THE PRICING OF LIABILITIES IN AN INCOMPLETE MARKET USING DYNAMIC MEAN-VARIANCE HEDGING WITH REFERENCE TO AN EQUILIBRIUM MARKET MODEL RJ THOMSON SOUTH AFRICA

The ‘Pricing’ of a Liability the price at which the liability would trade if a complete market existed (a fiction) the price at which the liability would trade if a complete market existed (a fiction)

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

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 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 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 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

Mean–Variance 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 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

Mean–Variance 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 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. The variance of the surplus is minimised.

Dynamic Mean–Variance Hedging mean–variance 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

Thesis If a stochastic asset–liability 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.

Pricing Method At the start of a year, the price of the liabilities equals: the price of the hedge portfolio for that year the price of the hedge portfolio for that year

Pricing Method At the start of a year, the price of the liabilities equals: the price of the hedge portfolio for that year the price of the hedge portfolio for that yearplus: the (negative) price of the remaining exposure to undiversifiable risk the (negative) price of the remaining exposure to undiversifiable risk

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

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

market portfolio standard deviation MM EMEM remaining exposure mean capital market line Pricing the Remaining Exposure

Formulation of the Problem Let X t Let X t be the p-component state vector of the stochastic model at time t. The model defined the conditional distribution:

Formulation of the Problem Let X t Let X t 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 X t : C t = the institution’s net cash flow at time t; V  t = the market value at time t of an investment in asset category  = 1,..., A per unit investment at time t – 1; f t +1 = the amount of a risk-free deposit at time t + 1 per unit investment at time t.

We denote by L t the market value of the institution’s liabilities at time t after the cash flow then payable.

Suppose that, in order to minimise the variance of the difference between (C t + L t ) and the value of its hedge portfolio at time t given X t – 1 = x, the institution would invest an amount of g ,t–1 in asset category  and h t-1 in the risk-free asset (together comprising the hedge portfolio).

Let and

Then where  t, being the undiversifiable exposure, is independent of V t, E(  t ) = 0, and g t is such that is minimised.

Let and Then where  t, being the undiversifiable exposure, is independent of V t, E(  t ) = 0, and g t is such that is minimised. Now to get the same expected return on the hedge portfolio as on the liability, we require:

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 L 0 given that L N = 0 (where N is the last possible cashflow date).

Besides the hedge portfolio, the institution has an exposure to  t. 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 k t-1, of which l t-1 is in the market portfolio and (k t-1 – l t-1 ) is in the risk-free deposit.

Besides the hedge portfolio, the institution has an exposure to  t. 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 k t-1, of which l t-1 is in the market portfolio and (k t-1 – l t-1 ) is in the risk-free deposit. Then: and

Solution of the Problem In order to minimise, we require:

Solution of the Problem In order to minimise, we require: The resulting value of is:

In order to get   t = 0, we require:

i.e.:

i.e.:Also:

i.e.:Also: And hence:

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

Conclusions Bias can be avoided by allowing for cash flow to be at the start and end of the year. Bias can be avoided by allowing for cash flow to be at the start and end of the year.

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

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

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

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

Further Research the reduction of the computational demands associated with the large number of components of the state-space vector; 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 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 analysis of the stochastic processes followed by liabilities prices for the determination of capital adequacy

Further Research the reduction of the computational demands associated with the large number of components of the state-space vector; 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 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 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 in DB fund models of the probability of the insolvency of the employer

Further Research the reduction of the computational demands associated with the large number of components of the state-space vector; 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 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 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 in DB fund models of the probability of the insolvency of the employer; the inclusion of higher-order moments the inclusion of higher-order moments

Further Research the reduction of the computational demands associated with the large number of components of the state-space vector; 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 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 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 in DB fund models of the probability of the insolvency of the employer; the inclusion of higher-order moments; and the inclusion of higher-order moments; and the adaptation of the method to a multi-currency world. the adaptation of the method to a multi-currency world.