Trade Cost Optimisation II: Tracking Error and the Cutting Plane Algorithm

Summary

• This blog article builds on our first blog article about trade cost optimisation approaches. We discuss some weaknesses of the simple approach presented in the first article and make suggestions for extending and improving the trade cost optimisation towards a more sophisticated and powerful algorithm.
• Distances between portfolios can be measured with different metrics: We compare turnover and tracking error as distance measures.
• Drawbacks of turnover-based trade cost optimisations in terms of indifferences and suboptimalities are explained and illustrated.
• An extended and more sophisticated trade cost optimisation using tracking error is introduced and formulated in terms of a mixed integer quadratic program (MIQP).
• We explain how the cutting plane algorithm can be used to solve mixed integer quadratic programs (MIQPs) sequentially with solvers for mixed integer linear programs (MILPs).
• The superiority of the tracking-error-based trade cost optimisation is shown by revisiting the momentum strategy backtest from the first trade cost optimisation blog article and we provide comparative results.

1. Measuring the Distance Between Weight Vectors: Turnover vs. Tracking Error

In the last blog article, we already got to know different measures for the distance of two portfolio weight vectors:
Trade count

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

and turnover

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

Both measures have a direct link to some kind of transaction costs, i.e., fixed costs and variable costs, respectively. In the context of a trade cost optimisation, distance measures for portfolios are relevant at multiple occasions. The main purpose of a trade cost optimisation is to handle the trade-off between cost efficiency and optimality of a portfolio with respect to an ideal target portfolio. In this sense, the optimality also has to be measured in terms of the distance of two portfolio weight vectors. However, the turnover is not necessarily a well-suited similarity measure for the proximity of a portfolio to its target portfolio.

1.1. Turnover vs. Tracking Error

In the following, we will demonstrate the suboptimality of turnover as distance measure using a simple example of three assets. We consider the bond ETF TLT and the two equity ETFs IWM and EEM. Let us assume that we hold an equal-weights portfolio ${\mathit{x}}_0=(\frac{1}{3},\frac{1}{3},\frac{1}{3}{)}^{\mathrm{\prime }}$
bm{x}_0 = (
frac{1}{3},
frac{1}{3},
frac{1}{3})' and consider three different target portfolios

$${\tilde{\mathit{x}}}_1=\left(\begin{array}{c}0\\ 0.5\\ 0.5\end{array}\right),{\tilde{\mathit{x}}}_2=\left(\begin{array}{c}0.5\\ 0\\ 0.5\end{array}\right),{\tilde{\mathit{x}}}_3=\left(\begin{array}{c}0.5\\ 0.5\\ 0\end{array}\right)\mathrm{.}$$
tilde{
bm{x}}_1 =
left(
begin{matrix}0

0.5

0.5
end{matrix}
right),

quad

tilde{
bm{x}}_2 =
left(
begin{matrix}0.5

0

0.5
end{matrix}
right),

quad

tilde{
bm{x}}_3 =
left(
begin{matrix}0.5

0.5

0
end{matrix}
right).

The turnover distances are all equal, i.e.,

$$TO($$
bm{x}_0,
tilde{
bm{x}}_1) = TO(
bm{x}_0,
tilde{
bm{x}}_2)
= TO(
bm{x}_0,
tilde{
bm{x}}_3) = 0.3333.

Even measuring exactly the same distance, it is rather unlikely that an investor or portfolio manager would assign the same utility to all three portfolio distances. To give an impression, we consider an investor holding the four different portfolios with fixed weights (i.e., daily rebalancing to the fixed weights) and analyse the differences of realised portfolio returns. The studied sample period is the same as in the previous blog article, i.e., from Jan-2008 till Dec-2018. Denoting the portfolio return corresponding to weight $\mathit{x}$
bm{x} by $r_{$
bm{x}}, we obtain the following sample volatilities (standard deviations) for daily percentage return differences:

$$\begin{array}{rl}{\displaystyle \widehat{\sigma }(r_{{\mathit{x}}_0}-r_{{\tilde{\mathit{x}}}_1})}& {\displaystyle =0.7320,}\\ {\displaystyle \widehat{\sigma }(r_{{\mathit{x}}_0}-r_{{\tilde{\mathit{x}}}_2})}& {\displaystyle =0.3950,}\\ {\displaystyle \widehat{\sigma }(r_{{\mathit{x}}_0}-r_{{\tilde{\mathit{x}}}_3})}& {\displaystyle =\mathrm{0.5385.}}\end{array}$$
begin{aligned}

hat{
sigma}(r_{
bm{x}_0} - r_{
tilde{
bm{x}}_{1}}) &= 0.7320,



hat{
sigma}(r_{
bm{x}_0} - r_{
tilde{
bm{x}}_{2}}) &= 0.3950,



hat{
sigma}(r_{
bm{x}_0} - r_{
tilde{
bm{x}}_{3}}) &= 0.5385.

end{aligned}

These volatilities are often called tracking error. The tracking-error as volatility of the return differences provides a measure for the dispersion of two portfolios in terms of performance differences. In contrast to the turnover, which is not distinguishing between weight differences in different assets, the tracking error is a dispersion measure which is sensitive with respect to weight differences in different assets. Therefore, tracking error is better suited for measuring the distance of a portfolio from its ideal target.

Tracking error can also be written as a quadratic form in the variance-covariance matrix of the assets, i.e.,

$$\text{Var}(r_{{\mathit{x}}_0}-r_{{\tilde{\mathit{x}}}_1})=\text{Var}({{\mathit{x}}_0}^{\mathrm{\prime }}\mathit{r}-{\tilde{\mathit{x}}}_1^{\mathrm{\prime }}\mathit{r})=({\mathit{x}}_0-{\tilde{\mathit{x}}}_1{)}^{\mathrm{\prime }}\mathrm{\Sigma }({\mathit{x}}_0-{\tilde{\mathit{x}}}_1),$$
text{Var}(r_{
bm{x}_0} - r_{
tilde{
bm{x}}_{1}})
=
text{Var}({
bm{x}_0}'
bm{r} -
tilde{
bm{x}}_{1}'
bm{r})
= (
bm{x}_0 -
tilde{
bm{x}}_{1})'
Sigma (
bm{x}_0 -
tilde{
bm{x}}_{1}),

with $\text{Cov}(\mathit{r})=\mathbf{\Sigma }$
text{Cov}(
bm{r}) =
bm{
Sigma} and $\mathit{r}$
bm{r} denoting the vector of asset returns. Therefore, a definition of tracking error $TE($
cdot,
cdot) as distance between two portfolios in terms of weight vectors is given by

$$TE($$
bm{x},
tilde{
bm{x}})
:=
left(
text{Var}(
bm{x}'
bm{r} -
tilde{
bm{x}}'
bm{r})
right)^{
frac{1}{2}}
=
left((
bm{x} -
tilde{
bm{x}})'
bm{
Sigma} (
bm{x} -
tilde{
bm{x}})
right)^{
frac{1}{2}}.

1.2. Relative Tracking Error

Let us consider another portfolio weight vector ${\tilde{\mathit{x}}}_4=(0.59,0,0.41)$
tilde{
bm{x}}_4 = (0.59, 0, 0.41) for which $TO($
bm{x}_0,
tilde{
bm{x}}_4) = 0.3333 and $\widehat{\sigma }(r_{{\mathit{x}}_0}-r_{{\tilde{\mathit{x}}}_4})=0.5413$
hat{
sigma}(r_{
bm{x}_{0}} - r_{
tilde{
bm{x}}_{4}}) = 0.5413. So it has equal turnover distance compared to the one between ${\mathit{x}}_0$
bm{x}_0 and ${\tilde{\mathit{x}}}_3$
tilde{
bm{x}}_3 and almost equal tracking error distance. However, the target portfolios have empirical volatilities of $\widehat{\sigma }(r_{{\tilde{\mathit{x}}}_3})=0.7001$
hat{
sigma}(r_{
tilde{
bm{x}}_{3}}) = 0.7001 and $\widehat{\sigma }(r_{{\tilde{\mathit{x}}}_4})=0.7840$
hat{
sigma}(r_{
tilde{
bm{x}}_{4}}) = 0.7840. Therefore, while the absolute dispersion in terms of tracking error from the current portfolio, ${\mathit{x}}_0$
bm{x}_0, is very similar, it might be more sensible to consider the tracking error distance relative to the volatility of the target portfolio. This gives rise to the definition of relative tracking error $TE_{rel}($
cdot,
cdot)

$$TE_{rel}($$
bm{x},
tilde{
bm{x}})
:=
left(
frac{
text{Var}(
bm{x}'
bm{r} -
tilde{
bm{x}}'
bm{r})}{
text{Var}(
tilde{
bm{x}}'
bm{r})}
right)^{
frac{1}{2}}
=
left(
frac{(
bm{x} -
tilde{
bm{x}})'
bm{
Sigma} (
bm{x} -
tilde{
bm{x}})}{
tilde{
bm{x}}'
bm{
Sigma}
tilde{
bm{x}}}
right)^{
frac{1}{2}}.

For the exemplary portfolio vectors that are considered we get the following relative tracking error distances

$$\begin{array}{rl}{\displaystyle {\widehat{TE}}_{rel}({\mathit{x}}_0,{\tilde{\mathit{x}}}_1)}& {\displaystyle =0.4430,}\\ {\displaystyle {\widehat{TE}}_{rel}({\mathit{x}}_0,{\tilde{\mathit{x}}}_2)}& {\displaystyle =0.4319,}\\ {\displaystyle {\widehat{TE}}_{rel}({\mathit{x}}_0,{\tilde{\mathit{x}}}_3)}& {\displaystyle =0.7682,}\\ {\displaystyle {\widehat{TE}}_{rel}({\mathit{x}}_0,{\tilde{\mathit{x}}}_4)}& {\displaystyle =0.6905,}\end{array}$$
begin{aligned}

hat{TE}_{rel}(
bm{x}_0,
tilde{
bm{x}}_{1}) &= 0.4430,



hat{TE}_{rel}(
bm{x}_0,
tilde{
bm{x}}_{2}) &= 0.4319,



hat{TE}_{rel}(
bm{x}_0,
tilde{
bm{x}}_{3}) &= 0.7682,



hat{TE}_{rel}(
bm{x}_0,
tilde{
bm{x}}_{4}) &= 0.6905,

end{aligned}

where ${\widehat{TE}}_{rel}(\cdot ,\cdot )$
hat{TE}_{rel}(
cdot,
cdot) denotes the ex-post observed tracking error based on the empirical variance-covariance matrix $\widehat{\mathrm{\Sigma }}$
hat{
Sigma}. We can see that in terms of relative tracking error the distances of ${\mathit{x}}_0$
bm{x}_0 to ${\tilde{\mathit{x}}}_3$
tilde{
bm{x}}_3 and ${\mathit{x}}_0$
bm{x}_0 to ${\tilde{\mathit{x}}}_4$
tilde{
bm{x}}_4 are no longer evaluated approximately the same. At the same time it is also interesting to note that in terms of relative tracking error, the distances of ${\mathit{x}}_0$
bm{x}_0 to ${\tilde{\mathit{x}}}_1$
tilde{
bm{x}}_1 and ${\mathit{x}}_0$
bm{x}_0 to ${\tilde{\mathit{x}}}_2$
tilde{
bm{x}}_2 are quite similar, which is caused by the differences in the volatility of the target portfolios. It is important to point out that this relative assessment of distances is not symmetric, i.e., it matters which of the two portfolios is the target portfolio and in general we have $TE_{rel}($
bm{x},
tilde{
bm{x}}) ≠ $TE_{rel}($
tilde{
bm{x}},
bm{x}) .

2. On Turnover-Based Trade Cost Optimisation

In our first blog article about trade cost optimisation, we introduced the optimisation problem

$$\underset{\mathit{x}}{\min }C({\mathit{x}}_0,\mathit{x})\text{s.t.}TO(\mathit{x},\tilde{\mathit{x}})\le \gamma ,$$
min
limits_{
bm{x}} C(
bm{x}_0,
bm{x})

qquad
text{s.t.}
qquad
TO(
bm{x},
tilde{
bm{x}})
leq
gamma,

where $C($
bm{x}_0,
bm{x}) are the costs to trade from ${\mathit{x}}_0$
bm{x}_0 to $\mathit{x}$
bm{x} and $TO($
bm{x},
tilde{
bm{x}}) is the turnover distance of the post-trade weights, $\mathit{x}$
bm{x}, to the ideal weights, $\tilde{\mathit{x}}$
tilde{
bm{x}}. In this approach, the trade-off between cost efficiency and optimality of the new portfolio is handled by enforcing an upper bound for the turnover distance to the ideal portfolio $\tilde{\mathit{x}}$
tilde{
bm{x}} and at the same time trading costs are minimised. While this approach performs very well in implementing systematic portfolio strategies cost-efficiently, it also comes at some drawbacks that we would like to discuss in the following sections.

2.1. Indifferences Between Different Optimal Solutions

One problem of turnover distance is its indifference with respect to different assets, i.e., for turnover distance it does not really matter in which asset a weight difference occurs. As a result, it can happen that there exist multiple non-unique solutions to the trade cost optimisation problem. As an illustration, we again use the three-asset case and for the current portfolio weight we assume ${\mathit{x}}_0=(0.4,0.3,0.3{)}^{\mathrm{\prime }}$
bm{x}_0 = (0.4, 0.3, 0.3)' and as target portfolio weight $\tilde{\mathit{x}}=(0.5,0.25,0.25{)}^{\mathrm{\prime }}$
tilde{
bm{x}} = (0.5, 0.25, 0.25)'. For $\gamma =0.025$
gamma = 0.025 optimal solutions of the optimisation problem are

$${\mathit{x}}_1=\left(\begin{array}{c}0.4750\\ 0.2500\\ 0.2750\end{array}\right),{\mathit{x}}_2=\left(\begin{array}{c}0.4750\\ 0.2750\\ 0.2500\end{array}\right),{\mathit{x}}_3=\left(\begin{array}{c}0.4750\\ 0.2625\\ 0.2625\end{array}\right),$$
bm{x}_1 =
left(
begin{matrix}0.4750

0.2500

0.2750
end{matrix}
right),

quad

bm{x}_2 =
left(
begin{matrix}0.4750

0.2750

0.2500
end{matrix}
right),

quad

bm{x}_3 =
left(
begin{matrix}0.4750

0.2625

0.2625
end{matrix}
right),

with

$$\begin{array}{rl}{\displaystyle TO({\mathit{x}}_1,\tilde{\mathit{x}})}& {\displaystyle =TO({\mathit{x}}_2,\tilde{\mathit{x}})=TO({\mathit{x}}_3,\tilde{\mathit{x}})=0.025,}\\ {\displaystyle C({\mathit{x}}_0,{\mathit{x}}_1)}& {\displaystyle =C({\mathit{x}}_0,{\mathit{x}}_2)=C({\mathit{x}}_0,{\mathit{x}}_3)=c_{fix}\cdot 3+P_{PF}\cdot 2\cdot c_{var}\cdot \mathrm{0.15.}}\end{array}$$
begin{aligned}
TO(
bm{x}_1,
tilde{
bm{x}})
&= TO(
bm{x}_2,
tilde{
bm{x}}) = TO(
bm{x}_3,
tilde{
bm{x}}) = 0.025,


C(
bm{x}_0,
bm{x}_1) &= C(
bm{x}_0,
bm{x}_2) = C(
bm{x}_0,
bm{x}_3)
= c_{fix}
cdot 3 + P_{PF}
cdot 2
cdot c_{var}
cdot 0.15.

end{aligned}

We see that all these solutions have the same fit value and there even exists an infinite number of solutions with this fit value. Two issues arise: First, even providing the same fit, investors and portfolio managers might not be indifferent to the choice between the different solutions of the optimisation problem. Second, from a numerical optimisation point of view stability issues can arise when solving an optimisation problem with an infinite number of optimal solutions. Depending on the algorithm used for solving the optimisation problem, different solutions can be obtained if computations are repeated or done on different machines. The non-uniqueness can make it difficult or impossible to obtain reproducible results. Both issues can be tackled by using tracking error instead of turnover to measure the distance to the target portfolio. While indifferences are very likely with turnover, they are only possible in edge cases with tracking error.

2.2. Suboptimal Solutions to the Multi-Objective Optimisation Problem

Theoretically, investors face a multi-objective problem when handling the trade-off between cost efficiency and optimality. In combination with turnover as a metric, we could even face the situation where the trade cost optimisation problem has multiple non-unique solutions with the same fit value (in terms of costs) but different turnover distances to the target portfolio. If we set $c_{var} = 0$ and reconsider the example above, it holds

$$C($$
bm{x}_0,
bm{x}_1) = C(
bm{x}_0,
bm{x}_2) = C(
bm{x}_0,
bm{x}_3)
= C(
bm{x}_0,
tilde{
bm{x}})
= c_{fix}
cdot 3.

This means that we can even achieve a turnover distance of $TO($
tilde{
bm{x}},
tilde{
bm{x}}) = 0 with the same fit value, i.e. costs, as the other three stated solutions with $TO($
bm{x}_{i},
tilde{
bm{x}}) = 0.025, $i = 1,2,3$. Due to the fact that the distance to the ideal portfolio is not part of the objective function, one of the indifferent solutions will be arbitrarily chosen by solvers for the optimisation problem. To tackle this issue directly, we would need to translate the trade-off into a utility function that is directly relating both distances in some way. This utility can then either be used as objective function or is made a part of the constraints.

2.3. A Two-Step Trade Cost Optimisation

As an alternative to the construction of a utility function we can use a two-step approach which saves us the formalisation of the trade-off in terms of utility. The basic idea behind it is:

1. Use a trading threshold to determine the need for rebalancing.
2. Use a first trade cost optimisation to find the costs that need to be spent in order to get close enough to the target portfolio.
3. Use another trade cost optimisation to spend the determined costs as efficient as possible.

As trading threshold, we use the relative tracking error

$$TE_{rel}($$
bm{x}_0,
tilde{
bm{x}}) >
phi.

Then we apply our standard trade cost optimisation to find

$$\widehat{\mathit{x}}=\arg \underset{\mathit{x}}{\min }C({\mathit{x}}_0,\mathit{x})\text{s.t.}TO(\mathit{x},\tilde{\mathit{x}})\le \gamma ,$$
hat{
bm{x}} =
arg
min
limits_{
bm{x}} C(
bm{x}_0,
bm{x})

qquad
text{s.t.}
qquad
TO(
bm{x},
tilde{
bm{x}})
leq
gamma,

determining our trade count limit $\psi =T{C}_1({\mathit{x}}_0,\widehat{\mathit{x}})$
psi = TC_{1}(
bm{x}_0,
hat{
bm{x}}). The final weight vector, $\mathit{x}$
bm{x}, is then obtained by solving the optimisation problem

$$\underset{\mathit{x}}{\min }T{E}_{rel}(\mathit{x},\tilde{\mathit{x}})\text{s.t.}T{C}_1({\mathit{x}}_0,\mathit{x})\le \psi \mathrm{.}$$
min
limits_{
bm{x}} TE_{rel}(
bm{x},
tilde{
bm{x}})

qquad
text{s.t.}
qquad
TC_{1}(
bm{x}_0,
bm{x})
leq
psi.

The two-step approach can be summarised as follows: The first optimisation is used to determine an optimal trade count budget and the second optimisation tries to optimally use those trades in order to minimise the tracking-error distance to the target portfolio. Indifferences are also possible in the second step but of a different kind, i.e., there might now be a solution with the same distance to the target as the optimal solution, but with lower costs. There is, however, never the situation that a smaller distance could be achievable with same costs as the optimal solution. If the budget is chosen with care, e.g., by restricting the number of trades to a low number, this indifference might be less severe. In our case the budget is not fixed over time, but determined by the first-step trade cost optimisation.

3. Applications: The Cost-Efficient Implementation of a Relative Strength Momentum Strategy

Like in our first blog article about trade cost optimisation, we would like to showcase the benefits of the presented trade cost optimisation using a backtest of a systematic portfolio strategy. To allow for direct comparison, we again consider the relative strength momentum strategy (Faber 2010) for a US investor rotating across asset classes using the following ETFs: SHY, TLT, RWR, IWM, IWB, GLD, EFA, EEM, DBC. Our ten-year sample period spans from Jan-2008 to Dec-2018.