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  gt1Vt  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  gt1Vt X t 1  x 
is minimised.
 g1t 


Let gt     and
g 
 At 
 V1t 
 
Vt    
V 
 At 
Then Ct  Lt  gt1Vt  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  gt1Vt X t 1  x 
is minimised.
Now to get the same expected return on the hedge
portfolio as on the liability, we require:
et  ECt  Lt  gt1Vt  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 1Mt  kt 1  lt 1  f t  0
and
2
s e2t  lt21s 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  gt1 σ CVt  σ CLt 
In order to get et = 0, we require:
Ct   Lt  gt1 μVt  ht 1 f t  0
In order to get et = 0, we require:
Ct   Lt  gt1 μVt  ht 1 f t  0
i.e.:
1
ht 1  Ct   Lt  gt1 μVt 
ft
In order to get et = 0, we require:
Ct   Lt  gt1 μVt  ht 1 f t  0
i.e.:
1
ht 1  Ct   Lt  gt1 μVt 
ft
Also:
s et   Mt 

kt 1  
 1
s Mt  f t

In order to get et = 0, we require:
Ct   Lt  gt1 μVt  ht 1 f t  0
i.e.:
1
ht 1  Ct   Lt  gt1 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.