## Author

Jin Zhang^{1}

## Keywords

Information Deficit, Clustering Technique, Efficient Portfolio, Differential Evolution

## Review Status

General review by COMISEF Wiki Admin, 30/11/2008

# Abstract

This paper presents a new optimization approach which combines a clustering technique with traditional asset allocation methods to improve portfolio performance. The optimization procedure is subject to maximizing the in-sample Sharpe Ratio of a portfolio from the clustered market. According to experimental results, for different asset allocation methods one can find that the clustering the market improves portfolio risk-adjusted performance, return stability, and optimization overfitting in the out-of-sample period. Furthermore, clustering the market reduces the estimation error impact on efficient weights when investors pursue a robust portfolio.

# 1 Introduction

This paper proposes an approach that combines existing asset allocation approaches with a clustering technique, to search for an optimal cluster partition by using a heuristic method. The approach firstly classifies the market into a series of disjoint clusters, secondly it applies a traditional asset allocation approach to compute efficient asset weights based on the cluster member returns, then it applies the same asset allocation approach to construct the final portfolio based on the cluster returns. The evolutionary procedure of searching an optimal clustering partition is subject to optimizing a performance evaluation of the final portfolio. Traditional clustering algorithms employ criteria that minimize a measure of dissimilarity between the objects inside clusters, while maximizing the dissimilarity between clusters. This paper considers a different clustering criterion from the similarity measuring. A suitable criterion can be a performance measure in the literature such as the Sharpe Ratio or the Information Ratio. In order to study the impact of clustering on portfolio performance, the total cluster number is specified manually. Regarding the asset allocation approaches, the so-called naive $1/\tilde{N}$ allocation, the Markowitz minimum variance portfolio (MVP) allocation and the modified Tobin portfolio allocation are considered. To evaluate portfolio performance, the risk-adjusted performance measure, a return distribution test, a robustness and an overfitting measure are applied to analyze the final portfolio performance.

# 2 The Model

# 2.1 The problem

The optimization problem is to find a partition $\mathcal{C}=\{\mathcal{C}_1,\mathcal{C}_2,...,\mathcal{C}_g\}$, such that the final portfolio from such a partition yields the highest in-sample Sharpe Ratio when one applies an asset allocation to cluster members and clusters respectively. The optimization of the clustering problem can be written as follows:

(1)where $\operatorname{SR}$ is the Sharpe Ratio, $\mathcal{C}$ is the optimal partition, $\bar{r}_p$ is the average return of the portfolio, $\bar{r}_f$ is an estimate of the risk-free return over the evaluation period, and ${\sigma_p}$ is a measure of the portfolio's estimated total risk or variability (i.e., portfolio's standard deviation) over the in-sample period.

# 2.2 The constraints

In the following, $\mathcal{C}_{g}$ stands for the $g$th asset cluster, $\mathcal{M}$ is the market, $G$ represents the total cluster number applied to the market. The total number of market assets is denoted as $N$, thus $N = \sharp \{ \mathcal{M} \}$. The union of segmented markets contains all market assets, and there is no intersection between two different asset clusters, which are described by using the following equations:

(2)To avoid cases in which one single cluster contains too many assets or an empty cluster, a cardinality constraint is imposed on each cluster, which relate to the total cluster number $G$. Let $\tilde{N}^{\min}$ and $\tilde{N}^{\max}$ denote the minimum and maximum constraints to the clusters, which can be modeled by using the following mathematical expressions:

(3)where $I_{s \in \mathcal{C}_g}$ is an indicator function, returning $1$ if $s \in \mathcal{C}_g$ and $0$ otherwise.

In addition to the cardinality constraints, one should consider weight constraints on cluster members and clusters. The sum of asset weights in a same cluster should be equal to 1, and the sum of cluster weights should be 1 as well. $w_g$ is the weight of the $g$th cluster $\mathcal{C}_g$, and $w_{g,s}$ is the weight of asset $s$ in the cluster $\mathcal{C}_g$, the above weight constraints are described by using the following equations:

(4)The above optimization problem is NP-hard, which can hardly be solved by traditional numerical methods. The paper employs Differential Evolution to tackle the above problem.

# 3 Results

Regarding the asset allocation approaches, three traditional portfolio allocations are considered. All results in the followung tables and figures show the best solution after 100 independent restarts.^{2}

# 3.1 Portfolio Instability

The reduced Instability in the two allocations implies that the final efficient portfolio of the clustering design is less sensitive to changes in the asset return distribution, or to the estimation errors in the asset returns than those of the non-clustered market.

# 3.2 Portfolio Performance Indicators

Table 2 lists the average values of the in-sample and out-of-sample ($\operatorname{SR}$), the average values of the $p$-value, and the average values of Shrinkage which are computed by using random grouping patterns with the simulated asset returns in different $G$s and the non-clustered case. Table 2 shows the performance indicators which are computed by using the optimized clustering partitions and simulated out-of-sample returns. In the comparison of Table 2 with Table 3, the out-of-sample Sharpe Ratios based on the optimized clustering are close to the average values from the random clustering method in the $1/\tilde{N}$ and MVP allocation. Also, one can find that the $p$-values from the random clustering are higher than those from the optimized clustering, implying the Sharpe Ratio optimization criterion reduces the portfolio return stability.

After the analysis based on the simulated out-of-sample returns, the performance indicators using the optimized clustering partition and the actual FTSE 85 asset returns are presented in Table 4. Table 5 shows the performance indicators based on the actual DJIA 65 daily returns. The clustering effect is clearly revealed by the experimental results. According to the tables, the in-sample and out-of-sample Sharpe Ratios are higher than those in the non-clustered market using the three allocations. The $1/\tilde{N}$ allocation has the highest $p$-values, implying its returns are the most stable among the three allocations. Furthermore, the $1/\tilde{N}$ allocation outperforms the other two allocations due to the lowest Shrinkages regarding the overfitting severity.

# 4 Conclusion

In this paper, we discuss a new approach that combines a clustering technique and traditional asset allocation to improve portfolio performance. We search for the optimal clustering partitions subject to maximizing the in-sample Sharpe Ratios in different cluster number cases by using the Differential Evolution. In the comparison with the in-sample and out-of-sample Sharpe Ratios, the $p$-values, the Instability and Shrinkages in the non-clustered and clustered markets, one can find that clustering improves portfolio risk-adjusted performance, stability, robustness and overfitting. However, the clustering effect fades when the total cluster number $G$ increases to a certain threshold, since the clusters degenerate to individual assets when $G$ increases to the number of market assets $N$, due to the non-empty cardinality constraint. According to the experiments results, the $1/\tilde{N}$ and MVP have a similar portfolio performance based on the FTSE 100 market, while the $1/\tilde{N}$ allocation outperforms the MVP allocation in terms of the out-of-sample Sharpe Ratios, $p$-values and Shrinkages in the DJIA 65 case.

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Bibliography
1. Gilli, M., D. Maringer and P. Winker (2008). Applications of Heuristics in Finance. In: D. Seese, C. Weinhardt, F. Schlottmann (Eds).
Handbook on Information Technology in Finance. Springer.2. Gilli, M. and P. Winker (2008). A review of heuristic optimization methods in econometrics. Research Paper Series 08-12, Swiss Finance Institute, available on SSRN.
3. Maringer, D. (2005).
Portfolio Management with Heuristic Optimization. Springer.4. Price, K. V., R. M. Storn and J. A. Lampinen (2005).
Differential evolution: A Practical Approach to Global Optimization. Springer. |

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