Financial Engineering and Actuarial Science In the Life Insurance - - PowerPoint PPT Presentation

Financial Engineering and Actuarial Science

In the Life Insurance Industry Presentation at University of California at Santa Barbara February 21, 2014

Frank Zhang, CFA, FRM, FSA, MSCF, PRM Vice President, Risk Management Pacific Life Insurance Company

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Disclaimer

This presentation contains the current opinions of the author and not of my employer Pacific Life Insurance Company or the Society of Actuaries. Any such opinions are subject to change without notice. This presentation is distributed for educational purposes only.

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Table of contents

 Traditional insurance and risk management  Variable annuities as a newer generation of hybrid products  Financial engineering and actuarial science as a new practice for variable annuities

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Traditional insurance based on averages

 Traditional insurance products with mortality, longevity, morbidity, etc. may be diversified  Most products can be safely priced based on actuarial expected values (averages) and standard deviation  Typical actuarial risk management techniques ⇒ Test sensitivities of assumptions ⇒ Test of distributions and tails

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A flaw of traditional averages – extreme cases

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Law of large numbers

 To reduce the uncertainty (risk or standard deviation) of the expected outcome ⇒ Sell a large number of small amount of insurance  The old and true statistical principle of “law of large numbers” ⇒ Statistically speaking, larger samples reduce “sample error”

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Worked well with tail risk management, until …

 Risk mitigations ⇒ Cut off the tails with caps in insurance coverage (to make sure no single case is too large to kill like the crossing river example) ⇒ Sell more to diversify ⇒ Be prepared for rare but extreme pandemic events (systemic)  Actuarial risk analysis ⇒ Stochastic simulations and principle-based valuation ⇒ Look at more than the averages but also the distributions ⇒ Look at percentiles and use CTEs for tail risk  Worked well with traditional life and annuity products, until recently ⇒ Core insurance expertise is to pool diversifiable risks ⇒ How about newer generation of products, such as variable annuities?

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Table of contents

 Traditional insurance and risk management  Variable annuities as a newer generation of hybrid products  Financial engineering and actuarial science as a new practice for variable annuities

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Variable annuity (VA)

 Life insurance products are increasingly derivatives oriented and many of the same derivatives valuation techniques apply  The hybrid products also create unique challenges and opportunities to actuaries and financial engineers  Variable annuity is a retirement investment account sold by life insurers

⇒ Underlying investments are generally “mutual funds” of various asset classes ⇒ Contract holders pay insurer and mutual fund manager fees over time ⇒ The account value can go higher or lower due to investment results ⇒ Guarantee payoffs = f( guaranteed amount - total basket value of mutual funds) ⇒ Death benefit is paid when

a) The account value is lower than principal and b) Policyholder died

⇒ Living benefits can be paid without having to die, based on different designs such as accumulation guarantees (wait for 10 years), withdrawal benefits (guaranteed withdrawal amounts, regardless of investment performance), etc. ⇒ Policyholders keep upside potential of the account performance – and insurance company provides the downside guarantees (put options!) ⇒ Many VA contracts have much more exotic benefits ⇒ Traditionally, policyholder may or may not exercise optimally

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Life insurance or derivatives?

Variable annuity (VA) guarantees blur the boundary between derivatives products and traditional life insurance products: Living or dying!

Variable Annuities

Derivatives Life Insurance Mutual Funds

Diversifiable Law of large numbers Non-diversifiable Derivatives pricing Multiple underlying assets

Dynamic Policyholder Behavior

Path Dependence

VA contracts invest in mutual funds, paying fees to insurer, and getting guarantee benefits GMDB (Guaranteed Minimum Death Benefit) => Payable at death VAGLB (Variable Annuity Guaranteed Living Benefit) => Payable Under Predefined Conditions:

GMAB (Guaranteed Minimum Accumulation Benefit) for account value guarantee GMIB (Guaranteed Minimum Income Benefit) for annuitized payouts guarantee GMWB (Guaranteed Minimum Withdrawal Benefit) for withdrawals guarantee

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Sample VA GMDB designs

 Different strikes for different designs Return of premium: strike = initial AV = initial premium deposits Ratchet: discrete look back strike = max (sample AVs during the contract life) Rollup: increasing strikes at an annual rate x: strike t = (1+x)t Combinations: strike = max of ratchet and rollup

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GMDB pricing

Benefit paid upon death

Death benefit paid upon death Rate of mortality based on law of large numbers Mortality rates increase quickly at older ages

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GMDB pricing

Benefit paid only If GMDB contract stays in force at death

Not all contracts initially issued still in force in later years People could lapse the contract or annuitize (decrements)

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How to price VA embedded derivatives

Risk neutral valuation using stochastic simulations GMDB is paid only If GMDB is in the money and still In force at death Price = sum of all future possible death payoffs on surviving contracts (by simulations)

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Table of contents

 Traditional insurance and risk management  Variable annuities as a newer generation of hybrid products  Financial engineering and actuarial science as a new practice for variable annuities

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Capital market risk management

 Capital markets represented by Wall Street and portfolio investments  Traditional investments in fixed income (bonds), equity (stocks), and modern derivatives (options)

⇒ Fixed income portfolio management focuses on interest rate risk and credit risk ⇒ Equity investment focuses on systematic risk (measured by beta) ⇒ Derivatives are priced using financial engineering techniques

 Financial engineering is based on “law of one number” or no arbitrage  Wall Street has used financial engineering to price and manage the risks for derivatives

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Annuity derivatives vs. mortgage backed securities

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Annuity derivatives pricing challenges

Comparison - Variable annuities

• Variable annuities are sold to individual investors who pay money to insurance

company.

• VAs pass through mutual fund performance BUT add derivatives guarantees
• There is no active secondary market who collect the investments from the investor

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Annuity derivatives pricing challenges

Comparison - Mortgage backed securities

• Mortgages are sold to banks/institutional investors who pay money to

fund houses.

• The funding needs created secondary MBS markets
• MBS are created to pool mortgages.

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 Dynamic policyholder behavior modeling is critical & difficult

⇒ Key driver for pricing but options not always exercised optimally ⇒ Mortality risk managed by pool of large numbers but living benefits much more challenging ⇒ Behavior very difficult to predict and with little or no experience ⇒ Policyholder dynamics causing significant gamma exposure ⇒ Capital market risks not diversifiable as insurance risks

 MBS prepayment vs. annuities dynamic policyholder behavior modeling

⇒ MBS prepayments based on real world experience or expectations but validated by active capital market MBS prices, unlike annuities ⇒ Risk neutral pricing standard in financial engineering, but transition from actuarial expectations to risk neutral pricing caused confusions about probability distributions and stochastic simulations ⇒ MBS markets not usually concerned with nested stochastic projections that mix risk neutral world and risk neutral valuations, unlike annuities

Annuity derivatives pricing challenges

Dynamic policyholder behavior modeling

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Insurance or mortgage cash flows vs. derivatives averaging

Be careful when using cells to average the derivatives

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Derivatives, VA and fixed indexed annuities

 Options are capital market instruments  Puts and calls have non-linear payoffs

⇒ Only when stock price ends below or higher than the put/call strike respectively ⇒ The options are highly leveraged with small option premium for potentially large payoffs

 VA writers sold short put options in GMDB and VAGLB as embedded benefits  Fixed indexed annuity (FIA) writers sold short of call/call spread options

Small premium Potentially large payoff

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Some things cannot be averaged

Source: Simon Proctor

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Derivatives cannot be averaged

 Selling 10% in-the-money put and 10% out-of-money put is not the same as selling an at-the-money put  Shorting call and put at the same strike might offset the directional stock movements but still leaves exposure to market volatility

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Capital market law of one number (price)

 Financial engineering is based on “law of one number” or no arbitrage  No arbitrage means no risk-free way of making money and there is only one price that is the market price

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Capital market one price – Risk neutral valuation

 Derivatives can be replicated using risk free instruments  It turns out risk neutral pricing is a very convenient trick to simplify derivatives calculations  Risk neutral replications is dynamic hedging  Derivatives pricing using risk neutral does NOT assume risk free market outside this mathematical exercise

Martingale or risk neutral measure: Ht = EQ(HT|Ft) Girsanov theorem and Radon-Nikodym derivative: Relationship between risk neutral and real world measures

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Derivatives cannot be diversified

 Diversification by derivatives product offering is very limited ⇒ Shorting 100,000 put options with varying strikes and maturity dates



Add, but not reduce, the main exposure to the block – systemic exposure to equity ⇒ Selling VA and FIA



Might offset to some degree of the directional exposure to equity but not the volatility  Derivatives risks are systemic ⇒ Time diversification is not very reliable – still the same market exposure

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Derivatives are hedged but not diversified

 Sound risk management practice is ⇒ To neutralize the systemic capital market exposure by hedging (hope we have priced adequately) ⇒ The best “diversification” for derivatives is hedging  Hedging with risk neutral valuation is the most reliable way to offload the risks ⇒ Risk neutral valuation of derivatives ensures consistent results ⇒ No matter what happens - real world or risk neutral path or good/bad scenario ⇒ Without taking any bets of market directions Hedging is to create opposite economic payoffs of the liability guarantees

• Through matching the sensitivities (Greeks) of the VA/FIA
• Greeks are capital market drivers of equity, interest rates, volatility, etc.

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Annuity derivatives challenges

Stochastic simulations

 Simulations are often the only choice

⇒ No closed form solutions ⇒ Path dependency ⇒ Amortizing options ⇒ Multiple underlying assets ⇒ Very complex rules ⇒ Individual modeling ⇒ Option premiums (fees) collected over time ⇒ Policies are not uniform, i.e. everything is customized by individual investments

 A lot of exciting challenges and opportunities ahead

⇒ Most existing theoretic researches can’t deal with path dependency ⇒ Passport optionality ⇒ American optionality ⇒ Lattice approach rarely used ⇒ Large scale grid computing (i.e. thousands of CPUs) typical ⇒ Model efficiency critical

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A dynamic hedging program (a “trading book”)

Formatted Data Feed For Hedge Production

Database

Trading Model

► Collect market data ► Calculate asset/liability position mismatch ► Determine trading positions ► Produce trading instructions ► Store transaction history

VA Derivatives Pricing Engine Derivatives Trades Execution Liability Policy Admin System and Fund Data

Hedge performance attribution

► Capital market drivers and Greeks ► Actuarial policyholder behavior and decrements ► Compare difference between actual and expected ► Manage P/L

Reporting Hedge Accounting & Financial Reporting

Daily Market Data (Bloomberg, etc.)

► Interest rate curve ► Implied volatility surface ► Implied dividend yield curve ► Volatility surface calibration ► Correlations ► Hedge ratios/fund mapping

• There may be hundreds of thousands or millions of derivatives contracts in a book
• No closed form solutions but stochastic simulations for path dependent, long-dated, and basket options in VA book.
• Need very large computing grid for overnight simulation runs
• Maybe active derivatives trading to hedge the Greeks

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Dynamic replications – Whether or not market goes up or down

 Example of GMAB block with $5M option value

 Two paths of market performance of replicating strategies  Whether or not market goes up or down – the hedging is to track against the change in liability with small net P/L

• 20,000

40,000 60,000 80,000 100,000 120,000 140,000 160,000

• 5,000,000

10,000,000 15,000,000 20,000,000 25,000,000 30,000,000 100 200 300 400 500

Account Value

Dynamic Hedge Performance (Hedging Target vs. Hedge Assets)

Option Value Cum Futures G/(L) + PV of Option Premium Account Value

Weeks

Hedged target vs. Hedging Assets

AV Hedged Target = GMAB Liability Hedging Assets = Prem + P/L

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 Higher realized volatility increases cost of dynamic hedging program ⇒ Hedge cost increases during periods when underlying funds moves sharply, resulting in elevated transaction costs and increased “buy-high-and-sell-low” round trip trades ⇒ Volatility equals cost of options

Dynamic hedging boils down to gamma risk

• 2. Why volatility matters
• 20,000

40,000 60,000 80,000 100,000 120,000 140,000 160,000 50 100 150 200 250 300 350 400

Account Value and Dynamic Hedging

Account Value

Time

Buy at higher and higher prices, AFTER market is up Sell at lower and lower prices, AFTER market is down

One up-and- down-cycle

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VA Risk Management Strategies

 Goals

#1 Reduce the expected cost of future guarantees – through product designs #2 Reduce the uncertainty of cost of providing actual future guarantee claims – through hedging

Time 0 Hedging to lock in the cost of providing guarantees Initial option value sold = Future expected cost Final Payoff Uncertainty Difference between target and actual net of hedge is breakage

Goal #1 Goal #2

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Many factors affecting VA guarantee profits/risks

VA profits/risks

Risk-free rate Hedging transaction costs Expected asset growth rates Asset correlations Implied volatility Basis risks/ replication cost Realized volatility Capital needs STAT/GAAP accounting Complex product designs Liquidity/ funding costs Policyholder behavior

These are some of the most exotic, super-long dated, and hybrid derivatives ever created!

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Long term financial and hedging projections

There is something better than back testing

Long term financial projections are core of actuarial technology Stochastic-on-stochastic real world and risk neutral projections For projecting VaR and economic capital into the future and testing different techniques for efficient financial and capital managements under different accounting rules and capital rules Required for regulatory capital and reserve CTE calculations To test optimized hedge strategies and derivatives positions under different market scenarios “Realistic” projections of the future more dynamic and comprehensive than simple back testing or stress testing Calculate option values and Greeks along the paths of real world financial projections

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Summary – Financial engineering and actuarial science

 Law of large numbers does not work effectively with derivatives ⇒ Capital market risks are now dominate risks in many life insurance companies  Law of one number is critical for insurance derivatives products ⇒ Derivative contracts will be priced and risk is mitigated individually (not by average) ⇒ Derivatives are priced with no-arbitrage and risk neutral hedging replication ⇒ Dynamic hedging is used to manage the derivatives risks sold  Variable annuities need both law of large numbers and law of one numbers ⇒ New practice requires both financial engineering and actuarial science ⇒ The non-market components are somewhat diversifiable ⇒ Capital market derivatives exposure long dated and path dependent ⇒ Imperfectly defined derivatives with many moving parts ⇒ New practical and theoretical challenges and opportunities

Contact: Email: frank.zhang@pacificlife.com LinkedIn: www.linkedin.com/in/zhangfrank