Empirical tests

Question 1

what is linear regression?

Answer 1

is a technique of fitting a line to data, so we will want to estimate parameters alpha and beta for example.
y = a + BX + e, where a and B are constants and epsom is a random variable.
CAPM CAN BE WRITTEN SIMILAR TO REGRESSION.

Question 2

What is the covariance of 2 constants? How to find a/B in CAPM regression?

Answer 2

a) 0
b) use the y = a + Bx + e, which is line of best fit.

Question 3

When using the method of ordinary least squares and we get estimates of alpha and beta, we know they are subject to noise, so what do we say?

Answer 3

they are correct on average

Question 4

What t statistic would we use to calculate this?

Answer 4

if e.g. t = 2 then it means that a ( estimated alpha) and alpha ( real alpha from data) are to standard deviations apart . When t is too large we reject the null hyphothesis.

Question 5

What does R^2 mean?

Answer 5

R-squared assesses the regression model’s fit by comparing predictions to actual outcomes. High R-squared values indicate better explanation of the dependent variable’s variation, while low values suggest limited explanation.

Question 6

What is the main regression equation we will be testing for stock I?

Answer 6

rMt - rf = excess return on market

Question 7

Interpret this?

Answer 7

1) on average a 1% increase in the S&P monthly return causes a 1.4693% increase in MSFT monthly return.
2) in months when the S&P 500 doesn’t move, MSFT’s return tends to be -4.24%.

Question 8

If CAPM is true then all stock returns should be on the security market line, what is expected returns of stocks? what is capm saying ?

Answer 8

So essentially CAPM predicts their should be a linear relationship between CAPM and betas and there should be no alphas.

Question 9

How can be estimate the risk free rate, Market risk premium and betas?

Answer 9

Question 10

What does this cloud of observations relate to?
How do we calculate beta of a portfolio?

Answer 10

relates to the return of the stock we are interested in and the return of the market portfolio, so we can fit the line of best fit. The slope of this line is the beta coefficient.

Question 11

What is the CAPM REGRESSION WE NEED TO KNOW ( what type of regression this and what is the intercpt of this the same as ?

Answer 11

Time series
The

Question 12

Suppose an investor finds a stock with negative alpha, what will investor do and what happens to e(r) also outline an important relationship? Is this an arbitrage opportunity?

Answer 12

If an asset has a negative alpha, its expected return is below the CAPM’s fair risk compensation. Investors will sell the stock, causing its price to drop and expected return to rise, eventually restoring CAPM equilibrium. ( inverse relationship between expected return and price)
Arbitrage strategies are riskless but in the equation we have here we have expected probabilities.

Question 13

According to CAPM is this stock over or underpriced?

Answer 13

Question 14

What are 3 things we can test for the implications of CAPM?

Answer 14

1) Expected excess returns are linear in beta
2) Slope of the SML line is market risk premium.
3) Expected returns depend only on beta.

Question 15

There are 2 main types of tests of testing CAPM which are what?

Answer 15

Time series approach
Two-pass approach.
Both are related to the same equation
( You take expectations of both sides of the SML and get this)

Question 16

What is the time series approach?

Answer 16

we estimate a bunch of regressions for different stocks and see whether the alpha = 0, if a = 0 then capital asset pricing model is satisfied.

1) Calculate excess returns: Find the difference between the asset’s returns and the risk-free rate (e.g., returns on government bonds) for each period. Do the same for the market’s returns.

2) Perform time-series regression: Run a regression with the asset’s excess returns as the dependent variable and the market’s excess returns as the independent variable.

3) Interpret alpha and beta: The regression gives you alpha and beta estimates. A positive alpha means the asset performed better than expected, while a negative alpha means it underperformed. Beta indicates how sensitive the asset’s return is to market movements.

Question 17

What is a caveat of the time series approach?

Answer 17

Lets say we have 3000 stocks and you run regressions on these 3000 stocks and it turns out that alphas are very close to 0 for most stocks but 100, so does that mean, do we reject the null hypthoesis for these 100 stocks? We don’t know, it might just be because of noise in retruns of stock to why we have alphas deviating from zero.

Question 18

What is the two pass approach?

Answer 18

1) Run the time series regression firstly and collect betas from this. First pass: Calculate the “beta” for each asset by running a time-series regression of the asset’s excess returns (returns above the risk-free rate) against the market’s excess returns (market returns above the risk-free rate) over a specific period. Then compute sample average risk premium.
2) we then run the second stage regression: we run a regression of the average risk premium for the individual stocks on betas we estimated from the first stage. ( basically we are testing the linear relationship between excess returns and betas, which CAPM tells us,

Question 19

For this second pass regression, if CAPM holds what should be gamma and gamma1?

Answer 19

Question 20

Do you remember what we are doing here?

Answer 20

We are essentially running regression on 2904 stocks, then we can compute average excess return, then after we regress each stock on a value weighted index ( portfolio) to get betas and use 1 month T-bills as the riskless asset.
1st stage

Second stage is what is linked we have average excess return regressed on betas and we can clearly see their is a clear relationship between expected returns and betas

Question 21

Whats happening here?

Answer 21

Looking at how estimated value deviates from actual value, both null hypthoesis’s are rejected. As we see the Empirical line tends to be flatter than theoretical line.

Question 22

So far what have our tests shown about CAPM?

Answer 22

we have shown that expected returns depend on beta but we know this isn’t the case ( we explore next dec, there are other factors that explain the cross section of expected returns.)
Risk return relation is flatter than what CAPM predicts- ( high beta stocks have lower returns relative to the CAPM)
-( Low beta stocks have higher returns relative to the CAPM) ( we see this in the imagine.

Question 23

What did Fama-Macbeth (1973) test and what did they find? ( TEST OF CAPM)

Answer 23

they did a two pass regression - They found that average returns and betas are linearly relate but weaker positive than CAPM predicted, the expected return on a portfolio with beta = 0 is higher than 1-month t-bill returns. Therefore the risk premium is smaller than the CAPM would predict.

Question 24

What is another way to write r2?

Answer 24

Question 25

What is another way of writing covariance in terms of their betas?

Answer 25

Question 26

Var ( Ri) = what?
Cov ( Ri, Rj) = what?
Risk adjusted excess return = what?

Answer 26

1) Var( Ri) = Var (ri-rf) = Var(ri)
2) Cov ( Ri, Rj) = Cov ( ri-rf, rj - rf) = Cov(ri,rf)
3) = abnormal return(a) deviation from model prediction.