Hierarchical Risk Parity as an Alternative to Conventional Methods of Portfolio Optimization: (A Study of Tehran Stock Exchange)

Document Type : Original Article

Authors

1 Ph.D. Candidate in Financial Engineering, Faculty of Economic, Management and Accounting, Yazd University, Yazd, Iran.

2 Assistant Prof., Department of Accounting and Finance, Faculty of Humanities and Social Sciences, Yazd University, Yazd, Iran.

Abstract

One of the most critical investment issues faced by different investors is choosing an optimal investment portfolio and balancing risk and return in a way that, maximizes investment returns and minimize the investment risk. So far, many methods have been introduced to form a portfolio, the most famous of the Markowitz approach. The Markowitz mean-variance approach is widely known in the world of finance and, it marks the foundation of every portfolio theory. The mean-variance theory has many practical drawbacks due to the difficulty in estimating the expected return and covariance for different asset classes. In this study, we use the Hierarchical Risk Parity (HRP) machine learning technique and compare the results with the three methods of Minimum Variance (MVP), Uniform Distribution (UNIF), and Risk Parity (RP). To conduct this research, the adjusted price of 50 listed companies of the Tehran Stock Exchange for 2018-07-01 to 2020-09-29 has been used. 70% of the data are considered as in-sample and the remaining 30% as out-of-sample. We evaluate the results using four criteria: Sharp, Maximum Drawdown, Calmer, Sortino. The results show that the MVP and, UNIF approach within the in-sample and, the UNIF and HRP approach out-of-sample have the best performance in sharp measure.

Keywords


Introduction                                                                        

The problem of securities optimization is a significant financial problem, and the issue of choosing the optimal stock portfolio has long occupied the minds of investment professionals. One of the basic assumptions in finance is that due to a lack of resources, all economic options face some exchange. When deciding on an investment, a fundamental issue that a rational investor face is choosing between the amount of return he wants to make and the amount of risk he is willing to accept according to that return. As a result, an essential step in the investment process is to see how to allocate your financial resources optimally (Bechis et al., 2020)

In the 1950s, Markowitz, the father of modern portfolio theory (MPT), proposed that investors act rationally in the most efficient way when deciding on resource allocation. If it cannot wholly eliminate the portfolio's variance, but there is a law that allows the investor to diversify his funds among all these securities that have the maximum expected return, this principle assumes that the portfolio of both Provides the maximum expected return and the minimum variance.

Markowitz's theory is now known as the modern portfolio theory and lays the foundation for all investment literature and securities optimization methods. This method succeeded in formulating an optimal approach to allocating resources among risky securities where people are only interested in the average and variance of stock returns. MPT provides a formal yet acceptable way to find optimal portfolios called the efficient frontier, which shows the maximum expected return for a given level of risk or the lowest risk for a given expected return level(Bechis et al., 2020).

The problem with the Markowitz variance-mean method, estimation errors, and inconsistencies led to the development of several other academics' attempts to find possible portfolio solutions that would lead to optimal asset allocation. The Minimum Variance theory has many practical drawbacks due to the difficulty of estimating the expected return and covariance for different asset classes. Portfolio diversification and securities performance during the 2008 credit crunch created the need for the asset management industry to develop new theoretical frameworks with strong empirical results. The new models are risk-based, meaning that they try to estimate risk factors instead of expected returns that are not predictable. The weight of the new portfolio does not take into account the expected returns and depends only on the specific risk factors affecting each security in the portfolio.

Despite the prevailing view that diversification has failed in the recent credit crunch, risk-equalization strategies have performed better than traditional portfolios.

The key to risk equality is to diversify asset classes that behave differently in economic environments: In general, stocks perform well in high-growth and low-inflation environments, bonds perform well in inflation or recession. Moreover, commodities usually perform best in inflationary conditions. Therefore, creating a balanced basket can lead to much more robust returns. Equity risk portfolios typically invest more in low-volatility securities than traditional asset allocation strategies. Some of the essential risk-based models are:

Equal Risk Contribution Portfolio (ERC), Risk Parity Portfolio (RP), Global Minimum Variance (GMV), Maximum Diversification Portfolio (MDP), Maximum Sharpe Ratio Portfolio (MSP), Inverse Volatility Strategy (IV), Market-Capitalization-Weighted Portfolio (MCWP).

In this research, we implement a Hierarchical Risk Parity (HRP) approach based on clustering methods and compare the results with the three Minimum Variance methods, uniform distribution, and Risk Parity (RP). The data set used for this research is the top 50 companies of the Tehran Stock Exchange. In the next section, we will discuss the theoretical literature and research done in this regard, and in the third section, HRP theory will be explained and finally, the implementation of a portfolio using this approach for the Top 50 companies of Tehran Stock Exchange in two We deal in-sample and out-of-sample courses.

Literature Review

The portfolio is a process in which investors choose how to allocate assets. Markowitz's portfolio theory not only reveals the determinants of portfolio risk but more importantly, it concludes the critical conclusion that "the expected return on an asset is determined by asset risk." Therefore, the price of an asset is determined by its variance or standard deviation.

Investment managers can achieve investment portfolio performance in the three activities that make up the portfolio management process: investment policy, portfolio selection, and market timing. Studies of large US retirement plans show that the overall return on investment policy is 93.6%, and therefore investment policy is the most crucial part of portfolio management, often referred to as strategic allocation (Brinson et al., 1986). Investment policy, or strategic allocation, determines which asset categories are selected and by what weight to achieve the investment goal (Brinson et al., 1986). Given the asset class and its weight, since each asset class is related to its risk and return, the investment manager must decide on the risk tolerance, investment horizon, and level of investment risk (Cochrane, 1999).

Portfolio management can be divided into active and inactive management (Al-Aradi & Jaimungal, 2018)،(Sharpe, 1991).

The issue of portfolio diversification has long been of interest to researchers. To better understand the subject, the keyword "Diversified Portfolios" was searched in Scopus on April 9, 2021, and the following outputs were received using the bibliometrics package in R software. The bibliometric method process is performed as described by(Börner et al., 2003).  The general science mapping workflow was introduced by (Börner et al., 2003).

Table 1. Descriptive statistics of research

Description

Results

MAIN INFORMATION ABOUT DATA

 

Timespan

1957:2021

Sources (Journals, Books, etc)

477

Documents

771

Average years from publication

10/3

Average citations per document

18/72

Average citations per year per doc

1/585

References

24153

DOCUMENT TYPES

 

article

603

article in press

1

book

3

book chapter

48

business article

4

conference paper

58

conference review

2

editorial

2

letter

1

note

6

review

42

short survey

1

DOCUMENT CONTENTS

 

Keywords Plus (ID)

1758

Author's Keywords (DE)

1835

AUTHORS

 

Authors

1603

Author Appearances

1789

Authors of single-authored documents

167

Authors of multi-authored documents

1436

AUTHORS COLLABORATION

 

Single-authored documents

187

Documents per Author

0/481

Authors per Document

2/08

Co-Authors per Documents

2/32

Collaboration Index

2/46

 

According to the results obtained from the Scopus portal, 447 studies have been conducted between 1957 and 2021, of which 603 have been reported in the form of articles.

According to Fig 1, the peak of research was conducted in the years 2015 to 2020.

The following figure shows the importance of portfolio diversification in research. In this network, the relationship between the authors, keywords as well as the subject of the research have been examined. By examining this form, research fields of interest to researchers can be obtained. Portfolio selection, portfolio diversification, risk management are some of the essential keywords in this research.

 

 

Fig 1. Annual Scientific production

 

 

Fig 2. Financial network, the relationship between authors, keywords, and research topics

The Hierarchical structure of complex financial systems was first examined by Nobel Laureate Herbert Simon in 1991. In the famous article (Simon, 1991) " The architecture of complexity," the author states that "with a complex system, it means that it is composed of many parts that interact with each other in a non-simple way." In such systems, the total is greater than the sum of the components.

He argues that complex financial systems have a hierarchical organization by which the entire system is broken down into distinct subgroups that can be more easily analyzed. A hierarchical system means a system consists of interconnected systems, each of which has a hierarchical structure when we reach the lowest level of the original subsystem. Thus, a hierarchical structure can help solve complex problems and divide them into smaller, simpler subgroups, after which all those solutions are grouped.

To predict the covariance matrix of size N, we need at least  the expected i.i.d return (Independent and identically distributed). However, there is sufficient evidence that asset returns have cluster fluctuations and variance heterogeneity and have an unstable correlation structure over long periods, leading to severe errors that can lead to diversification benefits Destroy the portfolio.

To overcome this problem, Marcos Lopez de Prado was the first researcher to propose a hierarchical model for portfolio construction in his famous paper “Building Diversified Portfolios that outperform out-of-sample” in 2016. The Spanish author uses network theory and machine learning to build a diversified portfolio with hierarchical equity of risk (HRP) approach that differs significantly from risk-based portfolio optimization models. The HRP method prevents the inversion of the covariance matrix. The securities relationship in the portfolio is organized as a hierarchy in which similar asset clusters are created using the correlation coefficient. Replacing the traditional covariance structure with a hierarchical structure makes possible three main goals: First, it makes full use of covariance matrix information. Second, it covers weight stability. Third, unlike most traditional methods of risk-based asset allocation, there is no need to invert the covariance matrix (Bechis et al., 2020).

Researchers such as (Barziy & Chlebus, 2020), (Snow, 2020), (Molyboga, 2020), (Jaeger et al., 2021) used the HRP approach in their research. (Lohre et al., 2020) In their paper, they examine diversification strategies based on hierarchical clustering. (Raffinot, 2018) Their results show that HERC portfolios based on descending risk criteria perform statistically better than CDaR criteria for risk adjustment. (Jain & Jain, 2019) This paper examines the effect of incorrect choice of covariance matrix on the performance of different allocation methods. It then examines whether the performance of HRP based on machine learning methods is better than portfolios based on traditional risk adjustment methods.

Research Methodology and Research Findings

This research was modeling in terms of type and descriptive in terms of method and applied in terms of purpose. The subject area of this research is the application of machine learning in selecting the optimal portfolio. The statistical population of this research is the top 50 companies in the stock market index, which because some of these companies were initial public offers and, did not have enough data, 5 of them were removed and with the remaining 45 during the period 2018-07-01 to 2020-09-29 (554 trading days) were examined. 70% of the data (381) were considered as in-sample, and 30% of the data were considered as out-of-sample (165 days).

Below is a table of descriptive statistics data:

Table 2. Descriptive statistics of data

 
 

count

mean

std

min

25%

50%

75%

max

Akhaber

545

6123.485

6470.783

1459.24

1929.909

3434.15

6671.68

32260

Barekat

545

6883.84

8724.086

934.026

1247.5

2893.62

8606.85

44610

Beterans

545

16936.42

17777.2

4779.44

6191.295

7438.34

20200.66

72566.02

Fakhas

545

14661.34

11812.88

3511.24

6713.042

9364.77

19094.42

53550

Fakhos

545

13763.72

12214.39

4453.69

6753.285

8539.37

13123.51

53040

Fameli

545

7792.054

9848.934

1339.56

2277.366

3241

6696.61

37410

Fars

545

3811.114

4312.852

872.434

1316.657

1730.02

3848.374

16924.5

Foulad

545

5120.75

5729.813

1168.90

1874.428

2574.97

4463.076

23391.63

Ghegol

545

2816.467

3465.898

289.974

370.88

1225.74

3238.29

14140

Hamrah

545

17315.85

15064.97

5688.56

7798.533

11754.7

16277.04

72850

Hekeshti

545

12973.14

18455.14

2450.22

3366.398

4258.37

13119.04

87063.28

Jam

545

17071.44

12098.51

6210.16

10290.05

11695.59

15953.46

53830

Kachad

545

6556.218

6064.017

1553.72

2930.63

3883.17

6591.567

25340

Kama

545

45091.25

72151.71

3909.84

9612.19

14767.12

28620.93

327980

Kave

545

6261.674

6132.945

1347.69

2300.487

3398.507

6536.528

25100

Kegol

545

6898.812

6030.581

1962.28

3402.227

4402.132

6286.819

26300

Khebahman

545

5799.647

9410.09

631.879

943.547

1761.692

3734.866

41650

Khegostar

545

1650.015

2817.325

87.328

165.513

330.29

1132.024

12700

Khepars

545

3298.322

4645.156

639.059

910.808

1234.319

2541.303

24010

Khesapa

545

834.7744

1174.499

157.005

249.561

337.701

623

5740

Khodro

545

912.8746

1559.381

102.186

142.097

287.186

587

7660

Mobin

545

11872.34

8139.767

3487.71

6732.788

8356.223

13335.14

35310

Pars

545

59278.17

44298.39

15455.7

27718.84

39255.43

69881.83

173221.8

Parsan

545

8952.406

9327.385

2086.03

3554.254

4903.584

8011.135

40630

Petrol

545

3498.383

4126.808

840.036

1266.882

1807.387

3220.61

19942.12

Ranfor

545

18394.66

11523.81

7787.28

10389.1

13398.58

21238.87

52390

Remapna

545

8678.869

8492.709

1348.34

2777.863

5384.298

9649.043

34524.33

Shabriz

545

15173.71

14181.17

2528.56

6053.296

9578.63

15351.89

65520

Shapdis

545

32132.43

31645.24

7013

12818.04

19286.93

29676.35

126650

Shapna

545

5843.769

7531.354

953.476

2058.955

2969.642

4184.428

34008.95

Shebandar

545

12478.3

13144.92

2797.77

5106.101

7055.387

11252.62

59370

Shetran

545

5203.401

6446.049

860.476

1780.879

2871.752

4215.715

30091.75

Shiran

545

10929.11

12417.95

1365.96

1994.264

5752.336

13925.4

47900

Tapiko

545

5101.302

5794.381

1094.07

1791.914

2437.90

4466.57

24055.96

Vabank

545

4262.278

4470.639

1011.48

1485.989

2127.77

3979.102

19543.15

Vabemellat

545

1902.984

1969.678

216.324

621.719

1045.68

1923.279

6550

Vabsader

545

983.5722

1162.3

241.572

381.818

443.257

712.858

4870

Vakharazm

545

4072.053

5145.334

674.902

995.731

1473.14

3557.585

20518.81

Vamaaden

545

5546.136

5091.121

1194.64

2718.218

3253.99

5487.507

20420

Vaomid

545

11706.54

12930.25

2370.24

4481.546

5955

11233

52630

Vapars

545

1047.476

1554.882

145.495

180.007

382.971

778.256

5940

Vaqadir

545

4852.133

5507.711

981.109

1554.92

2311.42

4639.76

25060

Vasandogh

545

5828.791

6246.521

1251.1

2006.539

2948.05

5683.21

26510

Vasapa

545

3916.885

5566.175

621.913

882.458

1336.66

2831.052

24910

Vatejarat

545

867.0455

1073.945

103.806

320.342

408.606

651.869

4420

In the following, we will explain the primary model of work, which is the HRP method.

The concept of HRP is based on graph theory and machine learning techniques and can be divided into three main stages: tree clustering, quasi-diagonalization, and recursive bisection. In the following, we will describe each step in more detail. The first step involves breaking down portfolio assets into different clusters using a hierarchical tree clustering algorithm. For the two assets i and j, the correlation matrix is converted to the correlation-distance matrix D as follows (Burggraf, 2020):

(1)

 

 

 

 

Fig 3. Correlation clustering matrix

     

The second step in the tree clustering step involves calculating the Euclidean distance between all the columns in a pair-wise manner, which gives us the distance matrix :

 

 

(2)

 

The main difference between the above equation and the equation we calculated in the previous section is that the former calculates the distance between the two securities i and j in the portfolio while the latter calculates the distance between those pairs of assets. And then is a function of the whole correlation matrix. The next step involves creating the first cluster . Pairs that are at least spaced apart can be returned:

 

 

(3)

 

Where U is a set of clusters. Next, we need to update the  matrix through a path called "linkage criterion". The distance between the first cluster and the other clusters i is calculated as follows:

 

 

(4)

 

This step is repeated for each stock in the portfolio. Each time a new cluster of assets is formed, the distance matrix algorithm updates until only one cluster remain (Bechis et al., 2020)

 

 

 Fig 4. Stock clustering tree

Next, the quasi-diagonalization covariance matrix is used, which sorts the data to sort the intrinsic clusters. The rows and columns of the covariance matrix are organized so that similar assets are stacked, and different investments are separated. Thus, large covariances are located along the diagonal of the covariance matrix, while smaller covariances are located around this diagonal; hence it is called quasi-diagonal.

 

 

Fig 5. Covariance matrix

Bisection is the last step of the HRP algorithm and is the essential step as it defines the final weight of the securities in the portfolio. Here it takes advantage of the portfolio feature that "inverse allocation is optimal for the diagonal covariance matrix".

Following the tree-clustering process, the algorithm divides each cluster into two sub-clusters  and , starting with the final cluster . According to the weight given to the portfolio, ,  the variance of each sub-cluster is calculated as follows:

 

                                 (5)

where:

 

    (6)

Due to these two weight factors, the algorithm updates the weight of portfolios for each sub-cluster. Therefore, only the assets in each cluster are considered for the final portfolio allocation. The weights  and for these two sub-clusters are as follows:

 

(7)

This top-down assignment of weights is an advantage of HRP over other allocation algorithms. Only the assets of one group compete for allocation instead of all the portfolio assets competing with each other. The whole algorithm ensures that  and .

The following diagram shows the price and return of data, respectively.

 

Fig 6. Price chart

 

Fig 7. Return chart

 

The dendrogram diagram of the companies is as follows.

 

Fig 8. Dendrogram diagram

Next, using four methods of HRP and Min-Var, uniform distribution, and risk Parity (RP), calculate the optimal portfolio weight and evaluate the four methods using in-sample and out-of-sample data.

 

 

Fig 9. The weight of each stock according to four types of methods

The following Table show Weight of each stock based on four types of methods.

Table 3. Weight of each stock according to four types of methods

 

MVP

HRP

UNIF

RP

Akhaber

0.0001

0.0394

0.0222

0.0263

Barekat

0.0021

0.0137

0.0222

0.0131

Beterans

0.0435

0.0234

0.0222

0.0175

Fakhas

0.0933

0.0380

0.0222

0.0184

Fakhos

0.0013

0.0147

0.0222

0.0286

Fameli

0.0434

0.0135

0.0222

0.0262

Fars

0.0424

0.0293

0.0222

0.0327

Foulad

0.0005

0.0264

0.0222

0.0288

Ghegol

0.0572

0.0126

0.0222

0.0120

Hamrah

0.3231

0.0680

0.0222

0.0508

Hekeshti

0.0001

0.0278

0.0222

0.0188

Jam

0.1610

0.0635

0.0222

0.0413

Kachad

0.0005

0.0150

0.0222

0.0255

Kama

0.0004

0.0163

0.0222

0.0152

Kave

0.0001

0.0104

0.0222

0.0176

Kegol

0.0041

0.0257

0.0222

0.0302

Khebahman

0.0003

0.0216

0.0222

0.0180

Khegostar

0.0002

0.0095

0.0222

0.0132

Khepars

0.0069

0.0095

0.0222

0.0169

Khesapa

0.0041

0.0087

0.0222

0.0155

Khodro

0.0010

0.0154

0.0222

0.0150

Mobin

0.0002

0.0215

0.0222

0.0273

Pars

0.0079

0.0255

0.0222

0.0288

Parsan

0.0002

0.0268

0.0222

0.0248

Petrol

0.0001

0.0198

0.0222

0.0189

Ranfor

0.0950

0.0497

0.0222

0.0369

Remapna

0.0002

0.0123

0.0222

0.0160

Shabriz

0.0004

0.0136

0.0222

0.0154

Shapdis

0.0091

0.0166

0.0222

0.0265

Shapna

0.0264

0.0181

0.0222

0.0195

Shebandar

0.0002

0.0167

0.0222

0.0188

Shetran

0.0009

0.0179

0.0222

0.0193

Shiran

0.0001

0.0204

0.0222

0.0147

Tapiko

0.0001

0.0122

0.0222

0.0229

Vabank

0.0002

0.0189

0.0222

0.0301

Vabemellat

0.0407

0.0215

0.0222

0.0104

Vabsader

0.0063

0.0230

0.0222

0.0214

Vakharazm

0.0001

0.0166

0.0222

0.0159

Vamaaden

0.0004

0.0269

0.0222

0.0250

Vaomid

0.0130

0.0211

0.0222

0.0268

Vapars

0.0003

0.0292

0.0222

0.0173

Vaqadir

0.0001

0.0202

0.0222

0.0263

Vasandogh

0.0002

0.0145

0.0222

0.0273

Vasapa

0.0003

0.0101

0.0222

0.0141

Vatejarat

0.0122

0.0241

0.0222

0.0143

 

In the following, we test the portfolio performance generated by the algorithms by examining the results in-sample and out-of-sample.

 

Fig 10. in-sample

As can be seen, the performance of the weighting method based on uniform distribution is better than other methods in both time samples in-sample and out-of-sample.

To evaluate the four estimated methods, we use the Sharp, Calmer, Sortino, and Maximum DrawDown for both in-sample and out-sample sections:

 

Table 4. Evaluation of in-sample optimization methods

 

Sortino

Sharpe

Maxdd

Calmar

MVP

4/937

3/515

-0/133

6/105

HRP

4/460

3/289

-0/200

4/498

UNIF

4/658

3/302

-0/223

4/312

RP

4/401

3/149

-0/215

4/194

According to the results obtained from the in-sample evaluation table, we find that based on Sortino, Sharp, Calmer and Maximum DrowDown, the min-var approach and uniform distribution performed better than HRP methods and Risk parity.

 

Fig 11. out-of-sample

 

Table 5. Evaluation of out-of-sample optimization methods

 

Sortino

Sharpe

Maxdd

Calmar

MVP

7/783

4/159

-0/373

12/988

HRP

9/609

5/511

-0/312

11/044

UNIF

10/212

5/648

-0/332

10/362

RP

9/716

5/440

-0/301

10/489

According to the results obtained from the out-of-sample evaluation table, it is perceived that based on the ratio of Sortino, Sharp, Calmer and maximum DrawDown, uniform distribution method and Risk parity, better performance than HRP and min-var.

Conclusion

Selecting stocks and forming an optimal portfolio have long been one of the essential concerns of investors. For this purpose, many methods have been created and introduced regarding how to choose a portfolio. In this research, we use the Hierarchical Risk Parity (HRP) machine learning technique, and compare the results with the three methods of Minimum Variance, Uniform Distribution and, Risk Parity (RP) in two time periods in-sample and out-of-sample for the top 50 companies the stock exchange. The results show the Minimum Variance approach within the in-sample and the Uniform Distribution approach out-of-sample have the best performance. It should be noted that such optimization methods can show different ranking outputs by changing the period, changing the amount inside and outside the sample. Therefore, portfolio managers should have an active approach in evaluating each of these methods according to the conditions and situations in which they are.

 

Declaration of Conflicting Interests

 The authors declared no potential conflicts of interest concerning the research, authorship and, or publication of this article.

 

 Funding

The authors received no financial support for the research, authorship and, or publication of this article.

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Volume 5, Issue 4
November 2021
Pages 1-24
  • Receive Date: 09 June 2021
  • Revise Date: 10 August 2021
  • Accept Date: 25 August 2021
  • First Publish Date: 01 November 2021