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STUDENT NUMBER – 670058066



Estimating and Testing the Capital Asset Pricing Model (CAPM)

(Assignment 1)




The model applied to this project is the Capital Asset Pricing Model (CAPM) introduced by Jack Lawrence Treynor which has been built on the basis of the earlier works of Harry Markowitz. It is one of the most widespread approaches used for valuation of assets, forecasting the future rates of return of those assets, and constructing efficient investment portfolios with the relationship between risk and return on an asset.


The CAPM is explained as the method of determining the theoretically appropriate required rate of return of an asset, if that asset is to bead to an already well-diversified portfolio, given that asset’s non-diversifiable risk.


In this assignment, we use monthly data to estimate the `betas of stocks (equities) traded on the New York Stock Exchange, namely two: BOISE (forestry) and Citicorp (Banking). We study equations of the form:

rjt = ?j + ?jrmt + ujt

where rjt represents the actual return to holding company j’s stock in month t, rmt is the return on the market portfolio in month t and ujt represents other influences on returns of the stock j.

The assumption is that market traders are fully informed and efficiently maximises the utility of their portfolio, which depends positively on return and negatively on risk, i.e. shows the phenomenon of risk-averse and rationality. Further Assumptions taken are: Security markets are perfectly competitive and markets are frictionless (Trade without transaction or taxation costs). Also, Investors are short time traders and have homogeneous expectations. And the strong form of markets holds true.











On observation of the scatter plot, we can observe a relation between the two stocks, i.e. the correlation between Citi Corp and Boise with the Market, which shows a positive relation for Boise, while it’s fairly scattered for Citi Corp suggesting instability in the values. Upon further investigation of the graphs, we observe that both the stocks bear almost a direct relationship with the market with Boise being more volatile with respect to the market fluxes than Citi Corp. As we see the market crashes of 1987 (Black Monday) and world financial crash of 1978 leads to a fall in returns of the stocks. Whereas the Reagan boom of 1982 leads to a rise in return, especially for Boise as its more sensitive to the market.


Now further in this experiment, we try to establish an estimate of the regression of the returns of the market on the stocks of Citi Corp for the period of Jan-78 to Dec-82.


2.Estimation of ? and ? using OLS Regression in TSM 4



BOISE = -0.00021       

BOISE =  0.87888


As per our initial observation, we see that ?BOISE has a strong positive relationship with the market outcome.

CITCRP = 0.00517

CITCRP = 0.44137


As per our initial observation, we see that ?CITCRP has a weakly positive relationship with the market outcome.

Now we test the null hypothesis of ? and ? against the alternative hypothesis. The null hypothesis is represented as H0 and the alternative hypothesis H1 and these determine the precision of the regression model.


H0: ?=0        H1: ??0

This is a two-tailed test with 5% significant level, from t-tables we get a critical value of |1.96|. The TSM output gives us a t-ratio of tCiticorp = 0.588 and BOISE = -0.019.

Therefore the respective confidence intervals for ? are:

Citi Corp: -0.01206 ?Citicorp ? 0.02240

Boise:  -0.02139 ?Boise ? 0.02181

Since the t-values are less than the critical values we do not reject the null hypothesis according to theory. As, | tasset| < |1.96|, we get the result that ?=0 which relates to the assumption of the CAPM model which states that the risk of an asset is only due to its systematic risk. Now we do the same for beta and check the hypothesis for the approximate hypothetical value of beta.                                          H0: ?= 1        H1: ?>1

This is a one-tailed test with 5% significant level, from t-tables we get a critical value of 1.67.

Therefore the respective confidence intervals for ? are:

Citi Corp: citicorp ? 0.67417

Boise: Boise ? 1.11513



t-value for beta =      . Thus, we compute the t-statistic for the both the stocks using the TSM output.


tboise =  =  -0.85672

tciticorp = =  -4.00739


Since both the t-statistic values for beta are less than tcritical of 1.67, thus we do not reject the null hypothesis and this suggests that the CAPM model is reliable for computation of risk-adjusted asset returns.


Note: Confidence Interval calculated using:  (t used is 1.67 for ? and 1.96 for ?)




The residual standard deviation is the standard deviation of points formed around a linear function and estimates the accuracy of the dependent variable being measured. Thus, measures the individual risk of the stock, and the R2 of the regression measures the proportion of the risk attributable to the market. Note: The lower the value of residual standard deviation compared to the sample the more accurate the model fitness is.



Residual SD = 0.0827

R-Squared = 0.3995



Residual SD = 0.0703

R-Squared = 0.1883


Thus, again in line with our initial investigation of the data we see that R2 of Boise is higher than that of Citi corp which suggests that the proportion of risk attributable to the market is higher for Boise (39.95%) as its more sensitive with the market outcomes than Citi corp (18.83%). The model can also be determined as a fit as both the residual standard deviations are low as compared to their individual sample standard deviations which is what CAPM theory predicts that the only attributable risk on an asset’s return is the Systemic risk and not the Idiosyncratic risk. As we see the individual risk associated with each asset is relatively very low.



The Chow test is used to determine whether the independent variables have a structural break and have different impacts on two subgroups of the population.

        From TSM output of Citi Corp., we observe the stability test equals 1.5342 is greater than the critical value of 5.99146 and the p-value of 0.004 is less 0.05 at 5% significance level.

11.2359 > 5.99146

0.004 < 0.05 The TSM output of BOISE computes the stability test equals 1.0443 and is less than the critical value of 5.99146 and the p-value of 0.593 is greater than 5% significance level. 1.0443< 5.99146 0.593 > 0.05


This implies Boise is stable over the 10-year period of the sample but Citi Corp is relatively unstable as we had implied from the scatter plots.


Note: Here our Ho is that our model is stable whereas H1 is that our model is unstable. Thus, we rejected Ho for Citi Corp and failed to reject for Boise.


5. Arbitrage Pricing Model (APM) & STRICT CAPM TEST


Under the APM model, we try to test the ability of macroeconomic variables to predict returns which is an alternative to CAPM and says that the risk and return of an asset depend on more factors other than the market. Thus, we run a variable addition test to test its significance.

Figure: Wald test for Citi Corp

Figure: Wald test for Boise


Thus, we test: Rasset,t = ?asset + ?asset. Rmarket + ?1RGIND + ?2RRINF + ?3ROIL + U asset, t

With Ho : ?1= ?2­ = ?3 =0 and H1 : that either of the betas is non-zero. Since the p values for both the assets are greater than 0.05 we do not reject it and hence assert CAPM is the appropriate model to be used as the macroeconomic variables have insignificant influences on the assets returned.





Using the CAPM equation we calculate the betas of the two assets used in this experiment:



Thus, using the means from TSM we compute that CITCRP = 0.50141 and BOISE = 0.77296. Since the above equation describes the subjective expectations of

market traders in equilibrium at date t, which are of course unobserved. We observe that indeed the betas are less than 1 and almost as predicted by the regression earlier. Since no model is absolutely accurate we observe a slight deviation from our approximated value and also because there are certain drawbacks to this approach. However, we see still CAPM predicts a better result than most other models.











This study presents some empirical tests of the Capital Asset Pricing Model (CAPM) using more robust statistical tests. The betas of securities were estimated by the OLS regression technique. However, it assumes that beta is changing systematically with the accounting measures of risk. There will always be some portfolio which is ex-post efficient and will bring about exactly observed linearity among ex-post sample mean returns and ex-post sample betas. If we do not know the composition of the market portfolio, we might by chance select a portfolio that is close to mean-variance efficient. In fact, it may be hard to find a highly-diversified portfolio that is sufficiently far inside the ex-post efficient frontier to permit the detection of statistically significant departures from mean return/beta exact linearity. Consequently, when testing the CAPM, the researcher is always testing the joint hypothesis that the CAPM is correct and that the portfolio used as the market proxy was the true market portfolio.



1 Markowitz, H. (1952). Portfolio Selection, Journal of Finance, 7, 77-91.

2Gupta, Nikhil. “The Size Effect and the Capital Asset Pricing Model.” minneapolisfed.org. minneapolisfed.org, n.d. Web. 19. Jan.2018..