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GREGORY LENOIR AND NILS S. TUCHSCHMID
INVESTMENT TIME HORIZON AND
ASSET ALLOCATION MODELS
Gregory Lenoir, Crédit Agricole Indosuez, Geneva, Switzerland
Nils S. Tuchschmid, BCV, CP 300 CH - 1001 Lausanne,
Switzerland, nils.tuchschmid@bcv.ch
This paper benefits from the financial support of the CTI
(Commission for Technology and Information # 4036.1) and
Synchrony Asset Management. The first version was written
when both authors were working at Synchrony.
We would like to thank Heinz Müller and Heinz Zimmermann
for their helpful comments and advice.
1. Introduction
Asset allocation models have been actively
studied for more than three decades. The well
known mean-variance strategy due to
MARKOWITZ (1953,) was originally devel-
oped to find the optimal allocation among as-
sets over a single period, that is, assuming a
constant investment opportunity set. Hence, a
mean-variance approach, even in a multi-
periodic framework, remains a sub-efficient
static method unless one is willing to accept
that the return and risk parameters character-
ising the financial assets are constant or at
least deterministic.
MERTON (1971) has extended the theory in a
continuous-time setting with constant, time-
dependent or even stochastic investment op-
portunity sets and taken explicitly into account
the investment time horizon in the optimisation
procedure. Unfortunately, absent analytical
solutions and the lack of computing power
made it infeasible to solve realistic problems
once it is assumed that the investment oppor-
tunity set evolves randomly. Only recently, a
few papers numerically implement the famous
MERTON’S optimisation problem (see for in-
stance BRENNAN et al. (1997), BREITLER et
al. (1998)). In the latter, the “economies” are
described by a small number of stochastic state
variables that drive the investment oppor-
tunities.
Hence, the objective of this paper is twofold:
first we wish to develop and to implement
an intertemporal or dynamic asset allocation
model based on the original MERTON’s prob-
lem for the Swiss financial market. The optimi-
sation procedure will thus take explicitly into
account the investor’s time horizon. Second,
we will analyse the performance of this model
in the scope of a practical implementation in
comparison with the traditional “mean-
variance” strategy.
The organisation of the paper is as follows. In
section 2, we present the theoretical founda-
tions of the optimal stochastic control. A par-
ticular attention is given to the definition and
the choice of state variables describing the
Swiss market and to the comparison of the re-
sulting dynamic optimal asset allocation model
with a standard mean-variance criterion. Sec-
tion 3 presents the innovations of the method:
© Swiss Society for Financial Market Research (pp. 76–93)
76 FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 15, 2001 / Number 1
that is, the impact of time horizon on the dy-
namic optimal allocation strategies and the be-
haviour of the hedging demand against the fu-
ture shifts of the investment opportunity set.
Section 4 is then devoted to deepen the analy-
sis of the statistical behaviour of the asset allo-
cation model by the use of simulations. In sec-
tion 5, we test and analyse the performance of
out-of-sample results based on realistic in-
vestment constraints and compare it to alter-
native standard strategies. Section 6 studies
the impact of transaction fees induced by a
practical implementation of the strategy. Fi-
nally, section 7 concludes by summarising the
main results and properties of the model.
2. Theoretical Foundations
The purpose of this article is not to provide an
exhaustive presentation on the optimal sto-
chastic control theory and its application to
dynamic asset allocation problems, but rather
to study practical problems linked to dynamic
asset allocation strategies and to analyse the
results in light of standard mono-periodic
strategies. For a complete view of the theoreti-
cal foundations behind the problem developed
in this section, one can refer to BREITLER
et al. (1999,). More technical details on the
optimal stochastic control theory can also be
found, for instance, in BAGCHI (1993),
KORN (1997) or MERTON (1990).
2.1 A Brief Explanation
In a traditional Mean-Variance optimisation
problem, the impact of the investor’s time ho-
rizon is neglected or simply not considered.
Indeed, it is assumed that the decision making
has to be made over a predefined time period,
(t0; T) where t0 is the starting date and T the
maturity date. Obviously, what could happen
after T is of no concern for the strategy. No-
tice however that it is also assumed that noth-
ing will change from the start of the strategy to
its end since changes in the allocation process
is not explicitly taking into account. Stated
otherwise, one implicitly supposes that pro-
portions invested among the different assets
will not change over time.[1] Theoretically, it
is well known that this will be the optimal
strategy to follow only if the return and risk
characteristics of the assets do not change over
time, that is, the asset expected returns, vari-
ances and covariances are constant over
time.[2] When the latter are not constant one
says that there are shifts in the investment op-
portunity set since return and risk parameters
are changing. Graphically speaking, it means
the shape of the efficient frontier will be modi-
fied thus in turn leading to changes in the op-
timal allocation strategy as of today.[3] Hence,
it should be clear that standard and static allo-
cation tools, like a Mean-Variance optimiser,
is unable to incorporate such changes in a cor-
rect manner unless one is willing to apply some
arguable rules of thumb as “the longer the time
horizon, the more one should invest in risky
assets as stocks”. Notice one can find exam-
ples of conditional asset allocation strategies
which incorporate possible changes in the es-
timation of the drifts and the risk parameters
of financial assets. More precisely, these mod-
els estimate period after period the inputs nec-
essary for the decision making letting them
changed over time. However, they remain un-
satisfactory because static by essence in the
sense that they do not explicitly incorporate
the investor’s time horizon.
In order to alleviate such difficulties, multi-
period asset allocation models have been de-
veloped in which both investor’s time horizon
and stochastic shifts in the investment oppor-
tunity sets are considered. Not surprisingly, it
can be shown that time horizon will then mat-
ter and that two investors having two different
time horizons will not implement the same
strategy despite the fact they may have the
Gregory Lenoir and Nils S. Tuchschmid: Investment Time Horizon and Asset Allocation Models
FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 15, 2001 / Number 1 77
same anticipation regarding the evolution of
the market over the coming period. To under-
stand this without formal development, let’s
assume a two period economy with two risk-
free bonds. The first bond will mature in one
period, the second one in two periods. To sim-
plify the reasoning, let’s also assume that their
expected returns for the coming period are the
same. It is therefore clear that a “one-period”
investor will only select the one-period bond
whatever could be his degree of risk aversion.
If the two instruments yield the same return,
only the first one is truly risk-free for a “one-
period” investor. Obviously, the same argu-
ment cannot be applied for an investor whose
time horizon is two periods from now. Unless
to assume a very high degree of risk aversion,
in which case the two-period bond is the only
candidate, it will be optimal to invest in the
two-period bond and in the one-period bond as
well. The latter offers a way to benefit from
shifts in bond prices stemming from potential
changes in the level of interest rate. Indeed, if
interest rate goes up between today and the
end of the first period, the price of the two-
period bond will go down. The investor will
therefore be less wealthy than expected. How-
ever, if he has also invested in the one-period
bond, the reimbursement will then be rein-
vested at an higher rate for the next period of
time. From a financial viewpoint, the investor’
optimal demand could thus be decomposed
into a traditional speculative demand and a
kind of hedging demand against possible shifts
in the investment opportunity set due to inter-
est rate modifications. The optimal asset allo-
cation strategy for the two period investor is
thus slightly more complex to handle and
clearly will differ from the one of the “static”
investor. Notice also that the “hedging de-
mand” is due to future bond price modifica-
tions that in turn are due to interest rate
changes. In this simple example, interest rate
represents the so-called “state variable”, that is
the variable which is driven by the economy
and against which the investor might try to
hedge. Remark also, and not surprisingly, that
the hedging demand is made of the asset that is
the most highly correlated with this variable,
the bond in our example. If we were to extend
our presentation to a more complex system,
the problem will remain the same. We will have
to specify the assets in which it is possible to
invest. We will then need to identify the po-
tential set of state variables that could affect
the return and risk parameters of these assets.
Finally, a dynamic asset allocation process,
that is the one designed in a multi-period set-
ting, will lead to an optimal demand decom-
posed as follows: a traditional speculative
component and an hedging component made of
a portfolio of assets the returns of which are
highly correlated with changes of the state
variables.
To summarise, taking only into account
changes in the risk and return characteristics of
financial assets is not enough to warrant opti-
mal asset allocation decisions. The investor’s
time dimension is one of the crucial variables
which should not be neglected. Indeed, in the
above example, we have shown that asset allo-
cation might differ once time horizon is con-
sidered even though the same set of parameters
are using.
2.2 The Choice of State Variables
We consider an investor able to trade a risk-
free asset and two risky asset classes: long-
term bonds B and stocks S. The latter are rep-
resented by two indices. In order to collect se-
ries long enough, we select the Pictet General
Index including only Federal and Cantonal
bonds and the DataStream Swiss stocks In-
dex[4] and assume that the investor will buy or
sell these entire markets. Hence, no unsystem-
atic risks need to be considered.
In the same vein as BRENNAN et al. (1997),
the Swiss financial market is described by three
Gregory Lenoir and Nils S. Tuchschmid: Investment Time Horizon and Asset Allocation Models
78 FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 15, 2001 / Number 1
state variables. Different studies have shown
that some financial variables have predictive
power as far as bond and stock expected re-
turns are concerned (see for instance FAMA
1981, KEIM et al. 1986). Among the most
commonly used state variables, one often finds
the term structure of interest rates, that is,
short- and long-term interest rates, the divi-
dend yield or the price-earning ratio.
In the following study, we choose a realloca-
tion period of one month. Thus, it seems rather
natural to select the one-month Swiss franc
Euro-rate as the short-term interest rate, r,
both as a state variable having predictive
power and as a measure of the risk-free inter-
est rate. The second state variable is chosen as
the long-term interest rate (Swiss franc 5-year
Euro-rate), l. Two reasons at least justify this
choice. First, the long-term bond price, B, is
inversely related to l. Therefore, the latter
should be able to partially predict bond price
movements. Second, changes in the long-term
interest rate are naturally related to changes in
the short-term interest rate, r. Thus, the two
variables need to be included in the same
model unless one assumes that the term struc-
ture of interest rates is driven by a single fac-
tor. Finally, the last state variable, δ, is taken
as the dividend yield of the DataStream index
and should help to predict expected changes in
the equity market.[5]
Summary statistics regarding the variables se-
lected are provided in Table 1 below. As ex-
pected, the volatility of the stock market
greatly exceeds the one of the bond market and
exhibits a much greater annualised mean per-
formance. Without formally testing for nor-
mality, it should be clear by looking at the
difference between mean and median that the
normal hypothesis will be rejected for stock
returns. One should nevertheless mention that,
if normality appears as a key assumption in the
use of a standard and static Mean-Variance
setting, it is not required within the inter-
temporal framework we are using. Volatility,
for instance, can be function of time or even be
stochastic.
In order to take into account the investment
time horizon in the optimisation procedure, we
have to determine the statistical dynamics of
the risky assets and those of the state vari-
ables. In a standard way for a model developed
in continuous-time, we consider that the latter
can be described by joint ITÔ’s processes.
Hence, to complete the definition of the model,
we must specify the functional form of the
drifts and volatilities of these stochastic
processes. We chose the following assump-
tions: first, the drifts are of linear form in the
state variables. Second, the volatilities of the
state variables are proportional to their levels.
Table 1: Summary Statistics
Stocks Bonds Div. Yield 3-month 5-year
Mean 15.36% 4.74% 1.79% 4.32% 5.10%
Median 22.30% 5.22% 1.73% 4.00% 5.00%
Maximum/month 18.39% 3.03% 2.89% 9.63% 8.13%
Minimum/month –27.31% –2.27% 0.89% 0.72% 2.38%
Std. Dev. 17.84% 2.96% 0.44% 2.49% 1.48%
The table provides summary statistics on the financial assets (stocks, bonds, cash) and the three state variables se-
lected (dividend yield, the 3-month rate and the long term 5-year interest rate). The sample covers the period from
01/01/1984 to 01/01/2000. Monthly observations are used. All the Figures are annualised but the maximum and the
minimum.
Gregory Lenoir and Nils S. Tuchschmid: Investment Time Horizon and Asset Allocation Models
FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 15, 2001 / Number 1 79
Thirdly, the volatilities of the risky assets are
constant.[6] The dynamics of the state variables
and the risky assets are thus given by the fol-
lowing system of stochastic differential equa-
tions:
σ+δ+++=
σ+δ+++=
δσ+δ+++=δ σ+δ+++= σ+δ+++=
δδ
SS20191817
BB16151413
1211109
ll8765
rr4321
dZdt)clcrcc(
S
dS
dZdt)clcrcc(
B
dB dZdt)clcrcc(d
ldZdt)clcrcc(dl
rdZdt)clcrcc(dr
(1)
where Zr, Zl, Zδ, ZB and ZS are standard
BROWNian motions and the coefficients ci and
σj are constants.
2.3 The Dynamic Optimal Asset Allocation
Model
Defining ωB and ωS as the proportions of
wealth invested in bonds and in stocks respec-
tively, the return of the investor’s total wealth
is the weighted average of the returns of the
assets that compose the portfolio:
S
dS
B
dB
rdt)1(
W
dW SBSB ω+ω+ω−ω−= (2)
Assuming that the investor’s preferences can
be represented by an isoelastic utility func-
tion[7], U, we have:
=γ
<γ
γ
=
γ
0),Wln(
1,
W
)W(U (3)
The innovation of the method compared to the
standard mean-variance criterion comes from
the fact the investor’s time horizon is explicitly
taken into account in the optimisation proce-
dure. More precisely, the investor is not simply
trying to maximise the return of his portfolio
over a single period for a given level of risk.
He is trying to maximise the expected utility of
his terminal wealth, over a given period of time
that can be short, medium or even long. As
opposed to a traditional “mean-variance” strat-
egy in which the problem is to find an optimal
portfolio composition, in a “dynamic” asset
allocation model, the solution gives the opti-
mal asset allocation path.
At time t, the investment opportunity set of the
investor is completely characterised by his
wealth, W and the state of the Swiss market (r,
l, δ). Defining by T the investor’s time hori-
zon, the optimal investment problem is thus
defined by:
))T(W(U(Emax)tT,,l,r,W(J t
),( SB Ω∈ωω
=−δ (4)
where J is known as the indirect utility func-
tion of the investor and Ω represents the in-
vestment constraints.
As mentioned before, the investor has thus to
find the dynamic optimal proportions of wealth
to allocate among the risky assets such that
expected utility of terminal wealth is maxi-
mised. This approach is therefore clearly dif-
ferent from the one pursued by a mean-
variance investor since the latter will not be
able to consider time in the maximisation pro-
cedure. In other words, the “optimal” strategy
of a mean-variance investor will be the same
whatever the investment time horizon is.
The optimal stochastic control (4) is solved
using the BELLMAN’S optimality principle
and a standard dynamic programming ap-
proach. Under the hypothesis of an investor
with an isoelastic utility function, it can be
shown that J is separable in wealth of the form,
J(W,r,l,δ,T – t) = U(W)H(W,r,l,δ,T – t).
The scalar function, H, is then solution of a
complex backward non-linear convection-
diffusion partial differential equation (see, for
example, BREITLER et al.).
Gregory Lenoir and Nils S. Tuchschmid: Investment Time Horizon and Asset Allocation Models
80 FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 15, 2001 / Number 1
Denoting by M the (2 x 2) covariance matrix
of the risky assets and by C the (2 x 3) covari-
ance matrix between the risky assets and the
state variables, the dynamic optimal allocation
in bonds and stocks is given in the absence of
investment constraints by:
)]tT,,l,r(H
),l,r(C
)tT,,l,r(H 1
),l,r([M
11
)tT,,l,r(
),l,r(
1*
−δ⋅
∇δ
−δ
+δη
γ−
=−δω
δ
−
(5)
where η is a vector composed of excess ex-
pected returns of the risky assets and 1 – γ is
the investor’s relative risk aversion parameter.
The dynamic optimal allocation (5) is com-
posed of the familiar speculative demand
(1/(1 – γ))M–1η corresponding to the “myopic”
demand of a mean-variance investor. The sec-
ond component of the right hand side of equa-
tion (5) is known as the hedging demand
against the shifts in the investment opportunity
set. The latter is due to changes in the state
variables which, in turn, affect the risk and
return parameters of the financial assets. From
a financial viewpoint, this demand is made of a
portfolio of financial assets such that the cor-
relation between the asset returns and changes
in the state variables is maximised. Of course,
the hedging demand vanishes if the risky assets
are uncorrelated with the state variables. In
particular, if one assumes that the drifts and
volatilities are constant, then one just ends up
with the speculative demand. In other words,
the traditional mean-variance strategy will be
the optimal one. The impact of investment time
horizon is thus included in this hedging de-
mand since it depends on the value of the state
variables, that is, their covariances with the as-
set returns, and the remaining investment time
horizon of the investor.
At this stage, it is worthwhile noticing that the
indirect utility function J of the investor pos-
sesses a useful financial interpretation. Indeed,
if we define the Certainty Equivalent (CE) as
the amount of wealth for which the investor is
indifferent between investing his wealth opti-
mally or receiving this amount for sure at ma-
turity, we have:
))T(W(U(EmaxJ)CE(U t
),( SB Ω∈ωω
== (6)
This measure represents a natural way to as-
sess the behaviour and efficiency of the strat-
egy. Indeed, if the hedging against future
changes of the market conditions – in other
words, against the shifts in the investment op-
portunity set – is perfect, the certainty equiva-
lent will be constant over time. Assessing the
degree of volatility of the Certainty Equivalent
is therefore the relevant metric to be used.
Hence, traditional performance measure, as the
volatility or SHARPE ratio, is not the right
“tool” to apply since the inter-temporal strat-
egy does not try to maximise it as opposed
to standard Mean-Variance asset allocation
strategy.
2.4 Model Calibration
Several estimation procedures can be envis-
aged as, for example, the indirect inference
method (see BROZE et al. 1998) or the maxi-
mum likelihood method in continuous-time
setting (PITERBARG 1998). The following re-
sults are obtained using a SUR (Seemingly Un-
related Regression) method which permits to
estimate the drift parameters and the variance-
covariance matrix of the joint ITÔ’S process
(1) in a single run. The process is discretized
using a standard EULER scheme in time and
calibrated using past series of observations of
the state variables and risky assets presented in
subsection 2.2.[8] For the “Swiss market
model” presented in equation (1), Table 2 pro-
vides the estimated annualised drift parameters
with their standard error while Table 3 gives
Gregory Lenoir and Nils S. Tuchschmid: Investment Time Horizon and Asset Allocation Models
FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 15, 2001 / Number 1 81
Table 2: Estimated Drift Parameters and their Standard Error given in Brackets for the Swiss Market
Jan. 1984 – Jan. 2000
Constant r l δ
dr –0.0144 –2.3596 3.0956 –2.7908
(0.0179) (0.5548) (0.8808) (1.3313)
dl 0.0251 0.4780 –0.8423 –0.2223
(0.0091) (0.2549) (0.4533) (0.6374)
dδ0.0002 –0.2293 0.5968 –1.2091
(0.0039) (0.1037) (0.1971) (0.2745)
dB/B –0.0257 –1.0529 1.8597 1.3724
(0.0393) (0.9558) (1.7897) (2.2640)
dS/S 0.3846 10.3931 –27.0356 40.2805
(0.2236) (5.5237) (10.5105) (12.6381)
the variance-covariance matrix of the stochas-
tic part. These results are obtained using
monthly observations covering the period from
January 1984 to January 2000.
We may observe that the changes in the short-
term interest rate, r, are negatively related to
its current level and positively related to the
level of the long-term interest rate, l. Thus, in
accordance with the results presented by
BRENNAN et al. (1997) for the US market
and obtained during another sample period, the
short-term interest rate tends to adjust towards
the long-term interest rate. Moreover, as ex-
pected, stock returns are positively related to
the dividend yield. As observed in previous
studies, some estimated parameters do not
appear to be
significant. Without a clear understanding
about the mechanisms driving the economy, we
have decided to keep the initial model unmodi-
fied. Indeed, the removal of insignificant vari-
ables could reinforce the estimation risk prob-
lem the model has to deal with.
3. The Impact of Investment Time
Horizon
An important work has been devoted to the
development of an efficient numerical proce-
dure to implement the optimal stochastic con-
trol, which requires to solve a complex non-
linear convection-diffusion partial differential
equation. In this paper, we do not attempt to
Table 3: Estimated Variance-Covariance Matrix of the Sate Variables and Risky Assets
Jan. 1984 – Jan. 2000
ZrZlZδZBZS
Zr 0.37347 – – – -
Zl 0.03930 0.03010 – – -
Zδ 0.00306 0.00696 0.03720 – -
ZB–0.00480 –0.00402 –0.00170 0.00087 -
ZS 0.00167 –0.00507 –0.02866 0.00129 0.02892
Gregory Lenoir and Nils S. Tuchschmid: Investment Time Horizon and Asset Allocation Models
82 FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 15, 2001 / Number 1
present the numerical aspects of the resolution
of the HAMILTON-JACCOBI-BELLMAN
equation (4). We only focus on the results of
the dynamic optimal allocation model taking a
practical viewpoint.[9]
Recall that the important innovation of the
model studied here compared to the mean-
variance criterion is its ability to take explicitly
into account the investment time horizon in the
optimisation procedure. To analyse its impact
on the dynamic optimal asset allocation strat-
egy, we estimate the parameters of the model
(1) over the period from January 1984 to Janu-
ary 2000 and fix the value of the state vari-
ables at 2.7% for the short-term interest rate,
4.2% for the long-term interest rate and 1.4%
for the dividend yield.[10] The investor’s risk
aversion parameter[11] γ is set to –20. Based
on these values, we then compute the dynamic
optimal allocation under different time hori-
zons. Figures 1 and 2 show, respectively, the
optimal demand in bonds and stocks. These
allocations are decomposed into the standard
speculative demand and the hedging demand
against shifts in the investment opportunity set
(see equation 5).
Firstly, we observe that the optimal bond de-
mand varies non-monotonically with time hori-
zon and can be lower or higher than the
speculative demand. Contrarily, the optimal
stock demand increases monotonically with
time horizon. Thus, the effect of a longer time
horizon is to invest more in stocks while it
could lead to invest more or less in bonds.
Figure 1: Optimal Bond Demand under Different Time Horizons
r = 2.7%, l = 4.2% and δ
δδ
δ = 1.4%
-20%
0%
20%
40%
60%
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Investment time horizon [year]
Proportion of wealth invested in bond
Speculative demand
Hedging demand against the future changes of the Swiss market
Optimal bond demand
Gregory Lenoir and Nils S. Tuchschmid: Investment Time Horizon and Asset Allocation Models
FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 15, 2001 / Number 1 83
Figure 2: Optimal Stock Demand under Different Time Horizons
r = 2.7%, l = 4.2% and δ
δδ
δ = 1.4%
-20%
0%
20%
40%
60%
0123456789101112131415
Investment time horizon [year]
Proportion of wealth invested in bond
Speculative demand
Hedging demand against the future changes of the Swiss market
Optimal stock demand
Secondly, for both bonds and stocks, the
hedging demand has a kind of asymptotic be-
haviour, that is, it seems to be stabilised for a
time horizon greater than 6 years. Thus, we
can define 3 classes of investors: short-term
investors (0–2 years), middle-term investors
(2–6 years) and long-term investors having
time horizon equal to or greater than 6 years.
Notice that these commonly accepted defini-
tions have been derived from the model with-
out additional assumption. Finally, the com-
parison of the speculative and the hedging de-
mand in stocks shows that their magnitudes
become equivalent for investment time hori-
zons greater than 2 years. Thus, the impact of
investment time horizon on the dynamic opti-
mal allocation cannot be neglected for middle-
term and long-term investors: classical mean-
variance models become really sub-optimal for
investment time horizons greater than 2 years.
Of course, the results presented here are only
valid for the specific state variable values
(r = 2.7%, l = 4.2% and δ = 1.4%). However,
the same conclusions are reached for other
levels of the state variables. While surprising at
first sight, the fact that the fraction invested in
stocks increases monotonically over time can
be easily understood. For a given starting
point, one has to consider that the stochastic
component of the stock price process will
more and more dominate as time horizon in-
creases. Stated otherwise, over long horizons,
deviations around the long term mean becomes
“pure noise” and the fact to set a maturity at
10, 15 or 20 years will not change the compo-
sition of the hedging demand as far as stocks
84 FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 15, 2001 / Number 1
Gregory Lenoir and Nils S. Tuchschmid: Investment Time Horizon and Asset Allocation Models
are concerned. Hence, the optimal weights will
be the same.
4. Simulations
Another way to evaluate the impact of invest-
ment time horizon on an optimal asset alloca-
tion strategy is to simulate the evolution of the
Swiss bond, stock and cash markets and to
compute the dynamic optimal strategy and the
mean-variance strategy for each path. Practi-
cally, we impose the following investment con-
straints: no short sales and a maximum of 50%
of wealth invested in stocks. The investor’s
risk aversion parameter γ is set to –20. We run
20 000 simulations of ten years each. These
different paths are computed based on the sto-
chastic differential equation system (1) with
the coefficients given in tables 2 and 3.
Figure 3 presents the probability density of the
annualised drifts, that is, the realised returns of
different strategies. The first results are the
ones obtained for the bonds and stocks mar-
kets. They obviously represent, based on
20 000 simulations, the annualised returns that
could have been achieved by somebody having
invested all his wealth in the bond or the stock
market respectively. Along with these results,
Figure 3 gives the probability density for the
mean-variance strategy and for the “optimal”
or dynamic strategy based on the assumption
that over each simulated path, the investor
starts with a ten-year time horizon.
The return of the stock market is very vola-
tile, going from 5% to 25% with an average of
Figure 3: Distribution of Annualised Drifts Based on 20 000 Simulations of Ten Years each
From 2000 to 2010 with a Maximum of 50% Invested in Stocks
0%
5%
10%
15%
20%
25%
30%
35%
0.0% 5.0% 10.0% 15.0% 20.0% 25.0%
Yearly drift
Probability density
Bonds Stocks Mean-variance strategy Optimal strategy
Gregory Lenoir and Nils S. Tuchschmid: Investment Time Horizon and Asset Allocation Models
FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 15, 2001 / Number 1 85
Figure 4: Distribution of the Annualised Drift Differences Between the Optimal Strategies and the
Mean-Variance Strategies Based on 20 000 Simulations of Ten Years each
From 2000 to 2010 with a Maximum of 50% Invested in Stocks
0%
1%
2%
3%
4%
5%
6%
7%
8%
-1.0% -0.5% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0%
Yearly drift difference
Probability density
15%. The return of the bond index is com-
prised between 2.5% and 7%, with an average
of 3.75%. The mean-variance strategy gives a
return ranging from 6% to 12.5% with an av-
erage of 9.1%. Finally the dynamic optimal
strategy return is comprised between 6% to
15%, with an average of 10.5%. Thus, even
with a relatively strong investment constraint
(a maximum of 50% can be invested in stocks),
the dynamic strategy yields an annualised aver-
age return of 140 basis points higher than the
one achieved by the sub-optimal mean-variance
strategy return. For a better understanding of
the situation, we can look at the probability
density of the difference of returns between the
dynamic optimal strategy and the mean-
variance strategy (Figure 4). As mentioned be-
fore, the dynamic optimal strategy return is on
average 1.4% higher than the mean-variance
strategy return. An interesting result is the fact
that the dynamic optimal strategy is beaten by
the “myopic” mean-variance strategy in 2% of
the cases only. In other words, within the
model developed here, the dynamic optimal
strategy beats the standard mean-variance
strategy with a probability of 98%. Therefore,
the impact of investment time horizon should
not be neglected in the tactical as well as stra-
tegic asset allocation process.[12]
5. Out-of-Sample Performance
In this section, out-of-sample results are pre-
sented from January 1990 to January 2000.
Each month till maturity, the coefficients of the
Gregory Lenoir and Nils S. Tuchschmid: Investment Time Horizon and Asset Allocation Models
86 FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 15, 2001 / Number 1
model (1) are estimated by using the available
monthly past observations of the state vari-
ables and the risky assets. More precisely,
starting in January 1990, the observations ex-
tending from 1984 to the end of 1989 are cho-
sen to estimate the coefficients. The latter are
then used to get the expected returns and risks
parameters necessary to implement both the
dynamic and the Mean-Variance strategies.
The window is moved by one month and the
process is thus repeated up to reach the fixed
maturity of January 2000.
By doing so, the optimal and mean-variance
strategies are implemented by using these same
parameters. As far as the dynamic optimal
strategy is concerned, the investor’s time hori-
zon is set to January 2000 and remains fixed.
Hence, as time elapses, one comes to closer to
maturity and month after month this procedure
implies to solve a complex optimal stochastic
control. Finally, the investor’s risk aversion
parameter, γ is set to –20 for both the dynamic
and the mean-variance strategy and the same
investment constraints as before are used, that
is, no short sales and a maximum of 50% in-
vested in stocks.
Figure 5 presents the evolution of the inves-
tor’s wealth during a ten-year period assuming
Figure 5: Wealth Evolution of the Long-Term Strategy, of the Mean-Variance Strategy and of the Pic-
tet LPP Index and the Evolution of the Certainty Equivalent of the Optimal Long-Term Strategy
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000
Realtive wealth
Optimal strategy Mean-variance strategy
Pictet LPP Index Certainty equivalent
Gregory Lenoir and Nils S. Tuchschmid: Investment Time Horizon and Asset Allocation Models
FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 15, 2001 / Number 1 87
Table 4: Summary Statistics on the Performance of Three Strategies
Optimal strategy Mean-variance strategy Pictet LPP Index
End Value 2.827 2.396 2.250
Returns
Mean 10.70% 8.99% 8.29%
Std. Dev. 7.26% 6.65% 5.58%
that he has pursued either the dynamic optimal
long-term strategy with a fixed investment time
horizon in January 2000 or the traditional
mean-variance strategy. As mentioned before,
these results are obtained “out-of-sample” in
the sense that only past observations and coef-
ficients stemming from them are used to im-
plement the strategies and that this process is
repeated month after month. The Pictet LPP
index is given as a benchmark even though the
latter includes investment in foreign securi-
ties.[13] Finally, the dashed line shows the
evolution of the certainty equivalent dynamic
optimal strategy as defined previously in sub-
section 2.3.
First, notice that since these results are ob-
tained in a true out-of-sample framework, they
could have been achieved. In terms of per-
formance, the dynamic strategy clearly out-
performs the “myopic” mean-variance strategy
and the Pictet LPP index. The dynamic long-
term strategy yields an average return of
10.7% per year, whereas the mean-variance
strategy return is equal to 9% close to the
Pictet LPP index, which yields a yearly return
of 8.3%. The annualised excess return of the
dynamic strategy compared to the mean-
variance strategy is therefore 1.7%, a signifi-
cant result from an economic viewpoint. Inter-
estingly, this result can also be compared to
the one obtained by simulations and presented
in the previous section. Indeed, we have found
that, on average, the excess return of the dy-
namic optimal strategy over the “myopic”
mean-variance strategy was, for a ten-year pe-
riod, equal to 1.4% per year. Thus the out-of-
sample behaviour of the method is very close
to the one obtained by simulations.
Finally, if we look at the behaviour of the cer-
tainty equivalent of the dynamic long-term
strategy, it is remarkably stable through the
whole allocation period. Hence, the hedging
component of the dynamic optimal demand
(see equation 5) is playing its role well.[14]
In the early nineties, the investor would
have been indifferent between following the
long-term strategy or to get, for sure at ma-
turity, 2.6 times his current wealth. Ten years
later, that is, at maturity, the investor has
effectively seen his wealth being multiplied by
a factor of 2.8.
Table 4 provides additional information re-
garding the performance of these strategies. In
particular, it shows that the dynamic optimal
asset allocation strategy yields the highest
mean return. The same strategy also appears to
be the riskiest one. At this stage, it is however
worthwhile mentioning that mean-variance
type of performance measure does not apply
since the “dynamic” strategy is not derived in
such a setting. In this sense, the certainty
equivalent behaviour is the correct metric to
analyse and its ability to “predict” the growth
of wealth over a ten year time horizon is cer-
tainly noticeable (see also BRENNAN et al.
1997 or BREITLER et al. 1998, 1999).
Gregory Lenoir and Nils S. Tuchschmid: Investment Time Horizon and Asset Allocation Models
88 FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 15, 2001 / Number 1
Figure 6: Asset Allocation of the Long-Term Strategy Based on Out-Of-Sample Results. From Jan.
1990 to Jan. 2000 (Investment Constraints: no Short-Selling and a Maximum of 50% in Stocks)
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000
Wealth allocations
Cash Bonds Stocks
Figure 6 presents the optimal asset allocation
implied by the dynamic long-term strategy and
its evolution over time. The average turnover
is 11% per month, which might appear quite
important. However, as shown in the last sec-
tion of this article, this strategy can be practi-
cally implemented without affecting the per-
formance even in the presence of transaction
fees.
Table 5 looks at the average proportions in-
vested in the different asset classes. Remark
that the optimal long-term strategy leads to in-
vest more in bonds and in stocks compared to
the standard mean-variance strategy. Never-
theless, it is interesting to point out that the
two strategies (the dynamic and the mean-
variance ones) have average proportions in-
vested in bonds and stocks that are lower than
the ones observed in the Pictet LPP index.
Table 5: Average Proportions Invested in the Different Asset Classes
Cash Bonds Stocks
Optimal strategy 43.43% 33.76% 22.81%
Mean-variance strategy 50.43% 31.95% 17.62%
Pictet LPP Index 0% 74.57% 25.43%
Gregory Lenoir and Nils S. Tuchschmid: Investment Time Horizon and Asset Allocation Models
FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 15, 2001 / Number 1 89
6. Transaction Fees
In this last section, we include transaction fees
and analyse their impact on the performance of
the out-of-sample dynamic strategy presented
in the preceding section. The objective is
therefore simple: could the introduction of re-
alistic brokerage fees affect the strategy so
badly that the latter could not be practically
implemented? To answer this question, we as-
sume that the strategy is implemented through
financial assets only.[15]
The considered brokerage fees are 10 basis
points for the Swiss stock market and 5 basis
points for the bond market. The Swiss federal
stamp is 7.5 basis points for both bonds and
stocks. Finally, a tax of 1 basis point is levied
by the Swiss exchange.
The initial investment in January 1990 is sup-
posed to be SFr 10 millions. Figure 7 presents
the evolution of wealth with and without
transaction fees.
Hence, brokerage fees have a very little impact
on the performance of the strategy. On aver-
age, the strategy with transaction fees has a
return of 10.41% per year, while without
transaction fees, the return is 10.70%. The
cost of implementing the dynamic optimal
strategy is thus only 29 basis points per year.
If we take into account the holding fees of the
financial assets (approximately 5 basis points
per year), the total cost is 34 basis points per
year, which is not a practical limitation of the
method.
Figure 7: Wealth Evolution of the Out-Of-Sample Dynamic Long-Term Strategy With and Without
Transaction Fees
SFr. 8'000'000
SFr. 10'000'000
SFr. 12'000'000
SFr. 14'000'000
SFr. 16'000'000
SFr. 18'000'000
SFr. 20'000'000
SFr. 22'000'000
SFr. 24'000'000
SFr. 26'000'000
SFr. 28'000'000
SFr. 30'000'000
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000
Wealth evolution
Optimal strategy without transaction fees Pictet LPP Index Optimal strategy with transaction fees
Gregory Lenoir and Nils S. Tuchschmid: Investment Time Horizon and Asset Allocation Models
90 FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 15, 2001 / Number 1
7. Conclusion
In this paper, we develop and analyse the per-
formance of a truly dynamic asset allocation
strategy and compare it to traditional approach
as based on Mean-Variance optimiser as well
as to standard benchmarks. The strategy is dy-
namic in both ways. First, it takes into account
the fact that investors may have different time
horizons and would therefore behave accord-
ingly. Second, it characterises the processes
followed by the prices of financial assets and
explicitly takes into account the relationship
between the latter and state variables that
might explain their evolutions. More precisely,
a stochastic control method is designed and
implemented in an intertemporal or dynamic
asset allocation model for the Swiss financial
market. The evolution of this market is de-
scribed by three (state-) variables: the short-
and the long-term interest rates and the divi-
dend yield that in turn drive the expected re-
turns and risk of financial assets. The investor
is thus supposed to be able to trade between
the Swiss stock market, the Swiss government
bond market and the cash market.
As opposed to the majority of asset allocation
models used even today, the solution we pro-
pose takes explicitly into account the inves-
tor’s time horizon. The optimal demand in
risky assets is then composed of a classical
speculative or mean-variance demand and a
hedging demand against future shifts in the in-
vestment opportunity set. The latter are due to
changes of the state variables that are describ-
ing the economy and which in turn affect the
return and the risk parameters of the financial
assets.
The possibility of including the investor’s time
horizon in the optimisation procedure signifi-
cantly modifies the optimal proportions in-
vested in the different asset classes. For in-
stance, the speculative and the hedging demand
in stocks see their magnitudes to become
equivalent for investment time horizons greater
than 2 years. Moreover, while the optimal allo-
cation in stocks increases with time horizon, it
starts to stabilise for time horizons longer than
6 years. The effect of time horizon on the op-
timal allocation in bonds is subtler. Indeed, the
optimal demand in bond can increase or de-
crease with time horizon.
Once implemented, out-of-sample performance
analysis of the method shows that the results
are totally coherent with the statistical behav-
iour of the simulated model of the Swiss
“economy”. Based on thousands of simula-
tions, the average excess return of the dynamic
optimal long-term strategy compared to a my-
opic mean-variance strategy is 1.4% per year
for a ten-year investment time horizon and
with investment constraints imposing no short
sales and an upper limit of 50% in stocks. The
out-of-sample excess return, under the same
conditions, was 1.7% per year. These results
are clearly significant from an economic view-
point.
Finally, the inclusion of transaction fees shows
that the strategy can be implemented without
affecting its performance. The transaction fees
amount to 29 basis point per year only.
To conclude, the results presented in this paper
clearly indicate two things. First, not including
the time dimension in the tactical asset alloca-
tion process is costly. Second, the develop-
ment and the implementation of truly dynamic
strategies are both possible and justified.
Gregory Lenoir and Nils S. Tuchschmid: Investment Time Horizon and Asset Allocation Models
FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 15, 2001 / Number 1 91
FOOTNOTES
[1] This should not be confused with a buy and hold
strategy. If the optimal asset allocation strat-
egy leads to invest for instance 60% in risky
bonds and 40% in stocks, these proportions
have to be maintained. Hence, as opposed
to a buy and hold strategy in which no trans-
actions are necessary, the investor will have to
transact dynamically to maintain these weights
constant.
[2] It is also well known that investors exhibiting log
utility function will behave as if they were “my-
opic” that is they will not consider potential
changes and will invest accordingly (see, e.g.
MERTON 1990), that is, as if they were in a
“traditional” mean-variance setting. This how-
ever does not mean that the optimal weights do
not change.
[3] Obviously, these shifts will really impact the as-
set allocation process if they are partially ran-
dom.
[4] The latter is made of more than one hundred
stocks thus covering almost the entire Swiss
stock market in terms of market capitalisation.
During the period considered in this study, the
correlation coefficient between the returns of
this index and the returns of the SPI amounts
for 0.95 based on monthly observations.
[5] Of course, one can think of other variables to
describe the Swiss financial market. For in-
stance, we also selected the inverse of the P/E
ratio or of the Price / Cash Flow ratio as well as
a credit spread measure. However, the inclusion
of these variables did not improve the predic-
tive ability of the model. We also increased the
number of state variables by adding the credit
spread to the initial three state variables (divi-
dend yield, short-term and long-term interest
rates). We did not notice significant improve-
ment.
[6] As mentioned previously, this assumption re-
garding the volatility is not necessary. The
methodology will remain the same if this pa-
rameter is assumed to be time dependant or it-
self stochastic.
[7] This utility function has a constant relative risk
aversion parameter, independent of wealth,
given by 1 – γ.
[8] Estimating the parameters of a continuous-time
stochastic process with a discrete-time ap-
proximation gives rise to a discretization bias.
BROZE et al. have proposed a quasi indirect
inference method to correct it. However, the
method becomes numerically heavy and unsta-
ble when the dimensionality of the stochastic
process and the number of unknown parameters
increase. For this reason, we choose to neglect
the discretization bias and to adopt a SUR
methodology.
[9] As far as numerical aspects are concerned, the
reader is referred to BRENNAN et al. (1997)
and BREITLER et al. (1998).
[10] These values correspond to the state of the
"economy" in June 2000.
[11] This large negative value of the risk parameter
γ may surprise. However, this can be explained
by the fact that the model has a time-varying
stochastic investment opportunity set that can
take extreme values. Thus, it is difficult to com-
pare the risk aversion parameter chosen here
with those generally used in standard mean-
variance models with a constant investment op-
portunity set. In addition to that, the sample pe-
riod selected is characterised by a pronounced
upward trendy stock market which in turn leads
to take extreme positions on the equity market
for an investor with moderate degree of risk
aversion. While it would have been difficult to
increase the number of observation for the Swiss
market, it could have been possible to constrain
the drifts. By doing so, the “calibrated” risk
aversion parameter would appear to be smaller.
However, we choose not to focus on this par-
ticular issue here.
[12] Notice that, even tough one compares the dy-
namic optimal strategy to a standard mean-
variance approach, the former is not designed
in a mean-variance setting. Hence, traditional
performance measures as SHARPE ratio will
not apply.
Gregory Lenoir and Nils S. Tuchschmid: Investment Time Horizon and Asset Allocation Models
92 FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 15, 2001 / Number 1
[13] Indeed, the Pictet LPP index is certainly not the
perfect benchmark to be used since it contains
investment in foreign markets. Nevertheless, it
is an asset allocation index with an upper limit
of 50% in stocks and approximately 80% in-
vested in Swiss securities or denominated in
Swiss franc. Therefore, the result of the optimal
strategy compared to the LPP index may be
used as a point of comparison.
[14] Indeed, recall that, if the investor were able to
perfectly hedge against future changes in the
investment opportunity set, the certainty equi-
valent should be constant over time.
[15] Different tests have been carried out by repli-
cating the strategy with future contracts. It has
been found that the less expensive strategy is
the one based on the assets themselves. Even
though this result might seem counter intuitive,
it can be easily explained if one notices that the
use of futures contracts implies to implement a
“rollover” strategy which ends up to be quite
costly. Indeed, since derivative instruments
have a maturity, the investor could be forced to
close positions and open new ones, and thus to
incur the fees associated with these operations,
even if these positions, given by the “optimal”
long-term strategy, are stable over time.
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FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 15, 2001 / Number 1 93
