Portfolio Choice Under State Dependent Adjustment
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Transcript of Portfolio Choice Under State Dependent Adjustment
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Portfolio Choice with State-dependentAdjustments
Analyzing leveraged positions without parametric assumptions
Peter Farkas1
Central European University
Friday, April 25, 2014
1Corresponding author, Nador u. 9, Budapest 1051, Hungary; [email protected].
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Abstract
In this paper, we discuss a new method for solving the portfolio choice
problem which models state-dependent quantity adjustment using bound-
ary crossing events. This new method has several advantages: first, we can
solve for the optimal portfolio weights without parametric assumptions, by
deriving them directly from the data. Next, we can describe the full distri-
bution of the portfolios value, not just its moments. Finally, we can easilydeal with important practical issues, such as transaction costs, leveraged po-
sitions and no-ruin conditions, or the cost of margin financing. In particular,
the method allows us to analyze leveraged positions in discrete time under
zero ruin probability. Analyzing historical stock data suggests that histori-
cally, the log-optimal portfolio was a not too extensive leveraged purchase of
a diversified stock portfolio, therefore leveraging does not necessarily imply
risk-seeking behavior. We also show that depending on how much weight we
allocate to this diversified stock portfolio, the downside risk measured as 5%
VAR of the portfolios value may be decreasing or increasing over time. Con-
sequently, an objective functions which incorporates the VAR of the portfo-
lios value or operate with VAR constrains result in horizon-dependent port-
folio weights. We also present some evidence suggesting that the log-optimal
portfolio weights are time-dependent. Finally, irrespective of what weight we
chose for the diversified stock portfolio, it is log-optimal to reduce exposure
to the stock market if the predicted volatility is high, and increase it in low
volatility periods.
JEL: G11, G17, C14
Keywords: optimal portfolio choice, log optimality, GOP, transaction
costs, leveraged positions, horizon dependence, VAR constrains, non-
parametric, first exit time, boundary crossing counting, backtesting
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1 Introduction
The optimal portfolio choice is an equally important problem for theoreti-
cians, for financial practitioners as well as for any non-professional with an
investment decision to make. According the Brandts (2009) review, there
is a renewed theoretical interest for this problem which is motivated by the
fact that relatively recent empirical findings (predictability, conditional het-
eroscedasticity) have not yet been fully incorporated into the theory of port-
folio allocation. This review also states that the main direction of academic
research is to identify key aspects of the real-world portfolio problem andto understand how these aspects influence the decision of institutions and
individuals. Hopefully, such efforts will reduce the gap between the theory
and the practice of portfolio management.
In this paper, we aim to follow this general academic direction: we hope to
provide financial theory with some new results and at the same time we also
aim to provide practitioners with a useful tool. The essence of our innovation
is a new, state-dependent quantity adjustment mechanism which is based on
boundary crossing counting processes introduced by Farkas (2013). Plainly
speaking, we propose to change the number of quantities each time the ap-
propriately adjusted, weighted price index changes more than a predefined
limit and not with a constant frequency, not once a day. We show how to
calculate the portfolios return by counting the number of adjustments. The
biggest advantage of this approach is that it allows us to describe the portfo-
lios full return, not just its expected value without parametric assumptions
or simulations. Consequently, we can solve for the portfolio choice problem
under VAR constrains. Also, our approach makes it straightforward to deal
with certain practical issues, such as proportional transaction costs, or the
cost of margin financing upon leveraged positions and finally the issues re-
lated with no-ruin conditions. From a applied theoretical point of view, our
paper describes a direct, non-parametric estimation method. From a practi-
tioners point of view, we introduce a back-testing technique which is useful
for robust, real-world analysis of historical data.
The methodology in this paper differs from the typical agenda on portfolio
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choice. Usually, solving for the optimal portfolio consists of three steps. The
first step is to specify and estimate a parametric model describing the returns
of the risky assets, a step we want to avoid as it has been proven notoriously
difficult to come up with an accurate parametric model. Classical papers
such as Markowitz (1952), Merton (1969, 1971), Samuelson (1969), Malkiel
and Fama (1970), often assume that prices follow a Geometric Brownian
Motion and abstract away from financial frictions. Unfortunately, the GBM
hypothesis is often rejected in practice as discussed by Lo and MacKinlay,
(1999), by Cont, (2001) or by Cambell and Thomson, (2008). Additionally,
even if we assume a parametric model, we have to face with the fact that wedo not know the true parameters, analyzed by the literature on parameter
uncertainty as shown by Jobson and Korkie (1980), Best and Grauer (1991),
Chopra and Ziemba (1993). The next step is to solve the investors optimal
problem during which researchers often abstract away from frictions. This
is done simply to reduce the complexity of the problem, however financial
frictions proved to be important from a theoretical and a practical point of
view as well as discussed by Constantinides, (1986) or by Dumas and Luciano
(1991), and finally by Balduzzi and Lynch (1999). The last step is to plug-in the potentially biased parameter estimates to a potentially oversimplified
portfolio choice solution and infer the optimal portfolio weights from there.
An alternative path is to make use of the large amount of available data,
and try to obtain optimal portfolio weights directly, without parametric as-
sumptions. This approach is called alternative econometric approach in
Brands (2009) review, while practitioners often call it backtesting. This is
basically a direct attack on the problem. Here, where we try to obtain portfo-
lio weights directly from the data. Besides the obvious benefits of not having
to build a parametric model for the returns, an other important advantage
is the reduction in dimensionality and hence gain in degree of freedoms. An
important, but sometimes overlooked issue summarized below is related with
the fact that constant portfolio weights typically1 require us to occasionally
change the number of securities in the portfolio. The fundamental problem
is as follows: investors, big or small, are typically interested in controlling
1Unless we want to hold 100% of our wealth in one asset only
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for the weights of the risky assets while stock- and commodity exchanges do
not offer this option: only the quantity of risky securities can be controlled
for. Therefore, controlling for the weights require occasionally changing the
number of securities in the portfolio.
maxW
S(U(VT)) =
maxWS(U(VT(W,RF,RR))) continuous adjustment
maxWS(U(VT(Q, P,RR,tr)))) otherwise
(1)
where S(.) is a stochastic function, typically some combination of the vari-
ables expected value, variance, or some extreme value statistics, such as VAR
limits. In both cases, U(.) is a non-path dependent utility function having
a well-defined maximum2, and utility depends only on the terminal value,
VT. Note, that expanding this formulation to other, path-dependent forms is
possible yet not detailed here due to space constrains. Also, Wis the weight
of the risky assets, Q is the number of securities hold,RFand Pdescribes
the return of the risky assets, RRdescribes the return on the composite asset
and finallytr is the proportional transaction cost. The literature has offered
three solutions to this problem.
1. Continuous adjustment started by Merton (1969,1971) simply abstracts
away from this problem by saying that lets assume we can change the
number of securities continuously and hence we can control for the
variable which drives the portfolios value, Vt.
2. Time-dependent adjustment proposes to change quantities at a given
time-frequency. For example, assuming daily quantity adjustment is a
typically time-dependent mechanism frequently used in practice.
3. State-dependent adjustment proposes to calculate quantities based on
some state variable, for example assuming that quantities are changed
each time the appropriately weighted cumulative price change since
the last adjustment reaches some pre-defined critical level. This type of
2Not all utility function has a well-defined maximum, linear utility functions typicallytend to be problematic in this regard.
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adjustments often used in the literature on optimal inattention which
originate from Baumol (1952) and Tobin (1956) and later on, was taken
up to deal with transaction costs by Constantinides, (1986) or Dumas
and Luciano (1991).
The main innovation of this paper is to introduce a new way to model state-
dependent adjustments which has several advantages. Firstly, from a theo-
retical point of view, we show a new analytical, although not exact, solution
to the simple portfolio choice problem under log-utility. Similarly to Con-
stantinides (1986) or Dumas and Luciano (1991), we can allow for propor-
tional transaction costs. Next, we can analyze leveraged positions as well as
short-sellings without truncating the underlying distribution, because state-
dependent adjustment prevents ruin events to occur. This is especially im-
portant for todays economy since economic conditions lead to record-high
level in margin loans in the United States.
Figure 1: End of month figures based on New York Stock Exchange Factbook
Moreover, time-dependent adjustment may overestimate transaction costs
in case the adjustment is too frequent. Finally, the proposed state-dependent
adjustment describes the portfolios value using a discrete stochastic variable
which facilitates dealing with stochastic issues greatly. In particular, we can
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calculate not only the optimal weight and the corresponding expected re-
turns, but the full distribution of these returns as well, which is useful when
solving VAR-constrained problems. Overall, our paper is a useful complement
to other, non-parametric studies relying on time-dependent adjustment such
as Brandt (1999) or Brandt (2003), or Covers Universal Portfolio approach,
detailed by Cover (1991), which has been extended, for example, by Blum
and Kalai, (1999) to be able to incorporate transaction costs, and further
explored by Gyorfi and Vajda (2008) and by Horvath and Urban (2011).
The paper is structured as follows. The second section first explains how
to use boundary crossing counting processes (BCC processes) to analyzeportfolio choice problems. We continue by briefly discussing the theory behind
these stochastic processes and explain how they are related with the first
exit time distributions and the upper boundary crossing probabilities. As a
conclusion for this section, we put our method into perspective by solving
for the simple portfolio choice under state-dependent adjustment and by
comparing the results with Mertons continuous solution under Geometric
Brownian Motion and log-utility. In the third section, we apply our method
to actual security data and the last section concludes.
2 Portfolio Choice and Quantity Adjustment
2.1 State-dependent adjustment
The following readjustment scheme is similar in spirit to the one rec-
ommended by Dumas and Luciano (1991) for portfolio allocation or by
Martellini, L. and Priaulet (2002) for option pricing. Here, we basically as-sume that the number of securities hold are changed once the cumulative,
financing-adjusted weighted price change since the last adjustment reaches
some exogenously chosen, pre-defined levels. The first process of Figure 2
called restarted process shows the cumulative price changes between two
readjustments. The second processY ULt, first introduced by Farkas (2013),
called boundary-crossing counting process or BCC process, as its name
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suggests, simply counts the number of boundary-crossing events.
Figure 2: Boundary Crossing Counting Process
We differentiate between four different types of BCC processes.
1. Y Ut counts the number of upper crossing events: Y Ut = Y Ut + 1 if
Xt = U Bt and Xt+ = X0.
2. Y Lt counts the number of lower crossing events: Y Lt = Y Lt + 1 if
Xt = LBt and Xt+=X0.
3. Y U Lt = Y Ut + Y Lt counts the number of upper and lower crossing
events.
4. Y Dt=Y Ut Y Lt described the difference between the upper and the
lower crossing events.
For appropriately chosen Xt, we can describe the portfolios value in time
t TUL, where TUL is the set of boundary crossing moments, as follows:
Vt = V0 GUY Ut GLY Lt (2)
The stochastic elements areY Utand Y Ltdescribing the number of boundary
crossing events while GU > 1 and 0 < GL < 1 are exogenous constants
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describing the change in wealth upon boundary crossings. It is important to
highlight that stochastic variables are discrete. Assuming away from potential
liquidity constrains and price-discontinuities, no-ruin conditions only require
finite weights for the risky assets as worst case Vt = V0 GLk > 0 for any
k N. Without loss of generality, we can normalize3 the initial portfolios
value to one. The log of wealth is than equal to:
log(Vt) =Y Ut log(GU) + Y Lt log(GL) (3)
Since we chose GU and GL exogenously, we can set them in a way thatGU= 1/GL and equation further simplifies as:
log(Vt) =Y Dt log(GU) (4)
As detailed in the appendix, the following restarted process restricts the
change in the portfolios value two two discrete values.
Xt = (1 +
jwj pj
jwj
Pj
Pj trj+ dbs(t)1
jw
j trj(5)
where j is the number of assets in the portfolio, wj and w
j are the portfolio
weights, Pj are the prices at the last boundary crossings, or the initial prices,
Pj are the actual prices, tr is the proportional transaction costs and finally
dbst is the cost of financing expressed as percentage of portfolios value. This
structure ensures that the percentage change in the portfolios value takes
only two discrete values by balancing out the change due to the variation
in risky assets value and the change due to financing. Theoretically, we can
chose from an infinite number of potential boundaries. Here, we will not
analyze this choice in detail, but try to place them approximately seven
standard deviation distance, which has some desirable statistical properties
3It would also make sense to normalize the initial values to V0 =
|wi| (1 tr),which would then take into account the cost of entry. In this paper, we aim to analyzeannualized returns over long horizons, therefore we abstract away from these initial costs.
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as detailed in Farkas (2013).
Finally, the stochastic issues are straight-forward since BCC-distributions
are discrete. Assuming logarithmic utility as an illustration, calculating the
expected value for example can be done as:
E(log(Vt)) = log(GU) E(Y Dt) = log(GU) ii
p(Y Dt=i) i; (6)
Note, that the expected value ofY Dcan be well approximated byE(Y Dt)
Y Dcountt , whereY D
countt indicates the number of events observed in the data.
This approximation is considerably faster and, based on simulations, rela-
tively precise if we observe more then 30 crossing events. Calculating other
stochastic measures are also straightforward, for example calculating a 5%
VAR value can be done as:
V aR(log(Vt), 5%) = log(GU) V aR5ii
p(Y Dt=i) i (7)
whereV ar5i
i p(Y Dt = i) = 0.05. Overall, we have shown that if we can cal-
culate the boundary crossing counting distribution, then we can also charac-
terize the full distribution of the portfolios value as well as many stochastic
properties derived from the full distribution, such as the expected values or
VAR limits.
2.2 Financing costs
The formula for the restarted process includes the amount of interest collectedor paid on the composite asset, which will be discussed next. Financing costs
are assumed to be a piecewise linear function of the portfolio weights.
dbst =
ti=TulDBS(i) RF(i)
Vt1(8)
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In particular, deposit (D) is collected on the fraction of wealth that is not
invested in risky assets, if any.
D=
1
W(w >0) if
W(w >0) < 1
0 otherwise(9)
Borrowing (B) occurs if we decide to finance the purchase of risky assets by
margin loans.
B =
1
W(w >0) if
W(w >0) > 1
0 otherwise(10)
Finally in case we want to short-sell (S) some risky assets, then we have to
borrow which also assumed to be costly.
S=
W(w
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borrowing for short-selling costs 200 bps above4 the reference rate. Naturally,
the actual rate on margin loans depends on the brokerage firms and likely
to vary by clients even within one firm, and investigating these contracts are
well beyond the scope of this paper.
2.3 Theory and estimation of BCC distributions
The number of boundary crossing events can be calculated directly, simply by
counting the number of crossing events. Also, it can be calculated recursively
using first exit time distribution5
and upper boundary crossing probabilities.These concepts will be reviewed next. Note, that we will only provide the
definitions along with a brief discussion, and we also point to references where
interested reader may find the technical details.
Definition 1 Let first exit time T UL be defined as the time in which the
processXt crosses either boundaries:
T U L=
inf(t: Xt / (LB,UB) if t is finite
otherwise
(12)
Throughout the paper, we are going to assume that T U L is positive and
finite, which are non-elementary assumptions. The finiteness of the first-
exit time is a well-known property for martingales, which is typically proven
4We have assumed higher costs for short-selling as from the perspective of the brokeragefirm, shortselling is more risky than a leveraged purchase as the loss in case of leveragedpurchase is limited, however the loss in case of shortselling is unlimited
5There is no consensus in the literature on the terminology. First passage time orhitting time is typically used in situations, where there is only one boundary. Expectedfirst passage time describes the expected amount of time needed to reach that boundary.First passage time distribution aims to characterize the full distribution. The case oftwo boundaries is usually referred to as first exit time or double-barrier hitting timealthough the term first exit time is also used to describe first passage time, see forexample Wilmott (1998, p. 144). Exit times should not be confused with first rangetime, as range is generally used to describe the difference between the maximum and theminimum value. In this paper, we follow the terminology of Borodin and Salminen (2002).They use the name first exit time to describe the case of double boundaries, so we stickto this notation as well.
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using Doobs lemma (optimal sampling theorem) as discussed for example
by Medvegyev, (2007). As for non-martingales, finiteness is proven by first
converting the stochastic process to a martingales, as explained for example
in Karlin and Taylor, (1998) and then apply Doobs lemma. The assumption
that T U L > 0 is problematic only if the limit of the boundaries at the
starting points are equal to the starting value of the restarted process, a case
which we will avoid in this paper.
Definition 2 Let first exit time distribution f et(t,UB,LB,X0) be defined
as a probability distribution describing the probability that the first exit time
ist.
We assume that the first exit time distribution is stationary. The distribu-
tions can either be calculated analytically for certain parametric processes,
or estimated from data using kernel density estimation. The typical proce-
dure for the analytical work begins by subtracting the expected value from
the original stochastic process which results a martingale. Next, we make use
of the Doobs lemma and equate the initial value of this martingale with its
expected value at the first exit time. Finally, by rearranging this expected
value, we can obtain the Laplace transforms of the first exit times. The prob-
ability distribution functions can then be derived by inverting these Laplace
transforms. As for the non-parametric case, when estimating the first exit
time distribution, it is important to take into account not only the closing,
but the minimum and maximum values as well, otherwise we induce sampling
bias. As for the type of kernel, simulations suggest to use the one based on
normal distributions.
Definition 3 Let upper boundary crossing probability be defined as the prob-
ability that the stochastic process reaches the upper boundary before hitting
the lower one.
p= P(XTUL =U BTUL|UBt> Xt> LBt); 0< t < TU L; (13)
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Also, let us introduce the notationqfor lower-boundary crossing probability,
that is q = 1 p. We assume that upper boundary crossing probabilities
are constant. For analytical processes, the boundary-crossing probability is
typically expressed using scale functions, as explained for example in Karlin
and Taylor (1981).
P(XTUL= U B) =S(X0) S(lb)
S(ub) S(lb) (14)
whereS(x) = exp( x 2(y)2
(y)
dy) is the scale function andmu(.) and2(.) are
the infinitesimal moments. The lower limit of the integrals does not play a
significant role thus is omitted in accord with the literature. This equations
essentially shows that once the process has been appropriately scaled, then
the probability of upper (or lower) boundary crossing depends only on the
initial points relative distance from the lower and upper boundaries. For non-
parametric processes, the boundary crossing probability can be estimated by
the ratio of the number of upper crossings and the number of total crossings.
We characterize the upper and the lower boundary crossing counting dis-
tribution with the following matrix:
P U L=
P U L1(0) P U L2(0) P U LT(0)
P U L1(1) P U L2(1) P U LT(1)...
... . . .
...
P U L1(n) P U L2(n) P U LT(n)
(15)
where some P ULt(i) describes the probability that until period t, exactly i
boundary-crossing events have occurred, that is p(Y ULt = i) = P U Lt(i).Calculating the first row simply involves evaluating the first exit time distri-
bution at time t:
P U Lt(0) =
1 t0
f et(t)dt for continuous distributions
1 t
k=1 f et(k) for discrete distributions(16)
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Thus, we have been able to obtain the first column of the P U Lmatrix. Any
other column can be calculated recursively using:
P U Lt(j) =F2(j) F1 (17)
wherej indicates the number of columns. NowF1 andF2(j) matrix both can
be calculated recursively using first-exit time distributions as shown in Farkas
(2013), therefore first-exit time distribution fully characterizes the boundary-
crossing distribution for the case of lower and upper crossing both. For our
purpose, we also need Y Dt describing the difference between the number ofupper and the lower crossing events which can be obtained using the following
random-time binomial tree. We named it random time tree, because the time
needed to move from one node to the next is random.
Figure 3: Random-time binomial tree
In comparison to classical binomial trees where stochastic variable may
either go up or down, here we allow for three options: the variable may either
go up, down, or remain in that particular node. Such random-time binomial
tree could also be represented by a classical trinomial tree. A node of the
tree B(i, j) can be described by the number of boundary crossing events, i
is the number of upper crossings, j is the number of lower crossings. Note,
that the grid itself also changes dynamically as time changes, the diagram is
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essentially a snapshot taken at a given point of time.
Characterizing the grid can be done in two steps. The vertical location
can simply be described by the number of boundary crossings.
BV =
P U LT(0) P U LT(1) P ULT(T)
(18)
The horizonal location can simply be described by the boundary crossing
probabilities conditioned on the number of boundary crossings. Forj number
of boundary crossing:
BH(j) =
0...
pj
pj1 q...
qj
...
0
(19)
Since the horizonal and the vertical location is independent, the grid can be
characterized as:
B =BV BH (20)
Obtaining the distribution from B simply involved collecting terms where
i j are equal.
p(Y Dt=k) =T
l=k
B(l, l k) (21)
Overall, the portfolios value in some time t can be described using the Y Dwhich in turn can be calculated from the first exit time distribution and
upper boundary crossing probabilities. Both can be estimated directly from
the data using kernel estimations, or can also be calculated analytically in
certain cases.
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2.4 Simple Portfolio Choice under Geometric Brown-
ian Motion and Log Utility
We continue by illustrating the technique described above using simple port-
folio consists of a single risky asset whose price follows Geometric Brownian
Motion (GBM), and a single composite asset which does not pay interest
under logarithmic utility. Besides being a frequently used pricing model, the
GBM is a good starting point as it has analytical solution which always
helpful in calibrating, fine-tuning the BCC-based solution.
Let us begin with the case of state state-based adjustment. As we onlyhave one risky asset and abstracted away from financial costs, therefore Xt
follow a Geometric Brownian Motion with some drift and standard devi-
ation. As the composite assets do not pay interest, therefore boundaries are
constant. For GBM model under constant boundaries, first exit time distribu-
tion as well as the upper crossing probability can be calculated analytically.
The first exit time distribution for Geometric Brownian Motion with variance
normalized to one is given by Salminen and Borodin (2002, p. 295):
f et(t,lb,ub) =e2t/2(elb(ub,ub lb)dt + eub(lb,ub lb)dt) (22)
where ss(.) is the theta function. Substituting the scale density of the Brow-
nian motion results the formula for upper boundary crossing probability.
P(XGBMTUL =U B) = 1 exp(lb 2
2)
exp(ub 22
) exp(lb 22
)(23)
Overall, the first exit time distribution and the upper boundary crossingprobability both can be calculated analytically. Consequently, the BCC-
distributions and the corresponding log-returns can also be obtained ana-
lytically, without simulations.
In order to create a basis for comparison for these analytical results, let
us briefly discuss the case of continuous adjustment, introduced by Merton
(1971) put into perspective for example, by Peters (2011). The portfolio
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consisting of a risky asset and a risk-free composite asset follows:
dPt= (rr+ we)Ptdt + wPtdWt (24)
whererr is the return on the risk-free assets assumed to be zero for the pur-
pose of this exercise, rm is the market return, e =rm rr is the excess
return, w is the weight of the risky asset and finally Wt is the Wiener pro-
cess. This formulation implicitly assumes that investors can keep a constant
fraction of their wealth in the risky asset which would require continuously
adjusting the number of shares, unless w = 1. The question of interest islog-optimal value ofw. Using Its formula:
dln(Pt) = (rr+ we 1
2w22)dt + wdWt (25)
The expected value of the Wiener process is zero, the expected value of the
exponential growth rate can be expressed as:
E(g) =E(dln(Pt)
dt = (rr+ we 1
2w22) (26)
Solving for the optimal value and substituting for rr = 0 yields:
wopt =
2+
1
2 (27)
Figure below compares the expected growth rate for state-dependent and con-
tinuous adjustment using the diffusion parameters based on the closing prices
of the Dow Jones Industrial Average between 1928 and 2012, estimated bythe standard maximum likelihood method detailed, for example, in Gourier-
oux and Jasiak (2001) resulting in muML = 0.0437 and sigmaML = 0.0342.
When calculating state dependent adjustments, we placed the boundaries at
7 standard deviation distance.
Diagram reveals little difference between the continuous and state-based
adjustment under this setup without transaction costs, therefore the state-
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Figure 4: State-dependent and Continuous Adjustment Under GeometricBrownian Motion
based adjustment ceteris paribus (keeping other assumptions unchanged)
does not influence the results significantly. Hence, the new approach keeps
the intuition of the simpler model: Replacing the continuous adjustment of
the Mertons model with the boundary-crossing based adjustment ceteris-
paribus does not lead to different result. Therefore, results are not driven
by the state-dependent adjustment mechanism we just introduced. This is
good news, as we can extend the analysis to a variety of more complex, more
realistic financial models as well as to actual security prices without losing
the insights provided by the simpler case. The case involving transaction
cost is comparable with Dumas and Lucianos (1991) paper. Both solutions
are analytical yet they provide a closed form solution while here, we pro-
vide an algorithmic one. Both models assume that investors adjustments are
infrequent, state-based, yet we do not assume that investors are optimally
inattentive. Finally, we work with log-optimal investors while they assume a
more general utility function.
3 Empirical results
The method discussed so far can be used to analyze a large variety of in-
vestment strategies. Here, we will select only a few topic which may be of
general interest. The investor we have in mind in the one who seeks absolute
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return, therefore our interest is not to outperform certain benchmark index
by selecting its most lucrative components but to chose between different as-
set classes in order to maximize the objective function, which is chosen to be
the expected log of the portfolio value. We complement log-optimality with
VAR-based measurements as well. The first part of this section provides us
with some descriptive results of the past while the second part discusses the
merits of an active, volatility based investment strategy.
We chose log-optimality or growth-optimality because it is considered
to be an important benchmark case having many theoretically appealing
properties as detailed by the large number of papers starting from Kelly,(1956), and Breiman, (1961) and reviewed recently for example by Chris-
tensen, (2005) or in MacLean, Thorp, and Ziemba, (eds. 2011). In particular
it has been shown by Breiman, (1961) and by Long, (1990) that there exists
a portfolio (growth-optimal portfolio or numeraire portfolio or log optimal
portfolio) for which the price of any other portfolio denominated in the price
of the growth-optimal portfolio becomes supermartingale. Also, this portfolio
is the one that maximizes the expected logarithm of the terminal wealth as
shown by Breiman, (1961) or by Kelly, (1956). It also maximizes the expectedgrowth rate of the portfolios value as described in Merton and Samuelson,
(1992). Furthermore, growth-optimal strategy maximizes the probability that
the portfolio is more valuable than any other portfolio, therefore has a cer-
tain selective advantage as detailed by Latane, (1959). Among all admissible
portfolios, the growth-optimal portfolio minimizes the expected time needed
to reach, for the first time, any predetermined constant as shown by Merton
and Samuelson, (1992). If claims are discounted using the growth-optimal
portfolio, then expectation needs to be taken with respect to historical prob-
ability measures as explained in Long, (1990) or Bajeux-Besnainou and R.
Portait, (1997) therefore these portfolios may provide a unifying framework
for asset prices as shown by Platen, (2006).
Let us continue with general data-related issues. As we aim to work with
long time-series therefore we need to combine data series which are sampled
with different frequencies. As adjusting our theory to handle sampling is-
sues would create unnecessary complexities, therefore we interpolate lower
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frequency data to the highest frequency using Brownian bridge for risky as-
sets and step-functions for the composite assets. Note, that we could also
aggregate the lower frequency to the higher one, yet this solution would have
resulted in a loss of information. Estimating the BCC distribution requires
not only closing prices, but minimum and maximum values as well. We have
approximated these values using the minimum and maximum values of the
SP500 index between 1951 and 2014 in two steps. First, we calculated the
ratio of minimum price and closing prices as well as the ratio of maximum
price to closing prices. Next, we matched these ratios to the full periods clos-
ing prices randomly. Finally, multiplying the closing prices with these ratiosresulted in the estimated minimum and maximum prices. Naturally, this ap-
proach ignores certain dependencies, for example intra-day volatility is likely
to be higher in volatile markets. Once again, a more sophisticated approach,
or a Brownian approximation described in Mcleish (2002) would have in-
creased the complexity greatly and we did not see the additional benefits of
going down this path.
As for data selection, we work with three asset classes: stocks, bonds
and gold. As for stocks, we have used the SP500 gross total return indexwhich assumes that dividends are reinvested. For the period 1870 1988,
we have used Shillers data while from 1988 onwards, we have used the total
return index obtained from Chicago Board of Trade. We have abstracted
away from taxation as including it at this stage would increase complexity
significantly and we do not see any significant additional benefits of going
down this path. We are well aware that investing into theSPindex was not
possible before the introduction of ETFs, yet we feel that this index is still be
best approximation of a representative diversified investment. Naturally, our
results suffers from common known issues such as survival bias. As for bonds,
we use the Bank of America Merrill Lynch US Corp Master Total Return
Index Value as downloaded from FREDs homepage which can be considered
as a representative investment into corporate bonds. Finally, we used London
Bullion Market Associations daily fixing for modeling the return on a gold
investment. Naturally, gold cannot actually be traded on exactly these prices,
yet these data series represent well the prices of an average tradable gold-
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based instrument. Besides the usual issues, here we also abstract away from
certain commodity-related issues such as the problem of rolling forward the
future contracts. As for the composite asset, we either used the FED target
rate obtained from FREDs database or the long-term interest rate from
Shillers database. This latter is somewhat problematic, yet we could not
obtain short-term rates prior to 1954 and we felt that increasing the amount
of data is more important than accounting for the differences between the
short rates and long rates.
3.1 Descriptive results
The first few diagrams focus on the choice between SP-stocks and the com-
posite asset in the United States, between 1870 and 2013 assuming that the
composite asset is the long-term interest rate.
Figure 5: Log-optimal investment and VAR constrains in the United States
The diagram reveals that the log-optimal investment is a leveraged pur-
chase, therefore the use of margin loans does not necessarily implies risk-
taking behavior: risk-averse log-optimal investors may equally use this tool.
The gain from leveraging however is relatively modest: the log-optimal weight
of 1.65 implies only a 63 bps gain in comparison to the buy and hold return.
The VAR levels are decreasing in portfolio weights. It is interesting to no-
tice that a typical brokers recommendation, according to Mandelbrot and
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Hudson (2014), suggests to hold 25 percent cash, 30 percent bonds, and 45
percent stocks; this latter corresponds to the level where the five year 5%
VAR of the portfolios value is 1. Based on the diagram above, brokers may
recommend a level where the investment is likely to be recovered, at least
in nominal terms after five years. Of course, this may be only a coincidence,
but it may also suggests that the average investors have strong preference
against loosing money, and history taught brokers where this level may be.
It is also interesting to notice that downside-risk falls into two categories:
for moderate weights, the five year downside risk is larger than the ten year
downside risk, while for higher weights, it is the other way around. Therefore,there are downside-increasing and downside-decreasing allocations.
Figure 6: Evolution of downsize risk in time for various portfolio weights
This diagram complements to the debate on whether the optimal frac-
tion of wealth invested into stocks is horizon-dependent or not. Here, we
show that if investors preference includes downside, then investment rules
are horizon-dependent. This is in line with common wisdom, described by
Malkiel (1999), which states that the brokers typical recommendation is
a horizon-dependent one: The longer period over which you can hold on
to your investment, the greater should be the share of common stocks in
your portfolio. Early academic papers such as Samuelson (1969) and Mer-
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ton (1971) derived horizon-independent rules which goes against this com-
mon wisdom. Consequently authors, for example Brennan, Schwartz and
Lagnado (1997), Liu and Loewenstein (2002) and Brandt (2009) proposed
many adjustments to these early models, such as time-varying investment
opportunities, time-varying parameters, transaction costs, or predictability
of dividends growth, which may result in horizon-dependent rules. Here, we
complement these finding by noticing the difference in the downside risks
evolution: in order to explain horizon-dependence, it is sufficient to assume
that older investors are more concerned with preserving their wealth and put
more emphasize on VAR limits, while younger ones are more focused on thepotential upside.
So far, we have assumed that optimal weight is time-independent which
may or may not be the case. In fact, certain active investment practices,
commonly named market-timing, assume that optimal investment decision
is time-dependent. These practices are built on two premises: First, they
assume that optimal portfolio weights are time-dependent and second they
assume that these weights are predictable. Here, we will focus on the first
premise. In the parametric realm, it is often assumed that parameters aretime-dependent, which can be translated into the non-parametric realm by
saying that let us assume that the data-generating process is time-dependent.
In fact, assuming a seven years holding period reveals that the optimal
portfolio weights appear to be highly time-dependent, ranging from zero to
over ten. Of course, part of the variation is due to the fact, that we rely
on significantly less data and hence the measurement is much more noisy.
Nevertheless, even without having a precise measure of the noise, it is still
plausible to say that log-optimal investment appears to be time-varying and
it may be log-optimal to occasionally hold leveraged positions.
Next, we continue by allowing for more type of assets and analyze the
decision of an investor who has access to three type of assets: stocks, corporate
bonds and gold, between 1975 and 2013 assuming that the composite asset
is the short-term interest rate. The weights for the log-optimal investment
consists of 0.5 for Gold, 4.55 for corporate bonds and 1.85 for stocks. The
overall optimal exposure defined as the sum of risky assets is 6.9.
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Figure 7: How Log-optimal investment over multiple assets change over time?
Figure 8: Log-optimal investment for a portfolio consists of gold, corporatebonds and stocks.
The general tendency is probably more of an interest then the actual val-
ues: it was log-optimal to borrow 200 bps over the short-term rate and invest
mostly into corporate bonds and stocks. The gain from leveraging appears
to be more significant than in the previous case. Once again, we have showed
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that margin purchase does not necessarily involve risk-seeking behavior and
a risk-averse, log-optimal investors may also rely on this financial tool. Let
us finish this section by briefly reviewing how lop-optimal investment varies
over time in this case.
Figure 9: Log-optimal investment for a portfolio consists of gold, corporatebonds and stocks.
It appears to be log-optimal to finance the purchase of corporate bonds
using short-term debt. Likewise, leveraged purchase of stocks typically in-
creases the portfolios expected growth rate. The leveraged purchase of Gold
has become log-optimal after the year 2000 which may be a structural is-
sue related with commodity markets, or may also be caused by the rumors
concerning the limited availability of the remaining global gold stocks.
3.2 Managing portfolio weights using predicted volatil-
ity
As already noted above, market timing is built on two premises. First, we
have to assume that optimal portfolio weights are time-varying. The previous,
descriptive, section provides some evidence on this possibility. In this section,
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we deal with the issue of predictability. Due to the practical interest of this
topic, there is such a large number of studies aiming to provide insights in
this regard, that we do not even have room nor to review, nor to test them.
Besides, the validity of these predicting rules are often questioned due to
data-snooping bias.
Here, we take an alternative route: first of all we do not apply optimiza-
tion, but presents a method which increases the growth rate of the portfolios
value regardless of chosen portfolio weights. The method we suggest is based
on facts that are accepted by overwhelming number of scientists and practi-
tioners as well. More specifically, we propose to change the portfolios weightsbased on the predicted volatility. This approach is built on two premises: first
of all, the possibility of predict volatility it is well known as detailed in the
large number of studies starting perhaps from Bollerslev (1987). Second, we
also know that under Geometric Brownian Motion, the optimal portfolio
weights are inversely related to the volatility. Hence, it is reasonable to as-
sume that the insights gained from the GBM model also carries over to actual
security data. Overall, intuition suggests that it makes sense to reduce the
portfolios weight if the predicted volatility is relatively large, and increaseit if the predicted volatility is low. We try to verify or falsify this intuition
assuming only one risky asset, the total return index of SP500 between 1988
and 2013 using the following algorithm:
1. We estimate a GARCH model using Matlabs GARCH package based
on the previous 2520 observations which corresponds to approximately
10 years of observation. When doing the estimation, we use the Glosten,
Jagannathan and Runkle (1993)s specification (further referred as
GJR model) which includes a leverage terms for modeling asymmet-ric volatility clustering. The first estimation period ranges from 1978 -
1988 end of May, and the estimated model predicts the volatility until
the end of June. The second estimation ranges from 1978 - 1988, end of
June, and the estimated model provides us with the estimated volatil-
ity until end of July. Therefore, we adopt a rolling method and hence
when making the predictions, we do not rely on any information which
has not been revealed previously.
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2. We calculate the BCC distributions using the total return indexes.
Each time there is a boundary-crossing, we set the quantities so that
the portfolio weights are equal to some base weights multiplied by
the ratio of the actual and the historical variance: W
= Wbase
max(min( 2
2predicted
, 4), 0.25). We restrict the variance correction factor
in order to ensure the stability and the precision of the measurement.
The Wbase ranges from 0.15 to 5.
3. Finally, we also calculate the BCC distributions and the corresponding
expected growth rate for the case of constant portfolio weights as well.
We plot the portfolios values expected growth rate as a function of the
average portfolio weights. The resulting diagram reveals that such volatility-
based adjustment results in a higher expected growth rate, regardless of the
chosen average portfolio weights. The log-optimal expected growth rate rep-
resents a considerable gain of 700 bps in comparison to the constant portfolio
weights.
Figure 10: Log-optimal investment for constant and volatility-dependentportfolio weights.
Also, the average portfolio weight for the log-optimal portfolio under
volatility-dependent weights is significantly higher than for the case of con-
stant weights. Overall, it appears to be log-optimal to take a leveraged pur-
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chase on diversified stock index, but the exposure needs to be reduced in
high-volatility periods. It is not difficult to see that such behavior, if acted
upon in practice, has serious implication for financial stability. It is often ar-
gued that the role of speculative capital is to provide liquidity, and liquidity
is needed most in high volatile periods. Yet, we found that is is log-optimal to
partially withdrawn from the market in these high-volatility period, therefore
speculative capital may disappear in times when it is the most needed.
4 SummaryIn social sciences, researchers have to rely on a few, more or less imperfect
technique: We can build an analytical model, study the past, look for natural
experiences, or draw conclusion from organized experiments. Each technique
has its merits and its weaknesses. In this paper, we have introduced a new,
non-parametric technique, which relies on studying the past: we have ana-
lyzed the problem of portfolio allocation using historical data. The novelty of
this paper is two fold: in one hand, we have introduced a new, non-parametric
technique, summarized first, and we have obtained some interesting result us-
ing this technique, summarized next.
1. As we solve the portfolio problem without parametric assumptions,
therefore we can incorporate many important features of financial data,
such as volatility-clustering or dependencies between the prices.
2. Also, our method describes the portfolios value using a discrete
stochastic variable, hence it allows us to calculate not only the ex-
pected value, but also many other stochastic properties, such as VARlevels without parametric assumptions or simulations.
3. The technique can be calibrated by analytically solving for the log-
optimal portfolio under Geometric Brownian Motion.
4. Finally, we can easily deal with many practical issues, such as propor-
tional transaction costs or the cost of margin financing.
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Regarding potential limitations and weaknesses, our approach requires com-
plete market, we have to assume that transactions can be done at the desired
price level. In other word, we abstract away from execution risk. In non-liquid
instruments, especially for large participants, execution risk may be signifi-
cant, therefore incorporating it into the framework may be meaningful and it
may be done in a forthcoming paper. Also, we implicitly assume that prices
are continuous, therefore at this stage we do not allow for discontinuities.
This is not a serious issue in the current paper since we have worked with
indexes, where such discontinuities are relatively rare. Incorporating jumps
would influence risk level as well as the no-ruin conditions, all may be dis-cussed in a forthcoming paper.
Regarding the actual results on historical data, we acknowledge that it
would be a great mistake to assume that the future will be like the past, yet
understanding what has happened should at least be indicative in figuring
our what may come in the future. One way to summarize historical results
is to translate them into stylized facts which will be done next.
1. Historical data suggests that not too extensive leveraged purchase of
common stocks is log-optimal therefore leveraged purchase does not
imply risk-seeking behavior: risk averse investors may also rely on this
technique if their risk-aversion is not too high.
2. Depending on the weight of the risky asset, downside risk measured
as 5% VAR of the portfolios value may be a decreasing or increasing
function of time. Consequently, objective functions which incorporate
the VAR of the portfolios value or operate with VAR constrains result
in horizon-dependent portfolio weights.
3. Optimal portfolio weight appear to be time-dependent even if we con-
sider relatively long investment horizon of seven years.
4. Irrespective of the chosen average portfolio weight, adjusting the expo-
sure to the risky asset in the function of the predicted volatility ceteris
paribus result in higher expected growth rate than constant portfolio
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weights. Therefore, it is log-optimal to reduce exposure to stock market
in high volatility periods and increase it in low volatility periods.
Interpretation of our finding can be done in a theoretical and in a policy
level as well. From a theoretical point of view, the term stability was im-
ported from natural sciences by Samuelson (1947), where it often referred as
Le Chatelier principle. This principle roughly states that if a closed system
is subjected to an external shock, then the system shifts to counteract this
shock. Analogically, one may argue that the financial system is a closed sys-
tem, at least in a short run6, which reacts to the real shocks. The stability
of the financial system in this context involves analyzing investors response
to these shocks. The following, simple, illustrative numerical example will
hopefully prove useful in explaining why leveraged positions may pose an
issue concerning financial stability.
Figure 11: The reaction to a positive real shock depending on the weight ofthe risky asset.
The table above compares the reaction given to a positive real shock de-
pending on the weight of the risky asset. In both cases, we have assumed that
the desired portfolio weights are unaffected by the real shock. The chronology
6Of course, the analogy is by far not perfect as equity market influences the real econ-omy in many ways, such as via equity withdrawal, via expectations, etc.
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of the though experiment is as follows: First, there is a positive shock which
result in a price increase. Second, investors observe the price increase and
react in order to restore the desired portfolio weights. Third, this reaction
influences the prices again. The question whether this third influence coun-
teracts or amplifies the original shock. The example reveals that an investor,
who holds an unleveraged position, is likely to counteracts the original real
shock. Therefore, she acts in line with the Le Chatelier principle. On the
other hand, an investor, who holds a leveraged position, is likely to react in
a way which amplifies the original shock, which is not in line with the Le
Chatelier principle. That is why, from a theoretical perspective, leveragedpositions may be an issue for the stability of the financial system. Of course,
this mechanism has been described by many, in a slightly different context,
starting from Bogen and Krooss (1960) and is sometimes named as pyramid-
ing. Our contribution is to show that there is a tendency in actual financial
market to favour those, who hold leveraged positions, and these leveraged
positions may also be held by risk-averse, log-optimal investors. On a policy
level, some authors, for example Shiller (2005), or Hardouvelis and Theodos-
siou (2002) proposes the Fed to return to more active margin policy, suchas the one between 1934 and 1974. Based on our finding, log-optimal invest-
ment strategies may involve leveraged purchases, therefore these policies may
effectively influence log-optimal investors.
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5 Appendix
The following section describes how to set up the boundary structure in a way
that change in portfolios value can only take two discrete values. Since we will
only consider two separate moments of time, therefore the variable describing
the second period are indicated with V
. The change in the portfolios value
between these two periods can be described as follows.
V =V + T R+ DBS (28)
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where is the profit or loss, T Ris the transaction cost while DB Sdescribes
the cost of financing. Let us substitute each elements.
=j
qj Pj =j
wjV
Pj Pj =V
j
wj pj (29)
where j is the number of assets in the portfolio and pj is the price change
since the last boundary crossing event, measured in percentage terms. Since
there is no change in the quantity between two boundary crossing events,
therefore the profit is the weighted average price change.
T R=j
abs(qj qj) P
j tr=j
(qj qj) P
j trj (30)
wheretrj =tr ifq
j > qj andtrj = trotherwise. Substituting out quantities
yields:
T R=j
(w
j V
Pj
wj V
Pj) Pj trj (31)
Simplifying results
T R= V j
w
j trj Vj
wjPjPj
trj (32)
Finally, financing can be calculated directly from the data and need to be
expressed as:
DBS=V dbs(t); (33)
combining yields
V (1 j
w
j trj) =V(1 +j
wjpj j
wjPjPj
trj+ dbs(t)) (34)
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Therefore the change in wealth between two boundary crossings are equal to:
V
V =
(1 +
jwjpj
jwjPjPj
trj+ dbs(t)
1
jw
j trj=X(t) (35)
Further, we will refer to X(t) as signal. Boundary crossing counting process
counts how many times the signal crosses the boundaries.
X(t) =
GU for upper crossings
GL for lower crossings(36)
For ease of calculation, it is advisable to chose GL= 1/GU. The algorithm
for calculating the boundary crossing counting is than as follows.
1. Initiate the signal by setting X[0] = (0).
2. Calculate the value of the signal for each consecutive observations. This
calculation involves two steps. One hand hand, we have to account
for the change in prices using minimum prices for lower crossings andmaximum prices for upper crossings. Also, we have to take into account
financing, during which we have assumed that interest on deposit is paid
after the period while interest on lending or on shortselling is collected
in advance.
3. Calculate the weighted percentage change in the portfolio values us-
ing minimum prices for lower crossings and maximum prices for upper
crossings.
4. updateY U=Y U+ k andY L= Y L + k for upper and lower crossings
respectively.
where k N is the largest natural number to which the inequality holds.
Most of the timek = 1 yet occasionally upon large changes, it may be greater
as discussed in Farkas, (2013).