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 LIABILITIES IN AN THE PRICING OF LIABILITIES IN AN INCOMPLETE MARKETINCOMPLETE MARKET

USING DYNAMIC MEAN-VARIANCE USING DYNAMIC MEAN-VARIANCE HEDGINGHEDGING

WITH REFERENCE TO AN EQUILIBRIUM WITH REFERENCE TO AN EQUILIBRIUM MARKET MODELMARKET MODEL

RJ THOMSONRJ THOMSONSOUTH AFRICASOUTH AFRICA

The ‘Pricing’ of a LiabilityThe ‘Pricing’ of a Liability

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


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

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


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

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

the price at which a prospective buyer or a seller who the price at which a prospective buyer or a seller who is willing but unpressured and fully informed would is willing but unpressured and fully informed would be indifferent about concluding the transaction, be indifferent about concluding the transaction, provided the effects of moral hazard and legal provided the effects of moral hazard and legal constraints would not be altered by the transactionconstraints would not be altered by the transaction

MeanMean–Variance Hedging–Variance Hedging The mean of the payoff on the assets covering the The mean of the payoff on the assets covering the

liabilities at the end of each period (conditional on liabilities at the end of each period (conditional on information at the start of that period) is equal to that information at the start of that period) is equal to that on the liabilitieson the liabilities

MeanMean–Variance Hedging–Variance Hedging The mean of the payoff on the assets covering the The mean of the payoff on the assets covering the

liabilities at the end of each period (conditional on liabilities at the end of each period (conditional on information at the start of that period) is equal to that information at the start of that period) is equal to that on the liabilities; andon the liabilities; and

The variance of the surplus is minimised.The variance of the surplus is minimised.

Dynamic MeanDynamic Mean–Variance–VarianceHedgingHedging

meanmean–variance hedging in which the time-–variance hedging in which the time-scale of measurement of returns and scale of measurement of returns and redetermination of hedge portfolios is redetermination of hedge portfolios is arbitrarily small in relation to the period to the arbitrarily small in relation to the period to the final payoff of the liabilityfinal payoff of the liability

ThesisThesis

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

Pricing MethodPricing Method

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



plus:plus: the (negative) price of the remaining exposure to the (negative) price of the remaining exposure to

undiversifiable riskundiversifiable risk



plus:plus: the (negative) price of the remaining exposure to the (negative) price of the remaining exposure to

undiversifiable riskundiversifiable risk

When all liability cashflows have been paid, the price of When all liability cashflows have been paid, the price of the liabilities is nil.the liabilities is nil.

Price of Remaining ExposurePrice of Remaining Exposure

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

market portfolio

standard deviation M

EM

remaining exposure

mea

n

capital market line

Pricing the Remaining ExposurePricing the Remaining Exposure

Formulation of the ProblemFormulation of the ProblemLet Let XXtt be the p-component state vector of the stochastic model at

time t. The model defined the conditional distribution:

1tX XxFt

Formulation of the ProblemFormulation of the ProblemLet Let XXtt 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 An ALM defines the following variables as functions of XXtt::

CCtt = the institution’s net cash flow at time = the institution’s net cash flow at time tt;;

VVtt = the market value at time = the market value at time tt of an investment in asset of an investment in asset

category category = 1,..., = 1,..., AA per unit investment at time per unit investment at time tt –– 1; 1;

fftt+1+1 = the amount of a risk-free deposit at time = the amount of a risk-free deposit at time tt + 1 per unit + 1 per unit

investment at time investment at time tt..

1tX XxFt

We denote by We denote by LLtt the market value of the institution’s liabilities at the market value of the institution’s liabilities at

time time tt after the cash flow then payable. after the cash flow then payable.

We denote by We denote by LLtt the market value of the institution’s liabilities at the market value of the institution’s liabilities at

time time tt after the cash flow then payable. after the cash flow then payable.

Suppose that, in order to minimise the variance of the difference Suppose that, in order to minimise the variance of the difference between (between (CCtt + + LLtt) and the value of its hedge portfolio at time ) and the value of its hedge portfolio at time tt

given given XXt t – 1– 1 = = xx, the institution would invest an amount of , the institution would invest an amount of gg,t,t–1–1

in asset category in asset category and and hhtt-1-1 in the risk-free asset (together in the risk-free asset (together

comprising the hedge portfolio).comprising the hedge portfolio).

LetLet and and

At

t

t

g

g

1

g

At

t

t

V

V

1

V

LetLet and and

ThenThen

where where tt, being the undiversifiable exposure, is , being the undiversifiable exposure, is

independent of independent of VVtt, E(, E(tt) = 0, and ) = 0, and ggtt is such that is such that

is minimised.is minimised.

At

t

t

g

g

1

g

At

t

t

V

V

1

V

ttttttt fhLC 11Vg

xXVg 112 Var tttttt LC

LetLet and and

ThenThen

where where tt, being the undiversifiable exposure, is , being the undiversifiable exposure, is

independent of independent of VVtt, E(, E(tt) = 0, and ) = 0, and ggtt is such that is such that

is minimised.is minimised.

Now to get the same expected return on the hedge Now to get the same expected return on the hedge portfolio as on the liability, we require:portfolio as on the liability, we require:

At

t

t

g

g

1

g

At

t

t

V

V

1

V

ttttttt fhLC 11Vg

xXVg 112 Var tttttt LC

0E 11 ttttttt fhLC Vg

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

A

tttt khgL1

111,1

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

The problem is to find The problem is to find LL00 given that given that LLNN = 0 (where = 0 (where NN is is

the last possible cashflow date).the last possible cashflow date).

A

tttt khgL1

111,1

Besides the hedge portfolio, the institution has an Besides the hedge portfolio, the institution has an exposure to exposure to tt. This exposure may be priced as an . This exposure may be priced as an

undiversifiable risk with reference to the risk-free undiversifiable risk with reference to the risk-free deposit and the market portfolio. Suppose the price deposit and the market portfolio. Suppose the price of the exposure is of the exposure is kktt-1-1, of which , of which lltt-1-1 is in the market is in the market

portfolio and (portfolio and (kktt-1-1 – l– ltt-1-1) is in the risk-free deposit. ) is in the risk-free deposit.

Besides the hedge portfolio, the institution has an Besides the hedge portfolio, the institution has an exposure to exposure to tt. This exposure may be priced as an . This exposure may be priced as an

undiversifiable risk with reference to the risk-free undiversifiable risk with reference to the risk-free deposit and the market portfolio. Suppose the price deposit and the market portfolio. Suppose the price of the exposure is of the exposure is kktt-1-1, of which , of which lltt-1-1 is in the market is in the market

portfolio and (portfolio and (kktt-1-1 – l– ltt-1-1) is in the risk-free deposit. ) is in the risk-free deposit.

Then:Then:

andand

0111 tttMttt flkl

221

2Mttt l

Solution of the ProblemSolution of the ProblemIn order to minimiseIn order to minimise , we require:, we require:2

t

CLtCVtVtt σσΣg

11

Solution of the ProblemSolution of the ProblemIn order to minimiseIn order to minimise , we require:, we require:

The resulting value of is:The resulting value of is:

2t

CLtCVttLtCLtCtt σσg 1222 2

CLtCVtVtt σσΣg

11

2t

In order to get In order to get tt = 0, we require: = 0, we require:

011 ttVttLtCt fhμg


i.e.:i.e.:

011 ttVttLtCt fhμg

VttLtCtt

t fh μg 11

1


i.e.:i.e.:

Also:Also:

011 ttVttLtCt fhμg

VttLtCtt

t fh μg 11

1

11

t

Mt

Mt

tt f

k


i.e.:i.e.:

Also:Also:

And hence:And hence:

011 ttVttLtCt fhμg

VttLtCtt

t fh μg 11

1

11

t

Mt

Mt

tt f

k

A

tttt khgL1

111,1

ConclusionsConclusions

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



Bias can be avoided by allowing for cash flow to be Bias can be avoided by allowing for cash flow to be 50-50 at the start and end of the year.50-50 at the start and end of the year.




Problem with the number of components in the state-Problem with the number of components in the state-space vector.space vector.




Problem with the number of components in the state-Problem with the number of components in the state-space vector.space vector.

Liability prices not additive.Liability prices not additive.

Further ResearchFurther Research the reduction of the computational demands associated the reduction of the computational demands associated

with the large number of components of the state-space with the large number of components of the state-space vectorvector


with the large number of components of the state-space with the large number of components of the state-space vector;vector;

the development of a new generation of stochastic the development of a new generation of stochastic actuarial models allowing for equilibrium conditionsactuarial models allowing for equilibrium conditions



the development of a new generation of stochastic the development of a new generation of stochastic actuarial models allowing for equilibrium conditions;actuarial models allowing for equilibrium conditions;

the analysis of the stochastic processes followed by the analysis of the stochastic processes followed by liabilities prices for the determination of capital liabilities prices for the determination of capital adequacyadequacy




the analysis of the stochastic processes followed by the analysis of the stochastic processes followed by liabilities prices for the determination of capital liabilities prices for the determination of capital adequacy;adequacy;

the inclusion in DB fund models of the probability of the inclusion in DB fund models of the probability of the insolvency of the employerthe insolvency of the employer





the inclusion in DB fund models of the probability of the inclusion in DB fund models of the probability of the insolvency of the employer;the insolvency of the employer;

the inclusion of higher-order momentsthe inclusion of higher-order moments





the inclusion in DB fund models of the probability of the inclusion in DB fund models of the probability of the insolvency of the employer;the insolvency of the employer;

the inclusion of higher-order moments; andthe 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.

Documents

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