The market is associated with risk factors that continually affect it. Such a component of risk that affects the entire market is the systematic risk. Systematic risk, also known as market risk or un-diversifiable risk refers to fluctuations of stock returns due to changes in the entire market return. Based on the modern finance theory, the Capital Asset Pricing Model (CAPM) is used for pricing an individual portfolio or security.

Theoretically, CAMP is used for estimating the expected rate of returns of an asset and the market, at a given risk level. This risk level is called the beta coefficient (Abonongo et al., 2017). CAMP, with the beta concept, is used as the measure of systematic risk; it has different applications in managing the portfolio. According to as explained by Zeng, Pan, Haidong & Guangqiang (2014), CAMP employs the relationship between systematic risk and the expected rate of return of the assets.

This study was designed to estimate the systematic risks of Digital Equipment Co. (DEC) and Motorola (MOTOR) stocks from January 1978 to December 1987 using the CAMP, following the time series model (TSM) approach. The statistical significance of the relationship between the systematic risks and the returns on the stock due to changes in the entire market returns were investigated.

**Materials and Method**

**Source of Data**

Data on the Digital Equipment Co. and Motorola stocks were obtained from the New York Stock Exchange databases using TSM. The data comprises of the monthly closing prices of the stocks of the companies over a period of 10 years, from January 1978 to December 1987.

**Methods of Data Analysis**

In this study, Time Series Modeling (TMS) ordinary least squares (OLS) was used to estimate the parameters. The beta risk over a period of ten years, from 1978 to 1987, was analyzed. The Capital Asset Pricing Model (CAPM) was used to estimate individual securities using the time series data extracted from the New York Stock Exchange database. The monthly data was used to estimate the betas of stocks traded within the period following this equation:

*r _{jt} *= α

*+ β*

_{j}*+*

_{j}r_{mt}*u*

_{jt}The following analyses were carried out:

- The scatter plot of the time series data set was created to examine how the returns across the periods, from January 1978 to December 1987. The scatter plot of
*r*against_{jt}*r*will be examined._{mt} - Using OLS, α and β, and the approximate 95% confidence level were estimated. A null hypothesis (
*H*of α = 0 and β = 1 was tested, while the alternative hypothesis (_{o})*H*was β > 1._{a}) - Adjusted R-squared of the regression and the standard deviation of the OLS residuals were estimated. Adjusted R-squared measures the risk proportion that is caused by the market, while OLS residual standard deviation measured the individual risk of the stock.
- Chow Stability test was conducted to examine the stability of the model over the period of years of the data sample.
- 5. The strict CAMP was tested against the alternative model of the Arbitrage Pricing Model (APM). This test was carried out using the rate inflation (RINF), the growth in industrial production (GIND), and changes in the real oil price (ROIL) to assess the ability of macroeconomic variables to predict returns. A further test,
*t*statistics, was carried out to measure the value of the beta changes due to these included variables, i.e. to their individual significance. In addition, the*F*test was used to test the joint significance of these variables in influencing stock returns.

**Results and Discussion**

The times series plots of the monthly returns of DEC (Figure 1), MOTOR (Figure 2), and the market (Figure 3) show that the returns of DEC, MOTOR, and market respectively have some trends across the months. Moreover, most of the monthly returns of the three variables increased over time until the October 1987 crash. Similar findings were reported in a related study by Abonongo et al. (2017, p.3).

The scatter plots of returns on DEC against the returns on the market (Figure 4) and returns on MOTOR against the market returns (Figure 5) show that most of their points or returns are scattered and spread around the regression lines. This is the evidence that returns of DEC and MOTOR are not moving at the same rate as the returns of the market. Moreover, the scattered points around the regression line showed that both DEC and MOTOR stocks have much risk in relation to the market. This is contrary to the report by Abonongo et al. (2017) and Zakamulin (2017).

Table 1 shows the market beta values of DEC and MOTOR stocks are significantly less than one (P < 0.05). This shows that both stocks are defensive stocks and that their returns move in the same direction with the market return. This is an indication that the market returns is vital in determining firms’ returns and/or prices. This finding is consistent with the findings of Jensen et al. (1972), Fama & MacBeth (1973), Blume (1975), Elton & Gruber (1995), Kothari et al. (1995), Koffie (2012). Koffie (2012) posited that investors investing in such securities are more likely to receive less compensation due to the lower systematic risk.

Moreover, the positive betas of both stocks are an indication that the prices of the stocks have a probability of increasing as the market price increases and vice versa. Abonongo et al. (2017, p.6) gave the same report. Sharpe & Cooper (1972) assessed the CAPM to determine whether a higher return could be associated with higher systematic risk over a period. The investigation was carried out over 60 months on the New York Stock Exchange. Based on their results, they showed a strong positive and linear relationship between the market return and beta. The alpha values for the two stocks, DEC and MOTOR, in this study is small, tend towards 0. The beta value is closer to 1 (Table 1); therefore, the null hypothesis, α = 0, and also β = 1 seems to hold. This hypothesis was tested by to tailed t-test in which the critical value for both stocks was 1.67: However, the lower confidence limits for DEC and MOTOR stocks were 0.1131 and 0.1315 respectively; the upper confidence limits were 6.2100 and 4.2060 respectively. These intervals fall within the upper (+1.67) and lower (-1.67) values. Hence, the null hypothesis is accepted. This support the hypothesis of CAMP that the CAMP equation intercepts should be equal to zero while the slope should be excess returns. This is in accordance with the Capital Asset Pricing Model (CAPM) asset pricing theory of Sharpe (1964) and Lintner (1965).

R-squared values for the two stocks are low, 39.90% for DEC and 22.05% for MOTOR, these show that the proportion of the risk attributable to the market is low. It implies that the influence of the market beta in determining the firms’ returns or prices is not significant. Coffie & Chukwulobelu (2012) gave the same report that the variation in the market return is low for explaining the variation in the stock returns or prices.

The standard deviations of the OLS residuals of the stocks are small, 0.00661 for DEC and 0.0769 for Motor (Table 2). It is the measure of the fit to the measured data, and the small values show that the individual risk on the stock is low in relation to the market return. Chudhary & Chudhary (2010) similarly revealed that the residual risk has no influence on the portfolios expected returns.

Using the Chow Stability test, the *F*-values of DEC and MOTOR are 4.297 and 4.870 respectively. These values are less than the *F*-critical value (5.992) as shown in Table 3. According to Chow (1960), the null hypothesis is accepted, which means that there is no break point in the stock data. This indicates that the model is unstable over the full 10-year period of the sample.

Table 4 shows the result of the test of the strict CAPM against the Arbitrage Pricing Model (APM). The P-values of FBRIND, CPI, and RPOIL show that the influence of these macroeconomic variables on the stock returns is not significant. The results of this test were further confirmed by the F-test, which test the joint significance of the variables. F-test confirmed that the macroeconomic variables had no effect on the stock returns. Thus, unlike CAMP, they cannot be used as predictors of the level of stock returns. This contradicts Arbitrage Pricing Theory (APT) of Ross (1976), who argued that macroeconomic variables can be employed in explaining stock returns. However, Nkoro & Aham (2013) reported that inflation, index of manufacturing output, government expenditure, and interest rate exerted significant influence on stock returns.

In conclusion, this study showed that the Capital Asset Pricing Model (CAPM) is a valid tool for estimating betas of stocks, and for assessing and measuring systematic risks. It is also efficient for predicting returns and/or prices of stocks in relation with the market returns. This information is useful for investors for making valid decisions.

Reference

Abonongo, J., Ackora-Prah J. & Kwasi, B., 2017. Measuring the Systematic Risk of Stocks Using the Capital Asset Pricing Model. *Journal of Investment and Management*, 6(1), pp. 13-21.

Blume, M. E., 1975. Betas and their regression tendencies. *The Journal of Finance*, 30(3), pp. 785–795.

Chow, G.C., 1960. Tests of Equality between Sets of Coefficients in Two Linear Regressions. *Econometrica*, 28, pp. 591-605.

Chudhary, K. & Chudhary, S., 2010. Investigating the relationship between stock returns and systematic risk based on CAPM in the Bombay stock exchange. *Eurasian Journal of Business and Economics*, 3(6), pp. 127–138.

Coffie, W. & Chukwulobelu. O., 2012. The Application of Capital Asset Pricing Model (CAPM) to Individual Securities on Ghana Stock Exchange. In: K. Menyah and J. Abor, ed. *Finance and Development in Africa (Research in Accounting in Emerging Economies, Volume 12*. Emerald Group Publishing Limited, pp. 121 – 147.

Elton, E.J. & Gruber, M.J. eds., 1995. *Modern Portfolio Theory and Investment Analysis*. 5th ed. New York: Wiley.

Fama, E. F. & MacBeth, J. D., 1973. Risk, return and equilibrium: Empirical test. *The Journal of Political Economy*, 81(3), pp. 607–636.

Jensen, M. C., Black, F., & Scholes, M., 1972. The capital asset pricing model: Some empirical tests. In C. M. Jensen. *Studies in the theory of capital markets.* New York, NY: Praeger. pp. 79–121.

Kothari, S. P., Shanken, J., & Sloan, R. G., 1995. Another look at the cross-section of expected returns. *Journal of Finance*, 50, pp. 185–224.

Lintner, J., 1965. The valuation of risk asset and the selection of risk investment in stock portfolio and capital budgets. *The Review of Economics and Statistics*, 47(1), pp. 13-37.

Nkoro, E. & Aham, K.U., 2013. A Generalized Autoregressive Conditional Heteroskedasticity Model of the Impact of Macroeconomic Factors on Stock Returns: Empirical Evidence from the Nigerian Stock Market. *International Journal of Financial Research*, 4(4), pp. 38-51.

Ross, S.A., 1976. The arbitrage theory of capital asset pricing. *Journal of Economic Theory*, 13(6), pp. 341-360.

Sharpe, W.F. & Cooper, G.M., 1972. Risk return classes of New York stock exchange common stocks, 1931-1967. *Financial Analysts Journal*, 28(2), pp. 46–81.

Sharpe, W.F., 1964. Capital asset prices: a theory of market equilibrium under conditions of risk. *Journal of Finance*, 19(3), pp. 425-442.

Zakamulin, V., 2017. *Timing the US Stock Market Using Moving Averages and Momentum Rules: An Extensive Study*. M. Sc. University of Agder.

Zeng, A., Pan, D., Haidong, Y. and Guangqiang, X., 2014 Applications of Multivariate Time Series Analysis, Kalman Filter and Neural Networks in Estimating Capital Asset Pricing Model. In: M. Ali, J.S. Pan, S.M. Chen & M.F. Horng, ed. 2014. *Lecture Notes in Computer Science*. Switzerland: Springer, pp. 507–516.

**Appendix**

**Figure 1. **Time series plot of DEC monthly returns

**Figure 2. **Time series plot of MOTOR monthly returns

**Figure 3. **Time series plot of Market monthly returns

**Figure 4. **Scatter Plot of DEC returns vs. Market returns

**Figure 5. **Scatter Plot of MOTOR returns vs. Market returns

**Table 1. CAMP estimates of DEC and Motor**

Stock | α_{j} | β_{j} | Std. error | t-ratio | P-value |

DEC | 0.0027 | 0.7025 | 0.1131 | 6.2100 | 0.0000 |

MOTOR | 0.0096 | 0.5533 | 0.1315 | 4.2060 | 0.0000 |

**Table 2. **CAMP estimates of DEC and Motor – R^{2}, confidence limit, and chow test

Stock | Residual | R adjusted | 95% confidence | Critical Value | Chow test | |||

SD | Lower Limit | Upper Limit | Stability | Chi-Square | ||||

DEC | 0.0661 | 0.3890 | -0.0146 | 0.0200 | 1.6716 | 4.2971 | 5.9915 | |

MOTOR | 0.0769 | 0.2205 | -0.0105 | 0.0297 | 1.6716 | 4.8793 | 5.9915 | |

**Table 3. **Test of the strict CAPM against the Arbitrage Pricing Model (APM)

Parameter | DEC | MOTOR | |

α_{j} | 0.3177 | 0.0180 | |

β_{j} | 0.6854 | 0.5582 | |

β_{j} Std. error | 0.1137 | 0.1333 | |

β_{j}_{ }P-value | 0.0000 | 0.0000 | |

P-value | FBRIND | 0.5980 | 0.3660 |

CPI | 0.0950 | 0.8090 | |

RPOIL | 0.4680 | 0.3560 | |

F-value | Estimated | 0.9823 | 0.6631 |

Critical | 2.7725 | 2.7725 |