INVESTMENT DECISIONS

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Is Expected Return of an Asset positively related to Risk?If you need a copy of the paper below in a "readable" format please contact us.

Statement of the hypothesis

HO: Expected return of an asset is not positively related to risk

HA: Expected return of an asset is positively related to risk

Testing the restriction a = 0

Testing the null or maintained hypothesis a = 0 against the alternative that a ≠ 0

H0: a = 0

H1: a ≠ 0

Testing hypothesis about β

Testing the null or maintained hypothesis β = 1 against the alternative that β ≠ 1.

H0: β = 1

H1: β ≠ 1

Description of Data and Sampling Method

For my paper I selected 32 firms in the Computer Hardware Industry. All the 32 firms are listed on pages 5 and 6 of this paper (http://biz.yahoo.com/p/cmptrsconameu.html). I used 32 firms because sample size is considered large if n > 30. A large sample is more significant than a small sample. Not all the firms I selected are members of the S&P 500. The entire Computer Hardware Industry, the number of S&P 500 members is less than 15. So in my paper I am using the S&P 500 (^SPX) just for illustrational purposes because not all the firms are members. After-all, I am not trying to prove anything. The model was already proven. Thanks to the Internet, I was able to download stock prices and S&P 500 prices for a 5 year period (January 1998 to December 2002) straight from Yahoo! Finance to my spreadsheet (www.geocities.com/venji98/betacalculation.xls). The historical stock prices of each of the 32 Computer Hardware Industry firms can be found by clicking on the following link: http://biz.yahoo.com/p/cmptrsconameu.html, click on any one of the companies, then click on historical prices. The S&P 500 (^GSPC) historical prices can be found at the link given below: http://finance.yahoo.com/q/hp?s=%5EGSPC&a=11&b=02&c=2002&d=10&e=15&f=2003&g=m . I collected the 91 days T-Bill rates from HUD website: http://cms.hhs.gov/medicaid/drugs/drug4.asp. T-Bill rates can also be collected from Yahoo! Finance and the Federal Reserve Websites. My study covers 60 months, from January 1998 to December 2002. I used 60 months because beta is not stable over time. I used the 3-month T Bill rates because shorter maturity T-Bills are rather volatile. So I converted the 3-month T Bill rates to monthly rates to match with closing stock prices and S&P Prices. Description of Statistical Testing Procedure

I will use the simplified single index model (Sharpe (1963)) to predict security returns. The major characteristic and primary shortcoming of the single index model is that the only factor influencing a security’s return is its sensitivity to changes in the market portfolio return (Martin and Klemkosky (1976)).

I will use the Excel regression tool. The statistical model will be as follows:

rAt = aA + βArMt + ε where rAt is the return at time t for a particular asset A, rMt is the market return, aA is the risk premium intercept, ßA is the market model beta

Testing the restriction a = 0

H0: a = 0

H1: a ≠ 0

Alpha is the intercept of the regression equation. To determine how big the estimated value of a needs to be in order to reject

the null hypothesis we use the t-statistics:

ta=0 = ( _ 0) / [SE( ) + έ]

where is the least squares estimate of a and SE( ) is the estimated standard error. The value of the t-statistic, ta=0, gives
the number of estimated standard errors that a is from zero. If the absolute value of ta=0 is much larger than 2 then the data
cast considerable doubt on the null hypothesis a = 0 whereas if it is less than 2 the data are in support of the null hypothesis

Testing hypothesis about β

H0: β = 1

H1: β ≠ 1

I will estimate Beta by using the Willow Solutions Inc tip # 2002-03. The tip link is

www.willowsolutions.com/tips/tips_2002_03.shtml.

The only difference is that I used monthly while they used weekly percentage changes. To determine how big the estimated

value of β must be in order to reject the null hypothesis we use the t statistic:

tβ=1 = ( - 1) / [SE( )]

which measures how many estimated errors the least squares estimate of β is from one. The null hypothesis is reject at the 5%

level, say if | tβ=1 | > |tT-2 (0.025)|. This is a two-sided test. In a one-sided test we reject the null hypothesis only if the

estimated value of β is much greater than one. The null hypothesis on this one is reject hypothesis at 5% if

tβ=1 < - tT-2 (0.05) where tT-2 (0.05) is the one-sided 5% critical value of the t-distribution with T-2 degrees of freedom.

The Statistical Assumptions underlying the Single Index Model are as follows:

1) Returns are normally distributed 2) News is neutral 3) Magnitude of typical news events is constant over time (homoskedasticity) 4) News is independent 5) News affecting asset i in time t is independent of news affecting asset j in time s 6) Any correlation between asset i and asset j is solely due to their common exposure to RMt throughout the values of βi and βj.

Statement of Results Company Name Regression Equation R2 ta=0 tβ=1 1 Apple Computer, Inc (AAPL) =-3.5926 + 0.0463RMT 0.50 -3.5108 -156.344 (1.0233) (.0061) 2 Concurrent Computer Corp (CCUR) =-3.5926 + 0.0717RMT 0.50 -3.5108 -98.7553 (1.0233) (.0094) 3 CRAY Inc (Cray) =-3.5926 + 0.1119RMT 0.50 -3.5108 -60.415 (1.0233) (.0147) 4 CSP Inc (CSPI) =-3.5926 + 0.1476RMT 0.50 -3.5108 -43.9381 (1.0233) (.0194) 5 DELL, Inc (DELL) =-3.5926 + 0.0638RMT 0.50 -3.5108 -111.452 (1.0233) (.0084) 6 Digi International Inc (DGII) =-3.5926 + 0.0497RMT 0.50 -3.5108 -146.2 (1.0233) (.0065) 7 Doctrinic, Inc (DOTX.OB) =-3.5926 + -1.3529RMT 0.50 -3.5108 -13.2037 (1.0233) (.1782) 8 ECC Internal Corp (ECC) =-3.5926 + 0.0279RMT 0.50 -3.5108 -262.73 (1.0233) (.0037) 9 Electronics For Imaging (EFII) =-3.5926 + 0.0549RMT 0.50 -3.5108 -131.264 (1.0233) (.0072) 10 Enpoint Technologies (ENPT) =-3.5926 + 0.2413RMT 0.50 -3.5108 -23.8585 (1.0233) (.0318) 11 GTSI Corp (GTSI) =-3.5926 + 1.6253RMT 0.50 -3.5108 2.921963 (1.0233) (.2140) 12 Gateway Inc (GTW) =-3.5926 + 11.3591RMT 0.50 -3.5108 6.903765 (1.0233) (1.5005) 13 Ingram Micro Inc (IM) =-3.5926 + 0.0622RMT 0.50 -3.5108 -114.366 (1.0233) (.0082) 14 IBM (IBM) =-3.5926 + 0.0241RMT 0.50 -3.5108 -304.969 (1.0233) (.0032) 15 Kontron Mobile Company (KMBC.OB) =-3.5926 + 0.1475RMT 0.50 -3.5108 -43.9433 (1.0233) (.0194) 16 Maxwell Technologies Inc (MXWL) =-4.0319 + 0.0721RMT 0.50 -4.0542 -106.655 (.9945) (.0087) 17 Merisel Inc (MSEL) =-3.5926 + -0.8304RMT 0.50 -3.5108 -16.7313 (1.0233) (.1094) 18 NCR (NCR) =-3.5926 + 0.0241RMT 0.50 -3.5108 -304.969 (1.0233) (.0032) 19 National Data Computer Inc (IDCP.OB) =-3.5926 + 0.4556RMT 0.50 -3.5108 -9.07333 (1.0233) (.0600) 20 Neoware Systems Inc (NWRE) =-3.5926 + 0.2793RMT 0.50 -3.5108 -19.5842 (1.0233) (.0368) 21 Nematron Corp (NMN) =-3.5926 + 0.2425RMT 0.50 -3.5108 -23.7461 (1.0233) (.0319) 22 Pinnacle Systems Inc (PCLE) =-3.5926 + 0.1063RMT 0.50 -3.5108 -63.8357 (1.0233) (.0140) 23 Sea Change International Inc (SEAC) =-3.5926 + 0.0688RMT 0.50 -3.5108 -102.33 (1.0233) (.0091) 24 Silicon Graphics Inc (SGI) =-3.5926 + 0.2224RMT 0.50 -3.5108 -26.5392 (1.0233) (.0293) 25 Steel Cloud Inc (SCLD) =-3.5926 + 0.2292RMT 0.50 -3.5108 -25.5232 (1.0233) (.0302) 26 Sun Microsystems Inc (SUNW) =-3.5926 + 0.0485RMT 0.50 -3.5108 -148.672 (1.0233) (.0064) 27 Tech Data Corp (TECD) =-3.5926 + 0.0609RMT 0.50 -3.5108 -117.388 (1.0233) (.0080) 28 Telebyte Inc (TBTI.OB) =-3.5926 + 0.7482RMT 0.50 -3.5108 -2.55635 (1.0233) (.0985) 29 Televideo Inc (TELV.OB) =-3.5926 + 0.3538RMT 0.50 -3.5108 -13.867 (1.0233) (.0466) 30 TransNet Corp (TRNT.OB) =-3.5926 + 0.1324RMT 0.50 -3.5108 -49.8621 (1.0233) (.0174) 31 Xata Corp (XATA) =-3.5926 + 0.2382RMT 0.50 -3.5108 -24.2611 (1.0233) (.0314) 32 Xybernaut Corp (XYBR) =-3.5926 + 0.1697RMT 0.50 -3.5108 -35.6352 (1.0233) (.0233) The results are based on 60 monthly observations for the period from January 1998 to December 2002; RMT = monthly return in the S&P 500 index;

Using the first company (Apple Computer Inc) as an example the estimated equation is

R^AAPL,t= -3.5926 + 0.0463RMT, the estimated standard errors ((1.0233) (.0061) respectively) are reported underneath the estimated coefficients. The estimated intercept is close to zero at -3.5926, with standard error of 1.0233 (= SE( )), and the estimated value of β is 0.0463, with a standard error of .0061 (= SE( )). Notice that the estimated standard error of β is much smaller than the estimated coefficient and indicates that β is estimated reasonably precisely. R2 = .50. This means that 50% of the estimation of expected return can be explained by the variability in a and β.

Testing the restriction a = 0, using Apple as an example, we find that the absolute value of ta=0 is much larger than 2, therefore we reject the hypothesis that a = 0. This is true for all companies.

Testing hypothesis about β, still using Apple as an example, we find that

| tβ=1 | > |tT-2 (0.025)|, therefore, we reject the hypothesis that β = 1. This is true for all companies.

Conclusion

From the regression equations of the 32 firms it can be seen that there is a positive relationship between risk and return. If you increase the risk, expected return also increases. If you decrease the risk, expected return also decreases. The beta is unchanging.

Since we also find out that alpha is not equal to zero, CAPM is not perfect. While CAPM is not perfect, it does seem to capture important elements of risk. It measures risk only relative to market return. CAPM should be regarded as rule of thumb and not as the final word on how the world works. Traditional asset methodologies such as those of Sharpe (1964), Lintner (1965), Merton (1973) and Ross (1976) show that the expected return on a financial asset is linear function of its betas or co-variances with some systematic risk factors.

The advantage of the Single Index Model is that it can be expanded to capture multiple factors. It can be used to estimate returns, variances and co-variances that are needed to implement portfolio theory. It is used as a model to explain the normal or usual rate of return on an asset for use in the analysis of mergers and acquisitions, damage assessment in liability cases, effects of regulatory change, and last but not least, announcements of macroeconomic variables. It is also used to evaluate the performance of mutual fund and pension fund managers.

References

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