Beta Based Portfolio Construction: : Stock Selection Based on Upside- and Downside Market Risk

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ÖREBRO UNIVERSITY School of Business

Economics, Master Thesis Supervisor: Niclas Krüger Examiner: Dan Johansson Fall 2017

Beta Based Portfolio Construction:

Stock Selection Based on Upside- and Downside Market Risk

Authors:

Andreas Johansson 1992-01-27 Robin Petersson 1992-01-21

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Abstract

In the traditional Capital Asset Pricing Model (CAPM), a stock’s beta represents an average of its upside- and downside market risk. As a result, it may not be an accurate reflection of how the stock tends to move in market upturns and downturns respectively. The purpose of this study is to investigate if it is possible to achieve a higher risk-adjusted return by selecting stocks based on their upside- and downside beta when constructing financial portfolios. Using weekly Swedish stock data from 1995-12-29 to 2017-06-30, and the Stockholm Benchmark Index (SBX) as a proxy for the market portfolio, upside- and downside stock betas are estimated over a rolling two-year period using a single-factor model. The upside- and downside betas, as well as the difference between the two measures, are ranked and the stocks are sorted into equally weighted quintile portfolios. Each portfolio is then rebalanced every six-month and tested out-of-sample against an equally weighted benchmark index. The results indicate that although there seems to exist an upside-downside beta asymmetry, the portfolio performance seems to be largely explained by the traditional CAPM beta.

Keywords: Capital Asset Pricing Model, conditional CAPM, upside beta, downside beta,

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Acknowledgements

We would like to thank everyone who has helped us with this thesis by providing feedback and proof-reading the text. We would also like to thank our supervisor, Niclas Krüger, for always being available and willing to discuss any difficulties, as well as providing valuable comments and suggestions on how to move forward with the thesis.

________________ ______________ Andreas Johansson Robin Petersson

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1. Introduction

In the traditional Capital Asset Pricing Model (CAPM), a stock’s beta represents an average of its upside- and downside market risk. It is also assumed to be static, i.e. unconditional upon the period of estimation. As a result, the beta may not be an accurate reflection of how the stock tends to move in market upturns and downturns respectively. The purpose of this study is to investigate if it is possible to achieve a higher risk-adjusted return by selecting stocks based on their upside- and downside beta when constructing financial portfolios.

Modern portfolio theory is based on the findings of Markowitz (1952) who showed that efficient financial portfolios can be constructed using mean-variance optimization. The framework was later extended by several authors whom, independently, derived the CAPM (Sharpe, 1964; Lintner, 1965; Mossin, 1966). The model illustrates that if investors optimize their portfolios using variance optimization, the aggregate market portfolio will be mean-variance efficient and only contain systematic risk. The risk-premium demanded by a rational investor therefore only depends on the asset’s market risk, as measured by its beta in the CAPM.

Since its inception, the CAPM has been subject to various forms of critique. Several studies have, for example, concluded that the assumption of normality is unrealistic when modelling asset returns. Markowitz (1959) himself suggested the use of semi-variance in mean-variance optimization since it better reflects an assets downside risk. Furthermore, Mandelbrot (1963) observed longer tails than under the condition of normality, suggesting that non-normal distributions with fatter tails better explain asset returns.

Several empirical studies also indicate that the correlation between stocks tend to increase during market turmoil (see King & Wadhwani, 1990; Longin & Solnik, 1995; Forbes & Rigobon, 2002), possibly increasing an asset’s beta. Bawa & Lindenberg (1977) suggested extending the CAPM to incorporate the possibility of asymmetric upside- and downside risk. A later empirical study by Ang, Chen & Xing (2005) uses a similar framework when analysing the risk-premium for downside risk in the U.S. stock market. They sort stocks by their CAPM-, upside- and downside beta1, as well as the difference between these measures. The authors

1_{Ang, Chen & Xing (2005) define a stock’s upside (downside) beta as the covariance between the stock’s return and the market}

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2 illustrate that high downside beta stocks have higher average returns, indicating that risk averse investors demand an additional risk premium for holding downside risk. Guy (2015) argues that if investors are risk-averse, they have incentives to take advantage of the upside- and downside beta asymmetry. By constructing portfolios that have a high upside risk and a low downside risk, an investor can earn a higher return in market upswings, while hedging some of the downside risk. Using monthly U.S. stock returns from the year 1980-2012, Guy estimates upside- and downside betas, conditioned on positive and negative market returns, over a rolling three-year period, and ranks the stocks by each measure. He then constructs a Target Date fund consisting, among other assets, of the lowest downside beta quintile, and the highest upside beta quintile, holds it for a year, and then rebalances it. The results indicate that an investor can achieve a higher risk-adjusted return using this approach of investing.

This study builds on the work of Guy (2015), but differs in a few aspects. First, upside- and downside beta portfolios are tested using Swedish stock data over the period 1995-12-29 to 2017-06-30. Second, since a large difference between a stock’s upside- and downside beta may indicate a greater upside risk relative to its downside risk, portfolios based on this difference are also tested to analyse its effect on risk-adjusted returns2. Third, shorter estimation periods with more frequent data are used to better account for changing betas while also increasing the sample size. Fourth, since zero transaction costs is an unrealistic assumption for an actual investor, rebalancing costs are added to analyse their effects on the portfolio returns. Finally, the skewness and kurtosis of each portfolio are analysed to examine if the method of selecting stocks has any visible effects on the return distribution of the portfolios. The results indicate that although there seems to exist an upside-downside beta asymmetry, the portfolio performance seems to be largely explained by the traditional CAPM beta where low beta portfolios outperformed high beta portfolios.

The rest of the thesis is organized as follows. Chapter 2 provides a theoretical background on the conditional CAPM and the upside- and downside stock beta. In Chapter 3, the data is introduced followed by the methodology of constructing portfolios in Chapter 4. Chapter 5 presents the results, which are discussed in Chapter 6. Chapter 7 concludes.

2_{Ang, Chen & Xing (2005) found that stocks with a small realized difference between their upside- and downside beta had}

higher average returns, indicating that a risk premium was required for holding stocks with a large downside risk relative their upside risk. The returns were, however, not risk-adjusted.

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2. Theoretical Background

2.1 The Conditional CAPM

In the traditional CAPM, the expected return of an asset is a linear function of the market excess return since all unsystematic risk can be eliminated through the process of diversification. As a result, a rational investor will only demand a risk premium for bearing market risk. In mathematical form, the model can be expressed as:

E r_i = r_f + β_i E r_M - r_f (2.1)

where

β_i = Cov ri - rf, rM - rf

Var r_M - r_f (2.2)

and E r_i is the expected return of asset i, r_f is the risk-free interest rate, β_i is asset i’s sensitivity to market risk, and E rM is the expected market return (Bodie, Kane & Marcus, 2014).

Furthermore, investors are assumed to plan for a single homogenous time-period, causing the asset’s beta and the market risk premium to be unconditional upon a given point in time, i.e. they are assumed to be constant. In reality, however, investors plan multiple periods ahead, suggesting that an asset’s beta and the market risk premium are conditional upon a given point in time (Ghysels, 1998). The difference between the unconditional- and conditional CAPM can be illustrated using the expression for the covariance:

Cov Y1, Y2 = E Y1Y2 - E Y1 E Y2 (2.3)

E Y₁Y₂ = E Y₁ E Y₂ + Cov(Y₁, Y₂) (2.4)

Substituting asset i’s market beta, β_i, for Y₁, and the market excess return, R_M, for Y₂, equation (2.4) becomes:

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4 where E β_iRM is the risk premium of asset i. If the unconditional model holds, the expected

beta, E β_i , and the market risk premium, E RM , are both constant, and their covariance,

Cov(β_i, R_M), is equal to zero. If, however, E β_i and E R_M covary positively over time, the unconditional beta will underestimate the asset’s market risk, and if they covary negatively, it will overestimate the market risk (Barinov, 2014).

As an illustration, consider the following example by Barinov (2014) using a single risky asset

i. Assume that throughout the business cycle, an expansion lasts three times longer than a

recession and that the asset’s beta and the market risk premium is equal to two thirds (β_iExp. =

"

#) and four percent (E(RM Exp.

) = 4%) respectively. During the recession, when the systematic risk increases, the asset’s beta and the market risk premium rises to two (β_iRec. = 2) and twelve percent (E(R_MRec.) = 12%) respectively. This suggests that the average market risk premium over the business cycle will be six percent (3

4E(RM Exp._{) +}1

4E(RM

Rec._{) = 6%), and the average beta}

according to the unconditional CAPM will be equal to one (#_$β_iExp. + %_$β_iRec. = 1). However, since the market risk premium and asset i’s beta covary positively, the actual risk premium of asset i is eight percent (3

4βi Exp._E(R M Exp._{) +}1 4βi Rec._E(R M

Rec._{) = 8%), resulting in a two percent}

difference compared to the unconditional case. According to the conditional CAPM, this two percent difference is not a positive alpha, but a result of the positive correlation between asset

i’s beta and the market risk premium.

2.2 The Upside- and Downside Beta

In most research on the conditional CAPM, the asset’s market beta is conditioned on the time period it was estimated, e.g. a rolling one-year period, or other risk factors, e.g. market capitalization. Usually, however, they do not condition on upside- and downside market movements within the sample period. Since the return distribution of financial assets is often asymmetric, and their covariance tends to increase during market downturns, an asset’s downside beta may be different from its upside beta. An asset’s upside beta may thus be conditioned on market returns over some predetermined value c3, and its downside beta may be

3_{Ang, Chen, and Xing (2005) tested the robustness of their results by setting c equal to the previous year’s average market}

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5 conditioned on market returns below this value. Mathematically, asset i’s upside beta, β_i+, and downside beta, β_i-, for period t can be expressed as (Ang, Chen & Xing, 2005):

β_i,t+ = Cov Ri,t, RM,t rM,t > c

Var R_M,t r_M,t > c (2.6)

β_i,t- = Cov Ri,t, RM,t rM,t ≤ c Var RM,t rM,t ≤ c

(2.7)

where R_i,t is the excess return of asset i over the risk-free interest rate during period t, R_M,t is the market excess return over the risk-free interest rate during period t, and rM,t is the market

return during period t. By separating the upside beta from the downside beta, and incorporating the potential risk asymmetry, an investor may, in other words, be able to construct more efficient financial portfolios.

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3. Data

3.1 Financial Market Data

The data consists of weekly observations over the total return indices of stocks listed on NASDAQ OMXS from 1995-12-29 to 2017-06-30. Given data availability, stocks with listing (delisting) dates after (before) 1995-12-29 (2017-06-30) were also included to increase the sample size and to reduce the effect of survivorship bias, resulting in a sample size of 515 stocks. In addition, weekly observations over the Stockholm Benchmark Index (SBX)4_{, and the}

6-month Swedish Treasury bill (SSVX6M) rate were used as proxies for the market portfolio and the risk-free interest rate. Table 3.1 provides descriptive statistics over the listed-, delisted- and total number of stocks, the SBX, and the risk-free interest rate. It includes the number of return indices in each category, their average return, standard deviation, skewness, and kurtosis. The stock data was collected from Bloomberg (2017), and the T-bill data was collected from the Riksbank (2017).

Table 3.1 Descriptive statistics over weekly returns.

Note: the table shows descriptive statistics over the weekly returns for the listed-, delisted- and total number of stocks, the SBX, and the SSVX6M over the period 1995-12-29 to 2017-06-30. It includes the number of indices in each category, the average return, standard deviation, skewness, and kurtosis. Max- and min values are presented in parentheses.

3.2 Data Frequency and Time Period

Weekly data was used in this study for several reasons. First, it contains less noise than daily data, making it more likely that the estimates represent “pure” signals. In addition, and although weekly data may contain more noise than monthly data, the occurrence of several market events

4_{The SBX is a free-float adjusted-, capitalization-weighted-, total return index, designed as an indicator of the Stockholm}

Stock Exchange (Bloomberg, 2017). Category Number of Indices Avg. Return % (Max/Min) Avg. SD % (Max/Min) Avg. Skewness (Max/Min) Avg. Kurtosis (Max/Min) Listed Stocks 303 0.37 (329 / -75) 5.96 (19.22 / 1.11) 0.99 (11.30 / -1.23) 12.21 (251.97 / -0.03) Delisted Stocks 212 0.26 (485 / -98) 7.42 (22.63 / 2.63) 1.55 (22.45 / -9.32) 17.81 (533.44 / -0.47) All Stocks 515 0.33 (485 / -98) 6.56 (22.63 / 1.11) 1.22 (22.45 / -9.32) 14.51 (533.44 / -0.47) SBX 1 0.25 2.99 -0.32 4.57 SSVX6M 1 0.05 0.03 0.03 -0.74

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7 within a given month may result in different market responses averaging out, not capturing their full effects. Second, stock betas tend to change over time (see Blume, 1971; Fabozzi & Francis, 1978), creating the need to estimate betas over relatively short periods of time. Guy (2015) used monthly observations over a rolling three-year period, but to increase the number of observations, while shortening the estimation period to account for changing betas, weekly data over a rolling two-year period was chosen in this study. Third, as argued by Momcilovic, Begovic, & Tomasevic (2014), weekly returns seem to provide more stable in-sample estimates of stock betas, further supporting the use of weekly data.

The starting date 1995-12-29 was chosen since it allowed the use of the SBX as a proxy for the market portfolio from the first week of its inception, week 1 of 1996, up until week 26 of 2017. It also resulted in a sufficient amount of out-of-sample observations when testing the portfolios, as well as an adequate number of stocks for each of them to be sufficiently diversified.

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4. Empirical Model

4.1 The Conditional CAPM and Portfolio Construction

In order to construct portfolios based on upside- and downside betas, as well as the difference between the two measures, upside- and downside stock betas were estimated over a rolling two-year period for each stock on the Stockholm Stock Exchange using ordinary least squares (OLS). To reduce the problem of survivorship bias, stocks that were listed and delisted over the sample period were also included, although with a couple of constraints. First, stocks that had not been listed for the entire two-year period were excluded due to their short history. Second, stocks that were delisted within the next six-month period were also excluded. This since it is often known a few months in advance if a stock will be delisted (Nasdaq 2016), and including these would increase the frequency and cost of rebalancing the portfolios. For each two-year period and stock, the upside- and downside beta was estimated according to:

R_i,t = α_i + β_i+ R_SBX,t + ε_i,t , ∀ r_SBX,t > 0 (4.1)

R_i,t = α_i + β_i- R_SBX,t + ε_i,t , ∀ r_SBX,t ≤ 0 (4.2)

where Ri,t is stock i’s excess return over the six-month T-bill rate during week t, αi is the

intercept, β_i+ (β_i-) is the upside (downside) beta, RSBX,t is the SBX excess return over the

six-month T-bill rate during week t, ε_i,t is stock i’s error term during week t, and r_SBX,t is the return of the SBX during week t.

The stocks were ranked lowest to highest according to their upside beta, downside beta, as well as the difference between their upside- and downside beta, resulting in three lists of stocks. Each list was then divided into equally weighted quintile portfolios, where the first quintile contained the 20 percent of stocks with the lowest beta values, the second quintile contained the 20 percent of stocks with the second lowest beta values, and so on. The portfolios were then held for a period of six months before new beta values were estimated over the most recent

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9 two-year period, and new quintile portfolios were constructed. This resulted in 15 time-series of portfolio returns, one for each quintile portfolio 5.

The portfolios were evaluated in three steps. First, to evaluate each portfolio’s relative performance, a benchmark index consisting of all eligible stocks, the same rebalancing date, and weighting technique as the quintile portfolio had to be constructed. However, since positive and negative observations of the market return are required to estimate both the upside- and downside beta of a stock using OLS, the composition of stocks for each benchmark will differ at some point in time. For example, if the total return index of a stock has periods of no observations, and the market return is negative (positive) for all existing observations during a given two-year period, the stock’s upside (downside) beta, and, hence, beta difference, cannot be estimated due to a lack of observations. Therefore, three benchmark portfolios had to be constructed, one for the upside beta portfolios, one for the downside beta portfolios, and one for the upside-minus-downside beta portfolios6. Second, to analyse the effect of transaction costs, rebalancing costs of 0.5-, 1-, and 2 percent of the total portfolio value were added each (semi-annual) rebalancing date. Finally, in order to examine if the method of selecting stocks had any visible effects on the return distribution of the portfolios, the skewness and kurtosis were calculated to sort out potential trends.

4.2 Performance Statistics

Performance statistics were used to evaluate each portfolio’s performance against its benchmark, but since different statistics measure different return characteristics, the ordinal ranking of the portfolios may change depending on the measure being used. Therefore, several widely-used statistics were calculated to provide a more comprehensive overview of each portfolio’s relative performance. First, to measure each portfolio’s reward-to-volatility trade-off, the Sharpe ratio was calculated. It is defined as the portfolio’s average excess return, R_p, divided by the standard deviation of the returns, σ_p, as illustrated in equation (4.3):

5_{That is, five portfolios based on upside betas, five portfolios based on downside betas, and five portfolios based on the}

difference between each stock’s upside- and downside beta.

6_{The SBX, or other market indices, were not used for evaluation since they are usually capitalization weighted, their}

composition of stocks differ, as well as their rebalancing dates. Using an existing index could therefore lead to incorrect inferences.

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10 Sharpe ratio = Rp

σp (4.3)

Second, to measure each portfolio’s reward-to-market risk, the Treynor ratio was calculated, which is defined as the portfolio’s average excess return divided by the portfolio’s beta, β_p, as illustrated in equation (4.4):

Treynor ratio = Rp

β_p (4.4)

Third, to measure how much of the portfolio’s return that is not explained by its exposure to the benchmark index, Jensen’s alpha was estimated. It is defined as the portfolios average excess return minus its expected excess return predicted by the benchmark, R_b, using a single-factor model, as shown in equation (4.5):

Jensen’s alpha = R_p - β_p R_b _(4.5)

Fourth, to risk-adjust each portfolio’s excess return over its benchmark, the Information ratio was calculated. It is defined as the portfolio’s alpha divided by the volatility of the excess return (i.e. the tracking error), σe, as illustrated in equation (4.6):

Information ratio = αp

σe (4.6)

Finally, two measures of downside risk were calculated, the weekly five percent Value at Risk (VaR) and the weekly five percent Expected Shortfall (ES). The five percent VaR is defined as the fifth percentile of portfolio returns, indicating that five percent of portfolio returns are expected to fall below this value. The five percent ES is defined as the expected return given that it falls below the fifth percentile (Bodie, Kane & Marcus, 2014; Hull, 2012).

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5. Results

Table 5.1 presents descriptive statistics for each of the five upside- (U), downside- (D) and upside-minus-downside (UMD) beta portfolios, their respective benchmark, and the SBX, over the out-of-sample period week 1 of 1998 to week 26 of 2017. It includes the ending value of one SEK invested at the beginning of the period, the geometric- and arithmetic averages, as well as the yearly standard deviation and tracking error7. As the table illustrates, a similar pattern is visible for both the upside- and downside beta portfolios, where portfolios with a lower upside- or downside beta tend to have a higher geometric average return, as well as a lower standard deviation and tracking error. For the upside-minus-downside beta portfolios however, another pattern is visible. The geometric average return tends to increase, and the standard deviation and tracking error tend to fall, the more moderate the size of the difference between the upside- and downside beta. The return of each portfolio also seems to be explained by its traditional CAPM beta, shown in Table A.1 of Appendix A, where lower beta portfolios tend to generate a higher geometric return, as well as a lower standard deviation. Overall, portfolio D1 had the highest ending value (25.72SEK), geometric- and arithmetic averages (18.04% and 21.94% respectively), and the lowest standard deviation (27.94%), while portfolio

UMD3 had the lowest tracking error (7.56%). The time-series of each portfolio is presented in

Figure A.1 to A.3 of Appendix A.

Table 5.2 presents the performance statistics, i.e. the annual Sharpe ratio, Treynor ratio, Jensen’s alpha, Information ratio, and the weekly five percent VaR and ES, of each portfolio, their respective benchmark, and the SBX. The same pattern as in Table 5.1 is visible, where lower upside- and downside beta portfolios, as well as portfolios with a more moderate difference between the two measures, tend to have a higher risk-adjusted return. As the table

7_{The return characteristics have been annualized from a weekly basis, but since none of the portfolios follow a geometric}

Brownian motion (see Appendix C), the standard method of scaling, t, would underestimate the annual volatility. As a result, the Hurst exponent of each portfolio was estimated and used to annualize the returns. Using a similar method as Hamed (2006), each time-series (N=1018) was initially divided into overlapping sub-periods, starting at 𝑛 = 10 observations. The length of each sub-period was then increased by a factor 2 until 𝑛 = 𝑁 4 was reached, yielding ten series of overlapping sub-periods containing n equal to 10, 14, 20, 29, 41, 58, 81, 115, 163, and 230 observations. The classical rescaled range and standard deviation were then calculated according to equation (D.1) and (D.2) in Appendix D, and the Hurst exponent was estimated using OLS according to equation (D.3). The average Hurst exponent for the quintile portfolios was 0.68, and the highest and lowest Hurst exponent was 0.70 and 0.66 respectively. See Appendix D for more details on the method.

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Table 5.1 Descriptive statistics over annual portfolio returns.

Note: the table shows descriptive statistics for each of the five upside- (U), downside- (D) and upside-minus-downside (UMD) beta portfolios, as well as their respective benchmark and the SBX, over the out-of-sample period week 1 of 1998 to week 26 of 2017. It includes the ending value of 1SEK invested at the beginning of the period, the geometric- and arithmetic averages, the standard deviation, and the tracking error. The values have been annualized from a weekly basis.

illustrates, portfolios U1-U3 and D1-D2 all have better performance statistics than their respective benchmark, although portfolios UMD2-UMD3 also show some indication of outperforming their benchmark. Again, portfolio D1 seems to be the best performing portfolio with the highest Sharpe ratio (0.71), Treynor ratio (0.48), VaR (-2.43%), and ES (-4.36%), while portfolio U1 had the highest Alpha (4.82%), and portfolio UMD1 and UMD3 had the highest Information ratios (0.42).

The results of adding rebalancing costs are presented in Appendix B. As Table B.1 and B.2 illustrates, with a 0.5 percent rebalancing cost, portfolios U1-U3, D1-D2, and UMD3 all outperformed their respective benchmark both in total- and risk-adjusted terms. With a 1

Portfolio Ending value of 1SEK Geometric Average % Arithmetic Average % Standard Deviation % Tracking Error % U1 (Low β+) 17.73 15.82 21.36 33.29 12.73 U2 14.98 14.83 20.20 32.78 11.65 U3 16.31 15.33 21.36 34.73 9.32 U4 6.32 9.88 17.18 38.23 9.31 U5 (High β+) 3.76 7.02 19.05 49.05 20.63

Benchmark (U) 10.96 13.01 19.50 36.02 n/a

D1 (Low β-) 25.72 18.04 21.94 27.94 13.49 D2 16.76 15.49 19.84 29.48 11.15 D3 10.37 12.69 18.91 35.28 9.66 D4 6.81 10.29 18.97 41.66 11.14 D5 (High β-) 3.29 6.28 20.01 52.42 20.16 Benchmark (D) 11.01 13.04 19.53 36.03 n/a UMD1 (Low β+- β-₎ _7.63 _10.94 _20.35 _43.39 _11.52 UMD2 12.01 13.54 19.78 35.33 9.75 UMD3 14.61 14.68 20.08 32.85 7.56 UMD4 11.59 13.33 18.89 33.34 8.24 UMD5 (High β+- β-₎ _8.80 _11.75 _19.78 _40.07 _11.36

Benchmark (UMD) 11.01 13.04 19.54 36.06 n/a

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Table 5.2 Risk-adjusted portfolio returns.

Note: the table shows descriptive statistics for each of the five upside- (U), downside- (D) and upside-minus-downside (UMD) beta portfolios, as well as their respective benchmark and the SBX, over the out-of-sample period week 1 of 1998 to week 26 of 2017. It includes the annual Sharpe ratio, Treynor ratio, Jensen’s alpha, and Information ratio, as well as the weekly five percent VaR and ES. Significance levels are not reported due to a lack of symmetry in the data as shown in Table C.1 to C.3 in Appendix C. The highest values are highlighted in bold.

Portfolio Sharpe Ratio (vs. Benchmark) Treynor Ratio (vs. Benchmark) Alpha % Information Ratio Weekly VaR(5%) Weekly ES(5%) U1 (Low β+) 0.58 (0.09) 0.43 (0.17) 4.82 0.38 -2.96 -4.67 U2 0.56 (0.07) 0.33 (0.07) 2.78 0.24 -3.21 -5.27 U3 0.56 (0.13) 0.31 (0.05) 3.15 0.34 -3.45 -5.36 U4 0.40 (-0.09) 0.20 (-0.05) -3.42 -0.37 -4.37 -6.41 U5 (High β+) 0.35 (-0.14) 0.17 (-0.09) -7.95 -0.39 -5.92 -8.49 Benchmark (U) 0.49 (n/a) 0.26 (n/a) n/a n/a -3.70 -5.61 D1 (Low β-) 0.71 (0.22) 0.48 (0.22) 4.80 0.36 -2.43 -4.36 D2 0.61 (0.12) 0.34 (0.08) 2.76 0.25 -3.10 -4.90 D3 0.48 (-0.01) 0.25 (-0.01) 3.13 0.32 -3.78 -5.85 D4 0.41 (-0.08) 0.21 (-0.05) -3.44 -0.31 -4.58 -6.64 D5 (High β-) 0.34 (-0.15) 0.19 (-0.07) -7.98 -0.40 -5.82 -8.66 Benchmark (D) 0.49 (n/a) 0.26 (n/a) n/a n/a -3.69 -5.61 UMD1 (Low β+- β-₎ _0.42 (-0.07) 0.27 (0.01) 4.80 0.42 -4.25 -6.56 UMD2 0.50 (0.01) 0.28 (0.02) 2.76 0.28 -3.61 -5.48 UMD3 0.55 (0.06) 0.28 (0.02) 3.14 0.42 -3.60 -5.42 UMD4 0.51 (0.02) 0.25 (-0.01) -3.43 -0.42 -3.75 -5.63 UMD5 (High β+- β-) 0.44 (-0.05) 0.24 (-0.02) -7.98 -0.70 -4.33 -6.41 Benchmark (UMD) 0.49 (n/a) 0.26 (n/a) n/a n/a -3.70 -5.62 SBX 0.38 (n/a) 0.26 (n/a) n/a n/a -4.81 -7.00

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14 percent rebalancing cost, as illustrated in Table B.3 and B.4, portfolios U1, U3, and D1-D2 still outperformed their respective benchmark in both total- and risk- adjusted terms, while portfolio

U2 only did so in risk-adjusted terms. Finally, with a 2 percent rebalancing cost, as illustrated

in Table B.5 and B.6, portfolio D1 was the only portfolio outperforming its benchmark in either total- or risk-adjusted terms. The time-series of each portfolio with transaction costs included are presented in Figure B.1 to B.9.

A Shapiro-Wilks test for normality, as well as tests for the skewness and kurtosis of each portfolio, are presented in Table C.1 to C.3 in Appendix C. As Table C.1 and C.2 illustrate, none of the portfolios, or the residuals of their excess returns, follow a normal distribution. This is further illustrated by the skewness and kurtosis statistics in Table C.3, which also show that there is no visible pattern between a portfolio’s quintile number or performance, and its level of skewness and kurtosis. Most portfolios have a high level of skewness and kurtosis, although the returns of portfolios U5, D5 and UMD1 do seem to be symmetric as the skewness is not significantly different from zero. Overall, portfolio D1 has the highest skewness (-0.96), and portfolio U2 has the highest level of kurtosis (10.27) compared to a normal distribution.

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6. Discussion

The results indicate that lower upside- and downside beta portfolios, as well as portfolios with a more moderate difference between the two measures, tended to perform better. This is not in line with theory, or previous research, for several reasons. First, the upside beta portfolios were expected to perform better the higher their upside beta since, according to the conditional CAPM, they are sensitive to upside market risk, and the market has mainly had a positive return over the sample period. Furthermore, Guy (2015) argues that one reason his Target Date fund, which partially consisted of stocks selected solely by their high upside beta, performed well was because it exposed investors to a lot of upside risk. However, since he did not evaluate pure stock portfolios based on different upside betas, it is difficult to determine whether the high upside beta stocks in his Target Date fund performed better on average than the low upside beta stocks. Second, and although one may argue that low downside beta portfolios should perform better since they offer a hedge during market downturns, the results are not in line with the findings of Ang, Chen, and Xing (2005). They found that high downside beta portfolios generated higher arithmetic average returns since investors demand an additional risk-premium for holding downside risk. The results of this study, on the other hand, suggest that there are no major differences between the arithmetic averages of the downside beta portfolios, but that their geometric averages differ substantially and are inversely related to their downside beta. Third, since a high upside-minus-downside beta indicates a high upside risk relative the downside risk, portfolios with a large beta difference were expected to perform better, which is not the case according to the results. However, since neither the upside- or downside beta portfolios performed according to theory, it is not surprising that the upside-minus-downside beta portfolios did not either. Portfolios with a high beta difference will by construction have a large upside beta relative their downside beta, which may result in a low average return since the negative effect from a high upside beta cancels out the positive effect from a low downside beta. The same reasoning applies for portfolios with a low upside-minus-downside beta.

For all three portfolio categories, the performance could potentially be explained by the traditional CAPM beta, which is inversely related to the returns. Although such a relationship is not predicted by the CAPM, it is a pattern well-documented by previous research, and could potentially be explained by behavioural factors such as lottery-ticket preferences (see Baker, Bradely & Wurgler, 2011; Bali, Brown, Murray & Tang, 2014). This would increase the relative demand for high beta portfolios, lowering their alphas, which is a pattern observed in the results.

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16 The high beta portfolios of this study could therefore have a higher market risk premium than low beta portfolios, but their low alpha, combined with the observed volatility drain, could have lowered their realized returns enough for the low beta portfolios to outperform in both total- and risk-adjusted terms. Beta values extreme by chance could also be part of the explanation since stocks with a large difference between their estimated- and real beta are more likely to end up in the top- or bottom quintiles.

The results of adding transaction costs indicate that the method of selecting stocks based on their upside-, downside-, or upside-minus-downside beta would have been profitable for an actual investor over the sample period. Even with a one percent rebalancing cost, several portfolios outperformed their respective benchmark in both total- and risk-adjusted terms. Portfolio D1 also outperformed its benchmark with a two percent rebalancing cost, although marginally in absolute terms. The high excess returns, combined with the large number of observations, also indicate that the results were not driven by mere chance and may persist some time into the future.

Although the results seem to be largely driven by the portfolios’ traditional CAPM beta, the fact that all 15 portfolios had a negative upside-minus-downside beta, and that the returns of the upside- and downside beta portfolios differed, suggest that their market risk is asymmetric. However, the results of the skewness and kurtosis tests indicate that there are no distinct patterns between the quintile numbers and the skewness or kurtosis of the portfolios. The method of selecting stocks would therefore be inefficient in creating portfolios with a low (negative) skewness and kurtosis. The results rather suggest that the main difference between the return distributions of the portfolios is that they are located differently alongside each other.

Overall, the findings are not in line with the traditional CAPM, which states that a financial asset should earn a risk-premium that is proportional to its market risk. They also suggest that an upside-downside risk asymmetry may need to be incorporated into the model for more efficient asset pricing, supporting the results of previous research. The credibility of these findings is affected by several factors. First, by including both listed and delisted stocks in the analysis, the sample size was increased, and the risk of survivorship bias was reduced, better reflecting the contemporary market faced by a Swedish investor. Second, no statistical tests of significance were performed since the necessary assumptions were not met, lowering the risk of incorrect inferences. Instead, several widely used performance statistics were calculated to

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17 provide a broad overview of each portfolio’s performance. Third, because the portfolio values did not follow a geometric Brownian motion, the standard method of scaling asset returns over time would underestimate the actual risk and was therefore not used. Instead, the Hurst exponent was estimated using rescaled range analysis, and although the method may result in some bias (see Appendix D), the annualized returns will be more accurate than under the assumption of a geometric Brownian motion. Finally, all market betas were estimated using a single-factor model, resulting in other risk factors being omitted. Ang, Chen & Xing (2005) included stocks historical market capitalization and book-to-market values as proxies for size and value. Attempts were made to control for additional risk factors, but data over these variables were inconsistent both over time and across stocks, which would substantially reduce the sample size and shorten the out-of-sample period.

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7. Conclusion

The purpose of this study was to investigate if it is possible to achieve a higher risk-adjusted return by selecting stocks based on their upside- and downside market risk, as measured by their CAPM beta conditioned on positive and negative market returns. Using weekly Swedish stock data from 1995-12-29 to 2017-06-30, and the SBX as a proxy for the market portfolio, upside- and downside stock betas were estimated over a rolling two-year period using a single-factor model. The stocks were ranked lowest to highest according to their upside-, downside-, and upside-minus-downside beta at the end of each two-year period and divided into quintiles, resulting in fifteen equally weighted portfolios. Each portfolio was rebalanced every six-months and tested out-of-sample against an equally weighted benchmark index consisting of all eligible stocks for that portfolio. The results indicate that although an upside-downside beta asymmetry seems to exist, the portfolios’ performance seems to be largely explained by the traditional CAPM beta rather than a market risk asymmetry.

There are two main conclusions from this study. First, the fact that all portfolios had a downside beta greater than their upside beta indicates that the market risk is asymmetric. This contradicts the traditional CAPM and suggests that a risk asymmetry may need to be incorporated into the model for more efficient asset pricing. Even if the skewness and kurtosis tests indicate that the method of selecting stocks was unable to produce portfolios with a low skewness and kurtosis, it may still be possible to exploit the risk asymmetry during portfolio construction, although with a different method. Second, whether the results of the low beta portfolios were driven by behavioral- or other effects, their high excess returns, combined with the large number of observations, indicate that the results were not driven by mere chance and may persist some time into the future. A Swedish investor could therefore relatively easy construct stock portfolios more efficient than the market using this approach of investing.

Since the results seem to be largely driven by the stocks’ traditional CAPM beta, a suggestion for future research would be to attempt controlling for the traditional beta when constructing financial portfolios based on market risk asymmetries. In addition, stocks that are more likely to yield inaccurate beta estimates, i.e. stocks with infrequent data over the in-sample period, could be excluded to reduce their stochastic effect on the results. The remaining stocks could also be weighted according to their beta values to investigate if it would increase the portfolio returns further.

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19

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Appendix A

Table A.1 Portfolio beta values.

Portfolio β β+ β- β+- β -U1 (Low β+) 0.45 0.33 0.65 -0.31 U2 0.54 0.41 0.74 -0.33 U3 0.62 0.50 0.77 -0.26 U4 0.78 0.68 0.88 -0.21 U5 (High β+) 1.00 0.92 1.11 -0.19 Benchmark (U) 0.67 0.56 0.82 -0.26 D1 (Low β-) 0.42 0.28 0.60 -0.32 D2 0.53 0.41 0.69 -0.29 D3 0.68 0.57 0.83 -0.26 D4 0.79 0.68 0.92 -0.24 D5 (High β-) 0.97 0.89 1.08 -0.19 Benchmark (D) 0.67 0.56 0.82 -0.26 UMD1 (Low β+- β-) 0.67 0.57 0.87 -0.30 UMD2 0.63 0.50 0.79 -0.28 UMD3 0.65 0.53 0.79 -0.25 UMD4 0.68 0.56 0.80 -0.24 UMD5 (High β+- β-) 0.75 0.65 0.88 -0.23 Benchmark (UMD) 0.68 0.56 0.82 -0.26

Note: the table shows beta estimates for each of the five upside- (U), downside- (D), and upside-minus-downside (UMD) beta portfolios, as well as for their respective benchmark, over the sample period week 1 of 1998 to week 26 of 2017. It includes the CAPM-, upside-, downside-, and upside-minus-downside beta. The SBX was used as a proxy for the market portfolio.

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Figure A.1 Time-series of upside beta portfolios.

Note: the figure shows the time-series of each upside beta portfolio (U), the upside beta benchmark portfolio, and the SBX over the out-of-sample period week 1 of 1998 to week 26 of 2017. Each time-series has been indexed at 100 in week 52 of 1997.

Figure A.2 Time-series of downside beta portfolios.

Note: the figure shows the time-series of each downside beta portfolio (D), the downside beta benchmark portfolio, and the SBX over the out-of-sample period week 1 of 1998 to week 26 of 2017. Each time-series has been indexed at 100 in week 52 of 1997.

0

600

1200

1800

Jul-97 Jul-02 Jul-07 Jul-12 Jul-17 U1 (Low β+) SBX Benchmark (U) 0 600 1200 1800

Jul-97 Jul-02 Jul-07 Jul-12 Jul-17

U2 SBX Benchmark (U) 0 600 1200 1800

U3 SBX Benchmark (U) 0 600 1200 1800

U4 SBX Benchmark (U) 0 600 1200 1800

Jul-97 Jul-02 Jul-07 Jul-12 Jul-17 U5 (High β+₎ _SBX Benchmark (U) 0 800 1600 2400

Jul-97 Jul-02 Jul-07 Jul-12 Jul-17 D1 (Low β-₎ _SBX Benchmark (D) 0 800 1600 2400

D2 SBX Benchmark (D) 0 800 1600 2400

D3 SBX Benchmark (D) 0 800 1600 2400

D4 SBX Benchmark (D) 0 800 1600 2400

Jul-97 Jul-02 Jul-07 Jul-12 Jul-17 D5 (High β-₎ _SBX

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24

Figure A.3 Time-series of upside-minus-downside beta portfolios.

Note: the figure shows the time-series of each upside-minus-downside beta portfolio (UMD), the upside-minus-downside beta benchmark portfolio, and the SBX over the out-of-sample period week 1 of 1998 to week 26 of 2017. Each time-series has been indexed at 100 in week 52 of 1997.

0

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Jul-97 Jul-02 Jul-07 Jul-12 Jul-17 UMD1 (Low (β+_{- β}-₎₎ _SBX Benchmark (UMD) 0 500 1000 1500

UMD2 SBX Benchmark (UMD) 0 500 1000 1500

Jul-97 Jul-02 Jul-07 Jul-12 Jul-17 UMD5 (High (β+_{- β}-₎₎ _SBX

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Appendix B

Table B.1 Descriptive statistics over annual portfolio returns with a 0.5% rebalancing cost.

Benchmark (U) 10.96 13.01 19.50 36.02 n/a

Benchmark (UMD) 11.01 13.04 19.54 36.06 n/a

SBX 5.59 9.19 14.89 33.78 n/a

Note: the table shows descriptive statistics for each of the five upside- (U), downside- (D) and upside-minus-downside (UMD) beta portfolios, as well as their respective benchmark, and the SBX over the out-of-sample period week 1 of 1998 to week 26 of 2017 with a 0.5 percent rebalancing cost included. It includes the ending value of 1SEK invested at the beginning of the period, the geometric- and arithmetic averages, the standard deviation, and the tracking error. The values have been annualized from a weekly basis.

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Table B.2 Risk-adjusted annual portfolio returns with a 0.5% rebalancing cost.

Portfolio Sharpe Ratio (vs. Benchmark) Treynor Ratio (vs. Benchmark) Alpha % Information Ratio Weekly VaR(5%) Weekly ES(5%) U1 (Low β+) 0.55 (0.06) 0.41 (0.12) 4.06 0.32 -3.00 -4.69 U2 0.52 (0.03) 0.31 (0.05) 2.02 0.17 -3.21 -5.29 U3 0.53 (0.04) 0.29 (0.03) 2.40 0.26 -3.47 -5.37 U4 0.37 (-0.12) 0.18 (-0.07) -4.17 -0.44 -4.37 -6.43 U5 (High β+) 0.33 (-0.16) 0.16 (-0.10) -8.71 -0.42 -5.92 -8.50 Benchmark (U) 0.49 (n/a) 0.26

(n/a) n/a n/a -3.70 -5.61

D1 (Low β-) 0.67 (0.18) 0.45 (0.18) 4.05 0.30 -2.43 -4.38 D2 0.57 (0.08) 0.32 (0.06) 2.00 0.18 -3.18 -4.91 D3 0.45 (-0.04) 0.24 (-0.02) 2.38 0.25 -3.83 -5.86 D4 0.38 (-0.11) 0.20 (-0.06) -4.19 -0.39 -4.58 -6.65 D5 (High β-) 0.32 (-0.17) 0.18 (-0.10) -8.74 -0.43 -5.82 -8.67 Benchmark (D) 0.49 (n/a) 0.26

(n/a) n/a n/a -3.69 -5.61

UMD1 (Low β+- β-₎ _0.40 (-0.09) 0.26 (-0.03) 4.05 0.36 -4.28 -6.59 UMD2 0.47 (-0.02) 0.27 (0.01) 2.01 0.21 -3.63 -5.50 UMD3 0.52 (0.03) 0.26 (0.02) 2.38 0.30 -3.67 -5.43 UMD4 0.47 (-0.02) 0.24 (-0.01) -4.19 -0.50 -3.75 -5.64 UMD5 (High β+- β-₎ _0.42 (-0.07) 0.22 (-0.04) -8.74 -0.78 -4.33 -6.42 Benchmark (UMD) 0.49 (n/a) 0.26

(n/a) n/a n/a -3.70 -5.62

SBX 0.38

(n/a)

0.13

(n/a) n/a n/a -4.81 -7.00

Note: the table shows descriptive statistics for each of the five upside- (U), downside- (D) and upside-minus-downside (UMD) beta portfolios, as well as their respective benchmark, and the SBX over the out-of-sample period week 1 of 1998 to week 26 of 2017 with a 0.5 percent rebalancing cost included. It includes the annual Sharpe ratio, Treynor ratio, Jensen’s alpha, and Information ratio as well as the weekly five percent VaR and ES. Significance levels are not reported due to a lack of symmetry in the data as shown in Table C.1 to C.3 in Appendix C. The highest values are highlighted in bold.

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Table B.3 Descriptive statistics over annual portfolio returns with a 1% rebalancing cost.

Benchmark (U) 10.96 13.01 19.50 36.02 n/a

D1 (Low β-) 17.55 15.76 19.82 28.49 13.45 D2 11.45 13.26 17.71 29.82 11.13 D3 7.08 10.51 16.80 35.46 9.49 D4 4.65 8.16 17.01 42.08 10.83 D5 (High β-) 2.25 4.22 18.03 52.55 20.12 Benchmark (D) 11.01 13.04 19.53 36.03 n/a UMD1 (Low β+- β-₎ _5.21 _8.79 _18.20 _43.37 _11.10 UMD2 8.20 11.35 17.67 35.57 9.58 UMD3 9.97 12.47 17.84 32.80 8.09 UMD4 7.91 11.14 16.81 33.67 8.54 UMD5 (High β+- β-) 6.01 9.59 17.66 40.17 11.21

Benchmark (UMD) 11.01 13.04 19.54 36.06 n/a

SBX 5.59 9.19 14.89 33.78 n/a

Note: the table shows descriptive statistics for each of the five upside- (U), downside- (D) and upside-minus-downside (UMD) beta portfolios, as well as their respective benchmark, and the SBX over the out-of-sample period week 1 of 1998 to week 26 of 2017 with a 1 percent rebalancing cost included. It includes the ending value of 1SEK invested at the beginning of the period, the geometric- and arithmetic averages, the standard deviation, and the tracking error. The values have been annualized from a weekly basis.

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Table B.4 Risk-adjusted annual portfolio returns with a 1% rebalancing cost.

Portfolio Sharpe Ratio (vs. Benchmark) Treynor Ratio (vs. Benchmark) Alpha % Information Ratio Weekly VaR(5%) Weekly ES(5%) U1 (Low β+) 0.51 (0.02) 0.39 (0.13) 3.31 0.26 -3.00 -4.71 U2 0.49 (0.00) 0.30 (0.04) 1.26 0.11 -3.25 -5.31 U3 0.49 (0.00) 0.28 (0.02) 1.64 0.18 -3.47 -5.39 U4 0.34 (-0.15) 0.17 (-0.09) -4.93 -0.52 -4.37 -6.45 U5 (High β+) 0.31 (-0.18) 0.15 (-0.11) -9.47 -0.46 -5.93 -8.51 Benchmark (U) 0.49 (n/a) 0.26 (n/a) n/a n/a -3.70 -5.61 D1 (Low β-) 0.63 (0.14) 0.43 (0.17) 3.29 0.24 -2.52 -4.41 D2 0.53 (0.04) 0.30 (0.04) 1.25 0.11 -3.18 -4.93 D3 0.42 (-0.07) 0.22 (-0.04) 1.62 0.17 -3.83 -5.87 D4 0.36 (-0.13) 0.19 (-0.07) -4.95 -0.46 -4.60 -6.66 D5 (High β-) 0.31 (-0.18) 0.17 (-0.09) -9.50 -0.47 -5.82 -8.68 Benchmark (D) 0.49 (n/a) 0.26 (n/a) n/a n/a -3.69 -5.61 UMD1 (Low β+- β-₎ _0.37 (-0.12) 0.24 (-0.02) 3.29 0.30 -4.28 -6.62 UMD2 0.44 (-0.05) 0.25 (-0.01) 1.25 0.13 -3.63 -5.52 UMD3 0.48 (-0.01) 0.25 (-0.01) 1.63 0.20 -3.68 -5.45 UMD4 0.44 (-0.05) 0.22 (-0.04) -4.95 -0.58 -3.81 -5.66 UMD5 (High β+- β-₎ _0.39 (-0.10) 0.21 (-0.05) -9.49 -0.85 -4.33 -6.44 Benchmark (UMD) 0.49 (n/a) 0.26 (n/a) n/a n/a -3.70 -5.62 SBX 0.38 (n/a) 0.13 (n/a) n/a n/a -4.81 -7.00

Note: the table shows descriptive statistics for each of the five upside- (U), downside- (D) and upside-minus-downside (UMD) beta portfolios, as well as their respective benchmark, and the SBX over the out-of-sample period week 1 of 1998 to week 26 of 2017 with a 1 percent rebalancing cost included. It includes the annual Sharpe ratio, Treynor ratio, Jensen’s alpha, and Information ratio as well as the weekly five percent VaR and ES. Significance levels are not reported due to a lack of symmetry in the data as shown in Table C.1 to C.3 in Appendix C. The highest values are highlighted in bold.

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Table B.5 Descriptive statistics over annual portfolio returns with a 2% rebalancing cost.

Benchmark (U) 10.96 13.01 19.50 36.02 n/a

Benchmark (UMD) 11.01 13.04 19.54 36.06 n/a

SBX 5.59 9.19 14.89 33.78 n/a

Note: the table shows descriptive statistics for each of the five upside- (U), downside- (D) and upside-minus-downside (UMD) beta portfolios, as well as their respective benchmark, and the SBX over the out-of-sample period week 1 of 1998 to week 26 of 2017 with a 2 percent rebalancing cost included. It includes the ending value of 1SEK invested at the beginning of the period, the geometric- and arithmetic averages, the standard deviation, and the tracking error. The values have been annualized from a weekly basis.

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Table B.6 Risk-adjusted annual portfolio returns with a 2% rebalancing cost.

Portfolio Sharpe Ratio (vs. Benchmark) Treynor Ratio (vs. Benchmark) Alpha % Information Ratio Weekly VaR(5%) Weekly ES(5%) U1 (Low β+) 0.45 (-0.04) 0.34 (0.17) 1.80 0.14 -3.16% -4.78 U2 0.42 (-0.07) 0.26 (0.09) -0.25 -0.02 -3.26 -5.37 U3 0.43 (-0.06) 0.24 (0.07) 0.13 0.01 -3.51 -5.44 U4 0.29 (-0.20) 0.14 (-0.03) -6.44 -0.65 -4.37 -6.48 U5 (High β+) 0.27 (-0.22) 0.13 (-0.04) -10.99 -0.54 -5.93 -8.55 Benchmark (U) 0.49 (n/a) 0.17 (n/a) n/a n/a -3.70 -5.61 D1 (Low β-) 0.54 (0.05) 0.38 (0.21) 1.78 0.13 -2.64 -4.49 D2 0.45 (-0.04) 0.26 (0.09) -0.27 -0.02 -3.22 -4.97 D3 0.36 (-0.13) 0.19 (0.02) 0.11 0.01 -3.83 -5.91 D4 0.31 (-0.18) 0.17 (0.00) -6.46 -0.59 -4.60 -6.70 D5 (High β-) 0.27 (-0.22) 0.15 (-0.02) -11.01 -0.55 -5.95 -8.72 Benchmark (D) 0.49 (n/a) 0.17 (n/a) n/a n/a -3.69 -5.61 UMD1 (Low β+- β-₎ _0.33 (-0.16) 0.21 (0.04) 1.79 0.16 -4.38 -6.68 UMD2 0.38 (-0.11) 0.22 (0.05) -0.26 -0.03 -3.63 -5.56 UMD3 0.42 (-0.07) 0.21 (0.04) 0.11 0.01 -3.68 -5.48 UMD4 0.38 (-0.11) 0.19 (0.02) -6.46 -0.73 -3.87 -5.70 UMD5 (High β+- β-₎ _0.34 (-0.15) 0.18 (0.01) -11.01 -0.98 -4.40 -6.48 Benchmark (UMD) 0.49 (n/a) 0.17 (n/a) n/a n/a -3.70 -5.62 SBX 0.38 (n/a) 0.09 (n/a) n/a n/a n/a -4.81 -7.00

Note: the table shows descriptive statistics for each of the five upside- (U), downside- (D) and upside-minus-downside (UMD) beta portfolios, as well as their respective benchmark, and the SBX over the out-of-sample period week 1 of 1998 to week 26 of 2017 with a 2 percent rebalancing cost included. It includes the annual Sharpe ratio, Treynor ratio, Jensen’s alpha, and Information ratio as well as the weekly five percent VaR and ES. Significance levels are not reported due to a lack of symmetry in the data as shown in Table C.1 to C.3 in Appendix C. The highest values are highlighted in bold.

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Figure B.1 Time-series of upside beta portfolios with a 0.5% rebalancing cost.

Note: the figure shows the time-series of each upside beta portfolio (U), including a 0.5 percent rebalancing cost, the upside beta benchmark portfolio, and the SBX over the out-of-sample period week 1 of 1998 to week 26 of 2017. Each time-series has been indexed at 100 in week 52 of 1997.

Figure B.2 Time-series of downside beta portfolios with a 0.5% rebalancing cost.

Note: the figure shows the time-series of each downside beta portfolio (D), including a 0.5 percent rebalancing cost, the downside beta benchmark portfolio, and the SBX over the out-of-sample period week 1 of 1998 to week 26 of 2017. Each time-series has been indexed at 100 in week 52 of 1997.

0

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Jul-97 Jul-02 Jul-07 Jul-12 Jul-17 U1 (Low β+) SBX Benchmark (U) 0 500 1000 1500

U2 SBX Benchmark (U) 0 500 1000 1500

U3 SBX Benchmark (U) 0 500 1000 1500

U4 SBX Benchmark (U) 0 500 1000 1500

Jul-97 Jul-02 Jul-07 Jul-12 Jul-17 U5 (High β+₎ _SBX Benchmark (U) 0 500 1000 1500 2000

Jul-97 Jul-02 Jul-07 Jul-12 Jul-17 D1 (Low β-₎ _SBX Benchmark (D) 0 500 1000 1500 2000

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Figure B.3 Time-series of upside-minus-downside beta portfolios with a 0.5% rebalancing cost.

Note: the figure shows the time-series of each upside-minus-downside beta portfolio (UMD), including a 0.5 percent rebalancing cost, the upside-minus-downside beta benchmark portfolio, and the SBX over the out-of-sample period week 1 of 1998 to week 26 of 2017. Each time-series has been indexed at 100 in week 52 of 1997.

Figure B.4 Time-series of upside beta portfolios with a 1% rebalancing cost.

Note: the figure shows the time-series of each upside beta portfolio (U), including a 1 percent rebalancing cost, the upside beta benchmark portfolio, and the SBX over the out-of-sample period week 1 of 1998 to week 26 of 2017. Each time-series has been indexed at 100 in week 52 of 1997.

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Figure B.5 Time-series of downside beta portfolios with a 1% rebalancing cost.

Note: the figure shows the time-series of each downside beta portfolio (D), including a 1 percent rebalancing cost, the downside beta benchmark portfolio, and the SBX over the out-of-sample period week 1 of 1998 to week 26 of 2017. Each time-series has been indexed at 100 in week 52 of 1997.

Figure B.6 Time-series of upside-minus-downside beta portfolios with a 1% rebalancing cost.

Note: the figure shows the time-series of each upside-minus-downside beta portfolio (UMD), including a 1 percent rebalancing cost, the upside-minus-downside beta benchmark portfolio, and the SBX over the out-of-sample period week 1 of 1998 to week 26 of 2017. Each time-series has been indexed at 100 in week 52 of 1997.

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Figure B.7 Time-series of upside beta portfolios with a 2% rebalancing cost.

Note: the figure shows the time-series of each upside beta portfolio (U), including a 2 percent rebalancing cost, the upside beta benchmark portfolio, and the SBX over the out-of-sample period week 1 of 1998 to week 26 of 2017. Each time-series has been indexed at 100 in week 52 of 1997.

Figure B.8 Time-series of downside beta portfolios with a 2% rebalancing cost.

Note: the figure shows the time-series of each downside beta portfolio (D), including a 2 percent rebalancing cost, the downside beta benchmark portfolio, and the SBX over the out-of-sample period week 1 of 1998 to week of 26 2017. Each time-series has been indexed at 100 in week 52 of 1997.

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Figure B.9 Time-series of upside-minus-downside beta portfolios with a 2% rebalancing cost.

Note: the figure shows the time-series of each upside-minus-downside beta portfolio (UMD), including a 2 percent rebalancing cost, the upside-minus-downside beta benchmark portfolio, and the SBX over the out-of-sample period week 1 of 1998 to week 26 of 2017. Each time-series has been indexed at 100 in week 52 of 1997.

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Appendix C

Table C.1 Shapiro-Wilks W test for normally distributed portfolio returns.

Portfolio Observations W z Prob. > z

U1 (Low β+) 1018 0.95 8.59 0.00 U2 1018 0.89 10.56 0.00 U3 1018 0.93 9.52 0.00 U4 1018 0.95 8.68 0.00 U5 (High β+) 1018 0.95 8.81 0.00 Benchmark (U) 1018 0.94 9.16 0.00 D1 (Low β-) 1018 0.94 9.09 0.00 D2 1018 0.93 9.49 0.00 D3 1018 0.93 9.43 0.00 D4 1018 0.96 8.17 0.00 D5 (High β-) 1018 0.94 9.09 0.00 Benchmark (D) 1018 0.94 9.16 0.00 UMD1 (Low β+- β-) 1018 0.93 9.42 0.00 UMD2 1018 0.95 8.85 0.00 UMD3 1018 0.94 8.92 0.00 UMD4 1018 0.94 9.03 0.00 UMD5 (Low β+- β-₎ ₁₀₁₈ _0.96 _8.31 _0.00 Benchmark (UMD) 1018 0.94 9.15 0.00 SBX 1018 0.96 8.15 0.00

Note: the table shows the Shapiro-Wilks W test for each of the five upside- (U), downside- (D) and upside-minus-downside (UMD) beta portfolios, as well as their respective benchmark, and the SBX over the out-of-sample period week 1 of 1998 to week 26 of 2017. It includes the number of observations, the Shapiro-Wilks test statistic, W, and the z statistic- and probability of normality. The null hypothesis is that the variables are from a normal distribution.

Beta Based Portfolio Construction: : Stock Selection Based on Upside- and Downside Market Risk