2007 Annual Meeting ● Assemblée annuelle 2007

Vancouver

2007 Annual Meeting ● Assemblée annuelle 2007

Vancouver

Canadian Institute

of Actuaries

Canadian Institute

of Actuaries

L’Institut canadien desactuaires

L’Institut canadien desactuaires

IP-41 Stochastic Modeling for Insurance Products

Stochastic Scenario Development for Inexperienced Users

Julia Wirch-Viinikka

Investment Products Pricing

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007

Stochastic Tools• Objectives and Constraints

• Scenario Generators• Start Simple:

• Fixed Income Scenarios• Equity Scenarios

• Picking a model:• Key Considerations• Calibration

• What can go wrong?

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007

Is stochastic modeling going to give you a better answer?

• What is your objective?• Appropriate Pricing (Products/Derivs)• Variability and Risk Management

• Duration, Convexity, Hedging• Aggregation across products• Reserve and capital calculation (tail results)

• What are your constraints?• Time, computation power• Analysis of results• Expertise

• What will you do with the results?

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007

When is Stochastic Important?• Models are dependent on

• Highly variable random variables• Complex correlations/ dependency structures• Models are highly sensitive to assumptions and

RVs

• Asymmetry of Results• Optionality / one extreme tail

• Low Frequency, High Severity• Fat tailed distribution

• Systematic/Non-diversifiable Risk

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007

P’s and Q’s: Stochastic Etiquette• P: Real World Probability

• Often based on economic theories and statistical data

• Historical data provides long-term averages• Full distribution of possible outcomes• Used for cash flow projection, tail events

• Q: Risk Neutral Probability• Replicates current market prices & implied

volatilities • No Arbitrage “free lunch”• Only the mean has significance• Used to price financial instruments, where

investors hedge risk and require no risk premium

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007

RN vs RW?• 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)

EXAMPLES:• 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

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007

REGIME 1 Low Volatility 1

High Mean 1

Correlation 1

Traditional Actuarial Stochastics

12p

21p

ttY 11

ttY 22 REGIME 2

High Volatility 2

Low Mean 2

High Correlation 2

• Wilkie (Cascade Structure)• Factor Models

More recent addition:• RSLN-2:

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007

Interest Rate Models• Yield Curve

• Spot / Yield / Forward curves• Starting yield curve• How many key rates do you need?• How to fill in the rest of the yield curve?

• General Characteristics• Initial yield curve should match actual• Mean Reversion (to a non-fixed target rate)• Long Periods of Relative Stability• Range of Shapes (Normal, Inverted, Humped)• Correlation between maturities and with Inflation• Non-negative• No exploding rates

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007

A Few Models• Mean Reversion

DRIFT: (Long-term mean – Current rate) dt• Hull & White (Vasicek with time dep params):

dr=theta(a(t)-r)dt+v(t)dz

• Non-NegativeRANDOM TERM: volatility * sqrt(current rate) dz

• CIR: dr=theta(a-r)dt+v sqrt(r) dz• As interest rates go toward zero, the random term diminishes and

the interest rate is pulled up toward the long term average

• Easier Calibration• BDT: mean reverting with lognormal distribution (mutually

dependent mean reversion and volatility terms): d(ln r)=(a(t)+(v’(t)/v(t))ln r)dt + v(t) dz

• Black-Karasinski: Lognormal – indep parameters (like BDT)• HJM: generalization - any volatility function

• Easier to parameterize when vol is indep of the forward rate

• Market Models• More flexible: separation of risk between volatility and correlation

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007

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)

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007

What to watch out for:

• Random Number Generators:• Are they really random? Do they generate

enough variables before repeating?

• Do you have Enough Scenarios?• Representative Sampling• Variance Reduction Techniques only helps you

get a quicker estimate of the mean.• Test robustness on much larger sample size

• Especially important for tail measures

• Are your Parameters Right?• Validation and Recalibration

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007

Enough Scenarios?Enough Scenarios?20

07 A

nnua

l Mee

ting

Ass

embl

ée a

nnue

lle 2

007

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

-$300

-$200

-$100

$0

$100

$200

$300

$400

$500

$600

1 -

1000

0

1 -

1000

1001

- 2

000

2001

- 3

000

3001

- 4

000

4001

- 5

000

5001

- 6

000

6001

- 7

000

7001

- 8

000

8001

- 9

000

9001

- 1

0000

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

Histograms & Box PlotsHistograms & Box Plots20

07 A

nnua

l Mee

ting

Ass

embl

ée a

nnue

lle 2

007

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

-2000

-1000

0

1000

2000

3000

Net

Inc

ome

Year

75/100 Guarantee - Income Projections

+

+ +

+

Maximum

75th Percentile

Median

25th Percentile

Minimum

Outliers

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% = -2.01Stochastic Results

The Small Print

• 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

Beware: Often there is a false sense of precision

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007

Questions?

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007

2007

Ann

ual M

eetin

g

Ass

embl

ée a

nnue

lle 2

007