Mean-Variance Portfolio Optimization with State-Dependent Risk
Aversion
Björk, Tomas; Murgoci, Agatha; Zhou, Xun Yu
Document Version
Final published version
Published in:
Mathematical Finance
DOI:
10.1111/j.1467-9965.2011.00515.x
Publication date:
2014
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CC BY-NC-ND
Citation for published version (APA):
Björk, T., Murgoci, A., & Zhou, X. Y. (2014). Mean-Variance Portfolio Optimization with State-Dependent Risk
Aversion. Mathematical Finance, 24(1), 1-24. https://doi.org/10.1111/j.1467-9965.2011.00515.x
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Mean–Variance Portfolio Optimization
with State Dependent Risk Aversion
∗
Tomas Bj¨ork
†
Agatha Murgoci
‡
Xun Yu Zhou
§
July 8, 2011
First version October 2009
Abstract
The objective of this paper is to study the mean–variance portfolio
optimization in continuous time. Since this problem is time inconsistent
we attack it by placing the problem within a game theoretic framework
and look for subgame perfect Nash equilibrium strategies. This particular
problem has already been studied in [2] w here the authors assumed a con-
stant risk aversion parameter. This assumption leads to an equilibrium
control where the dollar amount invested in the risky asset is independent
of current wealth, and we argue that this result is unrealistic from an eco-
nomic p oint of view. In order to have a more realistic model we instead
study the case when the risk aversion depends dynami cally on current
wealth. This is a substanti ally more compli cated problem than the one
with constant risk aversion but, using the general theory of time inconsis-
tent control developed in [4], we provide a fairly detailed analysis on the
general case. In particular, when the risk aversion is inversely proportional
to wealth, we provide an analytical solution where the equilibrium dollar
amount invested in the risky asset is proportional to current wealth. The
equilibrium for this model thus appears more reasonable than the one for
the model with constant risk aversion.
Key words: Mean–variance, time i nconsistency, time inconsistent control,
dynamic programming, stochastic control, Hamilton-Jacobi-Bellman equation
∗
The authors are greatly indebted to Ivar Ekeland, Ali Lazrak, Traian Pirvu, and Suleyman
Basak for very helpful discussions. We are also very grateful to two anonymous referees for a
numbe r of comments, which have improved the pape r considerably.
†
Department of Finance, Stockholm School of Economics, Box 6501, SE- 113 83 Stockholm,
SWEDEN. E-mail: tomas.bjork@hhs.se
‡
Department of Finance, Stockholm School of Economics, Box 6501, SE- 113 83 Stockholm,
SWEDEN. E-mail: agatha.murgoci@hhs.se
§
Mathematical Institute , University of Oxford, 24-29 St Giles’, Oxford OX1 3LB, UK, and
Dept artment of Systems Engineering & Engineering Management, The Chinese University of
Hong Kong, Shat in, Hong Kong. E-mail: zhouxy@maths.ox.ac.uk
1
This is the accepted version of the following article: Björk, T.,
Murgoci, A. and Zhou, X. Y. (2014), MEAN–VARIANCE
PORTFOLIO OPTIMIZATION WITH STATE-DEPENDENT RISK
AVERSION. Mathematical Finance, 24: 1–24., which has been
published in final form at doi:
http://dx.doi.org/10.1111/j.1467-9965.2011.00515.x
1 Introduction
Mean–variance (MV) analysis for opti mal asset allocation is one of the classical
results of financial economics. After the orig inal publi cation i n [14], a vast
number of papers have been published on this topic. Most of these papers deal
with the sing le period case, and there is a very good reason for this: It is very
easy to see that an MV optimal portfol io problem in a multi period framework
is time inconsistent in the sense that the Bellman Optimality Principle does not
hold. As a consequence, dynamic programming cannot be easily applied, a nd it
is in fact not at all clear what one should mean by the term “optimal”.
In the literature there are two basic ways of handling (various forms of)
time inconsistency in optimal control problems. One possibility is to study the
pre-committ ed problem, where “optimal” is interpreted as “opti mal from the
point of view of time zero”. Kydland and Prescott [10] indeed argue tha t a pre-
commi tted strategy may be economically meaningful in certain circumstances.
In the context of MV portfolio choice, [17] is probably the ea rliest paper that
studies a pre-committed MV model i n a continuous-time setting (although he
considers only one single stock with a constant risk-free rate), followed by [1].
In a discrete-time setting, [11] developed an embedding technique to change the
originally time-inconsistent MV problem into a stochastic LQ control problem.
This technique was extended in [21], along with an indefinite stochastic linear–
quadratic control approach, to the continuous-time case. Further extensions
and improvements are carried out in, among many others, [13], [12], [3], and
[20]. Markowitz’s probl em wi th transaction cost i s recently solved in [5]. Note
that in all these works only pre-committed strategies have been derived.
Another possibility is to take the time inconsistency more seriously and
study the problem within a game theoretic framework. This is i n fact the
approach of the present paper. One possible interpretation of the time incon-
sistency is that our preferences change in a temporally inconsistent way as time
goes by, and we can thus view the MV problem as a game, where the players
are the future incarnations of our own preferences. We then loo k for a subgame
perfect Nash equil ibrium point for this game.
The game theoretic approach to addressing general time i nconsistency via
Nash equilibrium points has a long hi story starting with [18] where a determin-
istic Ramsey problem is studied. Further work along this line in continuous and
discrete time is provided in [8 ], [9], [15], [16], and [19].
Recently there has been renewed interest in these problems. In the interest-
ing papers [6] and [7], the authors consider optim al consumption and investment
under hyperbolic discounting in deterministic and stochastic models from the
above game theoretic p oint of view. To our knowledge, these papers were the
first to provide a precise definition of the game theoretic equilibrium concept in
continuous tim e.
In the particular case of MV analysis, the game theoretic approach to time
inconsistency was first studied (in discrete and continuous time) in [2], where
the authors undertake a deep study of the problem within a W iener driven
framework. The case of multipl e assets, as well as the case of a hidden Markov
2
process driving the parameters of the asset price dynamics are also treated.
The authors derive an extension of the Hamilton–Jacobi–Bellman equation and
manag es, by a number of very clever ideas, to solve this equation explicitly for
the basic problem, and also fo r the above mentioned extensions. The method-
ology of [2] is, among other things, to use a “total variance formula”, which
partially extends the sta ndard iterated expectations formula. This works very
nicely in the MV case, but drawback of this particular approach is that it seems
quite hard to extend the results to other objective functions than MV.
The first pa per to treat the game theoretic approa ch to time inconsistency
in more general terms was [4] where the authors consider a fairl y general class
of (time inconsistent) objective functions and a very general controlled Markov
process. Within this framework the authors derive an extension of the standard
dynamic programming equation, to a system of equations (which i n the diffusion
case is a system of non linear PDEs). The framework of [4] is general enough
to include many previously known models of time i nconsistency. In particular,
[4] repro duces the result of [2] for the MV problem in a Black–Scholes market.
Going back to the MV po rtfolio optimization, there is a non trivial problem
connected with the solution presented in [2]. In the problem formulation of [2],
the MV objective function at ti me t, given current wealth X
t
= x, is given by
E
t,x
[X
T
] −
γ
2
V ar
t,x
[X
T
] ,
where X
T
is the wealth at the end of the time period, and where γ is a given
constant representing the risk aversion of the agent. For such a model it turns
out (see [2]) that, at time t and when total wealth is x , the optimal dollar
amount u(t, x) invested in the risky asset is of the form
u(t, x) = h (t)
where h is a deterministic f unction of time. In particular this i mplies that the
dollar amount invested in the risky asset do es not depend on current wealth x.
In our opinion this result is economically unreasonable, since it implies
that you will invest the same number of dollars in the stock if your wealth is
100 dollars as you would if your wealth is 100,000,000 dollars. See Section ??
for a more detailed discussion.
The deeper reason for this anomaly is the fact that the risk aversion param-
eter γ is assumed to be a constant, which is clearly unreasona ble. A person’s
risk preference certainly depends on how wealthy he is; and hence the obvi-
ous impli cation is that we should explicitly allow γ to depen d on current
wealth.
The main goal of the present paper is precisely to study MV problems
with a state dependent risk aversion. More explicitly we consider an objective
function of the form
E
t,x
[X
T
] −
γ(x)
2
V ar
t,x
[X
T
] ,
where γ is a deterministic function of present wealth x. This type of problem
cannot easily be treated within the framework of [2], but it is a simple special
case of the theory developed in [ 4].
3
The structure and main result of the present paper are as follows. In
Section 2 we present our formal model. We discuss the time inconsistency
of the mean variance problem and we place that problem w ithin the general
framework of [4]. The game theo retic problem is presented, bo th in informal
and in mathematically precise terms, and from [4] we cite the general theoretical
results that we need for our analysis.
In Section 3 we give a brief recapitulation of the MV problem with constant
γ studied in [2], and we use our general theory to derive the sol ution. In Section
4 we study the MV problem with state dependent risk aversion using the theory
developed in [4]. We start by deriving a surprisingly expl icit solution for the
case of a general risk aversion γ(x). We then specialize to the economically
natural case of γ(x) = γ/x and for this case we obtain an analytic solution.
More precisely, we show that the optimal dollar amo unt
ˆ
u
t
to invest in the
risky asset at time t is given by
ˆ
u
t
= c(t)x
where the deterministic function c solves an integral equation. This i s the main
result of the paper, and it shows that with the proper specification of the risk
aversion γ(x), the optimal solution is indeed economically reasonable.
We finish the paper by proving that the integral equation for c admits a
unique solution, and we also provide a numerical algorithm for computing c.
The algorithm is implemented for some natural parameter combinations and we
present graphs for illustrative purposes.
2 The basic framework
In this section we formulate the problem under consideration.
2.1 The model
Our basic setup is a standard Black-Scholes model for a risky stock with GBM
price dynamics and a bank a ccount with constant risk free short rate r. Denoting
the stock price by S and the bank account by B, the dynamics are as follows
under the objective probability measure P .
dS
t
= αS
t
dt + σS
t
dW
t
,
dB
t
= rB
t
dt.
Here W is a standard P -Wiener process, and the constants α, σ , and r are
assumed to be known. In the analysis below, we will study self-financing port-
folios (with zero consumption), based on S and B. Denoting the dollar value
invested in the risky asset at time t by u
t
, the value process X of the portfolio
is easily seen to have dynamics given by
dX
t
= [rX
t
+ (α − r) u
t
] dt + σu
t
dW
t
.
4
