Investment time horizon and asset allocation models
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ABSTRACT: In comparing an immediate life annuity with a payoutequivalent investment fund payout plan (selfannuitization), research to date has focused mainly on shortfall probabilities of selfannuitization. As an exception, Schmeiser and Post (2005) propose a family strategy where the chances of selfannuitization (i.e., bequests) are taken into consideration as well. In such a family strategy, potential heirs must bear shortfall risks, but in return have a chance of receiving a bequest. This paper analyzes under which conditions heirs will be willing to agree to a family strategy. The idea of a family strategy is integrated into a realistically calibrated intertemporal expected utility framework, taking into account risks arising from stochastic life span, asset returns, and nontradable labor income. A family strategy is shown to be accepted for many parameter combinations, especially in families with low marginal tax rates, if the heirs are wealthy, or in a case where the retiree has an average population life expectancy. We also work out how family selfannuitization decisions interact with asset allocation, saving decisions, and labor income risk. Under realistic conditions our results support two explanations for the empirically observable low demand for annuities (the socalled annuity puzzle), namely intrafamily risk sharing and high cost of marketannuitization.Financial Markets and Portfolio Management 09/2006; 20(3):265285.  SourceAvailable from: fmpm.ch[Show abstract] [Hide abstract]
ABSTRACT: In the last decades, the trend to globalization has been intensifying. The thereby increased interdependence between national economies translates into a higher covariance of traded assets and thus leads to a loss of diversification gains. The methodology of HESTON and ROUWENHORST is used to split the returns of G7 and Swiss stocks into a common, a sector, and a country component. The findings show that country diversification has been better in the last two decades. Nevertheless, a strong increase in the gains provided by sector diversification  mainly induced by the IT sector  can be observed in recent years.Financial Markets and Portfolio Management 05/2002; 16(2):234253.
Page 1
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 meanvariance
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
meanvariance approach, even in a multi
periodic framework, remains a subefficient
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
continuoustime 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
strategy due to
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
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 meanvariance criterion. Sec
tion 3 presents the innovations of the method:
traditional “mean
© Swiss Society for Financial Market Research (pp. 76–93)
76 FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 15, 2001 / Number 1
Page 2
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
outofsample 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 monoperiodic
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 MeanVariance 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 MeanVariance 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
Page 3
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 “oneperiod”
investor will only select the oneperiod bond
whatever could be his degree of risk aversion.
If the two instruments yield the same return,
only the first one is truly riskfree 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 twoperiod bond is the only
candidate, it will be optimal to invest in the
twoperiod bond and in the oneperiod 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 oneperiod
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 socalled “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 multiperiod 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
Page 4
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 longterm interest rates, the divi
dend yield or the priceearning ratio.
In the following study, we choose a realloca
tion period of one month. Thus, it seems rather
natural to select the onemonth Swiss franc
Eurorate as the shortterm interest rate, r,
both as a state variable having predictive
power and as a measure of the riskfree inter
est rate. The second state variable is chosen as
the longterm interest rate (Swiss franc 5year
Eurorate), l. Two reasons at least justify this
choice. First, the longterm bond price, B, is
inversely related to l. Therefore, the latter
should be able to partially predict bond price
movements. Second, changes in the longterm
interest rate are naturally related to changes in
the shortterm 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 MeanVariance
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 continuoustime, 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
15.36%
22.30%
18.39%
–27.31%
17.84%
Bonds
4.74%
5.22%
3.03%
–2.27%
2.96%
Div. Yield
1.79%
1.73%
2.89%
0.89%
0.44%
3month
4.32%
4.00%
9.63%
0.72%
2.49%
5year
5.10%
5.00%
8.13%
2.38%
1.48%
Mean
Median
Maximum/month
Minimum/month
Std. Dev.
The table provides summary statistics on the financial assets (stocks, bonds, cash) and the three state variables se
lected (dividend yield, the 3month rate and the long term 5year 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
Page 5
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+
l
++=
σ+δ+++=
δδ
SS2019 1817
BB 1615 1413
1211 109
ll8
c
765
rr4321
dZ dt)clcrc c (
S
dS
dZdt)clcrcc (
B
dB
dZdt)crc c (d
ldZ dt)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
W
=
γ
0 ),ln(
1,
W
)W(U
(3)
The innovation of the method compared to the
standard meanvariance 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 “meanvariance” 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(E max
,
B
ωω
) tT,, l , r ,W( J
t
)(
S
Ω∈
=−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 meanvariance 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 nonlinear 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
Page 6
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
δ
), l , r (
η
[M
1
1
−
) tT,, l , r (
), l , r (
1*
−δ⋅
∇δ
−
+δ
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 meanvariance 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 meanvariance 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(E max
,
B
ωω
J) CE(U
t
)(
S
Ω∈
==
(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 intertemporal strat
egy does not try to maximise it as opposed
to standard MeanVariance 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 continuoustime
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
Page 7
Table 2: Estimated Drift Parameters and their Standard Error given in Brackets for the Swiss Market
Jan. 1984 – Jan. 2000
Constant
–0.0144
(0.0179)
0.0251
(0.0091)
0.0002
(0.0039)
–0.0257
(0.0393)
0.3846
(0.2236)
rl
δ
dr–2.3596
(0.5548)
0.4780
(0.2549)
–0.2293
(0.1037)
–1.0529
(0.9558)
10.3931
(5.5237)
3.0956
(0.8808)
–0.8423
(0.4533)
0.5968
(0.1971)
1.8597
(1.7897)
–27.0356
(10.5105)
–2.7908
(1.3313)
–0.2223
(0.6374)
–1.2091
(0.2745)
1.3724
(2.2640)
40.2805
(12.6381)
dl
dδ
dB/B
dS/S
the variancecovariance 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 longterm 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
shortterm interest rate tends to adjust towards
the longterm 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 convectiondiffusion partial differential
equation. In this paper, we do not attempt to
Table 3: Estimated VarianceCovariance Matrix of the Sate Variables and Risky Assets
Jan. 1984 – Jan. 2000
Zr
Zl
–
Zδ
–
–
ZB
–
–
–
ZS




Zr
Zl
Zδ
ZB
ZS
0.37347
0.03930
0.00306
–0.00480
0.00167
0.03010
0.00696
–0.00402
–0.00507
0.03720
–0.00170
–0.02866
0.00087
0.001290.02892
Gregory Lenoir and Nils S. Tuchschmid: Investment Time Horizon and Asset Allocation Models
82 FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 15, 2001 / Number 1
Page 8
present the numerical aspects of the resolution
of the HAMILTONJACCOBIBELLMAN
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 shortterm interest rate,
4.2% for the longterm 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 nonmonotonically 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 δ δ δ = 1.4%
20%
0%
20%
40%
60%
012345
Investment time horizon [year]
6789 101112 1314 15
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
Page 9
Figure 2: Optimal Stock Demand under Different Time Horizons
r = 2.7%, l = 4.2% and δ δ δ δ = 1.4%
20%
0%
20%
40%
60%
0123456789 10 11 12131415
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: shortterm
investors (0–2 years), middleterm investors
(2–6 years) and longterm 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 longterm investors: classical mean
variance models become really suboptimal 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
Page 10
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
meanvariance 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
meanvariance strategy and for the “optimal”
or dynamic strategy based on the assumption
that over each simulated path, the investor
starts with a tenyear 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 Meanvariance strategyOptimal 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
Page 11
Figure 4: Distribution of the Annualised Drift Differences Between the Optimal Strategies and the
MeanVariance 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
~2%
15%. The return of the bond index is com
prised between 2.5% and 7%, with an average
of 3.75%. The meanvariance 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 suboptimal meanvariance
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 meanvariance
strategy return. An interesting result is the fact
that the dynamic optimal strategy is beaten by
the “myopic” meanvariance strategy in 2% of
the cases only. In other words, within the
model developed here, the dynamic optimal
strategy beats the standard meanvariance
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. OutofSample Performance
In this section, outofsample 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
Page 12
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 MeanVariance 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 meanvariance
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 meanvariance 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 tenyear period assuming
Figure 5: Wealth Evolution of the LongTerm Strategy, of the MeanVariance Strategy and of the Pic
tet LPP Index and the Evolution of the Certainty Equivalent of the Optimal LongTerm 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
199019911992 19931994 19951996 1997199819992000
Realtive wealth
Optimal strategy
Pictet LPP Index
Meanvariance strategy
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
Page 13
Table 4: Summary Statistics on the Performance of Three Strategies
Optimal strategy
2.827
Meanvariance strategy
2.396
Pictet LPP Index
2.250
End Value
Returns
Mean
Std. Dev.
10.70%
7.26%
8.99%
6.65%
8.29%
5.58%
that he has pursued either the dynamic optimal
longterm strategy with a fixed investment time
horizon in January 2000 or the traditional
meanvariance strategy. As mentioned before,
these results are obtained “outofsample” 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 outofsample framework, they
could have been achieved. In terms of per
formance, the dynamic strategy clearly out
performs the “myopic” meanvariance strategy
and the Pictet LPP index. The dynamic long
term strategy yields an average return of
10.7% per year, whereas the meanvariance
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”
meanvariance strategy was, for a tenyear pe
riod, equal to 1.4% per year. Thus the outof
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 longterm
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
longterm 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 meanvariance
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
Page 14
Figure 6: Asset Allocation of the LongTerm Strategy Based on OutOfSample Results. From Jan.
1990 to Jan. 2000 (Investment Constraints: no ShortSelling and a Maximum of 50% in Stocks)
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
1990199119921993199419951996199719981999 2000
Wealth allocations
Cash BondsStocks
Figure 6 presents the optimal asset allocation
implied by the dynamic longterm 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 longterm strategy leads to in
vest more in bonds and in stocks compared to
the standard meanvariance 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
43.43%
50.43%
0%
Bonds
33.76%
31.95%
74.57%
Stocks
22.81%
17.62%
25.43%
Optimal strategy
Meanvariance strategy
Pictet LPP Index
Gregory Lenoir and Nils S. Tuchschmid: Investment Time Horizon and Asset Allocation Models
FINANCIAL MARKETS AND PORTFOLIO MANAGEMENT / Volume 15, 2001 / Number 1 89
Page 15
6. Transaction Fees
In this last section, we include transaction fees
and analyse their impact on the performance of
the outofsample 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 OutOfSample Dynamic LongTerm 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 19911992199319941995 1996 199719981999 2000
Wealth evolution
Optimal strategy without transaction feesPictet 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
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