Asset pricing at the Oslo Stock Exchange. A Source Book

Transcription

1 Asset pricing at the Oslo Stock Exchange. A Source Book Bernt Arne Ødegaard BI Norwegian School of Management and Norges Bank February 2007 In this paper we use data from the Oslo Stock Exchange in the period 1980 to 2006 to calculate descriptive statistics relevant for asset pricing at the OSE. We give statistics for number of rms, the occurences of IPO's, dividend payments, trading volume, and concentration. Returns for various market indices and portfolios are calculated and described, including portfolios sorted on size, book/market, momentum and beta. Construction of the Fama French and Carhart factor portfolios using OSE data is shown and tested. We also show the well known calendar anomalies, the link between number of stocks in a portfolio and its variance and how mean variance optimal portfolios would be constructed from various empirical portfolios. BI Norwegian School of Management, Department of Financial Economics, N-0442 Oslo, Norway. The views expressed are those of the author and should not be interpreted as reecting those of Norges Bank.

2 Contents 1 Introduction New versions Data and data sources Can you get the indices? The various chapters Data and sample selection 5 3 Activity 7 4 Market portfolios Constructing market portfolios The equity premium Sharpe Ratios for market indices Distribution of market returns Some alternative portfolios A few large stocks 19 6 Dividends 20 7 Initial Public Oers 22 8 The Capital Asset Pricing Model (CAPM) CAPM Beta based portfolios Time series estimation of the CAPM Fama MacBeth analysis Advances in testing Further reading Crossectional portfolios Size base portfolios Book to market based portfolios Momentum based portfolios How many stocks are necessary for a well diversied portfolio? Methods Results for the whole period Subperiods How close do we get to a stock market index?

3 10.5 Conclusion Volatility Calendar eects Variations in daily returns over the week Variation in daily returns over the month Variation in monthly returns over the year Mean Variance Optimal Portfolios Beta portfolios Size portfolios B/M portfolios Momentum portfolios Factor Portfolios for Asset Pricing Fama French factors The Carhart factor Describing the calculated factors Do the factors work? International links Correlations between Norway and the rest of the world Interest Rates Writings on the Oslo Stock Exchange Work in english Work in norwegian

4 Chapter 1 Introduction This document is a source book for people doing empirical asset pricing using data from the Oslo Stock Exchange (OSE). The prime purpose of the paper is pedagogical, it is to be a useful resource for teaching nance in the Norwegian context. The same purpose is reected in the lack of discussion of the results, the focus is on the numbers themselves, and students are meant to ll in the details. Having said that, the paper may still be useful for researchers since it summarizes in one place various properties of stock returns on the Norwegian stock exchange. 1.1 New versions This paper will be updated with new data and additional analysis. The latest version will always be found at my homepage bernt. I am open for suggestions to additional descriptive statistics you'd like to see, but I make no promises. 1.2 Data and data sources The source data is is daily observations of prices and volume of all stocks listed on the OSE, as well as dividends and adjustement factors necessary for calculating returns. In addition to price data accounting data for all stocks listed at the OSE is used. The data starts in The data ends in December The data comes from two sources. All accounting and equity data are from OBI (Oslo BørsInformasjon), the data provider of the Oslo Stock Exchange. Interest rate date is from Norges Bank. 1.3 Can you get the indices? The data from the OSE used in constructing the various indices is governed by an agreement with the exchange that do not allow distribution of data. The raw data on indices and portfolio returns produced in this research is therefore only available to students and researchers at the Norwegian School of Management BI. 1.4 The various chapters Chapter 2 describes the data sample and the stocks used in the analysis. Chapter 3 has some numbers on the evolution of aggregate trading volume at the OSE. Chapter 4 shows return statistics for the whole market. Chapter 5 looks at the importance of a few large stocks. Chapter 6 has some numbers on dividends at the OSE. Chapter 7 looks at IPO's, and details the annual number of IPO's at the OSE. Chapter 8 3

5 investigates the CAPM. Chapter 9 looks at crossectional portfolios. Chapter 10 replicates the classical analysis of e.g. Wagner and Lau (1971) which looks at the link between the number of assets in a portfolio and the variance of the portfolio, illustrated with simulations on Norwegian data. Chapter 11 looks at the volatility of stocks at the OSE. Chapter 12 shows some calendar eects. Chapter 13 shows the construction of mean variance ecient portfolios using Norwegian data. Chapter 14 discusses construction of the factor portfolios of Fama and French (1992) and Carhart (1997). Chapter 15 shows some numbers for correlations between OSE and other stock markets. Chapter 16 details the interest rate data. Chapter 17 give references to empirical writings on the OSE. 4

6 Chapter 2 Data and sample selection The basic data for the empirical investigations in this paper are daily observations of all equities traded at the Oslo Stock exchange. The data contains end of day bid and oer prices, as well as the last trade price of the day, if there was any trading. The data also include the total trading volume at a given date. Not all stocks traded at the Oslo Stock Exchange should necessarily be used for empirical asset pricing investigations. In particular stocks which are seldom traded are problematic. In the following calculations we therefore require the stocks to have a minimum number (20) of trading days before they enter the sample. Low valued stocks (penny stocks) are also problematic since they will have very exaggerated returns. We therefore limit a stocks to have a price above NOK 10 before considering it in the sample. A similar requirement considers total value outstanding, which has a lower limit of NOK 1 million. 1 Table 2.1 provides some descriptive statistics for this ltering of the sample. 1 It should be noted that ltering such as this is very common for asset pricing investigations of this sort. See for example Fama and French (1992). 5

7 Table 2.1 Describing securities sample year number of average Number of securities with securites number of more than 20 trading days listed trading and price above 10 days and company value above 1 mill average The table provides some descriptive statistics for the sample of equities traded on the Oslo Stock Exchange in the period 1980 to The rst column lists the year. The second column lists the number of stocks listed during the year. The third column the average number of trading days for all listed stocks. The fourth column lists the number of stocks which traded for more than 20 days. The fth column adds the requirement that the stock did not have a price below NOK 10. The nal column additionally requires an equity market value above 1 mill NOK. 6

8 Chapter 3 Activity A simple measure of the activity at the OSE is the total trading volume in NOK. Figure 3.1 Monthly trading volume Monthly trade volume (mill) The plot shows monthly trading volume at the OSE, in mill NOK, using all stocks on the Exchange. 7

9 Chapter 4 Market portfolios The rst issue we consider is the evolution of the whole market at the Oslo Stock Exchange. 4.1 Constructing market portfolios A typical question is what what one would earn if one invested in stocks at the Oslo Stock Exchange. However, there are (at least two) dierent ways to answer that question. If one picks a random stock, one wants to nd the expected return for the typical stock, in which case an equally weighted average is the relevant measure. Alternatively, one can invest in the whole market, in which case a value weighted average is most relevant. Two indices are constructed to make this measurement. Stocks not satisfying the lter criterion discussed in chapter 2 are removed. Using the remaining stocks equally weighted and value weighted indices are constructed. The indices are constructed to include dividends and other distributions from the stocks. 1 In addition to these indices two market indices constructed by the Oslo Stock Exchange are used. The OBX is a value weighted index consisting of the thirty most liquid stocks at the stock exchange. This index was constructed to be the basis for derivatives contracts, and initiated at the beginning of In addition we consider a value weighted index of all stock on the exchange, termed TOT. The Oslo Stock Exchange has changed indices during the period, in the period up to 1999 the total index was called the TOTX. In 1999 this index was replaced by the All Share Index. TOT is constructed by splicing these two indices. Note that neither of these indices include dividends. Table 4.1 shows monthly average returns for the various indices for the whole period 1980 till 2006 and for various subperiods. An alternative view of the dierence between equally weighted and value weighted indices is shown in gure 4.1, which illustrates the growth of the two indices in the period from The indices do however not account for repurchases. 8

10 Table 4.1 Describing market indices at the Oslo Stock Exchange from 1980 Monthly returns Period index Returns Dividend Yield Capital Gains mean (std) min med max mean med mean med EW 1.92 (5.83) VW 2.10 (6.34) OBX 1.07 (6.77) TOT 1.48 (6.32) EW 3.04 (6.27) VW 2.34 (7.00) TOT 3.37 (6.24) EW 1.60 (5.80) VW 2.56 (6.83) OBX 2.41 (8.57) TOT 1.61 (6.98) EW 1.12 (6.83) VW 1.39 (6.67) OBX 0.24 (7.16) TOT 0.60 (6.79) EW 2.01 (4.93) VW 2.24 (5.52) OBX 1.13 (5.78) TOT 1.42 (5.55) EW 1.41 (5.37) VW 1.70 (5.71) OBX 0.45 (6.50) TOT 0.69 (5.79) EW 2.73 (3.68) VW 3.22 (4.91) OBX 2.57 (4.86) TOT 3.12 (5.07) The table describes two indices constructed from Norwegian equity market data, one equally weighted and one value weighted, using data starting in The numbers are percentage monthly returns. mean: (equally weighted) average. med: median. EW: equally weighted index. VW: value weighted index. TOT: Total index provided by OSE: : TOTX, Afterwards: All Share Index. OBX: Index provided by OSE. Contains the 30 most liquid stocks at the OSE. Note that OBX starts in January Returns are percentage monthly returns. The returns are not annualized. Data for stocks listed at the Oslo Stock Exchange during the period Calculations use the stocks satisfying the lter criteria discussed in chapter 2. 9

11 Figure 4.1 The evolution of market indicies vw ew The gures illustrates the growth of two OSE stock indices, one equally weighted and one value weighted, using data starting in Growth is shown by nding how much one NOK invested in January 1980 would have grown to. Data for stocks listed at the Oslo Stock Exchange during the period Calculations use the stocks satisfying the lter criteria discussed in chapter 2. 10

12 To see how dierent the indices are table 4.2 show the correlations of their returns. Table 4.2 Correlations between alternative market indices Panel A: Monthly returns Panel B: Weekly returns Panel C: Daily returns ew vw tot vw 0.88 tot obx ew vw tot vw 0.80 tot obx ew vw tot vw 0.83 tot obx The tables shows correlations between index returns for various market indices at the Oslo Stock Exchange. EW: equally weighted index. VW: value weighted index. TOT: Total index provided by OSE: : TOTX, Afterwards: All Share Index. OBX: Index provided by OSE. Contains the 30 most liquid stocks at the OSE. Note that OBX starts in January

13 4.2 The equity premium The equity premium is the return of a stock or a stock portfolio in excess of a risk free return er i = r i r f where r i is the return on a stock or a stock index, and r f a risk free rate. Whether the average return on equity is "too high" to be justied is a a long standing issue in nance. 2 In this section some estimates of the equity premium are calculated. Let us start by using monthly observations of stock index returns. Let r mt be the market return observed at date t. This is an ex post return calculated as r mt = xt xt 1 x t 1, where x t is the index level at time t. If this is a monthly return, the relevant risk free interest rate is the one month interest rate observed at date t 1, because this is the interest rate that can be guaranteed for the period t 1 to t. The excess return is thus calculated as er t = r mt r f,t 1 where r f,t 1 is the one month interest rate observed at date t 1. Table 4.3 shows estimates of this monthly excess market return. 2 See the literature starting with Mehra and Prescott (1985). A survey is provided in Kocherlakota (1996). 12

14 Table 4.3 Excess returns of market indices at the Oslo Stock Exchange from 1982 Monthly excess returns Period index Excess Returns Excess Dividend Yield Excess Capital Gains mean (std) min med max mean med mean med EW 1.15 (5.81) VW 1.55 (6.24) OBX 0.47 (6.79) TOT 0.80 (6.33) EW 2.19 (6.32) VW 2.36 (6.56) TOT 2.30 (6.24) EW 0.52 (5.83) VW 1.48 (6.85) OBX 1.31 (8.58) TOT 0.53 (6.99) EW 0.35 (6.84) VW 0.61 (6.68) OBX (7.17) TOT (6.80) EW 1.56 (4.94) VW 1.79 (5.52) OBX 0.68 (5.79) TOT 0.98 (5.55) EW 0.95 (5.43) VW 1.24 (5.76) OBX (6.55) TOT 0.24 (5.85) EW 2.52 (3.69) VW 3.00 (4.92) OBX 2.36 (4.86) TOT 2.91 (5.08) The table describes market indices for the Oslo Stock Exchange using data starting in (The risk free rate is only available from 1982.) The numbers are percentage monthly excess returns, returns in excess of the risk free rate. EW: equally weighted index. VW: value weighted index. OBX: Index provided by OSE. Contains the 30 most liquid stocks at the OSE. Note that OBX starts in January TOT: Total index provided by OSE: : TOTX, Afterwards: All Share Index. Returns are percentage monthly returns. The returns are not annualized. Data for stocks listed at the Oslo Stock Exchange during the period Calculations use the stocks satisfying the lter criteria discussed in chapter 2. 13

15 However, it it not clear that one want to use this high frequency data to estimate the longer term equity premium. One problem is that the monthly risk free rate is rather volatile. 3 An alternative is to use annual index returns and annual interest rates. Table 4.4 gives estimates of annual excess returns using a one year interest rate. Table 4.4 Annual excess returns Annual excess returns Index Period Average Annual Excess Return EW ( ) VW ( ) OBX ( ) 5.88 TOT ( ) 8.72 EW: equally weighted index. VW: value weighted index. OBX: Index provided by OSE. Contains the 30 most liquid stocks at the OSE. Note that OBX starts in January TOT: Total index provided by OSE: : TOTX, Afterwards: All Share Index. Note that the risk free rate series starts in See chapter 16 for some data on interest rates. 14

16 4.3 Sharpe Ratios for market indices The Sharpe ratio is a relative measure of how much return one gets per unit of risk, where risk is measured by the standard deviation. The Sharpe ratio is dened as SR i = E[r i] r f σ(r i r f ) The Sharpe Ratios in table 4.5 are estimated by replacing E[r i r f ] and σ(r i r f ) by their sample averages. Table 4.5 Sharpe ratios market indices at the Oslo Stock Exchange from 1980 Monthly returns EW VW The table shows ex post Sharpe ratios for market indices constructed from Norwegian equity market data. EW: equally weighted index. VW: value weighted index. OBX: Index provided by OSE. Contains the 30 most liquid stocks at the OSE. Note that OBX starts in January TOT: Total index provided by OSE: : TOTX, Afterwards: All Share Index. Returns are percentage monthly returns. The returns are not annualized. Data for stocks listed at the Oslo Stock Exchange during the period Calculations use the stocks satisfying the lter criteria discussed in chapter 2. 15

17 4.4 Distribution of market returns The statistical descriptions of the previous chapters does not give a complete picture of the distributional properties of the market returns. One way to show more detail is to plot the actual distributions. Figures 4.2, 4.3 and 4.4 shows histograms of respectively monthly, weekly and daily returns for the EW index. Figure 4.2 Histogram of monthly stock returns The gure shows the distribution of monthly stock return for the EW index. EW: equally weighted index. Data for stocks listed at the Oslo Stock Exchange during the period Calculations use the stocks satisfying the lter criteria discussed in chapter 2. 16

18 Figure 4.3 Histogram of weekly stock returns The gure shows the distribution of weekly stock return for the EW index. EW: equally weighted index. Data for stocks listed at the Oslo Stock Exchange during the period Calculations use the stocks satisfying the lter criteria discussed in chapter 2. Figure 4.4 Histogram of daily stock returns The gure shows the distribution of daily stock return for the EW index. EW: equally weighted index. Data for stocks listed at the Oslo Stock Exchange during the period Calculations use the stocks satisfying the lter criteria discussed in chapter 2. 17

19 4.5 Some alternative portfolios In addition to the broad market indices EW and VW a couple of alternative indices are constructed. If we want to make an investment in the market, but worry about transaction costs, one solution is to invest in a lower number of stocks. To look at how representative such portfolios are we construct indices using the 20 largest stocks at the OSE. Two such portfolios are calculated, 20EW and 20VW. For both indices we choose the 20 largest stocks at the beginning of the year. These stocks are then used to create portfolios for the next year, either equally weighted or value weighted. At each yearend the sample of stocks is changed to be the 20 largest stocks at that time. Table 4.6 Some special indices at the Oslo Stock Exchange from 1980 Average returns Period index Returns mean (std) min med max EW 2.15 (10.03) VW 2.15 (10.03) EW 1.73 (7.80) VW 1.73 (7.80) EW 1.95 (7.38) VW 1.95 (7.38) EW 1.08 (7.31) VW 1.08 (7.31) EW 3.88 (13.33) VW 3.88 (13.33) EW 2.29 (11.75) VW 2.29 (11.75) Correlations with other indices 20ew 20vw 20vw 0.79 ew vw tot obx The table describes indices for the Oslo Stock Exchange using data starting in The numbers are percentage monthly returns. 18

20 Chapter 5 A few large stocks The OSE has always had a few large companies which in terms of market capitalization have a dominant position on the exchange. For many years it was Norsk Hydro, but with the listing of the large, state dominated companies Telenor and Statoil this changed. To illustrate to what degree the exchange is likely to be aected by these large companies table 5.1 shows, for each year, the four largest companies, and each company's fraction of the value of the exchange. Table 5.1 The four largest companies each year year Largest 1980 Norsk Hydro 52.6 Saga Petroleum 9.4 Den norske Creditbank 4.5 Christiania Bank og Kreditkasse Norsk Hydro 33.6 Den norske Creditbank 5.3 Saga Petroleum 4.7 Actinor Norsk Hydro 29.3 Den norske Creditbank 6.9 Norsk Data 5.7 Storebrand Norsk Hydro 23.0 Norsk Data 7.4 Den norske Creditbank 4.7 Alcatel STK Norsk Hydro 13.2 Norsk Data 8.1 Den norske Creditbank 3.6 Alcatel STK Norsk Hydro 13.1 Norsk Data 7.6 Den norske Creditbank 4.0 Hafslund Norsk Hydro 11.9 Norsk Data 7.1 Den norske Creditbank 3.6 Christiania Bank og Kreditkasse Norsk Hydro 12.2 Hafslund 5.4 Bergesen d.y 3.8 Norsk Data Norsk Hydro 23.8 Hafslund 9.1 Bergesen d.y 5.8 NCL Holding Norsk Hydro 20.2 Bergesen d.y 6.4 Hafslund 5.3 Saga Petroleum Norsk Hydro 23.2 Saga Petroleum 7.2 Hafslund 6.0 Orkla Norsk Hydro 20.0 Hafslund 10.6 Saga Petroleum 7.7 Kværner Norsk Hydro 26.3 Hafslund 10.6 Saga Petroleum 7.8 Orkla Norsk Hydro 23.2 Kværner 8.1 Orkla 7.3 Hafslund Norsk Hydro 24.7 Kværner 5.5 Hafslund 5.0 Orkla Norsk Hydro 21.3 Hafslund 5.8 Orkla 5.3 Saga Petroleum Norsk Hydro 19.2 Orkla 5.2 Transocean Oshore 5.2 Den norske Bank Norsk Hydro 13.3 Transocean Oshore 5.5 Nycomed Amersham 5.4 Orkla Norsk Hydro 13.1 Royal Caribbean Cruises 10.3 Nycomed Amersham 7.5 Orkla Norsk Hydro 13.7 Royal Caribbean Cruises 9.4 Nycomed Amersham 4.8 Den norske Bank Norsk Hydro 15.1 Nycomed Amersham 6.9 Royal Caribbean Cruises 6.0 Orkla Statoil ASA 18.4 Norsk Hydro 13.7 Telenor 9.5 Nycomed Amersham Statoil ASA 22.8 Norsk Hydro 14.7 Telenor 8.5 Nycomed Amersham Statoil ASA 20.7 Norsk Hydro 13.9 Telenor 9.9 Nycomed Amersham Statoil ASA 21.3 Norsk Hydro 12.9 Telenor 9.8 Den norske Bank Statoil ASA 26.4 Norsk Hydro 14.0 Telenor 8.8 Den norske Bank Statoil ASA 23.7 Norsk Hydro 16.3 Telenor 13.1 Den norske Bank 7.7 The table lists the four largest companies on the exchange in terms of the market capitalization. For each company we list the name and the fraction of the market capitalization this company had at yearend. 19

21 Chapter 6 Dividends In this chapter we describe various aspects of dividend payments at OSE. First we look at the actual dividend amounts per security. Table 6.1 straties dividends amounts into four groups: no dividend payment, dividend up to NOK 5, dividend between NOK 5 and NOK 10, and dividend above NOK 10. The most striking feature of the table is the number of stocks which is not paying dividend at all, particularly in the early period. To further illustrate this particular point gure 6.1 shows the fraction of companies on the OSE which is not paying dividends. The gure clearly shows a regime change in dividend payments, where in 1985 close to 75% of the companies on the OSE did not pay dividends, which had fallen to less than 30% in This particular change is most likely a result of a tax change. In 1992 a new tax code was introduced. Under the new code dividends are much less tax disadvantaged. The huge increase in rms starting to pay dividends is most likely a result of this tax code change. Figure 6.1 What fraction of securities do not pay dividends? The plot shows what fraction of companies on the OSE does not pay dividend. Data for stocks listed at the Oslo Stock Exchange during the period Calculations use the stocks satisfying the lter criteria discussed in chapter 2. 20

22 Table 6.1 How much are companies paying in dividends? year d = 0 d (0, 5] d (5, 10] d > The table illustrates the amount paid in dividends by companies on the OSE. For each stock we nd the annual amount of dividend payment per stock. Each year we then calculate the number of stocks with dividends of zero, dividends between zero and ve, dividends between ve and ten, and dividends above 10. Data for stocks listed at the Oslo Stock Exchange during the period Calculations use the stocks satisfying the lter criteria discussed in chapter 2. 21

23 Chapter 7 Initial Public Oers In this chapter we give some details on new listings at the OSE. Table 7.1 shows how many rms are introduced at the OSE each year. Table 7.1 New listings per year Year Number of new listings

24 Figure 7.1 New listings per year

25 Chapter 8 The Capital Asset Pricing Model (CAPM) 8.1 CAPM In nance, the historically most important formalization of the idea that expected returns should be increasing in risk is known as the Capital Asset Pricing Model (CAPM), developed by Sharpe (1964), Lintner (1965) and Mossin (1966). The intuition which leads to the CAPM is that the only relevant risk for decisions is the systematic risk. Diversiable, non-systematic risk should not be priced. In a two period model where the consumption in the second period is the result of the return on the stock market, it should be obvious that the relevant risk for an asset is its covariability with the stock market. In empirical terms, the capital asset pricing model gives the crossectional prediction that expected returns should be linearly related to the stock's beta, E[r it ] = r f + β i (E[r m ] r f ) where beta measures the stock's sensitivity to market movements: 8.2 Beta based portfolios β i = cov(r i, r m ) var(r m ) For people wanting to use the CAPM, an obvious implication is that returns should be increasing in beta. If we have an estimate of a stocks beta, we should observe that stocks with higher beta have higher returns. A simple test of the implications of the CAPM is to rst estimate beta, and then see if portfolios with higher estimated betas have higher subsequent returns. 1 Since one do not observe beta it needs to be estimated. A common strategy is to use historical returns on a stock market index and stock returns to estimate β it = cov(r it, r mt ), var(r m ) and then see if in the crossection, this beta estimate is related to future return. Going forward, one reestimate beta and redo portfolio formation. Figure 8.1 illustrates the idea. In implementing this, we need to choose a time period to do this estimation over, and a returns frequency. Common choices are monthly returns using ve years of data. Alternatively one can use weekly or daily returns. To test the predictive ability of the CAPM redo this beta calculation every month, and see whether 1 A classical example of this style of analysis applied to US data is Sharpe and Cooper (1972). 24

26 Figure 8.1 Beta portfolios } {{ } β i,t 36,t 1 } {{ } β i,t 36,t 1 } {{ } β i,t 36,t 1 on average estimated low beta stocks have low returns. Using data for the Oslo Stock exchange, in the following we show the results from constructing 5 portfolios sorted on the estimated beta. Results are shown for the whole period and for various subperiods. We also consider two dierent methods for estimation of beta: One use 5 years of monthly returns data, another use 3 years of daily returns data. Table 8.1 show the results for the whole period, calculating average returns of beta-sorted portfolios. Table 8.1 Beta portfolios Using monthly returns in the beta estimation Returns Number of securities Portfolio mean (std) min med max min med max 1 (smallest) 1.60 (4.5) (5.1) (5.8) (7.0) (8.8) Using daily returns in the beta estimation Returns Number of securities Portfolio mean (std) min med max min med max 1 (smallest) 1.93 (4.6) (5.0) (6.4) (7.1) (8.6) Returns are percentage monthly returns. The returns are not annualized. Data for stocks listed at the Oslo Stock Exchange during the period Calculations use the stocks satisfying the lter criteria discussed in chapter 2. What one should observe is that the average return is increasing with beta, which does seem to be the case for the case. 25

27 To avoid the confounding eect of time varying interest rates table 8.2 shows excess returns for the same portfolios. Table 8.2 Beta portfolios Excess returns Using monthly returns in the beta estimation Excess Returns Number of securities Portfolio mean (std) min med max min med max 1 (smallest) 0.93 (4.5) (5.2) (5.8) (7.0) (8.8) Using daily returns in the beta estimation Excess Returns Number of securities Portfolio mean (std) min med max min med max 1 (smallest) 1.25 (4.6) (5.0) (6.5) (7.1) (8.6) Returns are percentage monthly returns. The returns are not annualized. Data for stocks listed at the Oslo Stock Exchange during the period Calculations use the stocks satisfying the lter criteria discussed in chapter 2. To consider the stability of these relations over time, tables 8.3 and 8.4 show similar results split by time period. While these are relatively short periods it may still be instructive to check the stability of the relations. Overall there seem to be a (slightly) positive relationship between beta estimates and subsequent returns. However, just looking at the realized portfolio returns like this does not constitute a formal test of a model. 26

28 Table beta portfolios in subperiods. Monthly returns in beta estimation Returns Number of securities Portfolio mean (std) min med max min med max 1 (smallest) 1.84 (5.0) (6.0) (6.2) (7.4) (8.9) Returns Number of securities Portfolio mean (std) min med max min med max 1 (smallest) 1.46 (5.0) (5.4) (6.1) (7.0) (8.8) Returns Number of securities Portfolio mean (std) min med max min med max 1 (smallest) 1.59 (2.8) (3.6) (4.7) (6.6) (8.6) Returns are percentage monthly returns. The returns are not annualized. Data for stocks listed at the Oslo Stock Exchange during the period Calculations use the stocks satisfying the lter criteria discussed in chapter 2. 27

29 Table beta portfolios, subperiods. Daily returns in beta estimation Returns Number of securities Portfolio mean (std) min med max min med max 1 (smallest) 2.52 (5.1) (5.4) (7.4) (7.6) (8.0) Returns Number of securities Portfolio mean (std) min med max min med max 1 (smallest) 1.70 (5.0) (5.5) (6.4) (7.0) (8.8) Returns Number of securities Portfolio mean (std) min med max min med max 1 (smallest) 1.59 (2.8) (3.6) (4.7) (6.6) (8.6) Returns are percentage monthly returns. The returns are not annualized. Data for stocks listed at the Oslo Stock Exchange during the period Calculations use the stocks satisfying the lter criteria discussed in chapter 2. 28

30 8.3 Time series estimation of the CAPM Let r it be the return on stock i at time t, r mt the return on the market and r ft the risk free rate. For stock i the CAPM imposes the relationship writing this in excess return form results in E[r it ] = r ft + β i (E[r mt r ft ) E[r it ] r ft = β i (E[r mt r ft ) If we collect a time series of stock returns on a given stock or portfolio i, and run the time series regression (r it r ft ) = a i + b i (r mt r ft ) + ε it, the CAPM predicts that in this regression a i = 0. The Black, Jensen, and Scholes (1972) paper is the prototypical paper that performs such a regression. In implementing the Black et al. (1972) method one will typically apply it at the portfolio level to reduce noise in the estimation. Table 8.5 BJS results Beta sorted portfolios a pval b pval R 2 n (0.02) (0.00) (0.06) (0.00) (0.41) (0.00) (0.34) (0.00) (0.24) (0.00) (0.69) (0.00) (0.33) (0.00) (0.02) (0.00) (0.42) (0.00) (0.15) (0.00) The table shows results from estimation of a and b in the regression (r it r ft ) = a i + b i(r mt r ft ) + ε it on 10 beta sorted portfolios. Portfolio 1 is the portfolio of the 10% stocks with lowest estimated beta, portfolio 10 contains the 10% with the highest estimated beta. Betas are updated every month. The rst column is the portfolio number, column 2 the coecient estimates for a, column 3 the probability value for a test of the estimated a coecient being dierent from zero, column 4 the coecient estimates for b, column 5 the probability value for a test of the estimated b coecient being dierent from zero, column 6 the R 2 value for the regression, and the last column contains n, the number of observations. The pvalues are calculated as OLS estimates. Data for stocks listed at the Oslo Stock Exchange during the period Calculations use the stocks satisfying the lter criteria discussed in chapter 2. Table 8.5 shows results from running the analysis on 10 portfolios sorted on beta. That is, the stocks are pre-sorted based on beta estimates using 3 years of prior daily returns data, and the calculated beta is used to place a stock in a portfolio. This procedure may seem to introduce some noise into the estimation, but it actually is designed to do the opposite, since it is will induce crossectional variation in the portfolios. In this regression standard OLS estimates are used to calculate the p values. What does the table say about the CAPM? Well, given the prediction that a should be equal to zero the results are supportive of the market portfolio being the only determinant of stock returns. Except for two cases, the null a = 0 is not rejected at the 5% level. However, in terms of economic signicance if we compare the portfolios with the largest positive and negative returns, the dierence is signicant. With a long-short portfolio of the beta portfolios 1 and 8, an investor would earn an annual return dierence of approx 12 (0.006 ( 0.005)) = 13.2%. But this calculation does not account for the sampling error in the estimates of a i. To see whether the pre-sorting into portfolios is important we do a similar time-series analysis for single stocks. That is, for each stock we run the regression (r it r ft ) = a i + b i (r mt r ft ) + ε it, and look at the estimates of a and b. Table 8.6 summarises the results of running the regression for all stocks listed at the OSE in the period. Note that most of the stocks are not on the exchange for the whole period. In the 29

31 Table 8.6 BJS results a b R 2 mean pvalue test di zero no signif(5%) n 449 The table summarizes the results of running the regression (r it r ft ) = a i + b i(r mt r ft ) + ε it on single securities at the Oslo Stock Exchange in the period The rst row gives means of the estimates of a i, b i and R 2 for the individual securities. The second row gives p-values for the test of the null hypotheses a i = 0 and b i = 0. The third row counts the number of estimates of respectively a i and b i that are signicant at the 5% level. n is the number of dierent individual securities used in the estimation. estimation we estimate the parameters a and b for each stock for the period the stock is listed, and then calculate the crossectional average of a i and b i. Treating these as iid observations we can not reject that a = 0, but we do reject that b = 0. In fact, b is close to one, as it should be. We also count the number of times the individual estimates reject a = 0, which is 30. This is not much more than you would expect by pure chance (5% of ). Overall we are not rejecting the CAPM by this method. What is not clear is whether this is due to the noise in the data being so large that it is not possible to get signicance, or whether the CAPM seems to be a reasonable characterization of the data. 30

32 8.4 Fama MacBeth analysis An alternative to the time series regression of Black et al. (1972) is the rolling regression approach pioneered by Fama and MacBeth (1973). This method is closer to the method used to construct portfolios earlier, and is explicitly engineered to test the crossectional implications, that returns should be increasing in risk, as measured by beta. Figure 8.2 illustrates the method. At each date t a crossection of n stock returns is observed. At each date a crossectional regression r t = X tˆbt is run, where X t is a set of observable characteristics. When testing the CAPM, the observable characteristic is an estimate of the stock beta, ˆβ i, typically estimated using a few years of prior data, but the method allows for putting any observable characteristic as an explanatory variable. Figure 8.2 The method of Fama Macbeth Time: t 1 t t Stock/Portfolio 1 r 1,t X 1,t r 1,t+1 X 1,t r 2,t X 2,t r 2,t+1 X 2,t n r n,t X n,t r n,t+1 X n,t+1... r t = X tˆbt r t+1 = X t+1ˆbt+1... average(ˆb t ) To reduce the errors in variables problem that betas are estimated, the Fama MacBeth procedure is often carried out on portfolios. Let us rst do this on 10 portfolios, each period estimating er it = γ 0t + γ 1t β it where er it is the excess return for stock i at time t, β it the estimated beta for the stock, and γ 0t and γ 1t are estimated. Table 8.7 shows the results. The CAPM predict that γ 0 = 0 and that γ 1 > 0. We do get a positive estimate of γ 1, but it is not statistically signicant. The above analysis was carried out by pre-sorting the stocks into beta sorted portfolios. It can also be carried out for individual stocks, although this typically is more noisy. Analysis for single stocks is shown in table 8.8, both for the whole period and for two sub-periods. In statistical terms the results are actually stronger, we reject γ 0 = 0 in all three cases. We also get positive estimates of γ 1, although they are not signicant. 31

33 Table 8.7 Fama Macbeth for portfolios Daily returns in beta estimation Weekly returns in beta estimation Monthly returns in beta estimation mean pval γ γ n 250 mean pval γ γ n 249 mean pval γ γ n 245 Results from estimating in the crossection er it = γ 0t + γ 1tβ it and then taking the time series averages of the estimated parameters γ 0 and γ 1. The assets are 10 portfolios sorted by beta, where beta is calculated using daily data. The numbers in parenthesis are p-values, estimated using standard OLS assumptions. Data for stocks listed at the Oslo Stock Exchange during the period Calculations use the stocks satisfying the lter criteria discussed in chapter 2. Table 8.8 Results from Fama MacBeth regressions on single stocks Daily returns in beta estimation Weekly returns in beta estimation Monthly returns in beta estimation Determinants of er i constant (0.00) beta (0.82) Average R Number of Periods 287 Determinants of er i constant (0.00) beta (0.96) Average R Number of Periods 286 Determinants of er i constant (0.01) beta (0.15) Average R Number of Periods 282 Results from estimating in the crossection er it = γ 0t + γ 1tβ it and then taking the time series averages of the estimated parameters γ 0 and γ 1. The assets are individual stocks. For each stock β it is calculated using 3 years of daily returns. The row marked "constant" contains the average of estimated γ0t. The row marked "beta" contains the average of the estimates of γ 1t. The numbers in parenthesis are p-values, estimated using standard OLS assumptions. Data for stocks listed at the Oslo Stock Exchange during the period Calculations use the stocks satisfying the lter criteria discussed in chapter 2. 32

34 8.5 Advances in testing The two last sections have illustrated the use of the two "classical" approaches to testing of the CAPM, initiated by Black et al. (1972) and Fama and MacBeth (1973). There is an enormous literature in nance that builds on these approaches and develop tests that can more explicitly test the model, and ask whether the model is an sucient description, or other "factors", such as rm size, are signicant risk factors. Later versions of this paper will include some of these, for now they are left as an exercise for the reader. 8.6 Further reading Textbook discussions of testing the CAPM are in (Huang and Litzenberger, 1988, Ch 10), Campbell, Lo, and MacKinlay (1997), Cochrane (2005) and Singleton (2006). Such early tests of the CAPM as Black et al. (1972) and Fama and MacBeth (1973) has been expanded upon in terms of econometric methods. For the interested, the following are some interesting papers: Gibbons (1982), MacKinlay (1987), Gibbons, Ross, and Shanken (1989) and MacKinlay and Richardson (1991). See Cochrane (2005) and Singleton (2006) for further references. On beta estimation, see Dimson (1979). Some important papers that looks at more conceptual issues in testing the CAPM are Roll (1977) and Stambaugh (1982). 33

35 Chapter 9 Crossectional portfolios In nance a number of so called "anomalies" has been introduced, which shows links between the crossection of asset prices and an observable characteristic of the stock in question, such as rm size and book to market portfolios. Each of them earned the name anomaly because it could not be explained by the standard benchmark asset pricing model, the CAPM, which relates asset returns to one factor, namely the stock beta. We will here illustrate the best known such crossectional anomalies, namely the size eect, the book to market eect and the momentum eect. 9.1 Size base portfolios Sorting securities based on the equity value of companies outstanding is a popular investigation, starting with Banz (1981). Internationally, the typical result is that the portfolios of small stocks (ie stocks with a low total equity value) tend to have high returns relative to the portfolio of large stocks. We provide a number of dierent views of size sorted portfolios. We start by using the same method as was shown with the CAPM earlier, portfolio sorts. We simply split the relevant equities into portfolios based on rm size at the beginning of a period. We consider ve and ten portfolios. The sorting is done on an annual basis. At the beginning of each January, stocks are sorted based on their equity size at the end of last year, and placed in the relevant portfolio. Portfolios are re-balanced every year. Table 9.1 provide descriptive statistics for monthly returns for 5 and 10 size sorted portfolios for the whole period. The sample is split into sub-periods and shown for the case of ve portfolios in table 9.2. As the tables show that the smallest portfolio is the one with the highest return, and return is roughly decreasing with size. Oslo Stock Exchange thus has the typical pattern. Note also that the pattern is changing over time, in the most recent period the pattern is by no means that clear. 34

36 Table 9.1 Size based portfolios, monthly returns, EW portfolios 10 EW portfolios Returns Number of securities Portfolio mean (std) min med max min med max 1 (smallest) 3.10 (6.6) (6.6) (6.4) (6.8) (7.0) Returns Number of securities Portfolio mean (std) min med max min med max 1 (smallest) 3.43 (7.9) (7.3) (7.3) (7.3) (7.1) (6.9) (7.5) (7.1) (8.0) (7.1) Returns are percentage monthly returns. The returns are not annualized. Data for stocks listed at the Oslo Stock Exchange during the period Calculations use the stocks satisfying the lter criteria discussed in chapter 2. 35