Valuation and RiskManagement of EquityLinked Life Insurance Contracts Via Quantile Hedging

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Valuation and Risk-Management of Equity-Linked
Life Insurance Contracts Via Quantile Hedging

• Alexander Melnikov, University of Alberta
• Victoria Skornyakova, Mercer Investment Consulting

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IntroductionTheoretical Motivation

• Using Hedging for Pricing and Risk Management
  construct a strategy that exactly replicates the
  cash flows of a contingent claim
• Exact replication is not possible find a
  strategy with a cash flow close enough to the
  payoff of the contingent claim in some
  probabilistic sense
• Equity-linked life insurance contracts have a
  mortality component gt the exact replication is
  not possible

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IntroductionReferences on equity-linked life
insurance

• Brennan and Schwartz (1976, 1979)
• Boyle and Schwartz (1977)
• Bacinello and Ortu (1993)
• Aase and Person (1994)
• Ekern and Person (1996)
• Moeller (1998, 2001)
• Contracts with fixed or deterministic guarantees
• Reduced them to call/put options
• Apply perfect or mean-variance hedging to
  calculate prices

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IntroductionIn this paper we

• Consider three models describing a financial
  market
• Study equity-linked pure endowment contracts with
  a fixed and stochastic guarantee
• Use quantile hedging technique for pricing and
  risk-management of these contracts
• Illustrate our results with actual data
• The mortality risk cannot be hedged, but, for
  equity-linked pure endowment contracts, the
  younger the client, the greater is the
  probability that the payment will be due at
  maturity, and the greater should be the
  probability of successful hedging.
• How much should be the probability of successful
  hedging for a client of given age?

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Description of the Models Financial Setting,
Model 1

• Non-risky asset
  
• Risky asset on , prices
  follow the Black-Scholes model
• Market is complete, the unique risk-neutral
  probability has the density
• Admissible, self-financing, adapted to filtration
  portfolio the value of
  the portfolio is
• Strategy with the payoff at maturity

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Financial SettingModel 2

• Non-risky asset
  
• Risky assets on
  , prices follow the diffusion model
• Market is complete, the unique risk-neutral
  probability has the density
• Conditions
• Admissible, self-financing, adapted to the
  filtration portfolio
  
• with the value
• Strategy with the payoff at maturity

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Financial SettingModel 3

• Non-risky asset
  
• Risky assets on
  , prices follow the jump-diffusion model
• Market is complete, the unique risk-neutral
  probability has the density
• are the unique solutions to
• Conditions

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Description of the Models Insurance Setting

• on is the remaining
  life time of a person at age
• is a survival
  probability
• Assumption and
  are independent
• Mortality risk arises from the dependence of the
  payoff on the survival status of a client at
  maturity
• The payoff for Model 1
• The payoff for Models 2 and 3

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Fair Pricing and Hedging

• Fair price for Model 1
• Fair price for Models 2 and 3
• Perfect hedging is not possible due to a budget
  constraint
• Find a strategy that will hedge successfully with
  the maximal probability

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Quantile HedgingDefinitions

• Self-financing strategy has a budget
  constraint
• 
  is a successful hedging set
• is a quantile hedge if
• How to construct the quantile hedge and the
  successful hedging set

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Quantile HedgingMethodology

• Lemma (Foellmer and Leukert (1999))
• Let be a solution to the
  problem
• Then the quantile hedge
• does exist
• is unique
• is a perfect hedge for a modified claim
• The structure of a maximal successful hedging set
  is where a
  constant is defined by

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Equity-Linked Life InsuranceConnecting Financial
and Insurance Risks

• Due to the structure of the fair price, we apply
  quantile hedging to
• 
  and
• Consider
  and as
  bounds on the budget used to hedge
  and respectively
  
• From the definitions of perfect and quantile
  hedging

Key formulae connecting financial and insurance
risks
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Application of Quantile HedgingPreliminary
Calculations

• Maximal successful hedging sets
• The characteristic equation
• has the unique solution if
  and two solutions if
• Further analysis relies on properties of
  diffusion and jump-diffusion processes

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Main ResultsTheorem 1

• Financial market is described by Model 1
• Equity-linked life insurance contract with a
  fixed guarantee
• The characteristic equation has one solution
  if and two solutions
  if
• Then
• 
  if
• 
  
  if
• where
  are defined by

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Remark How to find Solutions to Characteristic
Equation ?

• is a probability of failure to hedge
  perfectly or
• If then
  and
• Using log-normality of prices,
• 
  is the unique solution
• If then
  can be found from

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Main ResultsTheorem 2

• Financial market is described by Model 2
• Equity-linked life insurance contract with a
  flexible guarantee
• The characteristic equation has one solution
  if and two solutions
  if
• Then
• 
  if
• 
  
  if
• where
  

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Remark How to find Solutions to Characteristic
Equation ?

• Under technical assumptions
• the characteristic equation has the unique
  solution. Then
• Using log-normality of
• If the characteristic equation has two solutions,
  they can be found from

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Main ResultsTheorem 3

• Financial market is described by Model 3
• Equity-linked life insurance contract with a
  flexible guarantee
• The characteristic equation has one solution
  on a set if
  
• Then
• 
  
• where
  

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Remark How to find Solutions to Characteristic
Equation ?

• Fix a probability of failure to hedge on
  each set
• If
• then the characteristic equation has the
  unique solution on each set
  and
• Using the log-normality of the conditional
  distribution,

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Diversification of Mortality Risk

• Pool homogenous clients of the same age, life
  expectance, investment preferences into the group
  of size , then the cumulative claim at
  maturity is , where has
  a binomial distribution with parameters
• is the level of financial risk, or the
  probability that the quantile hedge
• will fail to hedge
  perfectly
• is the level of insurance (mortality) risk,
  or the probability that the number of clients
  alive at maturity will be greater than expected
• Due to the independence of financial and
  insurance risks,
• therefore, using the quantile hedge, the
  company is able to hedge the cummulative claim
  with the probability at least

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Risk-Management Implementation
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Numerical IllustrationInputs

• Contracts with a fixed guarantee SP 500,
• Contracts with flexible guarantee Russell 2000
  and the SP 500
• Estimated parameters (from monthly observations
  from 01/1979 to 12/2004)
• Model 1
  Model 2
• Model 3
• is an initial investment
• are terms of the
  contracts
• is a risk-free rate
• Mortality data UP94_at_2015

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Numerical IllustrationModel 1

• Acceptable Financial Risk as Function of Clients
  Age

• Acceptable Financial Risk for Clients at
  Specified Ages

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Numerical IllustrationModel 1 Pricing
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Numerical IllustrationModel 2

• Acceptable Financial Risk as Function of Clients
  Age

• Acceptable Financial Risk for Clients at
  Specified Ages

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Numerical IllustrationModel 2 Pricing
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Numerical IllustrationModel 3

• Acceptable Financial Risk as Function of Clients
  Age

• Acceptable Financial Risk for Clients at
  Specified Ages

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Numerical IllustrationModel 3 Pricing
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Numerical IllustrationConclusions

• Quantile hedging works better for contracts with
  flexible guarantees and shorter duration
  contracts with fixed guarantees
• Contracts with a flexible guarantee has greater
  exposure to financial risk than similar contracts
  with a fixed guarantee
• Whenever the age of clients increases, the
  probability of successful hedging decreases and
  the company is becoming able to take greater
  financial risk exposure
• With longer contract maturities, the company is
  able to attract younger clients while maintaining
  the same financial risk exposure, as a survival
  probability is decreasing over time
• The reduction in prices was possible for two
  reasons, we took into account
• Mortality risk of an individual client
• Diversification of cumulative mortality risk

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Further Developments

• Mortality Modeling
• Use theoretical models of mortality (Gompertz,
  Weibull, Lee-Carter etc.)
• Allows to take into account new tendencies in
  mortality
• Modeling with other Risk Measures
• Conditional Tail Expectation
• Rockafellar and Uryasev (2002)
• If is a solution to
• then

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Further DevelopmentsModeling with other Risk
Measures

• Shortfall minimization problem for the claim
• 
  over all strategies with initial
  budget constraints
• solution to this problem
  (Foellmer and Leukert (2000)), where
  is a perfect hedge to a modified claim
• The function has the following
  structure
• where

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References

• Aase, K. and S. Persson, 1994. Pricing of
  Unit-Linked Insurance Policies. Scandinavian
  Actuarial Journal 1 26-52
• Bacinello, A.R. and F. Ortu, 1993. Pricing of
  Unit-Linked Life Insurance with Endegeneous
  Minimum Guarantees. Insurance Math. and
  Economics 12245-257
• Boyle, P.P. and E.S. Schwartz, 1977. Equilibrium
  Prices of Guarantees under Equity-Linked
  Contracts. Journal of Risk and Insurance 44
  639-680
• Brennan, M., and E.S. Schwartz, 1976. The Pricing
  of Equity-Linked Life Insurance Policies with an
  Asset Value Guarantee. J. Financial Economics 3
  195-213
• Brennan, M., and E.S. Schwartz, 1979. Alternative
  Investment Strategies for the Issuers of
  Equity-Linked Life Insurance with an Asset Value
  Guarantee. Journal of Business 52 63-93
• Ekern, S. and S. Persson, 1996. Exotic
  Unit-Linked Life Insurance Contracts. Geneva
  Papers on Risk and Insurance Theory 21 35-63
• Foellmer, H., and P. Leukert, 1999. Quantile
  Hedging. Finance Stochast. 3 251-273
• Foellmer, H., and P. Leukert, 2000. Efficient
  Hedging Cost Versus Short-Fall Risk. Finance
  Stochast. 4 117-146
• Moeller, T., 1998. Risk-Minimizing Hedging
  Strategies for Unit-Linked Life-Insurance
  Contracts. Astin Bulletin 28 17-47
• Moeller, T., 2001. Hedging Equity-Linked Life
  Insurance Contracts. North American Actuarial
  Journal 5 79-95
• Rockafellar, R.T. and S. Uryasev, 2002.
  Conditional Value-at-Risk for General Loss
  Distributions. J.BankingFinance 26 1443-1471