Portfolio Choice Under State Dependent Adjustment

7/22/2019 Portfolio Choice Under State Dependent Adjustment

1/39

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].


2/39

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


3/39

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

1


4/39

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

2


5/39

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.

3


6/39

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

4


7/39

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

5


8/39

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

6


9/39

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.

7


10/39

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)

8


11/39

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


12/39

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.

10


13/39

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)

11


14/39

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)

12


15/39

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

13


16/39

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.

14


17/39

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

15


18/39

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-

16


19/39

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

17


20/39

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

18


21/39

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-

19


22/39

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

20


23/39

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-

21


24/39

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.

22


25/39

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

23


26/39

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,

24


27/39

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.

25


28/39

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-

26


29/39

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.

27


30/39

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

28


31/39

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.

29


32/39

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.

References

[1] Isabelle Bajeux Besnainou and Roland Portait. The numeraire portfolio:

A new perspective on financial theory. The European Journal of Finance,3(4):291309, 1997.

[2] Pierluigi Balduzzi and Anthony W Lynch. Transaction costs and pre-

dictability: Some utility cost calculations. Journal of Financial Eco-

nomics, 52(1):4778, 1999.

30


33/39


34/39

[15] John Y Campbell and Samuel B Thompson. Predicting excess stock

returns out of sample: Can anything beat the historical average? Review

of Financial Studies, 21(4):15091531, 2008.

[16] Vijay Kumar Chopra and William T Ziemba. The effect of errors in

means, variances, and covariances on optimal portfolio choice. The Jour-

nal of Portfolio Management, 19(2):611, 1993.

[17] Morten Mosegaard Christensen. On the history of the growth optimal

portfolio. Preprint, University of Southern Denmark, 389, 2005.

[18] George M Constantinides. Capital market equilibrium with transaction

costs. Journal of Political Economy, 94:842862, 1986.

[19] Rama Cont. Empirical properties of asset returns: stylized facts and

statistical issues. Quantitative Finance, 1(2), 2001.

[20] Thomas M Cover. Universal portfolios. Mathematical Finance, 1(1):1

29, 1991.

[21] Bernard Dumas and Elisa Luciano. An exact solution to a dynamic port-folio choice problem under transactions costs. The Journal of Finance,

46(2):577595, 1991.

[22] Peter Farkas. Counting process generated by boundary-crossing events.

theory and statistical applications. Working paper (available at

http://http://econpapers.repec.org/paper/ceueconwp/), Central Euro-

pean University, 2013.

[23] Peter Fortune. Margin requirements, margin loans, and margin rates:Practice and principles. New England Economic Review, pages 1944,

2000.

[24] Peter Fortune. Margin lending and stock market volatility. New England

Economic Review, pages 326, 2001.

32


35/39

[25] Lawrence R Glosten, Ravi Jagannathan, and David E Runkle. On the

relation between the expected value and the volatility of the nominal

excess return on stocks. The Journal of Finance, 48(5):17791801, 1993.

[26] Christian Gourieroux and Joann Jasiak. Financial econometrics: Prob-

lems, models, and methods, volume 1. Princeton University Press Prince-

ton, NJ, 2001.

[27] Laszlo Gyorfi and Istvan Vajda. Growth optimal investment with trans-

action costs. In Algorithmic Learning Theory, pages 108122. Springer,

2008.

[28] Gikas A Hardouvelis and Panayiotis Theodossiou. The asymmetric re-

lation between initial margin requirements and stock market volatility

across bull and bear markets. Review of Financial Studies, 15(5):1525

1559, 2002.

[29] Mark Horvath and Andras Urban. Growth optimal portfolio selection

with short selling and leverage. Machine Learning for Financial Engi-

neering, eds. L. Gyorfi, G. Ottucsak, H. Walk, Imperial College Press,pages 151176, 2011.

[30] J. David Jobson and Bob Korkie. Estimation for markowitz efficient

portfolios. Journal of the American Statistical Association, 75(371):544

554, 1980.

[31] Samuel Karlin and Howard M Taylor. A second course in stochastic

processes, volume 2. Gulf Professional Publishing, 1981.

[32] Samuel Karlin and Howard M Taylor. An Introduction to Stochastic

Modeling. Academic Press, 1998.

[33] John L Kelly. A new interpretation of information rate. Information

Theory, IRE Transactions on, 2(3):185189, 1956.

[34] Henry Allen Latane. Criteria for choice among risky ventures. The

Journal of Political Economy, pages 144155, 1959.

33


36/39

[35] Hong Liu and Mark Loewenstein. Optimal portfolio selection with trans-

action costs and finite horizons. Review of Financial Studies, 15(3):805

835, 2002.

[36] Andrew W. Lo and A.C. MacKinlay. A Non-Random Walk Down Wall

Street. 1999.

[37] John Long. The numeraire portfolio. Journal of Financial Economics,

26(1):2969, 1990.

[38] Leonard C. MacLean, Edward O. Thorp, and William T. Ziemba. TheKelly capital growth investment criterion: Theory and practice, volume 3.

world scientific, 2010.

[39] Burton G. Malkiel and Eugene F Fama. Efficient capital markets: A re-

view of theory and empirical work*. The Journal of Finance, 25(2):383

417, 1970.

[40] Burton Gordon Malkiel. A random walk down Wall Street: including a

life-cycle guide to personal investing. WW Norton & Company, 1999.

[41] Benoit Mandelbrot and Richard L. Hudson.The Misbehavior of Markets:

A fractal view of financial turbulence. Basic Books, 2014.

[42] Harry Markowitz. Portfolio selection*.The Journal of Finance, 7(1):77

91, 1952.

[43] Lionel Martellini and Philippe Priaulet. Competing methods for option

hedging in the presence of transaction costs.The Journal of Derivatives,

9(3):2638, 2002.

[44] Donald L Mcleish. Highs and lows: Some properties of the extremes of

a diffusion and applications in finance. Canadian Journal of Statistics,

30(2):243267, 2002.

[45] Peter Medvegyev. Stochastic integration theory. Number 14. Oxford

University Press Oxford, 2007.

34


37/39

[46] Robert C. Merton. Lifetime portfolio selection under uncertainty: The

continuous-time case. Review of Economics and Statistics, 51(3):247

257, 1969.

[47] Robert C. Merton. Optimum consumption and portfolio rules in a

continuous-time model. Journal of Economic Theory, 3(4):373413,

1971.

[48] Robert C. Merton and P. A. Samuelson. Continuous-time finance. 1992.

[49] Ole Peters. Optimal leverage from non-ergodicity.Quantitative Finance

,11(11):15931602, 2011.

[50] Eckhard Platen. A benchmark approach to finance. Mathematical Fi-

nance, 16(1):131151, 2006.

[51] P. A. Samuelson. Foundation of economic analysis. 1947.

[52] P. A. Samuelson. Lifetime portfolio selection by dynamic stochastic

programming. Review of Economics and Statistics, (51):239246, 1969.

[53] Robert J. Shiller. Irrational exuberance. Random House LLC, 2005.

[54] James Tobin. The interest-elasticity of transactions demand for cash.

Cowles Foundation for Research in Economics at Yale University, 1956.

[55] Paul Wilmott. Derivatives. 1998.

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)

35


38/39

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)

36


39/39

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).

Portfolio Choice Under State Dependent Adjustment

Documents

Transcript of Portfolio Choice Under State Dependent Adjustment