THE PRICING OF LIABILITIES IN AN INCOMPLETE MARKET USING DYNAMIC MEANVARIANCE HEDGING WITH REFERENCE

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THE PRICING OF LIABILITIES IN AN INCOMPLETE
MARKETUSING DYNAMIC MEAN-VARIANCE HEDGINGWITH
REFERENCE TO AN EQUILIBRIUM MARKET MODELRJ
THOMSONSOUTH AFRICA
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The Pricing of a Liability

• the price at which the liability would trade if a
  complete market existed (a fiction)

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The Pricing of a Liability

• the price at which the liability would trade if a
  complete market existed (a fiction) or
• the price at which the liability would trade if a
  liquid market existed (a fiction)

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The Pricing of a Liability

• the price at which the liability would trade if a
  complete market existed (a fiction) or
• the price at which the liability would trade if a
  liquid market existed (a fiction) or
• the price at which a prospective buyer or a
  seller who is willing but unpressured and fully
  informed would be indifferent about concluding
  the transaction, provided the effects of moral
  hazard and legal constraints would not be altered
  by the transaction

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MeanVariance Hedging

• The mean of the payoff on the assets covering the
  liabilities at the end of each period
  (conditional on information at the start of that
  period) is equal to that on the liabilities

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MeanVariance Hedging

• The mean of the payoff on the assets covering the
  liabilities at the end of each period
  (conditional on information at the start of that
  period) is equal to that on the liabilities and
• The variance of the surplus is minimised.

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Dynamic MeanVarianceHedging

• meanvariance hedging in which the time-scale of
  measurement of returns and redetermination of
  hedge portfolios is arbitrarily small in relation
  to the period to the final payoff of the liability

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Thesis

• If a stochastic assetliability model (ALM) is
  adopted, and the market, though incomplete, is in
  equilibrium, and the ALM is consistent with the
  market, then a unique price can be obtained that
  is consistent both with the ALM and with the
  market.

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Pricing Method

• At the start of a year, the price of the
  liabilities equals
• the price of the hedge portfolio for that year

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Pricing Method

• At the start of a year, the price of the
  liabilities equals
• the price of the hedge portfolio for that year
• plus
• the (negative) price of the remaining exposure to
  undiversifiable risk

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Pricing Method

• At the start of a year, the price of the
  liabilities equals
• the price of the hedge portfolio for that year
• plus
• the (negative) price of the remaining exposure to
  undiversifiable risk
• When all liability cashflows have been paid, the
  price of the liabilities is nil.

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Price of Remaining Exposure

• equal to that of a portfolio, comprising the
  market portfolio and the risk-free asset, whose
  expected payoff at the end of the period is nil
  and whose variance is equal to that of the payoff
  on the liabilities

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Pricing the Remaining Exposure
mean
capital market line
EM
market portfolio
remaining exposure
sM
standard deviation
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Formulation of the Problem

• Let Xt be the p-component state vector of the
  stochastic model at time t. The model defined the
  conditional distribution

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Formulation of the Problem

• Let Xt be the p-component state vector of the
  stochastic model at time t. The model defined the
  conditional distribution
• An ALM defines the following variables as
  functions of Xt
• Ct the institutions net cash flow at time t
• Vat the market value at time t of an
  investment in asset category a 1,..., A per
  unit investment at time t 1
• ft1 the amount of a risk-free deposit at time
  t 1 per unit investment at time t.

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• We denote by Lt the market value of the
  institutions liabilities at time t after the
  cash flow then payable.

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• We denote by Lt the market value of the
  institutions liabilities at time t after the
  cash flow then payable.
• Suppose that, in order to minimise the variance
  of the difference between (Ct Lt) and the value
  of its hedge portfolio at time t given Xt 1
  x, the institution would invest an amount of
  ga,t1 in asset category a and ht-1 in the
  risk-free asset (together comprising the hedge
  portfolio).

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• Let and

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• Let and
• Then
• where et, being the undiversifiable exposure, is
  independent of Vt, E(et) 0, and gt is such that
• is minimised.

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• Let and
• Then
• where et, being the undiversifiable exposure, is
  independent of Vt, E(et) 0, and gt is such that
• is minimised.
• Now to get the same expected return on the hedge
  portfolio as on the liability, we require

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• The price of the liability comprises the price of
  the hedge portfolio plus the (negative) price of
  the exposure, i.e.

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• The price of the liability comprises the price of
  the hedge portfolio plus the (negative) price of
  the exposure, i.e.
• The problem is to find L0 given that LN 0
  (where N is the last possible cashflow date).

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• Besides the hedge portfolio, the institution has
  an exposure to et. This exposure may be priced as
  an undiversifiable risk with reference to the
  risk-free deposit and the market portfolio.
  Suppose the price of the exposure is kt-1, of
  which lt-1 is in the market portfolio and (kt-1
  lt-1) is in the risk-free deposit.

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• Besides the hedge portfolio, the institution has
  an exposure to et. This exposure may be priced as
  an undiversifiable risk with reference to the
  risk-free deposit and the market portfolio.
  Suppose the price of the exposure is kt-1, of
  which lt-1 is in the market portfolio and (kt-1
  lt-1) is in the risk-free deposit. Then
• and

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Solution of the Problem

• In order to minimise , we require

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Solution of the Problem

• In order to minimise , we require
• The resulting value of is

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• In order to get met 0, we require

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• In order to get met 0, we require
• i.e.

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• In order to get met 0, we require
• i.e.
• Also

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• In order to get met 0, we require
• i.e.
• Also
• And hence

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Conclusions

• Method consistent with option pricing because the
  undiversifiable risk tends to zero as the time
  interval tends to zero.

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Conclusions

• Method consistent with option pricing because the
  undiversifiable risk tends to zero as the time
  interval tends to zero.
• Bias can be avoided by allowing for cash flow to
  be 50-50 at the start and end of the year.

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Conclusions

• Method consistent with option pricing because the
  undiversifiable risk tends to zero as the time
  interval tends to zero.
• Bias can be avoided by allowing for cash flow to
  be 50-50 at the start and end of the year.
• Problem with the number of components in the
  state-space vector.

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Conclusions

• Method consistent with option pricing because the
  undiversifiable risk tends to zero as the time
  interval tends to zero.
• Bias can be avoided by allowing for cash flow to
  be 50-50 at the start and end of the year.
• Problem with the number of components in the
  state-space vector.
• Liability prices not additive.

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Further Research

• the reduction of the computational demands
  associated with the large number of components of
  the state-space vector

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Further Research

• the reduction of the computational demands
  associated with the large number of components of
  the state-space vector
• the development of a new generation of stochastic
  actuarial models allowing for equilibrium
  conditions

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Further Research

• the reduction of the computational demands
  associated with the large number of components of
  the state-space vector
• the development of a new generation of stochastic
  actuarial models allowing for equilibrium
  conditions
• the analysis of the stochastic processes followed
  by liabilities prices for the determination of
  capital adequacy

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Further Research

• the reduction of the computational demands
  associated with the large number of components of
  the state-space vector
• the development of a new generation of stochastic
  actuarial models allowing for equilibrium
  conditions
• the analysis of the stochastic processes followed
  by liabilities prices for the determination of
  capital adequacy
• the inclusion in DB fund models of the
  probability of the insolvency of the employer

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Further Research

• the reduction of the computational demands
  associated with the large number of components of
  the state-space vector
• the development of a new generation of stochastic
  actuarial models allowing for equilibrium
  conditions
• the analysis of the stochastic processes followed
  by liabilities prices for the determination of
  capital adequacy
• the inclusion in DB fund models of the
  probability of the insolvency of the employer
• the inclusion of higher-order moments

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Further Research

• the reduction of the computational demands
  associated with the large number of components of
  the state-space vector
• the development of a new generation of stochastic
  actuarial models allowing for equilibrium
  conditions
• the analysis of the stochastic processes followed
  by liabilities prices for the determination of
  capital adequacy
• the inclusion in DB fund models of the
  probability of the insolvency of the employer
• the inclusion of higher-order moments and
• the adaptation of the method to a multi-currency
  world.

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