advertisement

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 ‘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) 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 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 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 variance of the surplus is minimised. Dynamic Mean–Variance Hedging mean–variance hedging in which the timescale 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 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 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. 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 mean Pricing the Remaining Exposure capital market line EM market portfolio remaining exposure standard deviation sM 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: FX t x X t 1 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: FX t x X t 1 An ALM defines the following variables as functions of Xt: Ct = the institution’s 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; ft+1 = the amount of a risk-free deposit at time t + 1 per unit investment at time t. We denote by Lt the market value of the institution’s liabilities at time t after the cash flow then payable. We denote by Lt 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 (Ct + Lt) and the value of its hedge portfolio at time t given Xt – 1 = x, the institution would invest an amount of ga,t–1 in asset category a and ht-1 in the risk-free asset (together comprising the hedge portfolio). g1t Let gt and g At V1t Vt V At g1t Let gt and g At V1t Vt V At Then Ct Lt gt1Vt ht 1 f t e t where et, being the undiversifiable exposure, is independent of Vt, E(et) = 0, and gt is such that s e2t Var Ct Lt gt1Vt X t 1 x is minimised. g1t Let gt and g At V1t Vt V At Then Ct Lt gt1Vt ht 1 f t e t where et, being the undiversifiable exposure, is independent of Vt, E(et) = 0, and gt is such that s e2t Var Ct Lt gt1Vt X t 1 x is minimised. Now to get the same expected return on the hedge portfolio as on the liability, we require: et ECt Lt gt1Vt ht 1 f t 0 The price of the liability comprises the price of the hedge portfolio plus the (negative) price of the exposure, i.e.: A Lt 1 ga ,t 1 ht 1 kt 1 a 1 The price of the liability comprises the price of the hedge portfolio plus the (negative) price of the exposure, i.e.: A Lt 1 ga ,t 1 ht 1 kt 1 a 1 The problem is to find L0 given that LN = 0 (where N is the last possible cashflow date). 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. 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: et lt 1Mt kt 1 lt 1 f t 0 and 2 s e2t lt21s Mt Solution of the Problem In order to minimise s e2t , we require: gt 1 Σ 1 Vt σCVt σCLt Solution of the Problem In order to minimise s e2t , we require: gt 1 Σ 1 Vt σCVt σCLt The resulting value of s et is: 2 s e2t s Ct2 2s CLt s Lt2 gt1 σ CVt σ CLt In order to get et = 0, we require: Ct Lt gt1 μVt ht 1 f t 0 In order to get et = 0, we require: Ct Lt gt1 μVt ht 1 f t 0 i.e.: 1 ht 1 Ct Lt gt1 μVt ft In order to get et = 0, we require: Ct Lt gt1 μVt ht 1 f t 0 i.e.: 1 ht 1 Ct Lt gt1 μVt ft Also: s et Mt kt 1 1 s Mt f t In order to get et = 0, we require: Ct Lt gt1 μVt ht 1 f t 0 i.e.: 1 ht 1 Ct Lt gt1 μVt ft Also: And hence: s et Mt kt 1 1 s Mt f t A Lt 1 ga ,t 1 ht 1 kt 1 a 1 Conclusions Method consistent with option pricing because the undiversifiable risk tends to zero as the time interval tends to zero. 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. 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 statespace vector. 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 statespace vector. Liability prices not additive. Further Research 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 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 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 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 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 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.