, 2006

Where is Your Equity Risk?

• Assets backing Liabilities (LTC)

• Surplus

• Liabilities – Seg Fund, VA and UL Guarantees– Equity Linked Products

• Fee Income based on Fund Value

• Hedging Mismatch– Tracking error, basis risk

AEGON Canada 5

2006

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6

2006

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September 22, 2006

Why Manage Equity Risk?

• Regulatory requirement– Equity Limits, MCCSR requirements– DCAT testing

• Valuation– Reserves and Capital Requirements

• CGAAP, IFRS, US GAAP results– Income and Surplus volatility

• Risk Management objectives

AEGON Canada 6

2006

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6

2006

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September 22, 2006

Market risk vs. Insurance risk

• Traditional insurance risks, such as mortality and longevity are less risky when pooled together: each individual follows their own “scenario” and the insurance company pays off on the average

• Capital market risks don’t diversify: every policyholder follows the same market scenario at the same time

AEGON Canada 7

2006

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6

2006

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September 22, 2006

What Risks can be Managed?

• Risks that are identifiable and well understood

• Risks that are monitored and controlled• Risks where there is the knowledge and

expertise to effectively manage them.• Where the reward is sufficient for the

remaining risk• Where financial instruments and

methods are available to hedge or control risk

AEGON Canada 8

2006

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6

2006

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September 22, 2006

What is a Model

• Imitation/simplification of a real world system

• Tool that provides statistical estimates and not exact results

• Computational, statistical or judgment-based

• Helpful for product design and pricing, valuation, forecasting, risk management, financial reporting, and performance management.

• Understand how your liability value changes over time, when your liability value needs to be calculated stochastically

AEGON Canada 9

2006

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6

2006

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September 22, 2006

What is a Stochastic Model?

• A model that involves probability or randomness– Random inputs (Normal, Lognormal,

Uniform)– Generally run many times (1000, 10000+)

• Representative sampling (Yvonne Chueh)

– Distribution of outputs– Estimates of statistics (mean, %ile, std.dev)– Error estimates (direct or bootstrapping)

AEGON Canada 10

2006

Sem

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6

2006

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September 22, 2006

What is Model Risk?• Model risk: the possibility of loss or

error resulting from the use of models. • Model misspecification• Assumption misspecification• Inappropriate use or application• Inadequate testing, validation, and

documentation• Lack of knowledge or understanding, user

and/or management• Error and negligence

AEGON Canada 11

2006

Sem

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6

2006

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September 22, 2006

How do you Model Equity Risk?

• Flat Return(8%) with an Extreme MfAD(-30%)• Set of deterministic scenarios (stress tests)• Purchase sets of stochastic scenarios• Stochastic Scenarios:

– Normal/Lognormal Returns – Autocorrelated Returns (time series)– Regime Switching LogNormal (RSLN)

• One correlation matrix• Different correlation matrices for each regime

– Other stochastic model (Wilkie, Smith, Lognormal, Stoch Volatility, empirical)

– Risk Neutral or Real World

AEGON Canada 12

2006

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6

2006

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September 22, 2006

Yield Curve vs. Equity

• Are they related?– Direct relation shows zero correlation

• However…– Bond Funds and Equity Indices show 30%-

60% correlation– Duration analysis can explain 90%+ of bond

fund returns: an( in

t – int-1) = Bond Fund Return (t-1,t)

– One way to connect Yield Curves with Equity Returns

– Leads to interest rates driving equity returns

AEGON Canada 13

2006

Sem

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6

2006

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September 22, 2006

What Equity Risk do you model?

• Indices:– Stock Market Indices:

• North America: S&P500, TSX, NASDAQ• Europe: FTSE, DAX

• Industry specific? Company specific?• Public Equity / Private EquityDo you model:• Hedge Funds? Pass-through products?• Real Estate? REITs?• Credit Spreads/Counterparty Risk?• Currency Risk?

AEGON Canada 14

2006

Sem

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6

2006

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September 22, 2006

Is Equity Related to other Returns?

• NO Independent

• Correlation Matrix (Normal/Lognormal)

• Regime Switching Assumptions

• Time Series, Volatility Jumps

• Macro-Economic Drivers (Wilkie Model)

• Does it matter?– It depends on what you are trying to do

AEGON Canada 15

2006

Sem

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6

2006

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September 22, 2006

Scenario Generators

Issues:• Is the focus on the mean, median, or tail events?• Economic vs. Risk Neutral model• Calibration (current/historical data)• Numerous Scenario Generators to choose from

Desirable Characteristics to check for:• Integrated model • Incorporates the principle of parsimony• Flexible. A component approach.

Beware: Often there is a false sense of precision

AEGON Canada 16

2006

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6

2006

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September 22, 2006

Why “risk-neutral”?

• Financial derivatives: value depends on the value of another financial instrument

– Their prices do not depend on the particular risk-preferences of the purchaser…

… so we can assume any risk-preferences

• Mathematically convenient to assume purchaser is risk-neutral

• If you project market movements along a risk-neutral random walk and discount asset payoffs at the risk-free rate, you will obtain the “fair value” of that asset

AEGON Canada 17

2006

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6

2006

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September 22, 2006

“Fair Value”

• Two portfolios with identical payoffs must have the same price

”Arbitrage” - opportunity for profit: buy the less expensive portfolio and sell the more expensive portfolio

FOR INSURANCE Liabilities:• “No Arbitrage” doesn’t work perfectly: the market

cannot freely buy and sell the insurance liability

• Risk-neutral pricing tells you what it would cost to buy the same payoffs in the market. (not necessarily a good estimate of the expected cost of the guarantee if left unhedged)

AEGON Canada 18

2006

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6

2006

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September 22, 2006

Risk-Neutral Valuation

• A Random Walk:

• μ = expected risk free forward rate

• σ = implied volatility, ε = random error

Does “Risk-Neutral” = “Market-Consistent”?• If μ and σ are market-consistent, the prices that the

model produces are market consistent

• Both μ and σ can be functions of time

• σ is often considered to be a function of market levels (market volatility increases when market levels fall)

ttSttSS

S

),(),(

AEGON Canada 19

2006

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6

2006

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September 22, 2006

Real World Model• Random walk for the stochastic model:

– Drift rate: long term averages of historical returns for that stock (not the risk-free forward curve)

– Volatility: long term average or stochastic (GARCH, jump diffusion, regime-switching lognormal)

– Goal: to reflect a reasonable distribution of potential future returns

• Fewer expected payoffs of the embedded option than under risk-neutral valuation: on average, the stock market has a better return than risk-free investments

• Higher variability of profit by scenario

• The “bad tail” can be very bad

AEGON Canada 20

2006

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6

2006

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September 22, 2006

Rule of Thumb• Tail risk:

– Use real-world valuation to measure tail risk

• Average cost:

– Use “real world” inputs when you are willing to accept the “average” result with a high amount of variability

– Use risk neutral when you want results (e.g. a price or a profit measure) which you can be very confident can be realized (through hedging)

AEGON Canada 21

2006

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6

2006

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September 22, 2006

Who uses your Equity Models?

• Hedging (Financial Engineering)– Market-consistent pricing - RN

• Risk Management, Valuation and Pricing (Actuarial Modeling)– Tail exposures – RW– Volatility - RW– Averages – RW/RN– Static Hedging - RN– Dynamic Hedging – RW/RN

AEGON Canada 22

2006

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2006

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September 22, 2006

Regime Switching Models• Discrete time (e.g. daily, monthly)

• Any model with different parameters in each regime (Normal, AR(1), ARCH….)

• 2-Regime Lognormal Monthly – estimation software free from SOA website

• Very simple stoch vol model

• Tractable, intuitive, 2 Regimes are usually enough for monthly data - 6 parameters: {1, 2, 1, 2, p12, p21}

• Regime 1: Low Vol, High Mean, High Persistence (small p12)

• Regime 2: High Vol, Low Mean, Low Persistence (large p21)

AEGON Canada 23

2006

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6

2006

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September 22, 2006

REGIME 1 1

Low Volatility 1

High Mean 1

2-Regime LogNormal

12p

21p

ttY 11

ttY 22 REGIME 2 2

Low Volatility 2

High Mean 2

AEGON Canada 24

2006

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6

2006

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September 22, 2006

Simple Stochastic Model

• 3-year 100% Seg Fund Maturity Guarantee• MER = 3%

Regime 1 Regime 2

Fund LN1(11%,16%) LN2(-8%, 20%)

P(Switch) p12=4% p21=22%

% Time in Regime

Regime 1 84.6%

Regime 2 15.4%

Mean Std.Dev

8.4% 18.1%

AEGON Canada 25

2006

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2006

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September 22, 2006

Simple Stochastic Model: Scen 13-yr Maturity Guarantee: No death / lapse

Initial Deposit = $1; Top-up = $0

Time: 0 1 2 3

Uniform =RAND()=0.934 0.641 0.135 0.053

Regime 2 2 1 1

Normal =NORMSINV(RAND()) -0.1635 0.7642 0.9195

Return = (1+it)

exp[mu*t

+sigma*sqrt(t)*RN]0.89 1.26 1.29

Fund Fund(t-1)*Return 0.89 1.13 1.46

Fund Less MER

FundLessMER(t-1)*Return

*(1-MER)0.87 1.06 1.33

AEGON Canada 26

2006

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2006

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September 22, 2006

Simple Stochastic Model: Scen 23-yr Maturity Guarantee: No death / lapse

Initial Deposit= $1; Top-up = $0.19

Time: 0 1 2 3

Uniform =RAND()=0.649 0.039 0.827 0.154

Regime 1 2 2 1

Normal =NORMSINV(RAND()) 0.1635 -0.7642 0.3195

Return = (1+it)

exp[mu*t

+sigma*sqrt(t)*RN]0.95 0.79 1.17

Fund Fund(t-1)*Return 0.95 0.76 0.88

Fund Less MER

FundLessMER(t-1)*Return

*(1-MER)0.93 0.71 0.81

AEGON Canada 27

2006

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2006

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September 22, 2006

More Advanced Stochastic Models

Other Modeling Considerations:– Death and Lapse (dynamic lapse?)

– Death Benefits and Living Benefits

– Ratchets and Resets

– Policyholder Behaviour

– Commissions / Surrender Charges / DAC

– Reserves / Capital

– Net Income / Tax / Distributable Earnings

– Discount Rates for Present Values

– Illustrating Results

– Hedging Strategies

AEGON Canada 28

2006

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2006

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September 22, 2006

Summary Statistics

• Mean, Standard Deviation, Skewness, Kurtosis,…

• Percentiles (Quantiles)

– Confidence intervals: http://www.fenews.com/fen47/topics_act_analysis/topics-act-analysis.htm

• CTE 95%: Mean of worst 5% of results

– Variance Estimate: Hancock and Manistre NAAJ 9(2): 129-156

AEGON Canada 29

2006

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2006

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September 22, 2006

Box Plots

1 2 3 4 5 6 7 8 9 10-3000

-2000

-1000

0

1000

2000

3000

Net

Inco

me

Year

75/100 Guarantee - Income Projections

+

+ +

+

Maximum

75th Percentile

Median

25th Percentile

Minimum

Outliers

AEGON Canada 30

2006

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2006

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September 22, 2006

Histograms and CTE’s

0 50 100 150 200 250 300 350 400 450 500-7

-6

-5

-4

-3

-2

-1

0

1

2

3x 10

4

CTE 95% = -2.01

• Histogram of scenario outcomes

AEGON Canada 31

2006

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2006

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September 22, 2006

How Many Scenarios are Enough?

Present Value of Cash Flows ($M) under Various Scenario Sets

-$300

-$200

-$100

$0

$100

$200

$300

$400

$500

$600

1 -

10000

1 -

1000

1001 -

2000

2001 -

3000

3001 -

4000

4001 -

5000

5001 -

6000

6001 -

7000

7001 -

8000

8001 -

9000

9001 -

10000

Scenario Set

CTE(95) CTE(80) CTE(60) CTE(0)

• Convergence / Sampling error• Variance Reduction Techniques may help

• Many techniques work for averages not tails

AEGON Canada 32

2006

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2006

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September 22, 2006

Are you taking a Holistic Approach?

• ERM Approach: takes advantage of synergies across products– Consistent set of RW and/or RN scenarios

used for all lines of business– Projections aggregated by scenario across

lines of business– Yield curve and equity return assumptions

must be consistent– More difficult if two Tier Stochastic

simulation is required

AEGON Canada 33

2006

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2006

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September 22, 2006

1-Tier Stochastic Simulation

• Projected Liability Payouts

Time

V0

0 T

•Can determine t=0 reserve(CTE70-80% and TBSR (CTE95%)

•Can determine liability payout projections

•Can not accurately determine future reserve and capital projections (approximations: NPATH, Black-Scholes)

AEGON Canada 34

2006

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2006

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September 22, 2006

2-Tier Stochastic Simulation

• Projected Liability Payouts, Reserves, Capital, Net Income ….

Time

V0

0 T

•Can determine t=3 reserve for each stochastic scenario (CTE70-80%)

•Can determine future capital needs and net income projections

•Much more time consuming

AEGON Canada 35

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September 22, 2006

2-Tier Stochastic Simulation

• Projected Liability Payouts, Reserves, Capital, Net Income ….

Time

V0

0 T

Much more time consuming:

•1000 Tier 1 Scenarios

•10 time steps each

•1000*10 points to perform a second tier simulation

•500 scenarios at each point = 5,000,000 Tier 2 scenarios

AEGON Canada 36

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September 22, 2006

Insurance Options

• Embedded options in insurance liabilities are different from financial options– Sub-optimal exercise behavior

• FPDA: can pay surrender charges and get a new contract if new money rates rise– Evidence: PHs are inefficient in using this option– Some PH will not surrender their contracts no matter how

uncompetitive their renewal rate

• Segregated Funds (VA/VL) GMAB: should invest in the most aggressive funds available– CAPM: more risk implies more return– Evidence: PHs invest in conservative and balanced funds

AEGON Canada 37

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September 22, 2006

Stochastic Modeling Challenges• Option payoffs that depend on policyholder behavior will reflect:

– Historical behavior patterns– Actuarial judgment

• Path-dependent behavior (ie. lower lapses for in the money guarantees) can be modeled– Introduces uncertainty to valuation results– Practitioners have argued about the “proper” way to model behavior in

a risk-neutral framework • (library.soa.org/library-pdf/RRN0608.pdf by M. Evans)

• Long-term nature of liabilities:– Expected market forward rates past 30 years is needed for valuation– Instruments that will hedge the yield curve past 30 years or equity

risks past 10 years are illiquid or unavailable

• Computational Requirements– Distributed processing (AXIS, MatLab, ….)– 2-Tier Stochastic Analysis (Stochastic-in-Stochastic)

AEGON Canada 38

2006

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September 22, 2006

Conclusions

• Equity risk is not like traditional insurance risk.

• Stochastic Modeling is a tool that can help us understand complex dynamic processes.

• Start simple and build.

• Test uncertain assumptions.

• Develop expertise.