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On Frequency Dependent Log-Optimal
Portfolio with Transaction Costs
Chung-Han Hsieh∗ †∗and Yi-Shan Wong††
Department of Quantitative Finance,
National Tsing Hua University, Hsinchu 300044, Taiwan R.O.C.
arXiv:2301.02754v1 [q-fin.PM] 7 Jan 2023
Abstract. The aim of this paper is to investigate the impact of rebalancing frequency and
transaction costs on the log-optimal portfolio, which is a portfolio that maximizes the expected
logarithmic growth rate of an investor’s wealth. We prove that the frequency-dependent log-
optimal portfolio problem with costs is equivalent to a concave program and provide a version
of the dominance theorem with costs to determine when an investor should invest all available
funds in a particular asset. Then, we show that transaction costs may cause a bankruptcy issue
for the frequency-dependent log-optimal portfolio. To address this issue, we approximate the
problem to obtain a quadratic concave program and derive necessary and sufficient optimality
conditions. Additionally, we prove a version of the two-fund theorem, which states that any
convex combination of two optimal weights from the optimality conditions is still optimal. We
test our proposed methods using both intraday and daily price data. Finally, we extend our
empirical studies to an online trading scenario by implementing a sliding window approach.
This approach enables us to solve a sequence of concave programs rather than a potentially
computational complex stochastic dynamic programming problem.
Keywords: Portfolio Optimization; Transaction Costs; Control and Optimization; Log-
Optimal Portfolio; Rebalancing Frequency.
Classcodes: G11, G14, C61
1 Introduction
The takeoff point of this paper is to study the celebrated log-optimal portfolio, which calls
for maximizing the Expected Logarithmic Growth (ELG) of an investor’s wealth. This ELG
maximization idea was introduced by Kelly jr (1956) and is also known as the Kelly Criterion.1
Some earlier works related to ELG maximization and its possible applications in gambling and
trading can be found in Breiman (1961); Thorp (1975); Cover (1984); Algoet and Cover (1988);
Rotando and Thorp (1992); Browne and Whitt (1996). Many subsequent papers contributed
to the ELG problem and its various ramifications. For example, Thorp (2006) studied ELG
problems in blackjack, sports betting, and the stock market. Maclean et al. (2010); MacLean
∗∗ †Correspondingauthor: Chung-Han Hsieh. Email: ch.hsieh@mx.nthu.edu.
† †Email:
yishan.wong13@gmail.com. This paper was supported in part by the Ministry of Science and Tech-
nology, R.O.C. Taiwan, under Grants: MOST110–2222–E–007–005– and MOST111–2221–E–007–124–.
1 See Poundstone (2010) for a storytelling book that brought the Kelly criterion to the attention of practical
investors.
1et al. (2011) summarized the good and bad properties of maximizing ELG. MacLean et al.
(2016) studied the risky short-run properties of the ELG criterion. Cover and Thomas (2006);
Luenberger (2013) are textbooks that contain an introductory chapter on the ELG problem.
Kim and Shin (2017) demonstrated the superior return of the log-optimal portfolio compared
to a traditional mean-variance portfolio in the Korean stock market. Lo et al. (2018) connected
ELG results to population genetics and discussed testable findings using experimental evolution.
More recently, Wu et al. (2020) analyzed Kelly betting in finite repeated games. MacLean
and Zhao (2022) studied the ELG problem in a regime-switching market framework. Wang
and Hsieh (2022) proposed a data-driven log-optimal portfolio via a sliding window approach.
However, among all of these papers, the effects of rebalancing frequency have not been extensively
considered in previous literature.
1.1 Rebalancing Frequency Considerations
There are some existing results regarding rebalancing frequency in a log-optimal portfolio, as
found in Kuhn and Luenberger (2010); Das et al. (2014); Das and Goyal (2015), and Hsieh
(2021). Specifically, Kuhn and Luenberger (2010) considered a portfolio optimization with re-
turns following a continuous geometric Brownian motion, but only focused on two extreme cases:
High-frequency trading and buy and hold. On the other hand, Das et al. (2014) and Das and
Goyal (2015) studied log-optimal portfolio with the constant weight K selected without regard
for the frequency. However, when the same weight K is used to find an optimal rebalancing pe-
riod, the resulting ELG levels are arguably suboptimal. Lastly, in our prior work Hsieh (2021),
we formulated a discrete-time frequency-dependent log-optimal portfolio problem and derived
various optimality conditions, but we did not consider the effects of the transaction costs.
1.2 Transaction Costs Considerations
Transaction costs are known to significantly impact trading performance in practice; see Cornue-
jols and Tütüncü (2006); Bogle (2017). These costs may include execution commissions, bid-ask
spreads, latency costs, and the price impact of trading. As a result, an optimal trading policy
may no longer be optimal when the transaction costs are not zero; see Cvitanic and Zapatero
(2004); Hsieh et al. (2018a). Consequently, many previous papers have focused on addressing
the effect of transaction costs on a portfolio optimization problem.
One of the earliest models for expected execution costs was developed by Bertsimas and Lo
(1998), where dynamic programming techniques are applied. Almgren and Chriss (2001) studied
an optimal tradeoff between expected cost and risk. In the meantime, Magill and Constantinides
(1976); Shreve and Soner (1994); Muthuraman and Kumar (2006) have all studied proportional
transaction costs in continuous-time portfolio optimization problems, but none of these studies
considered the effects of rebalancing frequency. Other transaction cost models can be found in the
literature; e.g., Lobo et al. (2007) considered fixed transaction costs in a portfolio optimization
problem. Woodside-Oriakhi et al. (2013) studied fixed, and V-shaped variable transaction costs
in a mean-variance model. A recent empirical study; see Ruf and Xie (2020), analyzed portfolios’
performance in the presence of proportional transaction costs under various discrete rebalancing
frequencies, constituent list size, and renewing frequency. While it compared trading performance
under various classical portfolios arising in stochastic portfolio theory, it does not take portfolio
optimization into account.
Following the frequency-dependent portfolio optimization framework described by Hsieh et al.
(2018b); Hsieh (2021), this paper extends the formulation to incorporate proportional transaction
costs. When there are nonzero proportional transaction costs, the frequency-dependent log-
2optimal portfolio problem would be intractable since there may be no (i.e., because bankruptcy
might occur) or only a trivial solution (i.e., zero investments) for such a problem. To address
this issue, we propose an approximation approach. We also derive optimality conditions for the
approximate problem. In addition, we prove a version of the Dominance Theorem involving
proportional transaction costs, which shows under what circumstances a log-optimal investor
would invest all available funds.
1.3 Contributions of the Paper
In Section 2, we formulate the frequency-dependent log-optimal portfolio problem involving pro-
portional transaction costs. We prove that the problem is equivalent to a concave program (see
Lemma 2.1) and show a version of the Dominance Lemma 2.2 with cost considerations. We also
state a sufficient condition to invest all available funds in a single asset. Then, in Section 3, we
investigate bankruptcy issues when there are nonzero costs, see Lemma 3.1. Then by approximat-
ing the ordinary optimization problem with a concave quadratic program, we provide necessary
and sufficient optimality conditions for an approximate log-optimal portfolio; see Lemma 3.2.
We further prove a version of the Two-Fund Theorem 4.1, demonstrating that a combination of
two optimal weights is still optimal. Finally, in Section 6, we extend our theory to online trading
via a sliding window approach.
2 Problem Formulation
This section provides some background information and the formulation for the frequency-
dependent log-optimal problem with costs. Let N > 1 be a terminal stage. For stage k =
0, 1, . . . , N − 1, consider an investor forming a portfolio consisting of m ≥ 2 assets and assume
that at least one asset is riskless with a rate of return rf ≥ 0. That is, if an asset is riskless,
its return is deterministic and is treated as a degenerate random variable with value X(k) = rf
for all k with probability one.2 Alternatively, if Asset i is a risky asset whose price at time k
is Si (k) > 0, then its per-period return is given by Xi (k) = Si (k+1)−SSi (k)
i (k)
. In the sequel, for
T
risky assets, we assume that the return vectors X(k) := [X1 (k) X2 (k) · · · Xm (k)] have a known
distribution and have components Xi (·) which can be arbitrarily correlated.3 We also assume
that these vectors are i.i.d. with components satisfying Xmin,i ≤ Xi (k) ≤ Xmax,i with known
bounds above and with Xmax,i being finite and Xmin,i > −1. The latter constraint on Xmin,i
means that the loss per time step is limited to less than 100% and the price of a stock cannot
drop to zero.
2.1 Linear Policy and Unit Simplex Constraint
Consistent with the literature, e.g., Barmish and Primbs (2015); Hsieh et al. (2020, 2018a,b);
Zhang (2001); Primbs (2007), we consider a linear policy with a weight vector K ∈ Rm . Let
V (k) be the investor’s account value at stage k and the weight for Asset i is given by 0 ≤ Ki ≤ 1
represents the fraction of the account allocated to the ith asset for i = 1, . . . , m. Said another
way, the policy for the ith asset is of a linear form ui (k) := Ki V (k). Note that the number of
shares invested on the ith asset is ui (k)/Si (k). Since Ki ≥ 0, the investor is going long. In view
2 In practice, the actual distribution of returns may not be available to the investor, but one can always estimate
it and work with the empirical surrogate.
3 Again, if the ith asset is riskless, then we put X (k) = r ≥ 0 with probability one. If an investor maintains
i f
cash in its portfolio, then this corresponds to the case rf = 0.
3of this, and given that there is at least one riskless asset available, we consider the unit simplex
constraint
( m
)
X
m
K ∈ K := K ∈ R : Ki ≥ 0 for all i = 1, . . . , m, Ki = 1 (1)
i=1
which is classical constraint in finance; e.g., see Cvitanic and Zapatero (2004); Cover and Thomas
(2006); Luenberger (2013); Cuchiero et al. (2019); Hsieh (2021). With K ∈ K, we guarantee
that 100% of the account is invested.
Remark 2.1. In the finance literature, it is known that long-only constraints like (1) can be
used to mitigate the overconcentration of weight. These constraints can also assist in containing
volatility and trading performance; see Jagannathan and Ma (2003).
2.2 Frequency Dependent Account Value Dynamics with Transaction
Costs
Letting n ≥ 1 be the number of steps Pbetween rebalancings, at time k = 0, the investor be-
m
gins with initial investments u(0) = i=1 ui (0) with ui (0) := Ki V (0) with Ki being the ith
component of the portfolio weight K satisfying constraint (1). It is worth mentioning that the
investment level ui (0) can be converted to the number of shares by dividing it by the price Si (0);
i.e., ui (0)/Si (0). The investor then waits n steps
Pm in the spirit of buy and hold. When k = n, the
investment control is updated to be u(n) = i=1 Ki V (n). Continuing in this manner, a waiting
period of n stages is enforced between each rebalance.
To incorporate the transaction costs into the frequency-dependent framework, let ci ∈ [0, cmax ]
be a percentage transaction costs imposed on Asset i where cmax ∈ (0, 1) is a predetermined
maximum transaction cost.4 That is, at stage k = 0, if one invests ui (0) at Asset i, then the
associated transaction costs in dollar is ui (0)ci ≥ 0. Then the dynamics of account value at
stage n ≥ 1 is characterized by the following stochastic recursive equation:5
Xm Xm
ui (0)
V (n) = V (0) + (Si (n) − Si (0)) − ui (0)ci . (2)
S (0)
i=1 i i=1
In the sequel, we may sometimes write VK (n) instead of V (n) to emphasize the dependence on
portfolio weight K.
Remark 2.2 (Transaction Costs). If there are no transaction costs; i.e., ci := 0 for all i =
1, 2, . . . , m, then the account value dynamics (2) reduces to the existing formulation in Rujeer-
apaiboon et al. (2018); Hsieh et al. (2018b); see also the frictionless market setting discussed
in Merton (1992).
4 Nowadays, while some online brokerage services offer fee-free trades for certain exchange-traded funds (ETFs)
in the United States, transaction costs are typically required. For example, trading on the Taiwan Stock Exchange
typically incurs a transaction cost of α · 0.1425% of the trade value for some α ∈ (0, 1). As a second example,
using professional broker services such as Interactive Brokers Pro., may incur a fee of $0.005 per share, with a
minimum fee of $1 dollar and a maximum fee of 1% of the trade value.
5 At stage ℓ ≥ 0 and rebalancing period n ≥ 1, the stochastic recursion of account value becomes
m m
X ui (nℓ) X
V (n(ℓ + 1)) = V (nℓ) + (Si (n(ℓ + 1)) − Si (nℓ)) − ui (nℓ)ci .
S (nℓ)
i=1 i i=1
42.3 Frequency-Dependent Optimization Problem
Following previous research in Hsieh et al. (2018b); Hsieh (2021), to study the performance
which is dependent on rebalancing frequency, for i = 1, 2, . . . , m, we work with the n-period
compound returns for each asset i, call it Xn,i , defined as
Si (n) − Si (0)
Xn,i = .
Si (0)
Qn−1
It is readily verified that Xn,i = k=0 (1 + Xi (k)) − 1 and −1 < Xmin,i ≤ Xn,i ≤ Xmax,i where
Xmax,i := (1 + Xmax,i )n − 1 and Xmin,i := (1 + Xmin,i )n − 1 > −1 for all n ≥ 1. In the sequel, we
work with the random vector Xn having ith component Xn,i .
Now for any rebalancing period n ≥ 1, we define the expected logarithmic growth (ELG)
1 VK (n)
gn (K) := E log .
n V (0)
Our goal is to solve the following frequency-dependent stochastic maximization problem:
sup {gn (K) : K ∈ K} (3)
Xm Xm
ui (0)
s.t. V (n) = V (0) + (Si (n) − Si (0)) − ui (0)ci
S (0)
i=1 i i=1
where K is the unit simplex defined previously in Equation (1). The following lemma shows
that maximizing the frequency-dependent ELG with nonzero costs is indeed solving a concave
program.
Lemma 2.1 (ELG Optimization as a Concave Program). Fix n ≥ 1 and ci ∈ (0, 1). The
frequency-dependent ELG optimization problem (3) is equivalent to
1 h i
max E log(1 + K T Xen ) : K ∈ K . (4)
n
where Xen is a vector with the ith component given by Xen,i := Xn,i − ci . Additionally, Problem (4)
is a concave program.
Proof. We begin by observing that the account value dynamics
Xm Xm
ui (0)
V (n) = V (0) + (Si (n) − Si (0)) − ui (0)ci
S (0)
i=1 i i=1
Xm X m
Si (n) − Si (0)
= V (0) + Ki V (0) − ui (0)ci
i=1
Si (0) i=1
m
X m
X
= V (0) + Ki V (0)Xn,i − ui (0)ci
i=1 i=1
m
X
= (1 + K T Xn )V (0) − ui (0)ci
i=1
= (1 + K T Xen )V (0) (5)
5where Xen is a vector with the ith component given by Xen,i := Xn,i − ci . Hence, it follows that
1 VK (n) 1 h i
gn (K) = E log = E log(1 + K T Xen ) .
n V (0) n
n h i o
Therefore, the original Problem (3) reduces to max gn (K) = n1 E log(1 + K T Xen ) : K ∈ K .
The supremum operator is replaced by the maximum since gn (K) is continuous in K over a
compact domain K. Hence, the Weierstrass extremum theorem; see Rudin (1976), guarantees
that the maximum is attained. To complete the proof, it remains to show that Problem (4) is a
concave program. This is accomplished by a standard convexity argument. Since 1 + K T Xen is
affine in K, taking the logarithm function yields a concave function. Moreover, taking the expec-
tation and multiplying a scaling factor 1/n preserve
h the concavity;
i see Boyd and Vandenberghe
1 T e
(2004). Therefore, the objective function E log(1 + K Xn ) is a concave function in K. On
n
the other hand, K is a unit simplex which is a convex compact set. Therefore, the maximization
considered in Problem (4) is a concave function over a convex compact set, hence, is a concave
program.
Henceforth, we denote gn∗ as the optimal expected logarithmic growth associated with the
given rebalancing period of length n. A vector K ∗ ∈ K ⊂ Rm satisfying gn (K ∗ ) = gn∗ is called
a log-optimal weight. The portfolio that uses the log-optimal fraction vector is called frequency-
dependent log-optimal portfolio.
2.4 Dominance Lemma with Costs
In this section, a version of the dominance lemma with costs is stated below.
Lemma 2.2 (Dominance). Given a collection of m ≥ 2 assets, if Asset j satisfying
" #
1 + Xen,i
E ≤ 1,
1 + Xen,j
6 j with i, j ∈ {1, 2, . . . , m}, then, for all n ≥ 1, gn (K) is maximized by K ∗ = ej
for all i =
where ej is the unit vector in the jth coordinate direction.
Proof. To prove K ∗ = ej , it suffices to show that gn (K) ≤ gn (ej ) for K ∈ K. For notational
convenience, we work with the random vector R e n := Xen + 1 where 1 := [1 1 · · · 1]T ∈ Rm .
e n ]. Hence, by applying Jensen’s
Since K 1 = 1 for K ∈ K, it follows that gn (K) = n1 E[log K T R
T
6inequality to the concave logarithmic function, we obtain
" #
1 KT R en
gn (K) − gn (ej ) = E log
n Re n,j
" #
1 KT R en
≤ log E
n Re n,j
m
" #!
1 X Re n,i
= log Ki E
n e n,j
R
i=1
m
" #!
1 X 1 + Xen,i
= log Ki E
n i=1 1 + Xen,j
m
!
1 X
≤ log Ki · 1
n i=1
1
≤ log 1 = 0
n
h e i
1+X
where the second last inequality holds since E 1+Xen,i ≤ 1 and the last inequality holds
Pm n,j
since i=1 Ki = 1. Therefore, gn (K) ≤ gn (ej ).
Remark 2.3. Lemma 2.2 indicates that, under certain conditions, an optimal log-optimal in-
vestor must invest all available funds in a specific asset when transaction costs are present. This
result can be viewed as an extension of the Dominant Asset Theorem in Hsieh (2021) to include
transaction costs. To see this, consider the case where there are no costs; i.e., ci = 0 for all i,
then Xen,i = Xn,i . This implies that the ratio
" # "n−1 #
1 + Xen,i Y 1 + Xi (k)
E =E
1 + Xen,j k=0
1 + Xj (k)
n
1 + Xi (0)
= E ,
1 + Xj (0)
h e i
1+X
where the last equality holds since Xi (k) are i.i.d. in k. Thus, the condition E 1+Xen,i ≤ 1
h i n,j
1+Xi (0)
reduces to a much simpler condition E 1+Xj (0) ≤ 1, which is consistent with the Dominant
Asset Theorem proved in Hsieh (2021).
3 An Approximate Log-Optimal Portfolio Problem with
Costs
When the transaction costs are present, the corresponding fee-adjusted return is given by Xen,i =
Xn,i − ci . The next lemma provides a sufficient condition for ensuring that the trades survive up
to stage n.
Lemma 3.1 (Probability of Having Survival Trades under Transaction Costs). Fix n ≥ 1. If
1/n
Xmin,i > ci − 1 for all i = 1, 2, . . . , m, then the probability P (V (n) > 0) = 1.
7Proof. Let n ≥ 1 be given. Observe that
P (V (n) > 0) = P ((1 + K T Xen )V (0) > 0)
= P (1 + K T Xen > 0)
m
!
X
=P Ki (1 + Xen,i ) > 0
i=1
m n−1
! !
X Y
=P Ki 1+ (1 + Xi (k)) − 1 − ci >0
i=1 k=0
m n−1
! !
X Y
=P Ki (1 + Xi (k)) − ci >0 . (6)
i=1 k=0
Pm
where nthe third equality holds by invoking the fact
o n that i=1 Ki = 1.oNow note that the
Pm Qn−1 Qn−1
event i=1 K i k=0 (1 + X i (k)) − c i > 0 ⊇ k=0 (1 + Xi (k)) > ci . With the aids of
monotonicity of probability measure, Equality (6) becomes
n−1
!
Y
P (V (n) > 0) ≥ P (1 + Xi (k)) > ci .
k=0
Qn−1 1/n
Since k=0 (1 + Xi (k)) ≥ (1 + Xmin,i )n for all i = 1, 2, . . . , m and Xmin,i > ci − 1 for all
Qn−1
i = 1, 2, . . . , m, it follows that k=0 (1 + Xi (k)) > ci for all i. Therefore, we have P (V (n) > 0) =
1.
Remark 3.1. (i) To assure a survival trade, Lemma 3.1 indicates that the worst returns must
be large enough. Specifically, for n = 1, it requires Xmin,i > ci − 1 for all i. On the other hand,
if n → ∞, which corresponds to buy and hold, then we must have Xmin,i > 0 for all i. (ii) On
the Lemma 3.1, it is readily verified that if mini ci > 0 and P (V (n) > 0) = 1
the converse of P
for n ≥ 1, then m n
i=1 Ki ((1 + µi ) − ci ) ≥ 0 where µi := E[Xi (k)]; see Lemma A.1. This reveals
a gap in obtaining a necessary condition for survival trades in Lemma 3.1.
Lemma 3.1 implies that for a fixed c∗ := maxi ci ∈ (0, 1), there exists Xmin,i > −1 such
that V (n) ≤ 0 with positive probability. Said another way, the investor’s account may experience
a “survival issue” when the rebalancing frequency and costs are taken into consideration. In
addition, this survival issue may cause the gn (K) to become ill-defined. To address this issue,
we use a Taylor-based quadratic approximation of gn (K) around K = 0; see Casella and Berger
(2001) and write
h i 1 h i
1 T e T e e T
gn (K) ≈ K E Xn − K E Xn Xn K := gbn (K). (7)
n 2
It is well-known that such a quadratic approximation is accurate for small returns; see Pulley
(1983).6 Hence, in the sequel, we consider an approximate frequency-dependent log-optimal
portfolio problem with costs as follows:
max {b
gn (K) : K ∈ K} . (8)
6 Without K T Xn )
loss of generality, set ci := 0 for all i. Then the Taylor expansion of E log(1 + =
T
P∞ d+1 (K Xn ) d
E d=1 (−1) d
converges for all K ∈ K if |K T Xn | ≤ 1 with probability one.
8Remark 3.2. It is readily verified that the approximate problem (8) described above is a concave
quadratic program, which enables us to solve it in an efficient manner; e.g., see Diamond and
Boyd (2016).
3.1 Optimality Conditions
In this section, we investigate the optimality conditions for the approximate frequency-dependent
log-optimal problem (8).
Lemma 3.2 (Necessity and Sufficiency). Fix n ≥ 1. Given a percentage costs ci ∈ (0, 1) for i =
b ∗ ∈ K is optimal to the approximate frequency-dependent log-
1, 2, . . . , m, the portfolio weight K
optimal problem (8), if and only if
h i Xm h i h i h i
E Xen,i − b j∗ E Xen,i Xen,j = K
K b ∗T E Xen XenT K
b ∗T E Xen − K b ∗ , if K
b i∗ > 0 (9)
j=1
h i Xm h i h i h i
E Xen,i − b ∗ E Xen,i Xen,j ≤ K
K j
b ∗T E Xen XeT K
b ∗T E Xen − K
n
b ∗ , if K
b∗ = 0
i (10)
j=1
Proof. Let n ≥ 1 and ci ∈ (0, 1) for all i be given. We begin by considering an equivalent
constrained stochastic minimization problem described as follows:
h i 1 h i
min −K T E Xen + K T E Xen XenT K
K 2
s.t. K T 1 − 1 = 0;
− K T ei ≤ 0, i = 1, 2, . . . , m
where ei ∈ Rm is unit vector having one at the ith component and zeros on the other components.
Consider the Lagrangian
h i 1 h i
L(K, λ, µ) := −K T E Xen + K T E Xen XenT K + λ(K T 1 − 1) − µT K.
2
By the Karush-Kuhn-Tucker (KKT) conditions; e.g., see (Boyd and Vandenberghe, 2004, Chap-
b ∗ is a local maximum then there is a scalar λ ∈ R1 and a vector µ ∈ Rm with
ter 5), if K
component µj ≥ 0 such that, for i = 1, 2, . . . , m,
h i Xm h i
− E Xen,i + b ∗ E Xen,i Xen,j + λ − µi = 0
K j (11)
j=1
b ∗T 1 − 1 = 0
K (12)
b ∗ = 0.
µi K (13)
i
From Equation (11), we obtain, for i = 1, . . . , m,
h i Xm h i
µi = −E Xen,i + b j∗ E Xen,i Xen,j + λ.
K (14)
j=1
b ∗ = 0 for all i, we take weighted sum of Equation (14); i.e.,
Since µi K i
m
X h i h i
b ∗ = −K
µi K b ∗T E Xen XeT K
b ∗T E Xen + K b ∗ + λ = 0. (15)
i n
i=1
9h i h i
This implies that λ = K b ∗T E Xen XeT K
b ∗T E Xen − K b ∗ . Substituting this into Equation (14), we
n
have, for i = 1, 2, . . . , m,
h i Xm h i h i h i
µi = −E Xen,i + b ∗ E Xen,i Xen,j + K
K j
b ∗T E Xen XeT K
b ∗T E Xen − K
n
b ∗. (16)
j=1
b ∗ = 0, it follows that for i = 1, 2, . . . , m, if K
From Equation (16) and the fact that µi K b ∗ > 0,
i i
then µi = 0 and
h i Xm h i h i h i
E Xen,i − b ∗ E Xen,i Xen,j = K
K j
b ∗T E Xen XeT K
b ∗T E Xen − K
n
b ∗.
j=1
b ∗ = 0, then µi ≥ 0 and
On the other hand, if K i
h i Xm h i h i h i
e
E Xn,i − K b ∗T E Xen XenT K
b ∗T E Xen − K
b j∗ E Xen,i Xen,j ≤ K b ∗.
j=1
To prove sufficiency, let K b ∗ ∈ K and satisfies the conditions (9) and (10). Then it follows
that there exists λ ∈ R and µj > 0 such that the KKT conditions (11) to (13) hold at K b ∗ . Since
the constrained minimization problem is a convex optimization problem, it follows that the KKT
conditions are also sufficient for optimality. Hence, Kb ∗ is optimal; see Boyd et al. (2017).
Remark 3.3. Let K b ∗ be the optimum obtained by solving the approximate frequency-dependent
log-optimal portfolio problem (8) and K ∗ be the true log-optimum. Using Jensen’s inequality,
we have
" #
∗T e
∗ b ) = E log
∗ 1 + K X n
0 ≤ g(K ) − g(K
1+K b ∗T Xen
" T
#
1 + K ∗ Xen
≤ log E .
1+K b ∗T Xen
The right-hand side is approximately zero when K ∗ ≈ K b ∗ . As we will see later in this paper,
this is typically the case. More interestingly, Lemma 3.2 serves to compliment Lemma 2.2 by
characterizing the log-optimal weights; see Example 3.1 below.
Example 3.1 (Two-Asset Toy Example). To demonstrate the application of Lemmas 2.2 and 3.2,
we first consider a high-frequency investor who rebalances her portfolio at every period; i.e.,
n := 1. Specifically, consider a two-asset portfolio including a risk-free cash asset with zero
interest rate; i.e. X1 (k) := rf = 0 with probability
one and a risky asset with a binomial return
X2 (k) ∈ {− 21 , 21 } with probability P X2 (k) = 21 := p ∈ 21 + c2 , 1 . The transaction costs are
c1 = 0 for cash and c2 < 1/2 for the risky asset. If K b ∗ > 0, by Lemma 3.2, we have
2
b ∗ 1 b ∗ b ∗ 1 1 2
(1 − K2 ) p − − c2 − K2 (1 − K2 ) − 2c2 p − + c2 = 0.
2 4 2
b 2∗ =
This implies that K −(4c2 −4p+2)
Incorporating with Lemma 2.2, we conclude
.
4c22 +4c2 −8c2 p+1
i
Kb∗ 1 4c22 +8c2 +3
2 if p ∈ + c 2 ,
K2∗ := h22 4+8c
2 (17)
1 4c2 +8c2 +3
if p ∈ 4+8c2 , 1
10
and K1∗ = 1 − K2∗ . Note that if c2 = 0, then K2∗ = 2(2p − 1) for p ∈ 21 , 34 or K2∗ = 1 for
p ∈ 34 , 1 , which reduces to the classical ELG result in gambling; see Kelly jr (1956); Hsieh et al.
(2018a).
To see the effect of rebalancing period n > 1, we consider a second
example with n = 2; i.e.,
one rebalances the portfolio for every two periods. For c2 ∈ 0, 41 , applying Lemmas 2.2 and 3.2
yield
K b ∗ , if p ∈ 1 + c2 , − 4c2 −9 − 1 C
2
K2∗ := h2 8c2 +6 i 4
(18)
1, 4c2 −9
if p ∈ − 8c2 +6 − 14 C, 1 ,
√
b∗ = 16p2 +16p−16c2 −12 −256c42 +384c32 +640c22 −504c2 +81
where K 2 2 2 2
16c2 +24c2 +32p −16p−32p c2 −32pc2 +9
, and C := 4c2 +3 and
∗ ∗ ∗ 16p2 +16p−12 1 3
∗
3
K1 = 1 − K2 . If c2 = 0, we have K2 = 32p2 −16p+9 for p ∈ 2 , 4 and K2 = 1 for p ∈ 4 , 1 .
4 Feasible Region and Efficient Frontier
Similar to how the performance of a portfolio can be characterized by its expected return and
variance in the celebrated Markowitz framework, the performance of log-optimal portfolios can
be characterized by the expected logarithmic growth and variance of the logarithmic growth and
plotted on a two-dimensional diagram; see Luenberger (2013). The region mapped out by all
possible portfolios defines the feasible region. That is, for any fixed n ≥ 1, we consider
VK (n) VK (n)
K 7→ E log , var log ⊂ R2 .
V (0) V (0)
As demonstrated later in Example 4.1, the feasible region is convex to the left. This means that
if we take any two points within the region, the straight line connecting them does not cross
the left boundary of the feasible region. A similar idea about analyzing the efficient frontier
analytically can be found in Merton (1972).
4.1 A Version of The Two-Fund Theorem
In the approximate log-optimal portfolio problem, as defined in (8), the upper left-hand portion
at the boundary of the feasible region is referred to as the approximate efficient frontier. This
frontier is considered efficient in terms of expected logarithmic growth rate and its variance; see
also (Luenberger, 2013, Chapter 14). Then, with the aid of Lemma 3.2, we can obtain a version
of the two-fund theorem, which states that any convex combination of two optimal weights from
the optimality conditions is still optimal.
Theorem 4.1 (A Version of Two-Fund Theorem). Let K ′ , K ′′ ∈ K be two weights satisfying the
optimality conditions stated in Lemma 3.2. Define a convex combination K α := αK ′ + (1 − α)K ′′
with α ∈ [0, 1]. Then K α also satisfies the optimality conditions.
Proof. Take K ′ and K ′′ be two weights satisfying Equations (11) to (13), for all α ∈ [0, 1], we
must show that the convex combination of the two weights K ′ and K ′′ , K α := αK ′ + (1 − α)K ′′,
with the jth component K α,j , also satisfies the same optimality equations. In particular, we
begin by proving that K α satisfies Equation (12). Indeed, we observe that
(αK ′ + (1 − α)K ′′ )T 1 − 1 = αK ′T 1 + (1 − α)K ′′T 1 − 1 (19)
where 1 := [1 1 · · · 1]T ∈ Rm . Since K ′ , K ′′ satisfy Equation (12), it follows that K ′T 1 = 1
and K ′′T 1 = 1. Therefore, Equation (19) becomes (αK ′ + (1 − α)K ′′ )T 1 − 1 = α + (1 − α) = 1
11which proves that the convex combination K α satisfies Equation (12). To see it also satisfies
b ∗ = 0 for i = 1, . . . , m, we observe that
µi K i
µi (αKi′ + (1 − α)Ki′′ ) = αµi Ki′ + (1 − α)µi Ki′′
= α · 0 + (1 − α) · 0 = 0.
To complete the h proof,
i Pwe show that
h K α satisfies
i Equation (11). It suffices to show that for i =
m
1, . . . , m, −E Xen,i + j=1 K α,j E Xen,i Xen,j + λ = µi . Note that the left-hand side using K α
yields
h i Xm h i
− (α + (1 − α))E Xen,i + (αKj′ + (1 − α)Kj′′ )E Xen,i Xen,j + (α + (1 − α))λ
j=1
h i Xm h i h i Xm h i
= α −E Xen,i + Kj′ E Xen,i Xen,j + λ + (1 − α) −E Xen,i + Kj′′ E Xen,i Xen,j + λ
j=1 j=1
= αµi + (1 − α)µi = µi
which completes the proof.
Example 4.1 (Five-Asset Portfolio with Intraday Minute-by-Minute Data). This example il-
lustrates the feasible region, efficient frontier, and Two-Fund Theorem 4.1 using a five-asset
portfolio consisting of a bank account, Vanguard Total Stock Market Index Fund ETF (Ticker:
VTI), Vanguard Total Bond Market Index Fund ETF (Ticker: BND), Vanguard Emerging Mar-
kets Stock Index Fund ETF (Ticker: VWO), and Bitcoin to the USD exchange rate (Ticker:
XBTUSD). The portfolio is well-diversified, covering the large US-Euro stock market, the global
bond market, and cryptocurrency. Here, transaction costs ci = 0.001% are imposed on the
ETFs (i.e., i ∈ {VTI, BND, VWO}) and costs cXBTUSD = 0.1% on the XBTUSD.7 Besides, in-
vestors receive interest at a (per-minute) rate rf = 0.0001% if they keep their funds in the bank
account. The data used in this example spans from 09 : 30 : 00 AM to 15 : 59 : 00 PM on
December 3, 2021, where the associated price trajectories for the four risky assets are shown
in Figure 1.8 To derive the approximate log-optimal portfolio and examine its trading perfor-
mance, we split the entire data set into two parts: The first portion from 09 : 30 : 00 AM
to 12 : 29 : 00 PM is for the in-sample optimization, and the second portion 12 : 30 : 00 PM
to 15 : 59 : 00 PM is for the out-of-sample testing.9 n o
Fix n ≥ 1. We define the approximate feasible region H := gn (K), var log VVK(0)
b (n)
:K∈K .
Figures 2 and 3 show the points in H and the approximate efficient frontier for different rebal-
ancing periods n = 1 and n = 5, respectively. As predicted by Theorem 4.1, any convex
combination of two optimal weights K ′ and K ′′ satisfying optimality conditions 3.2, denoted
as K α = αK ′ + (1 − α)K ′′ with α ∈ [0, 1], satisfies the optimality conditions. Interestingly, it
also lies on the approximate efficient frontier due to the small scale of the minute-by-minute price
data10 ; see Figures 2 and 3 for an example with α = 0.5. Similar findings also hold for other
rebalancing periods n > 5.
7 According to the platform Binance binance.com/en, regular users are charged a transaction cost of 0.1% for
Bitcoin trades.
8 The price data for the four underlying risky assets (VTI, BND, VWO, XBTUSD) are retrieved from the
Bloomberg terminal (accessed on November 17, 2022).
9 This will be demonstrated later in Example 5.2 in the next section.
10 This phenomenon disappears when using daily data; see also Remark 4.1 for more information.
12VTI BND
232
83.8
230
83.6
228
83.4
226
83.2
10:00 12:00 14:00 16:00 10:00 12:00 14:00 16:00
Dec 03, 2021 Dec 03, 2021
VWO 10 4 XBTUSD
48 5.6
47.8 5.5
47.6 5.4
5.3
47.4
5.2
10:00 12:00 14:00 16:00 10:00 12:00 14:00 16:00
Dec 03, 2021 Dec 03, 2021
Figure 1: Intraday Minute-by-Minute Prices for VTI, BND, VWO, and XBTUSD.
Remark 4.1. While not pursued further in this paper, the optimality conditions derived in
Lemma 3.2 only consider the approximate logarithmic growth function gbn (K) without taking
into account the log-variance var(log Vn (K)/V (0)). As a result, to ensure that any convex com-
bination of two points on the approximate efficient frontier is still on the frontier, the log-variance
must be included in the optimization problem (8). This topic presents a promising research di-
rection.
5 Illustrative Examples
This section presents empirical examples to demonstrate the validity of our theory. In the first
two examples, we use the same intraday data set as Example 4.1 to compare the log-optimal
and approximate log-optimal results. We evaluate the impact of different rebalancing periods
and levels of costs on trading performance. The third example examines the capability of our
theory to handle the mid-sized portfolio case by considering a portfolio of thirty-two assets (with
a Bank account, Dow-30 stocks, and cryptocurrency) using daily historical price data.
Example 5.1 (Five-Asset Portfolio Revisited). This example demonstrates that the approxi-
mate optimal weights K b ∗ from Lemma 3.2 is sufficiently close to the optimal weights K ∗ . To
∗
demonstrate this, we choose the weights K on
the efficient
frontier that satisfy the logarith-
VK ∗ (n) VK
c∗ (n)
mic variance condition: var log V (0) ≡ var log V (0) . Figures 4 and 5 show the portfolio
weights of the two trading strategies: the approximate log-optimal weights K b ∗ , and the true
13Figure 2: An illustration of Feasible Set, Efficient Frontier, and Two-Fund Theorem (K α
with α = 0.5) using Rebalancing Period n = 1 (Minute).
log-optimal weights K ∗ with different rebalancing periods n = 1 and n = 5. The results show
b ∗ ≈ K ∗ , for all i = 1, 2, . . . , 5.
that the weights of the two strategies are nearly identical, i.e., K i i
This suggests that the approximate optimal weights K b ∗ are a good approximation of the true
optimal weights K ∗ . While not showing here, it is also worth mentioning that if the transaction
costs are sufficiently large, then both of the optima K ∗ and K b ∗ will tend to fully invest in the
∗ b ∗
bank account, meaning that KBank account ≈ KBank account ≈ 1.
Example 5.2 (Trading Performance with Different Rebalancing Periods and Costs). This exam-
ple illustrates the in-sample and out-of-sample trading performances using the solutions obtained
in previous Example 5.1. Specifically, let V (N ) be the account value at the terminal stage N .
The portfolio realized return in period k is Rp (k) := V (k+1)−VV (k)
(k)
. With the aid of this real-
ized return, we consider the following metrics to study the trading performance: The realized
cumulative rate of return V (NV)−V
(0)
(0)
, realized log-return log VV(N ) p
(0) , volatility σ := std(R (k)),
maximum percentage drawdown d∗ := max0≤k≤N Vmax (k)−V (k)
Vmax (k) with Vmax (k) := max0≤i≤k V (i),
√
and the N -period Sharpe ratio N · SR with SR being the per-period realized Sharpe ratio.11
Starting with initial account V (0) = $1, Figures 6 and 7 reveal the in-sample and out-of-
sample values of the trading account using the three trading strategies: The log-optimal portfolio
11 Given a sequence of the realized portfolio per-period returns {Rp (k) : k = 0, 1, . . . , N − 1}, the per-period
p
R −rf p 1 PN−1 p
Sharpe ratio is SR := where R := N k=0 R (k) is the sample mean return, rf is the per-period
qs
risk-free rate, and s := 1 PN−1
(R p (k) − Rp )2 is the sample standard deviation of portfolio returns. A
N−1 k=0
detailed discussion of this topic can be found in Lo (2002).
14Figure 3: An Illustration of Feasible Set, Efficient Frontier, and Two-Fund Theorem (K α
with α = 0.5) using Rebalancing Period n = 5 (Minutes).
with weight K ∗ , the approximate log-optimum K b ∗ , and buy-and-hold with equal weight K =
1/m, for the same five-asset portfolio considered in Example 4.1. Note that there are nonzero
transaction costs of 0.001% for trading ETFs and a cost of 0.1% for trading cryptocurrency.
From the figures, we see that the account value trajectory obtained using K b ∗ is similar to that
∗
obtained using K . Moreover, both of the portfolios outperform the equally-weighted buy-and-
hold strategy.
To see clearly the effect of transaction costs on trading performance, we consider an additional
scenario with zero costs for trading both ETFs and cryptocurrency; see Figures 8 and 9 for the in-
sample and out-of-sample account value trajectories under rebalancing period n = 1 and n = 5,
with zero costs. Both figures demonstrate that the account values are improved when there are
no costs.
Tables 1 and 2 provide an overview of the out-of-sample trading performance metrics of the
three trading strategies for different rebalancing periods n = 1 and n = 5, respectively. For
the case of n = 1, i.e., the portfolio is rebalanced every minute, we find that the zero costs
lead to better performance of the log-optimal portfolio in terms of the Sharpe ratio. When
nonzero transaction costs are imposed, the Sharpe ratios for K ∗ and K b ∗ become negative. This
suggests that transaction costs have a negative impact on trading performance especially when
rebalancing occurs frequently. On the other hand, for the case of n = 5, where the portfolio is
rebalanced every five minutes, the Sharpe ratios are positive and generally higher than those for
n = 1. This indicates that a longer rebalancing period incurs fewer costs and may lead to better
trading performance.
15Weights for each trading strategy (n=1)
0.8
0.7
0.6
0.5
Weight
0.4
0.3
0.2
0.1
0
D t I O D
BN un VT VW US
cco T
kA XB
n
Ba
Asset
b ∗ with Rebalancing Period n = 1 (Minute).
Figure 4: Portfolio Weights K ∗ versus K
Weights for each trading strategy (n=5)
1
0.9
0.8
0.7
0.6
Weight
0.5
0.4
0.3
0.2
0.1
0
D t I O D
BN un VT VW US
co T
Ac XB
nk
Ba
Asset
b ∗ with Rebalancing Period n = 5 (Minutes).
Figure 5: Portfolio Weights K ∗ versus K
16In-sample trading performance (n=1)
1
Account value
0.995
0.99
09:30 10:00 10:30 11:00 11:30 12:00 12:30
Time Dec 03, 2021
Out-of-sample trading performance (n=1)
1.002
Account value
1
0.998
0.996
0.994
0.992
12:30 13:00 13:30 14:00 14:30 15:00 15:30 16:00
Time Dec 03, 2021
b ∗ , and
Figure 6: Account Value Trajectories under Three Trading Strategies (Optimal K ∗ , K
Equally-Weighted K = 1/m) with Rebalancing Period n = 1 (Minute) and Nonzero Costs.
Table 1: Out-of-Sample Trading Performance Metrics with Different Transaction Costs with
Rebalancing Period n = 1 (Minute)
Costs of 0% for ETFs and cryptocurrency K∗ Kb∗ Buy and hold
Cumulative rate of return V (NV)−V
(0)
(0)
(%) 0.13 0.13 −0.38
Realized log-growth log VV(N )
(0) (%) 0.13 0.13 −0.38
Volatility σ (%) 0.01 0.01 0.05
Maximum percentage drawdown
√ d∗ (%) 0.16 0.16 1.20
Sharpe ratio N SR 0.74 0.75 −0.58
Costs of 0.001% for ETFs and 0.1% for cryptocurrency K∗ b∗
K Buy and hold
Cumulative rate of return V (NV)−V
(0)
(0)
(%) −0.05 −0.06 −0.42
Realized log-growth log VV(N )
(0) (%) −0.05 −0.06 −0.43
Volatility σ (%) 0.01 0.01 0.05
Maximum percentage drawdown
√ d∗ (%) 0.18 0.19 1.20
Sharpe ratio N SR −0.58 −0.65 −0.64
Example 5.3 (Mid-Sized Portfolio: Thirty-Two Assets with Daily Price Data). Our theory is
readily applied to a mid-sized (or large-sized) portfolio. As an example, we consider a portfolio
consisting of 32 assets involving a bank account, Dow 30 Stocks,12 and the Bitcoin-to-USD
exchange rate (Ticker: BTC-USD) over a one-year horizon from November 20, 2021 to November
12 Dow 30 Stocks consist of the thirty stocks that make up the Dow Jones Industrial Average.
17In-sample trading performance (n=5)
Account value 1
0.995
0.99
09:30 10:00 10:30 11:00 11:30 12:00 12:30
Time Dec 03, 2021
Out-of-sample trading performance (n=5)
1.002
Account value
1
0.998
0.996
0.994
0.992
12:30 13:00 13:30 14:00 14:30 15:00 15:30 16:00
Time Dec 03, 2021
b ∗ , and
Figure 7: Account Value Trajectories under Three Trading Strategies (Optimal K ∗ , K
Equally-Weighted K = 1/m) with Rebalancing Period n = 5 (Minutes) and Nonzero Costs.
Table 2: Out-of-Sample Trading Performance Metrics with Different Transaction Costs with
Rebalancing Period n = 5 (Minutes)
Costs of 0% for ETFs and cryptocurrency K∗ Kb∗ Buy and hold
Cumulative rate of return V (NV)−V
(0)
(0)
(%) 0.14 0.14 −0.38
Realized log-growth log VV(N )
(0) (%) 0.14 0.14 −0.38
Volatility σ (%) 0.02 0.02 0.05
Maximum percentage drawdown
√ d∗ (%) 0.16 0.16 1.20
Sharpe ratio N SR 0.88 0.88 −0.58
Costs of 0.001% for ETFs and 0.1% for cryptocurrency K∗ Kb∗ Buy and hold
Cumulative rate of return V (NV)−V
(0)
(0)
(%) 0.10 0.10 −0.42
Realized log-growth log VV(N )
(0) (%) 0.10 0.10 −0.43
Volatility σ (%) 0.02 0.02 0.05
Maximum percentage drawdown
√ d∗ (%) 0.17 0.17 1.20
Sharpe ratio N SR 0.61 0.61 −0.64
20, 2022.13
The one-year data is divided into two parts: The first 90 days are used for in-sample optimiza-
13 The data considered in this example is retrieved from Yahoo Finance. It is worth noting that the time
period considered for this example is significant because the third-largest cryptocurrency exchange, FTX, declared
bankruptcy on November 11, 2022, which had a significant impact on cryptocurrency markets.
18In-sample trading performance (n=1)
Account value 1
0.995
0.99
09:30 10:00 10:30 11:00 11:30 12:00 12:30
Time Dec 03, 2021
Out-of-sample trading performance (n=1)
1.002
Account value
1
0.998
0.996
0.994
0.992
12:30 13:00 13:30 14:00 14:30 15:00 15:30 16:00
Time Dec 03, 2021
b ∗ , and
Figure 8: Account Value Trajectories under Three Trading Strategies (Optimal K ∗ , K
Equally-Weighted K = 1/m) with Rebalancing Period n = 1 (Minute) and Zero Costs.
tion and the remainder is used for out-of-sample testing. Here, we consider different scenarios
for the transaction costs: zero costs, 0.01%, 0.1%, and 0.5% for trading stocks, and zero costs
and 0.1% fees for trading cryptocurrency. If investors retain their capital in the bank account,
they earn daily interest with a rate rf := 1%/365.
Fix n = 1, i.e., the portfolio is rebalanced on a daily basis. When costs for trading stocks
are 0%, 0.01% and 0.1%, we find that KCV ∗ b∗ 14
X ≈ KCV X ≈ 1. However, when the proportional
∗
cost is 0.5%, the approximate optimum becomes KBank account ≈ K b∗
Bank account ≈ 1, indicating
that it is optimal to hold all capital in the bank account. Table 3 summarizes the performance of
the three trading strategies under different levels of costs for trading stocks and cryptocurrency.
As expected, higher costs result in a significant decrease in investor revenue. The corresponding
account value trajectories are plotted in Figure 10.
Subsequently, we examine the effects of different rebalancing periods by setting the rebal-
ancing period to every five days, i.e., n = 5. In this case, we find that KCV ∗ b∗
X ≈ KCV X ≈ 1
for all four different levels of costs (0%, 0.01%, 0.1%, and 0.5%) for trading stocks. This is in
∗
contrast to the case with n = 1, where the optimal weights dictated Kbank account ≈ 1 when the
proportional cost was 0.5%. Figure 11 shows that the associated trading performance using K ∗
and K b ∗ are similar and outperforms the buy-and-hold strategy with equal weights 1/m over the
given time period. Table 4 provides a summary of the performance metrics under four different
levels of costs with rebalancing periods n = 5.
14 Note that, in this example, Chevron Corporation (Ticker: CVX) is the dominant asset since the estimated
1 PN 1+Xi (k)
dominance condition max1≤i≤32, i6=CV X N k=1 1+XCV X (k) = 0.998 < 1 for all the assets in the portfolio
except for CVX. Hence, according to Lemma 2.2, a log-optimal investor should invest all the available capital in
this asset.
19In-sample trading performance (n=5)
Account value 1
0.995
0.99
09:30 10:00 10:30 11:00 11:30 12:00 12:30
Time Dec 03, 2021
Out-of-sample trading performance (n=5)
1.002
Account value
1
0.998
0.996
0.994
0.992
12:30 13:00 13:30 14:00 14:30 15:00 15:30 16:00
Time Dec 03, 2021
b ∗ , and
Figure 9: Account Value Trajectories under Three Trading Strategies (Optimal K ∗ , K
Equally-Weighted K = 1/m) with Rebalancing Period n = 5 (Minutes) and Zero Costs.
∗
For an even longer rebalancing period, say n = 10 and n = 30, the optimal weight KCV X = 1
remains under proportional cost for stocks being 0.5%.
6 Online Trading with Sliding Window Approach
In previous sections, optimal weights K ∗ and its approximation counterpart K b ∗ were obtained as
fixed values based on the empirical distributions of returns, rather than true distribution, which
is typically unknown to the investor in practice. Moreover, these fixed weights cannot adapt to
the constantly changing information in a dynamic market. To address this issue, we apply a
data-driven sliding window approach that generates time-varying log-optimal weights online; see
also Wang and Hsieh (2022) for a similar idea for online trading.
The idea of the sliding window approach is as follows. For k = 0, 1, . . . , the investor first
declares a fixed window size M ≥ 1. With k = 0, 1, . . . , M −1, one solves the log-optimal portfolio
problem (4) to obtain K ∗ or the approximation counterpart (8) to obtain K b ∗ . These optimum
weights are then applied in the next stage. Having done that, one re-solves the log-optimal
portfolio problem again using the data from k = 1, 2, . . . , M . Repeating this procedure until the
end, one obtains a time-varying optimum K ∗ (k) or K b ∗ (k). This approach has a computational
advantage because it solves a sequence of concave optimization problems rather than a stochastic
dynamic programming problem. The details of this approach can be found in Algorithm 1 below.
Example 6.1 (Mid-Sized Portfolio Revisited: Online Trading via the Sliding Window Ap-
proach). To illustrate the sliding window approach, we conduct additional empirical studies
20In-sample trading performance (n=1)
Account value
1.4
1.2
1
Nov 2021 Dec 2021 Jan 2022 Feb 2022 Mar 2022 Apr 2022
Time
Out-of-sample trading performance (n=1)
1.1
Account value
1
0.9
Apr May Jun Jul Aug Sep Oct Nov Dec
Time 2022
b ∗ , and
Figure 10: Account Value Trajectories under Three Trading Strategies (Optimal K ∗ , K
Equally-Weighted K = 1/m) with Rebalancing Period n = 1 (Day) and Costs of 0.01% for
Stocks and 0.1% for Cryptocurrency.
using the daily price data considered in Example 5.3 with the costs of 0.01% for stocks and
0.1% for cryptocurrency. Here, we first fix the rebalancing period n = 1 day and consider three
different window sizes: M = 10, 20, 30 days. By solving the log-optimal and approximate log-
optimal portfolio problems, we obtain the resulting time-varying optimal weights K ∗ (k) and
the approximate log-optimum K b ∗ (k) for k = 1, 2, . . . , see Figure 12 for an illustration. The
associated account value trajectories of three portfolios with different weights (K b ∗ , K ∗ , and an
equally-weights K = 1/m) are depicted in Figure 13. See also Table 5 for a summary of the
trading performance metrics under three different window sizes M . It is interesting to note that
the portfolios with weights K ∗ and K b ∗ using the window size M = 30 outperform the buy-and-
hold strategy in terms of Sharpe ratio. This observation suggests that the window size M may
be an important factor in determining the overall trading performance. While this point is not
pursued further in this paper, it is worth considering in future work when implementing the
sliding window approach in practice.
Likewise, we also study the performance with different rebalancing periods n = 5 and n = 10
and with different window sizes M = 10, 20, and 30. These results are summarized in Tables 6
and 7. Similar to the n = 1 case, we see that for both n = 5 and n = 10, the best performance
is obtained with M = 30 and M = 20, respectively in this example.
7 Concluding Remarks
This paper focuses on incorporating rebalancing frequency and transaction costs into the log-
optimal portfolio formulation, which aims to maximize the expected logarithmic growth rate of
21Table 3: Out-of-Sample Trading Performance with Zero Costs and Different Nonzero Costs for
Stocks and Cryptocurrency with Rebalancing Period n = 1 (Day).
Costs of 0% for stocks and cryptocurrency K∗ b∗
K Buy and hold
Cumulative rate of return V (NV)−V
(0)
(0)
(%) 14.20 14.18 −6.47
Realized log-growth log VV(N )
(0) (%) 13.28 13.26 −6.68
Volatility σ (%) 2.20 2.20 1.37
Maximum percentage drawdown
√ d∗ (%) 24.88 24.89 20.42
Sharpe ratio N SR 0.60 0.60 −0.33
Costs of 0.01% for stocks and 0.1% for cryptocurrency K∗ b∗
K Buy and hold
Cumulative rate of return V (NV)−V
(0)
(0)
(%) 12.39 12.37 −6.49
Realized log-growth log VV(N )
(0) (%) 11.68 11.66 −6.71
Volatility σ (%) 2.20 2.20 1.37
Maximum percentage drawdown
√ d∗ (%) 25.06 25.07 20.42
Sharpe ratio N SR 0.54 0.54 −0.33
Costs of 0.1% for stocks and 0.1% for cryptocurrency K∗ Kb∗ Buy and hold
Cumulative rate of return V (NV)−V
(0)
(0)
(%) −2.68 −2.7 −6.65
Realized log-growth log VV(N )
(0) (%) −2.72 −2.73 −6.88
Volatility σ (%) 2.20 2.20 1.37
Maximum percentage drawdown
√ d∗ (%) 27.15 27.17 20.42
Sharpe ratio N SR 0.03 0.02 −0.34
Costs of 0.5% for stocks and 0.1% for cryptocurrency K∗ Kb∗ Buy and hold
Cumulative rate of return V (NV)−V
(0)
(0)
(%) 0.22 0.17 −7.34
Realized log-growth log VV(N )
(0) (%) 0.22 0.17 −7.63
Volatility σ (%) 3.9 × 10−5 4.7 × 10−5 1.37
Maximum percentage drawdown
√ d∗ (%) 0.02 0.03 20.42
Sharpe ratio N SR −4.45 −4.55 −0.38
an investor’s wealth. We demonstrate that solving a frequency-dependent optimization problem
with costs is equivalent to solving a concave program. Conditions under which a log-optimal
investor would invest all available funds in a specific asset are provided. We also consider the
issue of bankruptcy that can arise due to transaction costs in the frequency-dependent formu-
lation and propose an approximate solution using a quadratic concave program. Additionally,
a version of the two-fund theorem is proven, demonstrating that a convex combination of two
optimal weights is still optimal. We present various empirical studies to explore the effect of
considering percentage transaction cost and rebalancing periods from the small to mid-sized
portfolio optimization problems. Lastly, we extend our empirical studies to an online trading
scenario by implementing a sliding window approach, which allows us to solve a sequence of
concave programs rather than a complex stochastic dynamic programming problem.
Regarding further research, one possible continuation is to consider additional practical trad-
ing issues; e.g., allowing to short an asset, i.e., Ki < 0 for some i and/or modeling the impact
of dividend/taxes. Another feasible direction is to incorporate an risk term into the objective
function for the ELG maximization problem, which would mitigate the situation when the op-
22In-sample trading performance (n=5)
Account value
1.4
1.2
1
Nov 2021 Dec 2021 Jan 2022 Feb 2022 Mar 2022 Apr 2022
Time
Out-of-sample trading performance (n=5)
1.1
Account value
1
0.9
Apr May Jun Jul Aug Sep Oct Nov Dec
Time 2022
b ∗ , and
Figure 11: Account Value Trajectories under Three Trading Strategies (Optimal K ∗ , K
Equally-Weighted K = 1/m) with Rebalancing Period n = 5 (Days) and Costs of 0.01% for
Stocks and 0.1% for Cryptocurrency..
timum suggests betting all capital on a specific asset; e.g., see Davis and Lleo (2008). Another
important consideration is the potential for estimation error in the distribution of returns, which
is often unknown and must be estimated in practice. In this case, it may be useful to study
the robust counterpart of the problem considered in this paper. That is, instead of solving
supK E[log VVK(0)
(N )
], one seeks to solve a data-driven distributional robust log-optimal portfolio
problem
VK (N )
sup inf EP log
K∈K P ∈P V (0)
where P is the ambiguity set of probability distribution; e.g., see Mohajerin Esfahani and Kuhn
(2018); Wu et al. (2022) for an approach using Wasserstein metric to characterize the ambigu-
ity set.
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23Table 4: Out-of-Sample Trading Performance with Zero Costs and Different Nonzero Costs for
Stocks and Cryptocurrency with Rebalancing Period n = 5 (Days).
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