Pawel Kliber
Akademia Ekonomiczna w Poznaniu
A tree-criteria portfolio selection: mean return, risk
and costs of adjustment
Introduction
In most work in the portfolio theory, transaction costs are either completely
neglected or treated only qualitatively1 . In this paper we quantify them in a
case when returns are normally distributed. We do this by augmenting classic
Markowitz analysis for one more dimension, namely costs.
In the classic analysis rational investor makes decision by considering two
criteria - risk and return. A fundamental example is the Markowitz model, in
which investor maximizes average return for acceptable level of risk or minimizes
variance of portfolio return for given mean return. Investor thus constructs a
portfolio on the efficient frontier. This is only single-period analysis, because
after several periods the portfolio will not be efficient. The reason for this is that
the structure of the portfolio changes with time. Thus investor has to rebalance
the portfolio after some time.
There are two ways of introducing transaction costs into portfolio optimization. The first one consists in allowing investor to construct different portfolios at
different moments. But choosing this solution one have to abandon mean-variance
setting, because (as we will show in section 2 of this article) the problem becomes too complicated. The authors who choose this way usually assume that
investor wants to maximize utility gained from terminal wealth. Mathematically,
it is a problem of dynamic programming in stochastic setup. For examples of
this solution see: [1], [9] or [11].
Here we take a different approach. We assume that investor chooses portfolio only once. At the next moments investor only rebalance the portfolio, if its
structure differs from the one he or she has chosen. This setup was chosen in
several articles. In [7] the problem of finding the optimal portfolio with transaction costs was stated as a double-objective programming problem. In [10] there
was considered a problem of finding a portfolio on efficient frontier such that
the total transaction cost incurred by rebalancing is minimal. In [3] multistage
1
See for example [2], [6] or [13].
2
Pawel Kliber
stochastic programming was used to find the optimal portfolio in a case with a
finite sample space.
In this article we assume that returns of assets are normally distributed.
Based on this assumption we show the formulas for probability distribution of
future portfolio structure. We assume that transaction cost is a function of a
structure of portfolio and a structure chosen by investor. We use mean cost of
adjustment as a third criterion in Markowitz’s problem of portfolio selection and
we redefine the efficient portfolio according to it. We also show that mean cost
can be easily computed numerically with Monte Carlo method. At the end of the
article we show two examples with a V-shaped function of costs. The optimal
portfolios in these examples differ much from the optimal portfolios in classic
Markowitz analysis.
1. Probability distribution of future portfolio structure
Let us consider first the simplest case with two assets only. Let µ1 and
µ2 be the mean (logarithmic) rate of return of the assets, σ12 and σ22 be the
variations of the returns and ρ be the correlation between the rates of return 2 . At
the moment t = 0 investor buys a portfolio of these two assets. Let α 1 ∈ [0, 1]
be the percent of the wealth invested in the first asset. By X = (X1 , X2 ) we
denote the random variable that describes the logarithmic returns of the assets.
It is assumed to be normally distributed: X ∼ N ((µ1 , µ2 ), Ω), where
ρσ1 σ2
σ12
.
(1)
Ω=
ρσ1 σ2
σ22
Thus the joint density of the random variable X is
f (x1 , x2 ) =
"
1
p
·
2πσ1 σ2 1 − ρ2
(x1 − µ1 )2 σ22 + (x2 − µ2 )2 σ12 + 2 (x1 − µ1 ) (x2 − µ2 ) ρσ1 σ2
· exp −
2σ12 σ22 (1 − ρ2 )
# (2)
We are interested in the probability distribution of the structure of portfolio
after one period. Let W denote the value of the portfolio after one period.
Because we can assume that the initial value of the investments equals 1 without
loosing the generality, we have W = α1 eX1 + (1 − α1 )eX2 . The two-dimensional
2
In the subsequent analysis we always assume that rates of returns are sufficiently low so
that the logarithmic rates of returns are approximately equal to the ordinary rates of returns (i.e.
(t)
ln PP(t+1)
, where P (t) is the price of the asset at the moment t).
≈ P (t+1)−P
(t)
Pt
A tree-criteria portfolio selection: mean return, risk and costs of adjustment
3
random variable (S, W ) is a function of the variable X and there exists reciprocal
of this function, namely
SW
,
α1
(1 − S) W
.
X2 = h2 (S, W ) = ln
1 − α1
X1 = h1 (S, W ) = ln
The Jacobian for the function above equals (see e.g. [14])
∂ (X1 , X2 ) 1
∂ (S, W ) = S (1 − S) W .
(3)
(4)
(5)
We can use the formula for a density of a function of a random variable (see e.g.
[15]) to obtain the joint density of the variables S and W . It equals
f(S,W ) (s, w) =
1
f (h1 (s, w) , h2 (s, w)) ,
S (1 − S) W
(6)
where f is given by eqn. (2) and h1 , h2 are given by eqs. (3)-(4).
The probability distribution of S can be obtained by integrating function
f(S,W ) (s, w) with respect to w, however it is rather cumbersome. In [4] the
stochastic calculus was used to obtain following density of S:
"
2 #
1
α1
s
1
√ exp
fS (s) =
− ln
−µ
ln
, (7)
2σ 2
1−s
1 − α1
s (1 − s) σ 2π
p
where σ = σ12 + σ22 − 2ρσ1 σ2 and µ = µ1 − µ2 .
The problem can be extended to multidimensional case. Suppose that the
portfolio consists of n assets. Let µ be the vector of the mean logarithmic returns
of the assets and Ω be the covariance matrix of the returns. At the initial moment
t = 0 the structure of the portfolio is α = (α1 , α2 , . . . , αn ), i.e. the value αi is
the percent of the wealth invested in the asset i. Let S1 , . . . , Sn be the random
variables that describe the structure of the portfolio after one period of time. The
n-dimensional random variable X = (X1 , X2 , . . . , Xn ) describes the logarithmic
returns of the assets. The variable X has distribution N (µ, Ω) and its density
function is
1
n
1
f (x) = (2π)− 2 |Ω|− 2 exp − (x − µ)T Ω−1 (x − µ) .
(8)
2
Let S1 , . . . , Sn describe the structure of portfolio at the moment t = 1. Of
course, Sn = 1 − S1 − . . . , Sn−1 . Defining the n-dimensional random variable
4
Pawel Kliber
(S1 , . . . , Sn−1 , W ), where W is the value of the portfolio at the moment t = 1,
and assuming that the value of initial investments equals 1 we can express the
variables X1 , . . . , Xn in terms of S1 , . . . , Sn−1 and W , namely
Xi = hi (S1 , . . . , Sn−1 , W ) = ln
Si W
, for i = 1, . . . , n − 1
αi
(9)
(1 − S1 − . . . − Sn−1 ) W
.
αn
(10)
and
Xn = hn (S1 , . . . , Sn−1 , W ) = ln
The Jacobian for this function equals
∂ (X1 , . . . , Xn ) 1
∂ (S1 , . . . , Sn−1 , W ) = W S1 S2 · · · Sn .
(11)
The joint distribution of the variables (S1 , . . . , Sn , W ) is thus given by the following formula:
f (S1 ,...,Sn−1 ,W ) (s1 , . . . , sn , w) =
1
f (h1 (s1 , . . . , sn , w) , . . . , hn (s1 , . . . , sn , w)) ,
W S 1 S2 · · · S n
(12)
if s1 +. . .+sn = 1 and f(S1 ,...,Sn−1 ,W ) (s1 , . . . , sn , w) = 0 otherwise. The density
of (S1 , . . . , Sn ) can be obtained by integrating w out in the formula (12). In [5]
it was shown that the joint density of (S1 , . . . , Sn ) on the (n − 1)-dimensional
simplex is given by
1−n
1
f S (s1 , . . . , sn ) = (2π) 2 |Σ|− 2 ·
1
T −1
· exp − (g (s1 , . . . , sn ) − m) Σ (g (s1 , . . . , sn ) − m) ,
2
(13)
where Σ is a (n − 1) × (n − 1) matrix such that its element σij equals
σij = σn2 + ρij σi σj − ρin σi σn − ρnj σn σj
(14)
(σi2 is the variance of return of the asset i and ρij is the correlation between
returns of the assets i and j). The vector m consists of elements
mi = ln
αi
+ µi − µn , for i = 1, . . . , n − 1
αn
and the function g is given by the formula
sn−1
s1
.
g (s1 , . . . , sn ) = ln , . . . , ln
sn
sn
(15)
(16)
A tree-criteria portfolio selection: mean return, risk and costs of adjustment
5
2. Problem of portfolio adjustment
Let us consider how adjustment costs affect the optimal portfolio. Without
loosing the generality we can assume that the initial wealth of investor equals 1.
We assume that transaction cost is proportional to value of a transaction. Let γ/2
be the cost of buying or selling per monetary unit. Thus if we want to transform
our investments from some asset to some other one, we have to pay γ times the
value of our investment.
Consider T periods of time. At the period t = 0 we invest one monetary
unit in a portfolio α(0) (we can assume that the costs at this moment equal zero).
At each moment t = 0, . . . , T − 1 we invest in a portfolio α(t). Let X(t) be the
vector of returns of the assets at the period t. The rate of return of the portfolio
at the period t is thus α(t)T X(t). As a result of price changes in the period
t the structure of the portfolio changes from α(t) to S(t + 1), where S(t) is a
random variable dependent on α(t) and X(t + 1). At the moment t we change
our portfolio from S(t) to α(t). If the value of the portfolio at the moment t is
V (t) then the transaction cost at this moment equals
where kxk =
equals
(17)
C(t) = γkS(t) − α(t)kV (t),
Pn
i=1 |xi |.
Thus the value of the portfolio at the moment t + 1
V (t + 1) = (V (t) − C (t)) α (t)T X (t) =
= (1 − γkS (t) − α (t) k) α (t)T X (t + 1) · V (t) .
(18)
Solving this as a difference equation we obtain the following formula for the
value for the wealth at the final moment:
V (T ) =
TY
−1
t=0
(1 − γ kS (t) − α (t)k) α (t)T X (t + 1) ,
(19)
where S(0) = α(0). For t = 1, . . . , T − 1 the random variable S(t) is a function
of α(t − 1) and X(t). The values S(t) can be expressed as follows:
Si (t) =
αi (t − 1) (1 + Xi (t))
1 + α (t − 1)T X (t)
.
(20)
The (ordinary) rate of return for the sequence of portfolios α(0), α(1), . . . ,
α(T − 1) is thus given by
R (T ) =
TY
−1
t=0
(1 − γ kS (t) − α (t)k) α (t)T X (t + 1) − 1.
(21)
6
Pawel Kliber
In theory, we can adapt classic Markowitz portfolio analysis to find the optimal portfolio for investor who takes into account only mean and variance of the
rate of return. This is however extremally cumbersome from the computational
point of view. In the formula (21) each portfolio α(1), . . . , α(T − 1) (except
for α(0)) is a function of S(t − 1). To compute mean and variance of R(T )
numerically we have to substitute one vector (α(0)) and T − 1 vector functions
(α(1), . . . , α(T −1)) into eqn. (21). The problem of finding the optimal portfolio
is thus mathematically a problem of variational calculus and there is a little hope
to solve it in easy way, especially if the time horizon is long.
3. Costs of adjustments as a third criterion
As we have seen the problem of optimizing portfolio dynamically is very
difficult to handle even if transaction costs are proportional to value of a transaction (as in section 2). However we can simplify the analysis assuming that
investor wants to have a portfolio with a steady structure. Investor chooses his
portfolio α only once - at the moment t = 0. At each subsequent moments
t = 1, . . . , T − 1 he only adjust his portfolio.
The advantage of this approach is that we do not have to assume proportional transaction costs. Let c(α, S) be the function which describes the costs
(per unit of wealth) of adjusting the structure of the portfolio from S to α. In
this way we can augment the classic portfolio analysis for one more dimension costs of adjustments.
To show the problem explicitly let us consider a case where there are only
two assets. At the period t = 0 investor chooses a portfolio α = (α1 , 1 − α1 ).
The mean return of the portfolio is R(α) = αT µ and the variance of the return
equals V (α) = αT Ωα. The mean cost of adjustments can be computed using
the formula (7):
C (α) =
Z
1
0
c ((α1 , 1 − α1 ) , (s, 1 − s))
s (1 − s)
"
2 #
α1
s
1
− ln
−µ
ln
ds,
× exp
2σ 2
1−s
1 − α1
(22)
where µ and σ are as in eqn. (7). The function C(α) gives the cost for two-period
investments. If we consider investments horizon of the length T , then the cost is
(T − 1)C(α).
The integral (22) cannot be solved analytically but it can be very easily
A tree-criteria portfolio selection: mean return, risk and costs of adjustment
7
computed numerically using Monte Carlo method. Defining
Z=
S
α
ln 1−S
− ln 1−α
−µ
σ
and substituting it into eqn. (22), we obtain
Z ∞
1 2
1
C (α) =
c ((α1 , 1 − α1 ) , (s, 1 − s)) √
e− 2 z dz.
2πσ
−∞
(23)
(24)
Hence the random variable Z has standard normal distribution N (0, 1) and according to (23):
1
(25)
S=
1−α −µ−σZ .
1+ α e
In this way we obtain the following algorithm for calculating C(α): generate a
random variable Z with the standard normal distribution, calculate S from the
formula (25), calculate c(α, (S, 1 − S)). Repeat this many times (a thousand
should suffice) and take the mean of obtained costs as the value of C(α).
Let us consider an example. Take µ1 = 0.08, µ2 = 0.10, σ1 = 0.07,
σ2 = 0.12 and ρ = −0.5. The figure 1 shows the mean cost of adjustment
against α1 . The figure 2 contains the plot of the risk of the portfolio against
the mean return of the portfolio. As we can see, the well-diversified portfolios
have the highest costs of adjustments. We can consider a problem similar to the
problem of finding the best portfolio in Markowitz setup, namely
max R(α1 , 1 − α1 ) − λσ σ(α1 , 1 − α1 ) − λC C(α1 , 1 − α1 ),
α1 ∈[0,1]
(26)
where R(α) is the expected value and σ(α) is the standard deviation of return
for the portfolio (α1 , 1 − α2 ). The parameters λσ and λC measure the investor’s
aversion to risk and costs. The solution to the problem (26) can differ significantly
from the optimal portfolio in the Markowitz setup. For our data if we neglect
adjustment cost (i.e. set λC = 0) and assume that λσ = 1, then the optimal
portfolio is (0.55, 0.45), while the solution to the problem (26) with λ σ = λC = 1
is (0, 1).
Let us consider now a portfolio consisting of many assets. Investor chooses
the structure of his portfolio only at the beginning of time horizon. The average
cost of portfolio adjustment is given by
Z
n−1
c (α, s)
(2π)− 2 |Σ|−1 ·
C (α) =
∆n s 1 · · · s n
(27)
1
T −1
· exp − (g (s1 , . . . , sn ) − m) Σ (g (s1 , . . . , sn ) − m) ds,
2
8
Pawel Kliber
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
Figure 1: Mean cost vs. the weighting of the first asset
0.0
0.2
0.4
0.6
0.8
1.0
where ∆n is a (n−1) dimensional simplex ∆n = x ∈ Rn+ : x1 + . . . + xn = 1
and the function g is given by (16).
As in the case with two assets we can simplify calculating the integral (27)
by using Monte Carlo method. Since the matrix Σ has a full rank and is positive
defined, we can write Σ−1 = D T D for some matrix D. Define a vector random
variable Z = (Z1 , . . . , Zn−1 as follows:
Z = D (g (S1 , . . . , Sn ) − µ) .
The derivatives of Z with respect to S1 , . . . , Sn−1 respectively equal
∂Zi
∂g (S) T
= Di
∂Si
∂Sj
(28)
(29)
A tree-criteria portfolio selection: mean return, risk and costs of adjustment
9
0.06
0.08
0.10
0.12
Figure 2: The variance of return for the portfolio vs. the mean return
0.08
0.09
0.10
0.11
0.12
0.13
0.14
i (S)
i (S)
= S1i , ∂g∂S
=
where Di is the ith row of the matrix D and ∂g∂S
i
j
i 6= j. The Jacobian for Z equals (see Appendix)
∂Z 1
1
= |D| · ∂g (S) = |Σ|− 2
.
∂S ∂S S1 S2 · · · S n
0.15
1
Sj
+
1
Sn
for
(30)
Substituting Z in (27) and using formula for a distribution of a function of a
random variable we can see that Z has the (n − 1)-dimensional standard normal
distribution N (0, I).
From eqn. (28) we obtain that
T
Sk =
αk eEk Z+mk
,
P
T
αn + ni=1 αi eEi Z+mi
(31)
10
Pawel Kliber
where Ei is the ith row the matrix Di−1 . Thus the mean cost of adjustments
can be calculated in the following way: generate a (n − 1)-dimensional standard
normal variable Z; compute S using the eqn. (31) and then compute c(α, S) and
add it to the sample. Repeat this many times and the mean from the sample
approximates expected cost of adjustments C(α).
Now we can augment the definition of effective portfolio. In our setup the
portfolio α is effective if there do not exist another portfolio β such that one of
the following three conditions holds: either
R(β) > R(α),
σ(β) ≤ σ(α),
C(β) ≤ C(α)
(32)
R(β) ≥ R(α),
σ(β) < σ(α),
C(β) ≤ C(α)
(33)
R(β) ≥ R(α),
σ(β) ≤ σ(α),
C(β) < C(α),
(34)
or
or
where R(α) is expected return of the portfolio α and σ(α) is standard deviation
of expected return. As we can see the portfolio is effective if and only if it is
Pareto-optimal with respect to three criteria: mean return, variance of return and
cost of adjustments.
The optimal portfolio in our setup is the solution to the following problem:
max R(α) − λσ σ(α) − λC C(α),
α∈∆n
(35)
where ∆n is an (n − 1)-dimensional simplex. It is straightforward to see that the
solution to the problem (35) is an effective portfolio. The parameter λ C measures
investor’s aversion to costs and the bigger length of the planned investment is,
the higher its value should be.
Let us take for example following data: µ1 = 0.09, µ2 = 0.12, µ3 = 0.15,
σ1 = 0.05, σ2 = 0.09, σ3 = 0.17 and let the correlation between returns of the
assets be ρ12 = −0.5, ρ13 = −0.8 and ρ23 = −0.4. The figure 3 depicts the
contours of average costs of adjustments with respect to α1 and α2 . If we neglect
the costs then the optimal portfolio for λσ = 1 is (0, 0.4, 0.6). Taking costs into
account with λC = 1 we get that undiversified portfolio (0, 0, 1) is optimal.
Summary
Bibliography
A tree-criteria portfolio selection: mean return, risk and costs of adjustment
11
0.0
0.2
0.4
0.6
0.8
1.0
Figure 3: Contours of average cost
0.0
0.2
0.4
0.6
0.8
1.0
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Appendix
In this appendix we prove that the Jacobian of the function g(s), given
∂si
1
1
1
1
i
. It is easy to check that ∂g
by (16), equals s1 ···s
∂si = si + sn and ∂sj = sn for
n
A tree-criteria portfolio selection: mean return, risk and costs of adjustment
i 6= j. Thus
∂g = Wn−1
∂s =
1
s1
+
1
sn
1
sn
1
sn
1
s2
1
sn
+
1
sn
...
...
1
sn
1
sn
1
sn
1
s3
1
sn
1
sn
+ s1n
...
1
sn
...
...
...
...
...
1
sn
1
sn
1
sn
...
1
sn−1 +
k
1
sn
.
13
(36)
Let sk = s1 s2 · · · sk sn and s̄ki = ssi = s1 · · · si−1 si+1 · · · sk sn .
By Wk we denote determinant like in (36) but containing only variables
s1 , . . . , sk and sn . We will show that Wk = s̄1k + s̄1k + . . . + s̄1k + s̄1k . We prove it
1
2
k
n
by induction. For k = 1 it is straightforward, because W1 = s11 + s1n = s̄11 + s̄11 .
n
1
For k > 1 we have
1
1
1
s + s1n
...
sn
sn
1 1
1
1
1
+ sn . . .
s
s
s
=
n
n
2
Wk = .
.
.
.
.
.
.
.
.
.
.
.
1
1
1 1
. . . sk + sn
sn
sn
1
1
1
s + s1n
. . . s1n s11 + s1n
. . . s1n sn
sn
1 1
1
1 1
1
1
1 1
.
.
.
+
.
.
.
+
s
s
s
s
s
s
s
s
n
n
n + 2
n
n
n =
2
= ...
. . . . . . . . .
...
. . . . . . ...
1
1
0
0
. . . s1k . . . s1n sn
sn
1
s
0 ... 0 1
1
0
1
.
.
.
0
s2
= 1 Wk−1 + 1 =
= Wk−1 + s
.
.
.
.
.
.
.
.
.
.
.
.
sk
s̄kk
k
1
1
. . . s1n sn
sn
!
1
1
1
1
1
1
1
1
1
+
+ . . . + k−1 + k + k = k + . . . k + k .
=
sk s̄1k−1 s̄2k−1
s̄
s̄
s̄k
s̄1
s̄k
s̄k−1
n
n
In this way we have proved that
∂g = Wn−1 = 1 + . . . + 1 = s1 + s2 + . . . sn =
∂s sn−1 sn−1
sn−1
s̄n−1
s̄n−1
n
1
1
s1 + . . . + s n
=
.
=
n−1
s
s1 s2 · · · sn