Optimal Vs 1/N Diversification and Portfolio Evaluation: A study on Indian Stock Market

 

V. Harshitha Moulya1, Abuzar Mohammadi1, Dr T. Mallikarjunappa2

1Research Scholar, Department of Business Administration, Mangalore University, Mangalagangothri, Konaje, Mangalore 574199 Karnataka

2Professor, Department of Business Administration, Mangalore University, Mangalagangothri, Konaje, Mangalore 574199 Karnataka

*Corresponding Author E-mail: harshuwhitetiger@gmail.com, abuzar.mohammadi@gcc.edu.in

 

ABSTRACT:

The Modern portfolio theory of Markowitz (1952) proposed maximisation of expected utility and minimisation of the risk of the optimal portfolio for the risk-averse investors. We used the linear programming technique to estimate the optimal portfolio weights for the mean-variance efficient optimal portfolio using rebalanced and non-rebalanced portfolios and compared the performances against the 1/N heuristic portfolio. We found that the minimum-variance optimal portfolio performed better than the 1/N heuristic portfolio.

 

KEYWORDS: Portfolio Optimisation, Markowitz portfolio, Portfolio rebalancing, portfolio return-risk, NSE

 

 


INTRODUCTION:

The Modern Portfolio Theory (MPT) of Markowitz (1952) proposed two axioms for portfolio formation and the efficient frontier of the optimal portfolio for investors viz., the expected utility maximisation and risk-minimisation. According to the MPT, investors are risk-averse and have certain beliefs for choosing a portfolio. The risk-averse investors try to maximise the discounted value of their expected returns generated by a maximum utility function where the investors invested all their funds in security that yield maximum anticipated returns. The axiom of expected utility maximisation alone didn’t generate anticipated returns from the optimal portfolio due to market risks and imperfections. The MPT proposed the second axiom viz., the EV rule (Expected returns–Variance in returns) or the Mean-Variance (MV) model according to which the investors allots weights on different assets or risky-securities to form a portfolio for earning the expected returns.

 

The investors attain expected returns by accepting a variance of returns vis-à-vis reduce the variance in expected returns by giving up expected returns through the process of asset allocation and diversification of securities in a portfolio. The asset allocation process is done by allotting the optimal weights of investments on risky assets. The weights are optimal if the proposed portfolio attains both the objectives of expected utility maximisation and risk-minimisation for the risk-averse investors.

 

There are various measures for measuring the portfolio risk viz., traditional measures– the standard deviation or variance of expected returns; advanced methods – Sortino ratio, the CVaR (Conditional Value at Risk); and the coherent risk measures using quantile-based methods, scenario-based methods (Artzner, Delbaen, Eber, & Heath, 1999) used in the computation of portfolio risk and returns. As the portfolio risk is a function of variance and co-variances of asset returns in the portfolio, an optimal portfolio for investors should have lower correlation/ co-variance in asset returns for various combination of assets for earnings the maximum expected returns.

In our study, we discuss the vast literature on the portfolio theory and empirically examine the performance of Markowitz’s MV model of optimal portfolio vis-à-vis the 1/N equal-weighted portfolio. We considered the single-asset case of risky-securities, i.e. the equities traded on the NSE for constructing the mean-variance efficient portfolio using the portfolio rebalancing technique for the single-period optimisation and multi-period optimisation scenarios. The stocks traded on the Nifty 50 index is considered for the optimal portfolio as the stocks represent more than 66% of market liquidity in trade and volume for the Indian market. The optimal weights for weight allocation on the equities were estimated using the linear programming technique. Our results show that periodic portfolio rebalancing gives higher expected portfolio returns for the given variance. The minimum-variance MV portfolio performed better than the maximum-return MV portfolio and the 1/N heuristic portfolio in terms of optimal portfolio returns, Sharpe ratio and minimum variance.

 

LITERATURE REVIEW:

The seminal works by Tobin (1958), Sharpe (1964), Lintner (1965), Merton (1969, 1973), and Black (1972) studied the mean-variance (MV) efficient portfolio frontier theory of Markowitz (1952) and found the results consistent with the doctrine of investments of expected utility maximization and risk-minimization by the risk-averse investors i.e. the investors maximized their expected portfolio returns by minimizing the portfolio variance. Tobin (1958) observed that the proportionate composition of investment made on the risky-assets was independent of the optimal cash holding of the risk-averse investors. He observed that the utility function of the risk-averse investors was quadratic, and the expected returns followed the multivariate normal distribution. Lintner (1965) established a relationship between the expected returns and the rational decision rules of the risk-averse investors given the cash constraints for investing. He postulated that the equilibrium properties of assets determine the optimal portfolio of risk-assets. The asset allocation for investors/ institutional investors depends on the factors like - the risk-free rate of return, the market price of the dollar (risk), variance in project’s present value and the aggregate present value returns. Sharpe (1964), Lintner (1965), Jensen (1968) and Black (1972) proposed the asset returns can be estimated as a linear function of market risk using the capital asset pricing model (CAPM) given the consumption of the risk-averse investors and the general equilibrium economy. Merton (1969, 1973) and Fama and French (1992, 1993, 1996) criticised the CAPM as it failed to explain the cross-sectional differences in the expected returns of portfolios even when systematic risk or market risks were absent. The follow-up studies by Merton (1969, 1973) proposed an intertemporal capital asset pricing model (ICAPM) to study the consumption problem and the problem of optimal portfolio selection using constant relative-risk aversion of investors. They got results consistent with both the expected utility maxim and the limited liability of assets propositions of Markowitz (1952) and Tobin (1958) using ICAPM. Fama and French (1992, 1993, 1996) used a multi-factor explanation for the cross-sectional returns and the return anomalies, and proposed a three-factor model that provided that along with CAPM beta, the non-risk firm factors like SMB (small minus big) and HML (high minus low) explained the cross-sectional variation in the expected returns.

 

The empirical studies on portfolio optimisation under the MV framework used various measures for risk and optimisation techniques for the efficient portfolio. A study by Chan, Karceski, and Lakonishok (1999) used a factor model to generate a global minimum variance portfolio to obtain factor loadings for the estimation of covariance matrices of asset returns. However, they failed to accurately forecast the asset return covariance matrix for the optimal portfolio. Lauprete, Samarov, and Welsch (2002) simulated a multivariate distribution data to address the estimation problem influenced by marginal heavy tails, applied robust alternative estimation techniques, and found that standard variance minimisation procedure had outlier problems due to marginal heavy tails and multivariate tail dependence, and suggested that alternative estimation techniques gave better results due to lower risks.

 

De Miguel, Garlappi, and Uppal (2009) performed a comparison of 14 different estimation models for MV optimal portfolio for the US stocks and found that no model performed better than the 1/N heuristic model in terms of Sharpe ratio, and the optimal diversification failed due to estimation errors of asset return moments. Lai, Xing, and Chen (2011) used the stochastic optimisation technique for estimating an efficient frontier and highlighted that the variability in portfolio returns occurred due to variability in plug-in weights used for optimisation and estimation of covariance matrices. Lim, Shanthikumar, and Vahn (2011) used a CVaR (conditional Value at Risk) as a measure of risk for estimating conditional VaR for portfolio optimisation. They suggested that the CVaR was weak due to the estimation errors for portfolio returns.

 

Studies by Baixauli-Soler, Alfaro-Cid, and Fernandez-Blanco (2011), Lwin, Qu, and MacCarthy (2017) used MODE-GL algorithm for the estimation of optimal weights for asset allocation at the stock index level viz., EURSTOXX50 and S&P indices, respectively, and found that the algorithm out-performed MV framework as it gave adequate and better results compared to linear programming techniques or smoothing techniques. Ranković, Drenovak, Urosevic, and Jelic (2016) found that mean-GARCH VaR optimisation model outper formed mean-multivariate GARCH VaR model for US stock data, as the results were robust in both low and high volatility samples. On the contrary, (Sukono, Sidi, Bon, & Supian, 2017) used mean-VaR framework over MV framework at the individual security level, and found an optimum solution using algebra approach.

 

The literature analysis shows that the studies have used a variety of optimisation techniques for the optimal portfolio under MV framework. However, none of the techniques was robust to estimation models and different asset classes. A comparative study on the estimation models showed that the heuristics performed better than any of the sophisticated techniques for the US stock market. All the studies assumed single-period portfolio scenarios for portfolio optimisation. As portfolio rebalancing is essential for active portfolio management, in our study, we used the single-asset case of using equities as risky-securities for constructing an optimal portfolio using portfolio re-balancing techniques. We performed the comparative study of the performance of heuristic equal-weighted portfolio vis-à-vis MV optimal portfolio. The optimal weights for the MV portfolio were estimated using the linear programming techniques for the Indian stock market.

 

DATA AND METHODOLOGY:

We used the monthly closing price data of the NSE listed stocks viz.., equities traded under the Nifty50 index for the period between January 2013 and December 2017. The data is referred from the NSE website.

 

The MV framework:

We used the single-asset case for the proposed portfolios viz., the MV portfolio and the equal-weighted portfolio, where Nifty50 stocks are used for asset allocation and portfolio diversification purposes. The equal-weighted 1/N heuristic portfolio is considered by allotting equal weights on the Nifty50 stocks.

 

The MV portfolio is the one which maximises the expected returns at given a variance vis-a-vis reduces variance for the given level of expected returns under the given conditions. The expected portfolio returns are measured as the weighted average discounted expected return of securities at a time ‘t’, and the portfolio risk is calculated as the covariance between securities’ returns at a time ‘t’. The portfolio return and risk of the proposed MV portfolio are mathematically represented as (1) and (2).

 

                                                                                           (1)

Where,

represents the expected portfolio returns;  is the return on the security ‘i’ at the time ‘t’;  is the discounting rate at time ‘t’;  refers to the weight of investment allotted on the security ‘i’ and n is the number of securities in the portfolio.

 

                              (2)

Where,  is portfolio variance;  represent returns on securities ‘i’ and ‘j’ respectively at a time ‘t.’

 

The MV framework (Markowitz, 1952) recommends diversification of assets, maximisation of expected utility for investors under the full-investment and long-only constraints, and minimisation of portfolio variance.

 

The MV portfolio is proposed under the two constraints viz., 1) full-investment constraint, where, all the investible fund is used for the asset-investment. The summation of allocation weights should be equal to unity. 2) The long-only constraint, where, all the allocation weights are positive, i.e. no short-selling is admissible. The full-investment and long-only constraints are mathematically expressed as (3) and (4), respectively.

 

                                                                                         (3)

Where represent the allocation weights for optimal portfolio,  refers to the full-investment constraint.

 

                                                                                          (4)

 

Estimation of optimal weights for the MV portfolio using linear programming technique

For the estimation of the optimal weights for the proposed MV portfolio, we used the linear optimisation technique. The linear programming achieves optimisation (maximisation/minimisation) under a linear objective function, subject to the linear equality and linear inequality constraints. The solution of the linear programming problem is the objective function that gives the optimal value of the linear expression, subject to the constraints expressed as inequalities.

 

The objective function of the portfolio optimisation problem should maximise expected returns  for given variance or minimise the variance  for the given expected returns for the given constraints of full-investment and long-only constraints. The objective functions and the constraints are mathematically represented in (5), (6), (7) and (8) respectively.

 

                              (5)

                                             (6)

 

Where  represents a vector of optimal weights, Q represents the estimated covariance matrix of asset returns.

 

Constraints:

,                                           (7)

, i=1,2…n                                                         (8)

 

Portfolio optimisation scenarios:

We carried out portfolio optimisation under two-scenario viz., a) Single-period optimisation, and b) Multiple-period optimisation. Under the single-period case, optimal weights are estimated at the beginning of the portfolio formation period for one time. Under the multiple-period case, the MV portfolio is re-balanced by altering the allocation weights on the assets at the beginning of every quarter. Thus, we carried out four optimisations of the MV portfolio under the given scenarios.

 

Risk-Return of MV portfolios and Portfolio evaluation:

The portfolio return and risk of the optimised portfolios (for both rebalanced and non-rebalanced) and the equal-weighted using equation (1) and (2). We used the Sharpe ratio (Sharpe, 1963) for comparing the return-risk performances of the optimised MV portfolios vis-à-vis the 1/N heuristic portfolio. The annualised Sharpe ratio is measured as the ratio of annualised portfolio return to the annualised portfolio risk. The higher value of the Sharpe ratio implies a higher risk-adjusted return for taking higher risk. We have assumed risk-free rate as zero in the study. The mathematical representation of the Sharpe ratio is given in (9).

               (9)

 

RESULTS AND DISCUSSION:

Table 1 provides the descriptive statistics of the Nifty50 stocks. We found that 29 stocks (57%) have average positive returns, 28 stocks (55%) have average returns higher than the cross-sectional average (0.01%). 13 stocks (25%) have a higher variance than the cross-sectional average variance (0.09%). The frequency distribution of the returns of the stock shows that the returns are highly leptokurtic and skewed to the right. The SBIN (0.17%) has the highest mean returns; MARUTI (-0.14%) has the lowest mean returns. SBIN (6.73%) has the highest average volatility, and HDFCBANK (1.26%) has the lowest average volatility.

 


 

Table 1: Descriptive statistics of asset returns


Equities

Mean

SD

Variance

Kurtosis

Skewness

SBIN

0.17%

6.73%

0.45%

1038.12

30.58

BANKBARODA

0.14%

5.26%

0.28%

702.94

22.68

ASIANPAINT

0.11%

6.67%

0.44%

1124.37

32.52

ICICIBANK

0.10%

4.92%

0.24%

879.78

27.02

GRASIM

0.08%

4.76%

0.23%

970.77

29.19

AXISBANK

0.07%

4.98%

0.25%

869.01

26.80

BHEL

0.07%

2.76%

0.08%

42.39

2.92

INFY

0.06%

3.28%

0.11%

332.08

15.66

TECHM

0.04%

4.32%

0.19%

879.83

27.10

ONGC

0.03%

2.23%

0.05%

75.45

4.24

YESBANK

0.03%

5.14%

0.26%

751.12

23.98

IDEA

0.02%

2.49%

0.06%

6.97

-0.68

LT

0.02%

2.35%

0.06%

115.46

6.83

MM

0.02%

2.62%

0.07%

478.58

17.11

SUNPHARMA

0.02%

2.67%

0.07%

318.55

12.86

WIPRO

0.02%

2.49%

0.06%

517.35

18.34

COALINDIA

0.01%

1.78%

0.03%

3.56

-0.08

ITC

0.01%

1.91%

0.04%

123.69

6.42

NTPC

0.00%

1.65%

0.03%

6.32

0.48

ACC

-0.01%

1.55%

0.02%

1.33

-0.19

DRREDDY

-0.01%

1.72%

0.03%

8.49

0.82

RELIANCE

-0.01%

2.51%

0.06%

469.37

16.87

TATAMOTORS

-0.01%

2.17%

0.05%

2.05

0.05

TATAMTRDVR

-0.01%

2.25%

0.05%

1.72

0.10

AMBUJACEM

-0.02%

1.75%

0.03%

2.21

0.09

BHARTIARTL

-0.02%

1.90%

0.04%

1.69

-0.32

BPCL

-0.02%

3.09%

0.10%

253.97

11.66

GAIL

-0.02%

2.04%

0.04%

45.33

2.79

LUPIN

-0.02%

1.77%

0.03%

11.55

1.18

BAJAJAUTO

-0.03%

1.48%

0.02%

2.56

-0.08

CIPLA

-0.03%

1.59%

0.03%

3.07

0.03

HCLTECH

-0.03%

2.64%

0.07%

428.99

15.92

TATASTEEL

-0.03%

2.29%

0.05%

2.73

0.01

INFRATEL

-0.04%

2.20%

0.05%

1.89

0.02

KOTAKBANK

-0.04%

2.53%

0.06%

467.50

16.79

POWERGRID

-0.04%

1.42%

0.02%

4.91

0.34

HINDALCO

-0.05%

2.50%

0.06%

1.37

-0.11

BOSCHLTD

-0.06%

1.63%

0.03%

2.58

-0.52

HDFC

-0.06%

1.69%

0.03%

1.65

-0.11

ULTRACEMCO

-0.06%

1.69%

0.03%

1.60

-0.09

TCS

-0.07%

1.46%

0.02%

2.40

0.19

HDFCBANK

-0.08%

1.26%

0.02%

4.04

-0.14

ZEEL

-0.08%

1.87%

0.03%

1.05

-0.16

ADANIPORTS

-0.09%

2.41%

0.06%

2.60

-0.13

AUROPHARMA

-0.09%

3.06%

0.09%

188.42

8.37

INDUSINDBK

-0.11%

1.82%

0.03%

3.32

0.30

MARUTI

-0.14%

1.62%

0.03%

2.94

-0.32

Source: Authors’ computation

 


Table 2 shows the estimated portfolio weights using a linear programming technique under the Single-period optimisation scenario. It is observed that 24 stocks (47%) have an estimated optimal weight ≥ 0.17%, TCS (13.28%) has the highest estimated optimal weight followed by POWERGRID (1.084%), HDFCBANK (10.31%), DRREDDY (7.60%) and BOSCHLTD (6.29%).

 

Table 2: Optimal weights for the MV Portfolio (Single-period Optimization)

Stock

Optimal weights estimated (%)

TCS

13.28%

POWERGRID

10.84%

HDFCBANK

10.31%

DRREDDY

7.60%

BOSCHLTD

6.29%

LUPIN

6.24%

INFRATEL

4.82%

COALINDIA

4.54%

ITC

4.54%

NTPC

4.47%

CIPLA

4.31%

BAJAJ_AUTO

3.99%

WIPRO

3.59%

BHARTIARTL

3.29%

HCLTECH

2.83%

ACC

2.43%

MARUTI

2.19%

INFY

1.38%

MM

0.92%

ZEEL

0.73%

IDEA

0.52%

RELIANCE

0.48%

SUNPHARMA

0.23%

GAIL

0.17%

Others

0.17% <  ≥ 0.01%

Source: Authors’ computation

 

 

 

Table 3 shows the return-risk measures and the Sharpe ratio for the optimised MV portfolios under the single-period scenario (non-rebalanced). It is observed that the maximum-return MV portfolio and the equal-weighted portfolio gave an optimal portfolio return of 0.06% with a portfolio variance of 1.021%, and Sharpe ratio of 0.006. The maximum-return MV portfolio has mimicked the performance of a 1/N heuristic portfolio of equal-weight allocation on the stocks. The minimum-variance MV portfolio has an estimated portfolio return of 0.035% with a portfolio variance of 0.755% and a Sharpe ratio of 0.046. The higher portfolio return and Sharpe ratio and lower portfolio variance values indicate that the minimisation of variance has better performance than the 1/N portfolio and the maximum-return MV portfolio.

 

Table 3: Optimal portfolio performance measures (Single-period optimisation scenario)

Optimal Portfolios

Portfolio Return

Portfolio Risk (std. dev in %)

Sharpe ratio

Minimum Variance

0.035%

0.755%

0.046

Maximum Return

0.006%

1.021%

0.006

Equal-weighted Portfolio

0.006%

1.021%

0.006

Source: Authors’ computation

 

Table 4 shows the estimated portfolio return-risk for the optimised MV portfolios using the quarterly portfolio-rebalancing technique. It is observed that the rebalanced risk for the minimum-variance MV portfolio has been increasing over the rebalancing periods, while the quarterly rate of return is decreasing. The lowest rebalanced risk is 0.35%, with a quarterly return of 0.64% recorded during the first quarter (January – March 2013). For maximum-returns MV portfolio, the highest quarterly return (1.33%) is recorded in the first quarter (January - March 2013) with a standard deviation of (-0.19%).


 

Table 4: Portfolio return-risk for MV portfolios using portfolio rebalancing under multi-period optimization scenario

Rebalancing Dates

Minimum Variance Portfolio

Maximum Return Portfolio

Rebalanced Risk (%)

Quarterly Return (%)

Rebalanced Return (%)

Quarterly Risk (%)

28-03-2013

0.35%

0.64%

-0.19%

1.33%

28-06-2013

0.58%

0.05%

-0.10%

0.05%

30-09-2013

0.69%

-0.03%

0.12%

-0.15%

31-12-2013

0.69%

0.15%

0.08%

0.19%

31-03-2014

0.67%

0.04%

0.34%

0.11%

30-06-2014

0.68%

0.24%

0.44%

0.27%

30-09-2014

0.68%

0.21%

0.16%

0.05%

31-12-2014

0.70%

0.04%

-0.24%

-0.10%

31-03-2015

0.72%

0.03%

-0.15%

-0.06%

30-06-2015

0.75%

-0.04%

0.01%

-0.06%

30-09-2015

0.78%

-0.03%

-0.28%

-0.15%

31-12-2015

0.78%

-0.04%

0.02%

0.03%

31-03-2016

0.80%

-0.07%

0.16%

-0.08%

30-06-2016

0.80%

0.11%

0.15%

0.12%

30-09-2016

0.79%

-0.05%

-0.05%

0.03%

30-12-2016

0.79%

-0.16%

-0.15%

-0.16%

31-03-2017

0.78%

0.07%

0.02%

0.15%

30-06-2017

0.77%

-0.11%

-0.09%

-0.04%

29-09-2017

0.77%

-0.01%

-0.24%

-0.11%

29-12-2017

0.76%

0.03%

-0.01%

0.09%

20-02-2018

0.76%

-0.08%

-0.12%

-0.14%

Source: Authors’ computation

 


Figure 4 provides a graphical representation of the estimated optimal weights for the MV portfolios under the multi-period re-balancing scenario.


 

 

Figure 4: Estimated optimal Weights for MV Portfolio under the multi-period scenario


 

Table 5 shows the annualised performance measures of optimised MV portfolios using the rebalancing technique. The minimum-variance MV portfolio has an annualised portfolio return of 3.727% with an annualised risk of 14.47% and a Sharpe ratio of 0.258. The maximum-return MV portfolio mimick the 1/N heuristic portfolio with an annualised return of 0.311%, annualised portfolio risk of 16.484% and a Sharpe ratio of 0.019.

 


 

Table 5: Portfolio return-risk performance of rebalanced portfolios

Optimal Portfolios

Annualized Portfolio Rebalancing Return

Annualized Portfolio Rebalancing Risk (std. dev in %)

Annualized

Sharpe ratio

Minimum Variance

3.727%

14.447%

0.258

Maximum Return

0.311%

16.484%

0.019

Equal-weighted Portfolio

0.311%

16.484%

0.019

Source: Authors’ computation

 


DISCUSSION:

The findings (Table 3 and Table 5) indicate that the multi-period optimisation of MV portfolio using the portfolio re-balancing technique has given an optimal annual return of 3.727% with the highest Sharpe ratio of 0.258 compared to the other portfolios. The 1/N heuristic portfolio and the maximum-return MV portfolios have lower return and Sharpe ratio and highest risk compared to the minimum-variance MV portfolio under all the scenarios. Therefore, our findings do not support the results of (DeMiguel et al., 2009). We found that the estimated optimal weights varied over the re-balancing period resulting in the variation of optimal portfolio returns. Our observation abides by (Lai et al., 2011) that the portfolio variance is a linear function of portfolio weights.

 

FINDINGS AND CONCLUSION:

The study empirically tested the Markowitz’s Mean-Variance optimal portfolio using the linear programming optimisation technique. The twin-objectives of portfolio optimisation is to achieve maximum expected utility for investors at minimum variance. We tested the objective functions of maximum-utility and minimum-variance for MV portfolios under the constraints of full-investment and long-only conditions. The portfolio optimisation was performed under the single-period scenario and the multi-period scenarios using the portfolio re-balancing technique. We found that the minimum-variance portfolio out-performed the 1/N heuristic portfolio in terms of optimal return-risk and Sharpe ratio in both scenarios of single-period and multiple-period optimisations. Our findings contrast (DeMiguel et al., 2009). The study showed that though the portfolio rebalancing gave better returns for given risk for minimum-variance portfolio, the variability in portfolio returns is attributed to the variability in the allocation of estimated weights, in support of (Lai et al., 2011).

 

The linear optimisation technique didn’t perform well as the technique didn’t optimise the allocation weights for the maximum-returns portfolio. The estimation The estimation failed due to the presence of asset returns properties  to the asset returns properties, i.e. the skewness and excess kurtosis in the asset returns and the associated marginal heavy tails (Lauprete et al., 2002). There is a lot of future research scope as the study can be improved by using sophisticated non-linear estimation techniques for estimating optimal weights for Markowitz optimisation, and estimation of asset returns moments. This study has contributed to the field of portfolio management by highlighting the importance of periodic portfolio rebalancing for achieving optimal returns.

 

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Received on 08.06.2019                Modified on 21.06.2019

Accepted on 10.07.2019           ©AandV Publications All right reserved

Asian Journal of Management. 2019; 10(3): 248-254.

DOI: 10.5958/2321-5763.2019.00039.8