Trade Cost Optimisation

3. Efficient Implementation of a Trade Cost Optimisation

In the following, we will present technical details and mathematical derivations necessary to implement the trade cost optimisation. Readers not interested in those technical details are encouraged to directly jump to either Section 3.4, which contains the backtest results with a no-fractional-dealing condition, or to continue reading with the concluding remarks in Section 4.

Section 3 is organised as follows. In Section 3.1 we introduce measures for the distance of two portfolio weight vectors and explain the relation of those distances to different types of transaction costs. Moreover, we give a mathematical formulation of a no-fractional-dealing condition. Section 3.2 contains the mathematical formulation of the trade cost optimisation problem and in Section 3.3 we provide derivations in order to transform the trade cost optimisation problem into a mixed integer linear program (MILP) for which standard solvers exist. Finally, in Section 3.4 we show simulation results for the previously studied momentum strategy but under consideration of a no-fractional-dealing condition.

3.1. Portfolio Weights, Distance Measures, Transaction Costs and Fractional Dealing

For simplicity, we restrict ourselves to the case of an existing portfolio with no deposits and $n$ assets. Our hypothetical investor intends to switch from one portfolio weight vector ${\mathit{x}}_0:=(x_{0,1},\dots ,x_{0,n}{)}^{\mathrm{\prime }}$
bm{x}_0 {:=} (x_{0,1},
ldots, x_{0,n})' to another weight vector $\tilde{\mathit{x}}:=({\tilde{x}}_1,\dots ,{\tilde{x}}_n{)}^{\mathrm{\prime }}$
tilde{
bm{x}} {:=} (
tilde{x}_1,
ldots,
tilde{x}_n)' in a trade-cost-efficient manner.

Measuring the Distance Between Weight Vectors

The distance between two portfolio weight vectors $\mathit{x}:=(x_1,\dots ,x_n{)}^{\mathrm{\prime }}$
bm{x} {:=} (x_1,
ldots,x_n)' and $\tilde{\mathit{x}}$
tilde{
bm{x}} can be measured with different metrics:

1. Trade count ${TC}($
   cdot,
   cdot)

$$TC($$
bm{x},
tilde{
bm{x}})
{:=}
sum_{i=1}^{n} 1_{
lbrace |x_i -
tilde{x}_i| > 0
rbrace},

where $1_{A}$ is the indicator function which is equal to one if $A$ is true and zero otherwise.
2. Trading volume $TV($
cdot,
cdot)

$$TV($$
bm{x},
tilde{
bm{x}}) {:=}
sum_{i=1}^{n} |x_i -
tilde{x}_i|.

3. Turnover $TO($
   cdot,
   cdot)

$$TO($$
bm{x},
tilde{
bm{x}}) {:=}
frac{1}{2}
sum_{i=1}^{n} |x_i -
tilde{x}_i|
=
frac{1}{2} TV(
bm{x},
tilde{
bm{x}})

Note that all three distances are symmetric.

Measuring Transaction Costs

Trading from ${\mathit{x}}_0$
bm{x}_0 to $\mathit{x}$
bm{x} comes at a cost. W.l.o.g. let the cash asset have index $i=1$. It is used as a residual asset with target weight zero. Different types of costs can be represented in terms of weight distances⁴.

1. Fixed costs of $c_{fix}$ per trade in an asset:

$$\begin{array}{rl}{\displaystyle C_{fix}({\mathit{x}}_0,\mathit{x})}& {\displaystyle =c_{fix}\sum _{i=2}^n{1}_{\{\mathrm{\mid }{x}_i-x_{0,i}\mathrm{\mid }>0\}}}\\ {\displaystyle }& {\displaystyle =c_{fix}\cdot T{C}_1({\mathit{x}}_0,\mathit{x})\mathrm{.}}\end{array}$$
begin{aligned}
C_{fix}(
bm{x}_0,
bm{x})
&= c_{fix}
sum_{i=2}^{n} 1_{
lbrace |x_i - x_{0,i}| > 0
rbrace}


&= c_{fix}
cdot TC_{1}(
bm{x}_0,
bm{x}).

end{aligned}

2. Variable costs of $c_{var}$ per traded volume:

$$\begin{array}{rl}{\displaystyle C_{var}({\mathit{x}}_0,\mathit{x})}& {\displaystyle =P_{PF}\cdot c_{var}\sum _{i=2}^n\mathrm{\mid }{x}_i-x_{0,i}\mathrm{\mid }}\\ {\displaystyle }& {\displaystyle =P_{PF}\cdot c_{var}\cdot T{V}_1({\mathit{x}}_0,\mathit{x})}\\ {\displaystyle }& {\displaystyle =P_{PF}\cdot 2\cdot c_{var}\cdot T{O}_1({\mathit{x}}_0,\mathit{x}),}\end{array}$$
begin{aligned}
C_{var}(
bm{x}_0,
bm{x})
&= P_{PF}
cdot c_{var}
sum_{i=2}^{n} |x_i - x_{0,i}|


&= P_{PF}
cdot c_{var}
cdot TV_{1}(
bm{x}_0,
bm{x})


&= P_{PF}
cdot 2
cdot c_{var}
cdot TO_{1}(
bm{x}_0,
bm{x}),

end{aligned}

where $P_{PF}$ denotes the total value of the portfolio.
3. Total costs

$$\begin{array}{rl}{\displaystyle C({\mathit{x}}_0,\mathit{x})}& {\displaystyle =C_{fix}({\mathit{x}}_0,\mathit{x})+C_{var}({\mathit{x}}_0,\mathit{x})}\\ {\displaystyle }& {\displaystyle =c_{fix}\cdot T{C}_1({\mathit{x}}_0,\mathit{x})+P_{PF}\cdot 2\cdot c_{var}\cdot T{O}_1({\mathit{x}}_0,\mathit{x})\mathrm{.}}\end{array}$$
begin{aligned}
C(
bm{x}_0,
bm{x})
&= C_{fix}(
bm{x}_0,
bm{x}) + C_{var}(
bm{x}_0,
bm{x})


&= c_{fix}
cdot TC_{1}(
bm{x}_0,
bm{x}) +
P_{PF}
cdot 2
cdot c_{var}
cdot TO_{1}(
bm{x}_0,
bm{x}).

end{aligned}

Portfolio Volumes, Weights and Fractional Dealing

For retail investors fractional dealing is often not possible. This implies that trade-cost efficient implementations need to handle integer constraints. If the price of asset $i$ is given by $P_i > 0$ and the number of volumes invested in asset $i$ by ${V_i$
in
mathbb{R}^{+}_{0}}, the total value of the portfolio can be computed as

$$P_{PF} =$$
sum_{i=1}^{n} V_{i}
cdot P_{i}

and the portfolio weight for asset $i$ is given by $x_i = V_i$
cdot
frac{P_i}{P_{PF}}. If fractional dealing is allowed, we have ${V_i$
in
mathbb{R}^{+}_{0}}, whereas with no-fractional-dealing condition ${V_i$
in
mathbb{N}_{0}} applies for all assets except cash ($i=2,$
ldots,n).

3.2. The Optimisation Problem: Minimising Transaction Costs

As explained in Section 1, a straightforward setup for a trade cost optimisation is to minimise the total costs under some constraint on the distance to the ideal portfolio weight. A trade event is triggered whenever $TO($
bm{x}_0,
tilde{
bm{x}}) >
delta. However, if the trading trigger $\delta$
delta is reached, we do not want to fully trade to our ideal weight and spend costs $C($
bm{x}_0,
tilde{
bm{x}}) = C_{fix}(
bm{x}_0,
tilde{
bm{x}}) + C_{var}(
bm{x}_0,
tilde{
bm{x}}). Instead, the trade cost optimisation should deliver a new weight vector, $\mathit{x}$
bm{x}, with minimum trading costs, but sufficiently close to the ideal weight vector (the maximum allowed distance is $TO($
bm{x},
tilde{
bm{x}}) =
gamma, with $\gamma \in (0,\delta )$
gamma
in (0,
delta)).

In mathematical terms, the optimisation problem can be written as

$$\underset{\mathit{x}}{\min }{C}_{fix}({\mathit{x}}_0,\mathit{x})+C_{var}({\mathit{x}}_0,\mathit{x})\text{s.t.}\{\begin{array}{l}{\sum }_{i=1}^n{x}_i=1\\ {\mathbf{0}}_n\le \mathit{x}\le {\mathbf{1}}_n\\ TO(\mathit{x},\tilde{\mathit{x}})\le \gamma \end{array},$$
min
limits_{
bm{x}} C_{fix}(
bm{x}_0,
bm{x}) + C_{var}(
bm{x}_0,
bm{x})

qquad
text{s.t.}
qquad

begin{cases}

sum_{i=1}^{n} x_i = 1



bm{0}_n
leq
bm{x}
leq
bm{1}_n


TO(
bm{x},
tilde{
bm{x}})
leq
gamma

end{cases},

where we use the notation ${\mathbf{0}}_n$
bm{0}_n and ${\mathbf{1}}_n$
bm{1}_n for $(n$
times 1) column vectors of zeros and ones, respectively. Besides the already discussed turnover restriction, we added the standard portfolio budget constraint and a no-short-selling restriction. For simplicity, we also assume that trading costs are not directly deducted from the portfolio but externally charged, for example on a monthly basis from a reference bank account.

Standard solvers for convex problems are not well suited for such a problem as the $C_{var}($
cdot,
cdot) and $TO($
cdot,
cdot) contain the absolute value function and $C_{fix}($
cdot,
cdot) even the absolute value function within the indicator function. Therefore, we will now show how the optimisation problem can be linearised using an augmentation technique to obtain a mixed integer linear program (MILP) which can be easily solved using standard software packages for numerical optimisation (see for example Roncalli (2013) where such an augmentation is also applied for quadratic programs).

3.3. Augmentation of Optimisation Problems to Linearise Distances Measured with the Turnover Metric

For $i=1,$
ldots,n let $x_{0,i}$ be the current weight invested in asset $i$ and $x_i$ be the future, i.e., after-optimisation weight of asset $i$. By introducing appropriate auxiliary variables $x_i^+$
geq 0 and $x_i^-$
geq 0 for each asset respectively, we can obtain

$$x_i = x_{0,i} + x_i^+ - x_i^-.$$

Without further constraints, we could find infinitely many combinations of $x_i^+$ and $x_i^-$ that fulfil this equation. However, if we impose the additional constraint $x_i^+$
cdot x_i^- = 0, i.e., at least one of the variables $x_i^+$ or $x_i^-$ is necessarily equal to zero, $x_i^+$ and $x_i^-$ are uniquely defined and we can write

$$|x_i - x_{0,i}| = |x_i^+ - x_i^-| = x_i^+ + x_i^-.$$

$x_{i}^{+}$ can then be interpreted as the excess weight of the final weight $x_i$ with regards to the initial portfolio weight $x_{0,i}$. In other words: It measures how much the individual asset weight is increased. Similarly, $x_{i}^{-}$ represents a decrease in the weight of asset $i$.

We can now re-write the variable costs as a linear function in the vectors ${$
bm{x}^+} {:=} (x_1^{+},
ldots, x_n^{+})' and ${$
bm{x}^-} {:=} (x_1^{-},
ldots, x_n^{-})' as

$$\begin{array}{rl}{\displaystyle C_{var}({\mathit{x}}_0,\mathit{x})}& {\displaystyle =P_{PF}\cdot 2\cdot c_{var}\cdot T{O}_1({\mathit{x}}_0,\mathit{x})}\\ {\displaystyle }& {\displaystyle =P_{PF}\cdot 2\cdot c_{var}\sum _{i=2}^n\mathrm{\mid }{x}_i-x_{0,i}\mathrm{\mid }}\\ {\displaystyle }& {\displaystyle =P_{PF}\cdot 2\cdot c_{var}\sum _{i=2}^n(x_i^{+}+x_i^{-})\mathrm{.}}\end{array}$$
begin{aligned}
C_{var}(
bm{x}_0,
bm{x})
&= P_{PF}
cdot 2
cdot c_{var}
cdot TO_{1}(
bm{x}_0,
bm{x})


&= P_{PF}
cdot 2
cdot c_{var}
sum_{i=2}^{n} |x_i - x_{0,i}|


&= P_{PF}
cdot 2
cdot c_{var}

sum_{i=2}^{n} (x_i^+ + x_i^-).

end{aligned}

In a similar way, we also linearise the turnover distance between $\mathit{x}$
bm{x} and $\tilde{\mathit{x}}$
tilde{
bm{x}} by enforcing the identity

$$x_i =$$
tilde{x}_i +
tilde{x}_i^{+} -
tilde{x}_i^-,

with the auxiliary variables ${\tilde{x}}_i^{+}\ge 0$
tilde{x}_i^+
geq 0 and ${\tilde{x}}_i^{-}\ge 0$
tilde{x}_i^-
geq 0. Under the condition ${\tilde{x}}_i^{+}\cdot {\tilde{x}}_i^{-}=0$
tilde{x}_i^+
cdot
tilde{x}_i^- = 0, we get

$$|x_i -$$
tilde{x}_i| =
tilde{x}_i^+ +
tilde{x}_i^-.

We can now re-write the optimisation constraint $TO($
bm{x},
tilde{
bm{x}})
leq
gamma as a linear constraint in the vectors ${$
tilde{
bm{x}}^+} {:=} (
tilde{x}_1^{+},
ldots,
tilde{x}_n^{+})' and ${$
tilde{
bm{x}}^-} {:=} (
tilde{x}_1^{-},
ldots,
tilde{x}_n^{-})'

$$TO($$
bm{x},
tilde{
bm{x}})
leq
gamma

quad
Leftrightarrow
quad

frac{1}{2}
sum_{i=1}^{n}
tilde{x}_i^+ +
tilde{x}_i^-
leq
gamma.

The fixed costs additionally contain an indicator function. Therefore, we need to add further binary auxiliary variables $z_i$
in
{0,1
}, for $2$
leq i
leq n. Via inequality constraints, we will force the binary variable $z_i$ to be an indicator variable which is equal to one if and only if we trade asset $i$. Let ${\alpha }_0=10^{-5}$
alpha_0 = 10^{-5}, ${\alpha }_1=1$
alpha_1 = 1 and consider the inequality

$$\begin{array}{rlr}{\displaystyle z_i\cdot {\alpha }_0\le \mathrm{\mid }{x}_i-x_{0,i}\mathrm{\mid }\le z_i\cdot {\alpha }_1,}& {\displaystyle }& {\displaystyle i=2,\dots ,n\mathrm{.}}\end{array}$$
begin{aligned}
z_i
cdot
alpha_0
leq |x_i - x_{0,i}|
leq z_i
cdot
alpha_1,
&&i=2,
ldots,n.

end{aligned}

If the inequality is fulfilled, it guarantees that $z_i = 0$ when $|x_i - x_{0,i}| = 0$ and $z_i=1$ when $|x_i - x_{0,i}|$
geq
alpha_{0}. Like the variable costs, we can re-write these constraints in terms of ${\mathit{x}}^{+}$
bm{x}^+ and ${\mathit{x}}^{-}$
bm{x}^- as linear functions. Moreover, we can now also write the fixed costs as a linear function in $\mathit{z}:=(z_2,\dots ,z_n{)}^{\mathrm{\prime }}$
bm{z} {:=} (z_2,
ldots, z_n)'

begin{aligned}
C_{fix}(
bm{x}_0,
bm{x})
&= c_{fix}
cdot TC_{1}(
bm{x}_0,
bm{x})
= c_{fix}
sum_{i=2}^{n} 1_{
lbrace |x_i - x_{0,i}| > 0
rbrace}


&
stackrel{(
star)}{=} c_{fix}
sum_{i=2}^{n} 1_{
lbrace |x_i - x_{0,i}|
geq
alpha_0
rbrace}
= c_{fix}
sum_{i=2}^{n} 1_{
lbrace z_{i} > 0
rbrace}
= c_{fix}
sum_{i=2}^{n} z_i.

end{aligned}

Note that, due to technical reasons, we have to assume a minimum trade size per asset of ${\alpha }_0=10^{-5}$
alpha_0 = 10^{-5} such that $($
star) holds, i.e., either $|x_i - x_{0,i}|$
geq
alpha_0 or $|x_i - x_{0,i}| = 0$.

Putting things together, we obtain the optimisation problem

$$\begin{array}{rl}{\displaystyle }& {\displaystyle \underset{\mathit{y}}{\min }{c}_{fix}\sum _{i=2}^n{z}_i+P_{PF}\cdot c_{var}\sum _{i=2}^n(x_i^{+}+x_i^{-})}\\ {\displaystyle }& {\displaystyle \text{s.t.}\{\begin{array}{l}{\sum }_{i=1}^n{x}_i=1\\ \mathit{x}={\mathit{x}}_0+{\mathit{x}}^{+}-{\mathit{x}}^{-}\\ \mathit{x}=\tilde{\mathit{x}}+{\tilde{\mathit{x}}}^{+}-{\tilde{\mathit{x}}}^{-}\\ {\mathit{x}}^{+}\odot {\mathit{x}}^{-}={\mathbf{0}}_n\\ {\tilde{\mathit{x}}}^{+}\odot {\tilde{\mathit{x}}}^{-}={\mathbf{0}}_n\\ {\mathbf{0}}_n\le \mathit{x},{\mathit{x}}^{+},{\mathit{x}}^{-}\le {\mathbf{1}}_n\\ \frac{1}{2}{\sum }_{i=1}^n{\tilde{x}}_i^{+}+{\tilde{x}}_i^{-}\le \gamma \\ x_i^{+}+x_i^{-}\ge {\alpha }_0\cdot z_i,2\le i\le n\\ x_i^{+}+x_i^{-}\le {\alpha }_1\cdot z_i,2\le i\le n\\ \mathit{z}\in \{0,1{\}}^{n-1}\end{array},}\end{array}$$
begin{aligned}
&
min
limits_{
bm{y}}
;
c_{fix}
sum_{i=2}^{n} z_i
+ P_{PF}
cdot c_{var}
sum_{i=2}^{n} (x_i^+ + x_i^-)


&
text{s.t.}
qquad

begin{cases}

sum_{i=1}^{n} x_i = 1



bm{x} =
bm{x}_0 +
bm{x}^+ -
bm{x}^-



bm{x} =
tilde{
bm{x}} +
tilde{
bm{x}}^+ -
tilde{
bm{x}}^-



bm{x}^+
odot
bm{x}^- =
bm{0}_n



tilde{
bm{x}}^+
odot
tilde{
bm{x}}^- =
bm{0}_n



bm{0}_n
leq
bm{x},
bm{x}^+,
bm{x}^-
leq
bm{1}_n



frac{1}{2}
sum_{i=1}^{n}
tilde{x}_i^+ +
tilde{x}_i^-
leq
gamma


x_{i}^{+} + x_{i}^{-}
geq
alpha_{0}
cdot z_{i},
quad 2
leq i
leq n


x_{i}^{+} + x_{i}^{-}
leq
alpha_{1}
cdot z_{i},
quad 2
leq i
leq n



bm{z}
in
{0,1
}^{n-1}

end{cases},

end{aligned}

where $\mathit{y}:=({{\mathit{x}}^{+}}^{\mathrm{\prime }},{{\mathit{x}}^{-}}^{\mathrm{\prime }},{{\tilde{\mathit{x}}}^{+}}^{\mathrm{\prime }},{{\tilde{\mathit{x}}}^{-}}^{\mathrm{\prime }},{\mathit{z}}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}$
bm{y} {:=} ({
bm{x}^+}', {
bm{x}^-}', {
tilde{
bm{x}}^+}', {
tilde{
bm{x}}^-}',
bm{z}')' is a $((5$
cdot n - 1)
times 1) vector and $\odot$
odot denotes the Hadamard product.

The Non-Linear Constraints $x_i^+$
cdot x_i^- = 0 and ${\tilde{x}}_i^{+}\cdot {\tilde{x}}_i^{-}=0$
tilde{x}_i^+
cdot
tilde{x}_i^- = 0

The non-linear constraints $x_i^+$
cdot x_i^- = 0 and ${\tilde{x}}_i^{+}\cdot {\tilde{x}}_i^{-}=0$
tilde{x}_i^+
cdot
tilde{x}_i^- = 0 can be indirectly integrated. Lets skip the non-linear constraints for a moment. Assume $({$
bm{x}^+}', {
bm{x}^-}', {
tilde{
bm{x}}^+}', {
tilde{
bm{x}}^-}',
bm{z}')' solves the new (without ${x_i^+$
cdot x_i^-} = 0 and ${$
tilde{x}_i^+
cdot
tilde{x}_i^-} = 0 constraints) optimisation problem. Then w.l.o.g.

$$\begin{array}{rl}{\displaystyle }& {\displaystyle \mathrm{\exists }j\in \{1,\dots ,n\}:x_j^{+}\cdot x_j^{-}\mathrm{\ne }0}\\ {\displaystyle }& {\displaystyle \Rightarrow x_j^{+}\cdot x_j^{-}>0}\\ {\displaystyle }& {\displaystyle \Rightarrow x_j^{+}>0\wedge x_j^{-}>0.}\end{array}$$
begin{aligned}
&
exists j
in
lbrace 1,
ldots,n
rbrace: x_j^+
cdot x_j^-
neq 0


&
Rightarrow x_j^+
cdot x_j^- > 0


&
Rightarrow x_j^+ > 0
quad
wedge
quad x_j^- > 0.

end{aligned}

Define the new variables (for $i$
in
lbrace 1,
ldots, n
rbrace):

$$\begin{array}{r}{\displaystyle y_i^{+}:=\{\begin{array}{ll}{x}_i^{+}-x_i^{-},& x_i^{+}\ge x_i^{-}\\ 0,& x_i^{+}\le x_i^{-}\end{array}\text{and}{y}_i^{-}:=\{\begin{array}{ll}{x}_i^{-}-x_i^{+},& x_i^{-}\ge x_i^{+}\\ 0,& x_i^{-}\le x_i^{+}\end{array}\mathrm{.}}\end{array}$$
begin{aligned}
y_i^+ {:=}
begin{cases}
x_i^+ - x_i^-, & x_i^+
geq x_i^-


0, & x_i^+
leq x_i^-

end{cases}

quad
text{and}
quad
y_i^- {:=}
begin{cases}
x_i^- - x_i^+, & x_i^-
geq x_i^+


0, & x_i^-
leq x_i^+

end{cases}.

end{aligned}

It holds

$$\begin{array}{rl}{\displaystyle y_i}& {\displaystyle :=x_{0,i}+y_i^{+}-y_i^{-}}\\ {\displaystyle }& {\displaystyle =x_{0,i}+x_i^{+}-x_i^{-}=x_i,}\end{array}$$
begin{aligned}
y_i &{:=} x_{0,i} + y_i^+ - y_i^-


&= x_{0,i} + x_i^+ - x_i^- = x_i,

end{aligned}

with $y_i^+$
cdot y_i^- = 0 for $1$
leq i
leq n. Similarly, we can construct variables ${\tilde{y}}_i^{+}$
tilde{y}_i^+ and ${\tilde{y}}_i^{-}$
tilde{y}_i^- with $y_i {:=}$
tilde{x}_i +
tilde{y}_i^+ -
tilde{y}_i^- =
tilde{x}_i +
tilde{x}_i^+ -
tilde{x}_i^- = x_i and ${\tilde{y}}_i^{+}\cdot {\tilde{y}}_i^{-}=0$
tilde{y}_i^+
cdot
tilde{y}_i^- = 0. At the same time it holds

$$\begin{array}{rl}{\displaystyle \sum _{i=1}^n\mathrm{\mid }{y}_i-{\tilde{x}}_i\mathrm{\mid }}& {\displaystyle =\sum _{i=1}^n{\tilde{y}}_i^{+}+{\tilde{y}}_i^{-}}\\ {\displaystyle }& {\displaystyle =\sum _{i=1}^n\{\begin{array}{ll}{\tilde{x}}_i^{+}-{\tilde{x}}_i^{-},& {\tilde{x}}_i^{+}-{\tilde{x}}_i^{-}\ge 0\\ {\tilde{x}}_i^{-}-{\tilde{x}}_i^{+},& {\tilde{x}}_i^{-}-{\tilde{x}}_i^{+}\ge 0\end{array}}\\ {\displaystyle }& {\displaystyle \le \sum _{i=1}^n{\tilde{x}}_i^{+}+{\tilde{x}}_i^{-}\le 2\cdot \gamma ,}\end{array}$$
begin{aligned}

sum_{i=1}^{n} |y_i -
tilde{x}_{i}| &=

sum_{i=1}^{n}
tilde{y}_i^+ +
tilde{y}_i^-


&=
sum_{i=1}^{n}
begin{cases}

tilde{x}_i^+ -
tilde{x}_i^-, &
tilde{x}_i^+ -
tilde{x}_i^-
geq 0



tilde{x}_i^- -
tilde{x}_i^+, &
tilde{x}_i^- -
tilde{x}_i^+
geq 0



end{cases}


&
leq
sum_{i=1}^{n}
tilde{x}_i^+ +
tilde{x}_i^-

leq 2
cdot
gamma,

end{aligned}

i.e., $({$
bm{y}^+}', {
bm{y}^-}', {
tilde{
bm{y}}^+}', {
tilde{
bm{y}}^-}',
bm{z}')' solves the original problem, including the non-linear constraints $y_i^+$
cdot y_i^- = 0 and ${\tilde{y}}_i^{+}\cdot {\tilde{y}}_i^{-}=0$
tilde{y}_i^+
cdot
tilde{y}_i^- = 0 as well as the turnover maximum constraint.

The Trade Cost Optimisation Problem as Mixed Integer Linear Program (MILP)

Having shown how to get rid of the non-linear constraint, we can write our optimisation problem as a mixed integer linear program (MILP) with linear (in-)equality constraints

$$\underset{\mathit{y}}{\min }{\mathit{f}}^{\mathrm{\prime }}\mathit{y}\text{s.t.}\{\begin{array}{l}\mathit{A}\mathit{y}\le \mathit{b}\\ \mathit{G}\mathit{y}=\mathit{h}\\ \mathit{z}\in \{0,1{\}}^{n-1}\end{array},$$
min
limits_{
bm{y}}
;

bm{f}'
bm{y}

qquad
text{s.t.}
qquad

begin{cases}

bm{A}
bm{y}
leq
bm{b}



bm{G}
bm{y} =
bm{h}



bm{z}
in
{0,1
}^{n-1}

end{cases},

with appropriately selected matrices and vectors $\mathit{f}$
bm{f}, $\mathit{A}$
bm{A}, $\mathit{b}$
bm{b}, $\mathit{G}$
bm{G} and $\mathit{h}$
bm{h}.

Integrating a No-Fractional-Dealing Condition into the Trade Cost Optimisation Problem

Using the identity $x_i = V_i$
cdot
frac{P_i}{P_{PF}}, we can obtain post-trade volumes and corresponding buy and sell volumes as

$$\begin{array}{rl}{\displaystyle V_i}& {\displaystyle =\frac{P_{PF}}{P_i}\cdot x_i}\\ {\displaystyle }& {\displaystyle =\frac{P_{PF}}{P_i}{x}_{0,i}+\frac{P_{PF}}{P_i}{x}_i^{+}-\frac{P_{PF}}{P_i}{x}_i^{-}}\\ {\displaystyle }& {\displaystyle =:V_{0,i}+V_i^{+}-V_i^{-}\mathrm{.}}\end{array}$$
begin{aligned}
V_i &=
frac{P_{PF}}{P_i}
cdot x_i


&=
frac{P_{PF}}{P_i} x_{0,i} +
frac{P_{PF}}{P_i} x_i^+ -
frac{P_{PF}}{P_i} x_i^-


&=: V_{0,i} + V_i^+ - V_i^-.

end{aligned}

Defining $\tilde{\mathit{y}}:=({{\mathit{V}}^{+}}^{\mathrm{\prime }},{{\mathit{V}}^{-}}^{\mathrm{\prime }},{{\tilde{\mathit{y}}}^{+}}^{\mathrm{\prime }},{{\tilde{\mathit{y}}}^{-}}^{\mathrm{\prime }},{\mathit{z}}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}$
tilde{
bm{y}} {:=} ({
bm{V}^+}', {
bm{V}^-}', {
tilde{
bm{y}}^+}', {
tilde{
bm{y}}^-}',
bm{z}')' and $\tilde{\mathit{P}}:=(\frac{P_1}{P_{PF}},\dots ,\frac{P_n}{P_{PF}}{)}^{\mathrm{\prime }}$
tilde{
bm{P}} {:=} (
frac{P_1}{P_{PF}},
ldots,
frac{P_n}{P_{PF}})', we obtain the trade cost optimisation with no-fractional-dealing condition as

$$\underset{\tilde{\mathit{y}}}{\min }{\tilde{\mathit{f}}}^{\mathrm{\prime }}\tilde{\mathit{y}}\text{s.t.}\{\begin{array}{l}\tilde{\mathit{A}}\tilde{\mathit{y}}\le \mathit{b}\\ \tilde{\mathit{G}}\tilde{\mathit{y}}=\mathit{h}\\ \mathit{z}\in \{0,1{\}}^{n-1}\\ V_i^{+},V_i^{-}\in {\mathbb{N}}_0,2\le i\le n\end{array},$$
min
limits_{
tilde{
bm{y}}}
;

tilde{
bm{f}}'
tilde{
bm{y}}

qquad
text{s.t.}
qquad

begin{cases}

tilde{
bm{A}}
tilde{
bm{y}}
leq
bm{b}



tilde{
bm{G}}
tilde{
bm{y}} =
bm{h}



bm{z}
in
{0,1
}^{n-1}


V_i^+, V_i^-
in
mathbb{N}_{0},
quad 2
leq i
leq n

end{cases},

where $\tilde{\mathit{A}}:=({\mathbf{1}}_{r_{\mathit{A}}}\otimes {\mathit{Q}}^{\mathrm{\prime }})\odot \mathit{A}$
tilde{
bm{A}} {:=} (
bm{1}_{r_{
bm{A}}}
otimes
bm{Q}')
odot
bm{A} and $\tilde{\mathit{G}}:=({\mathbf{1}}_{r_{\mathit{G}}}\otimes {\mathit{Q}}^{\mathrm{\prime }})\odot \mathit{G}$
tilde{
bm{G}} {:=} (
bm{1}_{r_{
bm{G}}}
otimes
bm{Q}')
odot
bm{G}, with $\mathit{Q}:=({\tilde{\mathit{P}}}^{\mathrm{\prime }},{\tilde{\mathit{P}}}^{\mathrm{\prime }},{\mathbf{1}}_{3\cdot n-1}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}$
bm{Q} {:=} (
tilde{
bm{P}}',
tilde{
bm{P}}',
bm{1}_{3
cdot n-1}')'. Moreover, $r_{$
bm{A}}, $r_{$
bm{G}} denote the number of rows of $\mathit{A}$
bm{A} and $\mathit{G}$
bm{G}, respectively and $\otimes$
otimes denotes the Kronecker product.

3.4. The Momentum Strategy with a No-Fractional-Dealing Condition

We rerun our momentum strategy backtest to see the impact of a no-fractional-dealing condition applied to all nine ETFs. Table 1 summarises the results. We can see that, even with a no-fractional-dealing condition, the trade cost optimisation is capable of keeping the average daily turnover distances to the ideal portfolio on a similar level. At the same time the annual turnovers are almost unchanged, but we observe mildly increasing trade costs in terms of an increase in the annual trade counts. This means that even under the more complex integer constraint of a no-fractional-dealing condition, the trade cost optimisation approach allows us to implement the momentum strategy in a cost-efficient way while keeping the distance to the ideal portfolio within certain bounds.

Table 1: Relative Strength Momentum Strategy with Trade Cost Optimisation

4. Concluding Remarks

In this blog article we introduced an efficient and flexible trade cost optimisation approach. It can be combined with every portfolio optimisation strategy in a two-step setup. Tuning of the optimisation parameters allows customisation of the trade cost optimisation to individual cost structures and requirements with regards to proximity of realised weights to ideal weights.

We demonstrated the benefits of the presented trade cost optimisation approach in a backtest by showing that a momentum strategy with a yearly turnover of 200% can be implemented with an average of only 24 trades per year. The main challenge in the technical implementation of the trade cost optimisation is the reformulation of the optimisation problem as a mixed integer linear program (MILP). Our derivation of a MILP representation allows a straightforward implementation in every standard software package for numerical optimisation.