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Friday 9 August 2013

Dissertation proposal: Low-risk effect in the U.K. stock market

I.       INTRODUCTION

The efficient market hypothesis has been challenged for many decades and there has not been a consensus among academic researchers about whether investors can earn excess return over the market. Many studies have given evidence that some investment strategies can actually outperform the market. These include technical analysis, value investing, contrarian investing, or small firm effects, etc. Such strategies have also been adopted in real-world investment and proved to be successful.

Concerning risk and return in investment, investors are always risk-averse while preferring high return. The mean-variance portfolio theory, which is initiated by Markowitz (1952), proposes that risk and return have negative relationship. Investors cannot earn higher return if they are unwilling to take more risk, which is measured by volatility of stock prices. Total risk consists of systematic risk and unsystematic risk. However, as reasoned by Sharpe (1964), investors cannot be rewarded with higher return if they bear more unsystematic risk because such risk can be eliminated for free through diversification. Only systematic risk is relevant for investor decision. In the Capital Asset Pricing Model (CAPM) proposed by Sharpe (1964) and Lintner (1965), expected return of an asset is linearly related to its beta, which is the standardized covariance between the individual asset and the market portfolio. That relationship is represented by the security market line. In market equilibrium, all assets should be plotted on the security market line.

Although the CAPM together with the efficient frontier and the capital market line are written in most finance books and widely taught in investment courses, there have been many debates around them. The security market line in CAPM has been noticed to be much flatter than being proposed by the theory (Black, Jensen, & Scholes, 1972, Fama & MacBeth, 1973, or Fama & French, 2004), that is low-beta stocks earns higher alphas than high-beta stocks. More recently, some studies have given evidence that low-risk equity portfolios, as measured by volatility or beta, outperform high-risk portfolios in terms of risk-adjusted returns, and even absolute returns (Clarke, De Silva, & Thorley, 2006, Blitz & Van Vliet, 2007, Ang et al., 2009, or Baker, Bradley, & Wurgler, 2011).

The evidence of low-risk effect has only been documented mainly in the U.S. market, but has not been widely acknowledged among the investment community. If that effect really exists in real-world, investors will have many benefits when exploiting that anomaly. Motivated by the unanswered question, the paper aims to investigate if the low-risk effect is present in the U.K. stock market.

II.    RESEARCH QUESTION AND OBJECTIVE

Title of the dissertation: Low-risk effect in the U.K. stock market.
Research question: Do low-risk portfolios, as measured by standard deviation and beta calculated from lagged price volatility, generate higher risk-adjusted returns in the U.K. stock market?
Research objective:
-       This paper seeks to investigate performance of low-risk portfolios in the U.K. stock market and compare it with the performance of high-risk portfolios and the general market.
-       Other well-known effects such as firm size, value, or momentum will also be analyzed in order to identify whether the low-risk effect overlaps with them.
-       This paper will give some possible explanations about the anomaly and implications for investors.

III. LITERATURE REVIEW

1.      Traditional theory about the trade-off between risk and return

In the mean-variance portfolio theory initiated by Markowitz (1952), one of the major assumptions is that investors are risk-averse, which means that among a group of investment opportunities with similar expected returns, investors will seek for the one that has the lowest risk, and they will only accepted a riskier investment if they are compensated by greater expected return. Risks and returns are the only two factors being considered by investors in their investment decisions. Expected returns are subject to a probability distribution under different scenarios over an investment time period, and risks are measured by the variance or standard deviation of expected returns. A combination of different levels of expected returns for different levels of risks maps the indifference curves of investors.

Because of investors’ nature of risk aversion, they will only invest in portfolios which are deemed to be efficient. A portfolio is efficient if there is no other portfolio offering a higher expected return at the same or lower level of risk or requiring lower risk for the same or higher amount of return. Combining a set of efficient portfolios for different levels of risk will create the efficient frontier, which is exhibited in Figure 1. Once investors are on the efficient frontier, it is impossible to increase expected returns without increasing risk (Sharpe, 1970: 32). If the capital market is efficient and investors have homogeneous expectations about risk and return, the efficient frontier implies that risk and return will have a positive relationship with each other. Being on the efficient frontier, investors will only earn higher return if they are inclined to take more risk. Delineating the efficient frontier from a set of individual securities today is quite a simple task with the assistance of computer program. The general approach is to estimated expected returns and covariance matrix for individual securities. For each level of expected return, the computer program can determine the weight of each security in the portfolio so that the portfolio’s variance will be minimized (Clarke, De Silva, & Thorley, 2006).

Figure 1. The efficient frontier and the capital market line

Combining a risk-free asset with the risky portfolio on the efficient frontier will create the capital market line, as being shown in Figure 1. Investors at the point R­f invest 100% of their capital into the risk-free asset, while investors at the point M invest all their capital into the risky market portfolio. Theoretically, the market portfolio consists of all types of risky assets, including stocks, bonds, alternative investments, so it is completely diversified. When moving on the capital market line to the left of point M, investors invest part of their funds into the risk-free asset and the remaining in the market portfolio. When switching to the right of point M, investors borrow money to buy more of market portfolio. Risks increase as investors move along the capital market line.

The total risk (σ) of a portfolio or a security comprises of systematic risk and unsystematic risk. Theoretically, unsystematic risk can be diversified away by combining various securities which have low correlations. Therefore, in market equilibrium, investors are not compensated for bearing unsystematic risks because it can be removed by diversification at no cost. The only relevant source of uncertainty about the return of an asset is its systematic risk (Sharpe, 1970: 97). The expected return will depend on the systematic risk, which is the covariance between the return of the asset and the whole market.


With the above argument about systematic risk, Sharpe (1964) and Lintner (1965) develops the famed Capital Asset Pricing Model (CAPM), which is used to determine the value of an asset based on its systematic risk. The CAPM proposes that the expected return of an asset has positive linear relationship with its systematic risk. That relationship creates the security market line, which is exhibited in Figure 2.
Figure 2. The security market line

In Figure 2, the systematic risk of an asset is defined as βi, which is the standardized covariance between the return on a single portfolio or asset and the market return. In market equilibrium, all securities and portfolios, whether diversified or not, should be plotted on the security market line. When βi = 0, return on the asset is equal to the risk-free rate. When 0 < βi < 1, the asset has lower risk than the market portfolio, thus generating less return than the market, but higher than the risk-free rate. When βi = 1, the asset is perfectly correlated with the market, and its return is expected to be the similar the market. When βi > 1, the asset is considered to be riskier than the market and is expected to generate higher return than the market. The slope of the security market line is equal to the excess return on the market portfolio above the risk-free rate, that is E(Rm) – Rf .

In summary, it is generally believed that investors can expect higher return when investing in assets that have higher risk, as measured by variance in return. According to the CAPM, the only risk investors should concern about is the systematic risk, measured by beta. Investors should not expect additional return when bearing unsystematic risk because it can be fully eliminated by means of diversification.

2.      Empirical tests of the relationship between return and risk

Some early empirical tests partly support the traditional view of the risk-return relationship. One of the studies which are in favor of the CAPM is that conducted by Sharpe and Cooper (1972). They create decile portfolios from all New York Stock Exchange stocks ranked on their betas. The betas are estimated from 5-year historical data. The decile portfolios are held in one year and rebalanced annually from 1931 to 1967. They find that the portfolios comprising of higher beta stocks have produced higher returns. An increase of one unit in beta will generate annual excess return of 12.75%. In addition, the variance in returns of these portfolios is mainly explained by betas, which represent systematic risk. The coefficient of determination in regressing portfolio returns on market returns ranked from 0.87 to 0.94.

However, other tests of the CAPM do not have consistent results with the theory. The studies of Black, Jensen, and Scholes (1972) and Fama and MacBeth (1973) are regarded as standard in testing the theory of CAPM. They form portfolios ranked on their betas and conduct a test of the traditional CAPM:
Rit – RFt = αi + βi (RMt – RFt) + eit
In the test by Black, Jensen, and Scholes (1972), low beta portfolios have positive alphas, and high beta portfolios have negative alphas, while the CAPM suggests that the alphas should be zero in market equilibrium. When they divide their data from 1931 to 1965 into four sub-periods, they find that the slopes of the security market lines were not constant. The security market line became flatter over time. In the last period of 1957-1965, it even had the negative slope.

Fama and MacBeth (1973) found similar results to Black, Jensen, and Scholes (1972). They form 20 portfolios from stocks in the NYSE ranked on their beta and observe their returns from 1926 to 1968. Although there is a positive relationship between beta and average return, the security market line is too “flat” compared with what is proposed by the CAPM. The intercept of the security market line is consistently higher than the risk-free rate, and the slope coefficient of beta is smaller than the excess return of the market. However, they find some evidences that are consistent with what being suggested by the CAPM: assuming the market is efficient, the unsystematic risk has no effect on the expected return of a security, and while the slope of the security market line is “flat”, it shows a linear relationship between expected return and beta. Fama and French (2004) once again confirm the “flat” security market line using data of stocks listed on NYSE (1928-2003), AMEX (1963-2003), and Nasdaq (1972-2003).

More recently, studies by Blitz and Van Vliet (2007) or by Baker, Bradley, and Wurgler (2011) show that low-beta and low-volatility stocks outperform high beta and high volatility stocks. In the mean-variance portfolio theory, portfolios that are on the capital market line should dominate all other portfolios. However, Blitz and Van Vliet (2007), using data from 1986 to 2006 in the U.S., Europe and Japan markets, demonstrate that portfolios whose total volatility is lower than the market are more efficient than portfolios lying on the theoretical capital market line. They also argue that the security market line has a negative slope if that line is created by plotting actual returns of portfolios with different level of betas. When the Sharpe ratio is taken into account, there is no reason to invest into high-risk portfolio because they generate low risk-adjusted returns. The volatility effect is a separate effect and does not relate to other effect such as value, size, or momentum.

Especially, the results found by Baker, Bradley, and Wurgler (2011) are stunning. They sort 1,000 largest stocks in the U.S. market by their 5-year total volatility and betas. From the sorting, the stocks are put into five portfolios, ranked by their level of risks in terms of volatility and betas. Each month, the portfolios are rebalanced so that their levels of risk are maintained. They find that a U.S. dollar invested in the low-volatility portfolio grows to USD 53.81 over 41 years, while a dollar only grew to USD 7.35 if invested in the high-volatility portfolio. For beta, the numbers are USD 78.66 and USD 4.70, respectively. The return gap has widened since the early 1980s, with the only exception in the technology stock bubble.

Another approach to capture the low-risk anomaly is to form the minimum-variance portfolio. Clarke, De Silva, and Thorley (2006) select stocks from 1,000 largest stocks in the U.S. to form a portfolio that has the minimum variance and calculate its returns from 1968 to 2005. The mean annual return of the minimum-variance portfolio in excess of the risk-free rate is 6.5%, while return on the market portfolio is lower at 5.6%. The standard deviation of the minimum-variance portfolio is also lower than the market. High excess return and low standard deviation lead to significantly better Sharpe ratio. The concept of efficient frontier is invalid in this case because the minimum-variance portfolio lies outside the theoretical efficient frontier. When returns of the minimum variance portfolio are regressed on the returns of the market portfolio, the annualized alpha is 2.8% and beta is 0.65.

The CAPM proposes that idiosyncratic risk (or unsystematic risk) is not relevant in determining expected return of an asset. Investors should not expect higher return when taking additional unsystematic risk because it can be simply eliminated by diversification. However, it is only the case when the market is efficient and frictionless, that is no tax, no transaction cost, no restrictions in borrowing and short-selling, and all information is readily available at no cost. In environment of market imperfection, unsystematic risk may be related to expected return (Ang, Hodrick, Xing, & Zhang, 2009).

The effect of idiosyncratic risk is a real puzzle. Studies conducted by Tinic and West (1986), Lehmann (1990), Barberis and Huang (2001), or by Malkiel and Xu (2006) find that higher returns may be expected from greater amount of idiosyncratic volatility. However, Ang, Hodrick, Xing, and Zhang (2006, 2009) argue that low return is associated with both high systematic risk and indiosyncratic volatility in the U.S. market. When regressing stock returns against total volatility and some other factors affecting returns, they find that the regression coefficient of total volatility carries a negative sign, which means that stocks with high systematic risk would have low average returns. In addition to this, when quintile portfolios are formed by sorting idiosyncratic risks of stocks, the lowest volatility portfolio generates returns that are higher than the highest risk portfolio by an average of 1.06% per month over the 1963-2000 period. The result is robust after being controlled for the effects of size, book-to-market, liquidity risk, trade volume, and momentum effects. Changing the period for calculating the volatility or the holding period does not diminish the robustness of the result. There are similar results about idiosyncratic risk when the study is expanded to other developed markets. Their criticism about previous studies is that such studies do not investigate idiosyncratic risk at the level of firms and not group firms into portfolios sorted by their idiosyncratic volatility, thus failing to identify the negative correlation between unsystematic risk and expected return.

Motivated by the finding of Ang, Hodrick, Xing, and Zhang (2006), Bali and Cakici (2008) conduct similar research about the relationship between idiosyncratic risk and return. They acknowledge the results of Ang, Hodrick, Xing, and Zhang (2006). However, they argue that the results may be biased because the highest idiosyncratic quintile represent only 2% of total market capitalization. Those stocks are very illiquid, small and always have low price. After modifying the methodology so that it would not be biased by the effect of small and illiquid stocks, no significant relationship between unsystematic risk and return is found.

In summary, most empirical studies have found that the security market line, which represents the relationship between stock returns and betas, is much flatter than predicted by the CAPM. While early studies in the 1970s and 1980s acknowledge the positive relationship between stock returns and systematic risks (but flatter than proposed by the CAPM), some studies in the 2000s noticed that the relationship is negative. Stock returns are also negatively correlated to total volatility. The low volatility  anomaly has become more obvious since the mid-1980s, with the exception in the technology stock bubble. Regarding the idiosyncratic risks, there is mixed evidence about its effect on returns.

The inefficiency in the relationship between risk and return may be explained by the unrealistic assumptions in the mean-variance portfolio theory and CAPM, such as investors have homogeneous expectations and investment periods, information is readily available at no cost and risk-free borrowing, lending, and short-selling are unrestricted. The difficulties in correctly estimating expected returns, betas and covariance among securities may also be an obstruction against market efficiency. Besides, in the real world, it is impossible to determine the market portfolio, which theoretically consists of all risky assets in the market. Baker, Bradley, and Wurgler (2011) explain the mispricing of high-risk stock from the perspective of behavioral finance. There is an irrational preference for high volatility, which causes high-risk stocks to be overpriced and generate low returns. For institutional investors, they are unlikely to exploit the low-risk anomaly because of performance benchmarks (Blitz & Van Vliet, 2007). Portfolio managers have an incentive to invest in high-risk stocks, as it is easy to generate above-average returns. In addition, many investment mandates restrict the use of leverage, thus making low-risk stock not the choice of investment managers. Without leverage, it is hard to earn high return on low-risk stocks, which are generally believed to have low returns. The restrictions in the use of leverage by institutional investors make low-risk stocks underpriced.

IV. METHODOLOGY

This paper will examine the relationship between stock returns and risks, measured by volatility and betas, to investigate whether the volatility effect exists in the U.K. stock market, that is, low-risk portfolios may generate high risk-adjusted return.

Most studies about the relationship between risk and return use portfolios ranked on their risk level instead of calculating the correlations from individual securities. If returns are regressed against risks for each individual security, there will be autocorrelation problem because the residuals of the regression are not independent (Elton, Gruber, Brown, & Goetzmann, 2007). Besides, the estimation of beta for each security may be incorrect. Therefore, in order to avoid these problems, the standard in conducting this type of research is to form portfolios and rank them on their risks, rather than working with individual security. Then the relationship between risk and return of each portfolio is specified in each holding period. Since the portfolios represent diversification and incorporate the effect of any cross-sectional interdependencies, the betas estimated for portfolios are more precise and the problem of residual correlation is resolved.

1.      Data collection

Data of all stock constituents in the FTSE All-Share Index will be collected in 31 years, from 1980 to 2010. The main source of data will be Reuters database. The FTSE All-Share Index currently represents approximately 98% of the UK’s market capitalization. The large market share of the FTSE All-Share Index ascertains that the results of this study will not be biased.

In order to calculate the volatility and beta of stocks, stock prices are collected on a weekly basis. Weekly returns are log-transformed in order to make them additive over time:
Standard deviation of return is calculated using weekly return in the past three years (156 weeks):
The FTSE All-Share is considered to be the market portfolio. Beta of each stock is calculated by regressing stock weekly return on weekly return of FTSE All-Share Index:
Apart from price, other data that should be collected includes book value, market capitalization, industry sector of each stock and the risk-free rate.

2.      Portfolio construction and evaluation

Stocks are ranked on standard deviation and grouped into quintiles in order to create five equally weighted portfolios with different risk level. The first quintile consists of lowest volatility stocks, the fifth quintile consists of the highest volatility stocks. Portfolios are rebalanced at the beginning of each month so that the first portfolio always has the lowest level of volatility. Transaction costs are ignored during the analysis.

After portfolios have been constructed, monthly returns of each portfolio are calculated at the time of rebalancing. The monthly returns are tracked in the whole period of 31 years. In the end, the following measures will be calculated for each portfolio and the market (represented by FTSE All-Share Index):
-       Mean monthly returns and standard deviations.
-       Returns of portfolios in excess of the risk-free rate, which is determined by the yield of 3-month U.K. government bonds since such short-term bonds have the lowest risk among all other kinds of bonds.
-       The Sharpe ratio of portfolios:
-       Alphas (excess return over the market) and betas of portfolios, by using the CAPM model to regress the excess return of portfolios on the market risk premium.
                                                Ri – Rf = αi + βi (RM – Rf) + ei                                      (1)

Comparisons are made among the quintile portfolios and the market in order to determine whether investing in the low-volality portfolios will be more efficient and generate any excess return. The following hypothesis will be tested:
-       There is difference among mean returns of the low-volatility portfolios and the high-volatility portfolios and the market. A t-statistic will be used to test the difference in mean (Kaplan, 2009):
-       If the market is efficient, the alphas in (1) should be zero. If the alphas in (1) are significantly different from zero for a portfolio, that portfolio is believed to generate excess return over the market. To test that hypothesis, a t-statistic will be used:
-       There is difference among Sharpe ratios of the low-volatility portfolios and the high-volatility portfolios and the market. The method of Jobson and Korkie (1981) and Memmel (2003) is used to test the hypothesis. The test statistic follows a standard normal distribution:
Where SRi is the Sharpe ratio of portfolio i, ρi,j is the correlation between portfolio i and j, n is the number of observations.

To investigate the effect of beta, a similar procedure is conducted. Stocks are allocated into five portfolios following their beta ranking. Performance of the quintile portfolios is then examined and compared.

3.      Controlling for other effects

One may suspect that the low-risk effect may be related to other effects which have been acknowledged such as size, value, or momentum. If the low-risk effect still exists after being controlled for other effect, it can be determined that the low-risk effect is a separate and independent one.

3.1. Construct portfolio ranked on size, value and momentum

It has been well documented in several researches that returns on small firms tend to outperform large firms, firms having high book-to-market value tends to outperform low-book-to-market value, and past winning stocks are likely to continue making excess returns in the short-term.

In this part, a method which is similar to part 2 is used to construct portfolios. Stocks are ranked on market capitalization, book-to-market ratio, 12-month minus 1-month total return and allocated to quintile portfolios. For each strategy, portfolio performance is evaluated in terms of return, Sharpe ratio, alpha, and volatility. If performance measurements of the best-performing portfolios in each strategy are not better than the performance of low-risk portfolio, while their volatility is higher, the low-risk anomaly can be considered to be a separate effect.

3.2. Control for size and value effect using the Fama-French regression

In this section, alphas of the quintile portfolios ranked on volatility and beta in part 2 are determined by the model of Fama and French (1993). The regression model is as follows:
                                       Ri – Rf = αi + βi (RM – Rf) + si SMB + hi HML + ei                           (2)
Where:
-       SMB is the return of the portfolio composed of the 30% smallest firms minus the return of the portfolio of 30% largest firms with respect to market capitalizations in the markets. 
-       HML is the return of the portfolio consisting of the 30% highest book-to-market firms minus the return of the portfolio consisting of 30% lowest book-to-market firms in the markets.
The Fama-French model captures the effects of firm size and value. If the market is efficient, the alpha in (2) should be zero. If the alpha of the low-risk portfolio is signficantly higher zero, the low-risk portfolio is believed to generate excess return over the market, after being controlled for the effect of firm size and value.

3.3. Double sorting

Control for value: Every month stocks are first grouped into five quintiles with respect to their book-to-market ratios. Then in each value quintiles, stocks are sorted into five groups ranked by volatility. Finally, the lowest-volatility groups of each value quintile are allocated into to the lowest-volatility portfolio. Four other portfolios ranking on value, then by volatility are formed in the same way. The performance of each final quintile portfolio is evaluated in the same way as in part 2. If the low-risk portfolio after being controlled for value still outperforms the market, an anomaly can be identified.

Control for size and momentum: The double-sorting method is similar as above in order to investigate the low-risk anomaly.

V.    EXPECTED FINDINGS

It is expected to find that the lowest-risk portfolio in the U.K. stock market is more efficient than the market portfolio and has superior risk-adjusted return over the high-risk portfolio in terms of Sharpe ratio and CAPM alpha. However, the results may not be as expected since recent discovery of empirical evidence has mainly concentrated in the U.S. market. There has been little knowledge of the anomaly outside the U.S. market. Besides, the low-risk effect has not been widely acknowledged by both academics and practitioners.

If the anomaly really exists, investors can take advantage of it by increasing their leverage or reducing investments in fixed-income securities. Following that investment style, they can earn higher returns while bearing the same level of risks.

For more theory and case studies on: http://expertresearchers.blogspot.com/

REFERENCE

Ang, A., Hodrick, R. J., Xing, Y., & Zhang, X. (2009). High idiosyncratic volatility and low returns: International and further U.S. evidence. Journal of Financial Economics , 91 (1), 1-23.
Ang, A., Hodrick, R. J., Xing, Y., & Zhang, X. (2006). The Cross-Section of Volatility and Expected Returns. Journal of Finance , 61 (1), 259-299.
Baker, M., Bradley, B., & Wurgler, J. (2011). Benchmarks as Limits to Arbitrage: Understanding the Low-Volatility Anomaly. Financial Analysts Journal , 67 (1), 40-54.
Bali, T. G., & Cakici, N. (2008). Idiosyncratic Volatility and the Cross Section of Expected Returns. Journal of Financial & Quantitative Analysis , 43 (1), 29-58.
Barberis, N., & Huang, M. (2001). Mental Accounting, Loss Aversion, and Individual Stock Returns. Journal of Finance , 56 (4), 1247-1292.
Black, F., Jensen, M. C., & Scholes, M. (1972). The Capital Asset Pricing Model: Some Empirical Tests. In M. C. Jensen, Studies in the Theory of Capital Markets. New York: Praeger.
Blitz, D. C., & Van Vliet, P. (2007). The Volatility Effect. Journal of Portfolio Management , 34 (1), 102-113.
Clarke, R., De Silva, H., & Thorley, S. (2006). Minimum-Variance Portfolios in the U.S. Equity Market. Journal of Portfolio Management , 33 (1), 10-24.
Elton, E. J., Gruber, M. J., Brown, S. J., & Goetzmann, W. N. (2007). Modern Portfolio Theory and Investment Analysis. Hoboken: John Wiley & Sons.
Fama, E. F., & French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics , 33 (1), 3-56.
Fama, E. F., & French, K. R. (2004). The Capital Asset Pricing Model: Theory and Evidence. Journal of Economic Perspectives , 18 (3), 25–46.
Fama, E. F., & MacBeth, J. D. (1973). Risk, Return, and Equilibrium: Empirical Tests. Journal of Political Economy , 81 (3), 607-636.
Jobson, J. D., & Korkie, B. M. (1981). Performance Hypothesis Testing with the Sharpe and Treynor Measures. Journal of Finance , 36 (4), 889-908.
Kaplan. (2009). CFA Level 1 Book 1: Ethical and Professional Standards, and Quantitative Methods. La Crosse: Kaplan Schweser.
Lehmann, B. N. (1990). Residual risk revisited. Journal of Econometrics , 45 (1), 71-97.
Lintner, J. (1965). The Valuation of Risk Assets and The Selection of Risky Investments in Stock Portfolios and Capital Budgets. Review of Economics & Statistics , 47 (1), 13-37.
Malkiel, B. G., & Xu, Y. (2006). Idiosyncratic Risk and Security Returns. Working Paper, The University of Texas at Dallas.
Markowitz, H. (1952). Portfolio Selection. Journal of Finance , 7 (1), 77-91.
Memmel, C. (2003). Performance Hypothesis Testing with the Sharpe Ratio. Finance Letters , 1 (1), 21-23.
Miller, M. H., & Scholes, M. S. (1972). Rates of Return in Relation to Risk: A Re-examination of Some Recent Findings. In M. Jensen, Studies in the Theory of Capital Markets. New York: Praeger.
Sharpe, W. F. (1964). Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. Journal of Finance , 19 (3), 425-442.
Sharpe, W. F. (1970). Portfolio Theory and Capital Markets. New York: McGraw-Hill.
Sharpe, W. F., & Cooper, G. M. (1972). Risk-Return Classes of New York Stock Exchange Common Stocks, 1931-1967. Financial Analysts Journal , 28 (2), 46-81.
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    TOP FACTS OF WATER PURIFIER IN EVERYDAY LIFE-RO Service
    Do you and your family often suffer from water-borne diseases? Does your water look stale and smell rotten? And you want to be trying to find the simplest water purifier! Well, you’ve knocked the proper door, we are here at your rescue!
    Improve Your Health by Using the Best RO Water Purifier - Improve Your Health by Using the Best RO Water Purifier
    Importance Of Having Water Purifier And Knowing A Good RO Water Purifier Service Centre In Delhi, Gurgaon
    How Do I Locate The Best RO Service Center in Gurgaon?
    Importance of RO Water Purifiers in Delhi
    Service for RO Water Purifiers in Delhi- Pahuja Aqua Service
    Mineral Water Purifier Effective Uses- Gurgaon, Delhi

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