Index Mutual Fund Replication with Two Rebalance Strategies


Jin Zhang1


Passive Portfolio Management, Fund Tracking, Multi-Period Optimization, Differential Evolution

Review Status



This paper discusses the application of an index tracking technique to mutual fund replication problems. By using a tracking error (TE) minimization method and two tactical rebalancing strategies (i.e. the calendar based strategy and the tolerance triggered strategy), a multi-period fund tracking model is developed that replicates S&P 500 mutual fund returns. The impact of excess returns and loss aversion on overall tracking performance is also discussed in two extended cases of the original TE optimization respectively. An evolutionary method, namely Differential Evolution, is used for optimizing the asset weights. According to experimental results, it is found that the proposed model replicates the first two moments of the fund returns by using only five equities. The TE optimization strategy under loss aversion with tolerance triggered rebalancing dominates other combinations studied with regard to tracking ability and cost efficiency.

1 Introduction

In the last decade, individual holdings of corporate stocks have decreased while holdings through fund management institutions have correspondingly increased. According to the Investment Company Institute's official survey, the combined assets of U.S. mutual funds reached a peak of 12 trillion dollars in May 2008; although there was a great redemption pressure on the fund industry due to the recent credit crunch, the net asset value of the funds was in excess of 9 trillion dollars at the end of 2008. As the survey shows, approximately half of the fund holdings were claimed and managed by equity funds. The latter typically choose from three management styles, namely active management, passive management, or a blend of the two. The literature shows that most of actively managed equity funds underperform their passive benchmark portfolios after adjustments are made for fund management fees and expenses. The question of whether charging higher fund fees benefits investors has been discussed at great length. Furthermore, researchers have drawn attention to a confusing phenomenon in the fund market: while fund fees and expenses vary quite a lot, the return patterns of the funds typically show relatively small amounts of dispersion. Therefore, it is not necessary to use expensive funds if the performance of funds are similar.

2 Tracking Error Minimization and Multi-period Readjustment

2.1 The Optimization Problem for Tracker Construction

The TE optimization problem is described as

\begin{align} \min_{\mathbf{n}} TE = \sqrt{\frac{\sum_t (r_{P,t} - r_{I,t})^{2}}{T_{0} - T_{\varpi}}}\\ %- (1 - \lambda) \cdot ER \\ \intertext{subject~to ~~}\\ n_{i,T_{0}} \in \mathds{N}_{0}^{+} \\ t \in [T_{\varpi},T_{0}]\\ k^{\text{min}}<\sharp \mathcal{C}_g = \sum_{i=1}^{N}\emph{I}_{\mathcal{C}_g}(i) \leq k < k^{\text{max}}\\ x_g^{\ell} \leq \frac{n_{i,T_{0}} \cdot {S}_{i,T_{0}}}{P_{T_{0}}} \leq x_g^{u} \text{~~ for $i \in \mathcal{C}_g$}\\ TC_{T_0} = \sum_{i \in \mathcal{C}_g} \rho \cdot n_{i,{T_0}} \cdot S_{i,{T_0}} \leq \gamma \cdot P_{T_{0}}, \end{align}

with the notation:
$n_{i,T_{0}}$ number of shares of the $i$-th equity invested at time $T_0$; $\gamma$ the transaction cost limiting ratio;$P_t$ market value of the tracker at time $t$; $r_{P,t}$ tracker return at time $t$; $r_{I,t}$ index fund return at time $t$; $\mathcal{C}_g$ tracker equity set; $x_g^{\ell}$ minimum weight of each equity; $x_g^{u}$ maximum weight of each equity; $TC_t$ transaction cost at time $t$; $\rho$ transaction cost coefficient; $Cash_t$ cash reserve at time $t$;
$C$ cash reserve rate; $B_t$ sum of the tracker market value and cash reserve at time $t$; $S_{i,t}$ per-share market value of the $i$-th equity at time $t$; $N$ number of available equities in the equity market.

2.2 Tracker Rebalancing Stage

The optimization problem at this stage is summarized as follows.

\begin{align} \notag \min_{\delta (\mathbf{n})} TE = \sqrt{\frac{\sum_t (r_{P,t} - r_{I,t})^{2}}{T_j - T_{j-1}}} \\ \intertext{subject~to~}\\ \notag \delta(n_{i,T_j}) \in \mathds{Z} \\ \notag n_{i,T_j} \in \mathds{N}_{0}^{+}\\ \notag t \in [T_{j-1},T_{j}] \\ \notag k^{\text{min}}<\sharp \mathcal{C}_g = \sum_{i=1}^{N} \emph{I}_{\mathcal{C}_g}(i) \leq k < k^{\text{max}}\\ \notag x_g^{\ell} \leq \frac{(n_{i,T_{j-1}} + \delta(n_{i,T_j})) \cdot {S}_{i,T_j}}{P_{T_j}} \leq x_g^{u} \text{~~ for $i \in \mathcal{C}_g$} \\ \notag TC_{T_j} = \sum_{i \in \mathcal{C}_g} 2\cdot \rho \cdot |n_{i,T_{j}} - n_{i,T_{j-1}}| \cdot S_{i,T_{j}} \leq \gamma \cdot P_{T_j} \end{align}

2.3 Rebalancing Strategies

Two portfolio rebalancing strategies are usually adopted by market practitioners. One is portfolio readjustment at regular calendar interval (e.g. quarterly), which is referred to as calendar based rebalancing. The other is a tolerance triggered strategy which is based on waiting until triggers reach certain thresholds.

2.3.1 Calendar Based Rebalancing

  1. The model splits the future time horizon $[T_{0},T_{\omega}]$ into $M$ subintervals $[T_{0}, T_{1}]$, $[T_{1}, T_{2}],$ $\cdots,$ $[T_{M-1}, T_{\omega}]$ according to a fixed calendar interval $T_{\psi}$. The interval number $M$ is decided by the length of the rebalancing stage and the time interval: $M = \lfloor (T_{\omega}-T_{0})/T_{\psi} \rfloor$.
  2. At the rebalancing time $T_j$, the model
        • decides an optimal set of quantities $\delta\left( n_{i,T_j}\right)$ based on the market information over the time period $[T_{j-1}, T_{j}]$,
        • adjusts portfolio holdings $n_{i,T_j} = n_{i,T_{j-1}} + \delta(n_{i,T_j})$,
        • updates cash reserves ${Cash}_{T_j} = {Cash}_{T_{j-1}} - TC_{T_j}$, and
        • waits till the next planned rebalancing point $T_{j+1} = T_{j} + T_{\psi}$.
  3. The model repeats the second step until the end of the rebalancing stage $T_{\omega}$.

2.3.2 Tolerance Triggered Rebalancing

The second strategy employs a rolling window that starts moving from an historical time. There are $M$ check-points $T_j$ in the rebalancing stage $[T_{0},T_{\omega}]$, following $T_j = j\cdot\wp$, $j = 1, 2, ..., M$ and $M = \lfloor (T_{\omega} - T_{0})/\wp \rfloor$. $\wp$ is the rolling step. At each check-point, the model computes trigger values based on the tracker performance over the current window which starts at an historical time $T_{\varsigma,j} = T_j - W_L$, where $W_L$ is the window length. Three values are considered as trigger tolerances for the current problem: the TE tolerance $\xi_1$; the lower equity weight limit $x_g^{\ell}$; and the upper equity weight limit $x_g^{u}$.

  1. At each check-point $T_j$, the tracker has the starting point of the $j$-th window, $T_{\varsigma,j} = T_j - W_L$ with $j = 1, 2, ..., M$.
  2. If any one of the following conditions is violated: $\sqrt{\frac{1}{W_L} \cdot {\sum_{t=T_{\varsigma,j}}^{T_j} |r_{P,t} - r_{I,t}|^{2}}} < \xi_1$, $\frac{n_{i,T_j} \cdot {S}_{i,T_j}} {P_{T_j}} > x_g^{\ell}$, and $\frac{n_{i,T_j} \cdot {S}_{i,T_j}}{P_{T_j}} < x_g^{u}$, the model
        • finds an optimal set of $\delta(n_{i,T_j})$ based on the market information in the time period $[T_{\varsigma,j}, T_{j}]$,
        • adjusts portfolio holdings: $n_{i,T_{j}} = n_{i,T_{j-1}} + \delta(n_{i,T_j})$,
        • updates cash reserves: ${Cash}_{T_j} = {Cash}_{T_{j-1}} - TC_{T_j}$
  3. Otherwise the model keeps the holdings unchanged: $n_{i,T_{j}} = n_{i,T_{j-1}}$.
  4. The model waits till the next check-point $T_{j+1}=T_j+\wp$, and repeats the second step up till the end of rebalancing stage $T_{\omega}$.

2.4 Extensions of Traditional Tracking Error Optimization

2.4.1 Extension to Include Excess Return

We consider the following average positive deviations from market benchmarks, or the average excess return (ER)

\begin{align} \begin{array}{ll} \notag ER = \frac{1}{T_N}{\sum_t (r_{P,t} - r_{I,t})}, \hspace{0.2in} & \text{~for~} r_{P,t} \ge r_{I,t} \end{array} \end{align}

as a part of the index tracking objective, where ${T_N}$ represents the number of returns observed over the period. The model considers index fund return as the benchmark, and modifies the classic TE optimization objective as follows:

\begin{align} \notag \min \hspace{0.02in} \lambda \cdot TE - (1 - \lambda) \cdot ER. \\ \end{align}

$\lambda$ is a value between $0$ and $1$, representing the weighting difference of the measure between TE and ER.

2.4.2 Extension to Include Loss Aversion

The classic TE minimization objective cannot distinguish between positive and negative deviations of the tracker relative to its target, due to ignorance of the sign of return deviations. Loss averse investors tend to strongly prefer avoiding losses to acquiring gains, therefore the behaviour can be modelled by introducing an aversion coefficient $\vartheta$ to the TE measure with $\vartheta>1$.

\begin{align} \notag \widetilde{\Delta_r}= \left \{ \begin{array}{ll} r_{P,t} - r_{I,t} \hspace{0.2in} & r_{P,t} \geq r_{I,t}\\ (r_{P,t} - r_{I,t}) \cdot \vartheta \hspace{0.2in} & r_{P,t} < r_{I,t} \\ \end{array}. \right. \end{align}

Thus the original objective function at the construction and rebalancing stage is modified to:

\begin{align} \notag \min \hspace{0.02in} \widetilde{TE} = \sqrt{\frac{1}{T_N}{\sum_{t}(\widetilde{\Delta_r})^{2}}}. \end{align}

Differential Evolution (DE) is employed to solve the above optimization problems.

3 Experiments

3.1 Data

A total of 445 equities were used to track index mutual funds. The following five S&P 500 index funds were considered as targets in this study: E*TRADE S&P 500 Index; Vanguard 500 Index; USAA S&P 500 Index; UBS S&P 500 Index A; and TIAA-CREF S&P 500 Index Retire. The five index funds were traced by Tracker 1 to 5, respectively. The data comprise daily prices and the net asset values of the equities and funds in the period January 2004 to December 2007, downloaded from Datastream.

3.2 Out-of-Sample TE


3 Conclusion

In this paper, the authors develop a fund tracking model in order to track index mutual funds with constraints on the cardinality size, assets' weights, transaction costs, and integer constraints. For the first time, this paper proposes to decompose the index mutual fund replication into the traditional index tracking problem and a multi-period optimization problem. By employing a heuristic method to solve the optimization problem, an empirical study is performed to track five S&P 500 mutual funds dynamically. The regression results show that the model replicates the first two moments of index fund returns by using limited equities; moreover, the optimized tracker portfolios do not exhibit significant difference between the original and replicated Sharpe ratios. By setting the tracking error tolerance at twice the in-sample tracking error in the tolerance triggered rebalancing, the model produced the same tracking error magnitude as that using the calendar based rebalancing. Also, it has been shown that tolerance triggered rebalancing outperformed the calendar based rebalancing in terms of both tracking ability and cost-efficiency.

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