Econometrics assignment, Econ 134A

TIANYUE WANG & XIANGDI WANG

1. Running the linear regression on Excel by setting x to be Beta and y to be Expected return, we get

the linear equation: y=0.1262x+0.0083. Accordingly, the slope of the security market line is 0.1262

and the y-intercept is 0.0083.

2. Since the security market line has 2 variables (Beta and Expected return) which are similar to the

CAPM model, the CAPM equation could best fit into the regression line and to interpret the

regression. According to the CAPM equation(R = RF + ß (RM – RF), the slope of the security line

(0.1262) is demonstrated as (RM – RF), which is the difference between the risk-free return (RF) and

the expected return on the market (RM) defined to be the risk premium. When beta equals zero, the

intercept of the linear regression model is 0.0083. In other words, whiling fitting into the CAPM

equation, the intercept represents the risk-free rate. Knowing the risk-free rate and risk premium, we

can also get the expected return on the market from adding the two and gives us

0.0083+0.1262=0.1345.

For a simple linear regression model Yi=𝞪Xi+𝞫+𝟄i, the residue 𝟄i is assumed to be normally

distribtued and the expectation of 𝟄i is zero. The null hypothesis is 𝞫=0, in order to test the

uncorrelation between Xi and Yi. Assuming the null hypothesis is true, the simple linear regression

model follows a t distribution which is demonstrated in the following diagram.

For a 95% confidence interval, the alpha value is 0.025(=(1-95%)/2). Since there are 2 random

variables in the linear regression model, the degree of freedom is 3(=5-2). According to the t-statistic

table, 3.182 is obtained as t-statistics. Meanwhile, acquire from the LINEST function in Excel, the

Standard error of X is 0.005432448, the Standard error of constant is 0.005177952.

According to the following confidence interval formula:

confidence interverval=sample meant-statistic * standard error

Mathematically, the calculation shows as following:

§ 95% confidence interval of the slope (risk premium):

(0.126183-3.182*0.005432448,0.126183+3.182*0.005432448)

=(0.1088969505,0.1434690495)

=(0.11,0.14)

§ 95% confidence interval of the intercept (risk-free rate):

(0.008267304-3.182*0.005177952,0.008267304+3.182*0.005177952)

=(-0.00821392864,0.02474853664)

=(-0.008,0.02)

3. In order to see the accuracy of our regression estimates, we look up the current return on one-year

US government issued bonds on the OANDA fxTrade

(http://fxtrade.oanda.com/analysis/economic-indicators/united-states/rates/yield-curve), and obtain 0.12% as the

annual risk-free rate on March.10 2014.

4. Compared with our estimated confidence interval of the risk-free rate, the current risk-free rate

0.12% lies inside the range of the confidence interval (-0.008,0.02). Though we have a small sample

size that can lead to a very large range, it is a good estimation since the 0.12% is within the range.

Therefore, we can conclude that the regression is pretty accurate on estimation.

5. Based on our regression, the current expected market rate of return could be obtained using the

CAPM equation. In this case, since beta means how the returns of a stock relate to the market’s

return and the security is designed to be the market, the Beta value which interprets the relationship

between the market’s returns to the market’s returns should always be 1. Thus, we plug Beta equals 1

into the CAPM equation,

R = RF + ß (RM – RF) = RF + (RM – RF) = RM

and get the expected rate of return (R) equal the expected market rate of return(RM). According to

the estimated linear regression model, expected return equals 0.0083 (=0.1262+0.0083) when Beta

(or x) equals to 1. Thus, the current expected market rate of return is 0.0083.