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9

Yes it is a better way. Just take a look to figure 3, from Buss and Vilkov (2012, RFS):


7

I think the answer to this question must be yes, it is flawed indeed. The CAPM says that the expected return of a security or a portfolio equals the rate on a risk-free security plus a risk premium. Yet empirically measures of risk like volatility and beta do not generate a positive correlation with average returns in most asset classes. The best ...


6

Don't just run simple time-series regression to see if you get statistically significant betas. This procedure will not tell you if the factors are actually priced. You run a high risk of finding spurious correlations. There is a fairly well established standard program to test factor models, called the Fama-MacBeth method. It is based on two sets of ...


5

There is a rationale here and it has to do with the connection between diversification and the number of assets in a portfolio. Suppose we purchase an equal-weighted portfolio of n stocks. The variance of the return is then: $\sigma_{p}^2$ = $\sum$ $\sum$ $w_i$$w_j$Cov($R_i$,$R_j$) In the above notation, the sigmas are summing over $i$ and $j$ ...


5

You might want to read this: Size, Value, and Momentum in International Stock Returns by Fama and French (2011) Abstract: In the four regions (North America, Europe, Japan, and Asia Pacific) we examine, there are value premiums in average stock returns that, except for Japan, decrease with size. Except for Japan, there is return momentum ...


5

I would not necessarily call it a failure. CAPM explained ~70% of returns (on average) so this may quite be one of the 30% that could not be explained (see link). However, an improved approach or extension of the CAPM would be the the Famma-French factor model which explains roughly 90% of returns (see link). Again, the Famma-French is an extension of CAPM ...


5

Because you have CAPM therefore the following holds: $$r_i = r_f + \beta_i (r_M - r_f) + \epsilon_i$$ where $r_i$ is the expected return of stock $i$, $r_f$ is the risk free return and $r_M$ is the expected market return, and $\epsilon$ is an idiosyncratic return adjustment or an error. Now if you take the $\text{Var}[\cdot]$ operator over the ...


4

What you observe in your regression is not strange at all. The regression beta you estimated is $\beta_i = \frac {\mathrm{cov}(r_i,r_m)}{\mathrm{var}(r_m)}$ where $i$ represents the country/region (such as the USA or China) and $m$ represents the "market" (which you take to be the ACWI). Since the USA is itself such a large component of the ACWI (about ...


4

The risk-free rate is the y-intercept of the Security market line. If the risk free rate goes negative the y-intercept of the Security market line would simply be below the x-axis. So if the risk-free rate decreases the whole line shifts down. This just means people are willing to pay for safety. According to the formula for the SML: E(Ri) : expected ...


4

In academics, Roll's critique of the CAPM is discussed a lot, for a start see Wikipedia page of Roll's critique. It is more of a principled "theoretical" critique of the CAPM than an empirical one. It says basically that the CAPM cannot be tested because every mean-variance efficient portfolio satisfies the CAPM the market portfolio is unobservable


3

This is in essence the idea behind Andrea Frazzini's paper 'Betting Against Beta'. There are various ETFs that aim to exploit the premium. In R, you can do just do a linear regression using the lm(Y~X) which includes an intercept or using lm(Y~X+0) which regresses without an intercept. Assuming you've saved the model in variable lm.r, then to get the ...


3

The CAPM states that the expected return of an asset i is related to the expected market return by $$\mathbb{E}[R_i] = r_f + \beta_i (\mathbb{E}[R_M] - r_f) $$ If the CAPM is a correct description of risk and return, then the next period price Q should be given by $$ Q = P (1+r_f + \beta_i (\mathbb{E}[R_M] - r_f)) $$ In your formulation, the denominator ...


3

Focusing on intuition rather than theory, $\beta$ can also be thought of as the "risk premium" of that specific asset relative to the market. In general, market risk premium links two very important aspects of the world: Consumption & Return. So if we look at the world in two states, an "Up State" & "Down State", here is what we would see: ...


3

Beta as a measure of risk has serious drawbacks, particularly in emerging markets. You need to consider alternative risk metrics (cost-of-capital build-up method or volatility, for example), or if you do use beta consider what the market index refers to and the composition of that index. This paper actually happens to touch on beta estimation and uses ...


3

This simply suggests the linear model is a poor fit in high frequency. But is this that surprising, even before you crunch the numbers? I argue not, for the following reasons: Even at low frequencies (i.e. monthly or annually), it is known that the classical CAPM (which is what you're running, albeit at a much higher frequency) does not fit well. It'd be ...


2

Assuming those are arithmetic returns and covariances at the horizon, calculate a $9\times1$ vector containing the betas with respect to the world index using the covariance matrix, call it $\beta$. The covariance resulting from the world index can be described as $\beta\sigma_{world}^{2}\beta'$. The matrix ...


2

2) you only take trading days for your analysis because taking in account days on which no price changes took place would shift results in a wrong direction. For exmple, you mostly take 250 trading days p.a. 3) Your time interval up to 2007 is okay and excludes the financial crisis, which is a non-normal circumstance. Therefore, your time interval can be ...


2

In the following paper: "On the Cross-Section of Expected Stock Returns: Fama-French Ten Years Later" (by Chou, Chou, and Wang), the authors found, using the Fama-Mac Beth two-pass regression, that the size effect becomes insignificant during the post-1981 period, and the Book/Market effect becomes insignificant during the post-1990 period. It is important ...


2

Idiosyncratic volatility is NOT included in the regressors, so it should not be and actually cannot be part of your matrix X. Idiosyncratic volatility is the volatility (of Y) your matrix X (explanatory variables) cannot explain (i.e. remaining unexplained part), so it is the error term of your regression equation. Just compute the standard deviation of ...


2

Basically you have two equations as follows: -Regression: $$R_i = α + β \cdot R_m + e_i$$ -CAPM equation: $$E(R_i) = r_f + β\left[E(R_m) - r_f\right]$$ In CAPM sense, there is no α . It only exists as idiosyncratic return. You would need lot more than one year of data to estimate the coefficients for regression. You can check p-values to see if the ...


2

The answer is NO. It's mathematically incorrect. Simply look the correlation and covariance formulas. But here is a gedankenexperiment (thought experiment) that demonstrates that it's incorrect. Suppose, R1 = M. Then the claim Corr(M,R1) = Corr(M,R2) implies 1 = Corr(M,R2) for any R2, which is obviously wrong.


2

When we discuss CAPM it assumes a lot of things about the market and investor behaviour. There is enough literature on "CAPM doesn't hold". In fact most low beta stocks plot above the security market line (SML). So it would be a mistake to take CAPM so seriously in practice and I would cross question if CAPM works as it is? In theory, if there are negative ...


2

I think there is not too much to say. At first glance it looks good if the manager loses $10\%$ if the whole market loses $30\%$. But plugging the beta and the risk-free rate into the CAPM formula we see that we would have expected a loss of $2\%$ only. So the $10\%$ are much worse than expected. Note however that there are various reason's why CAPM just ...


2

The coefficients assuming they are statistically significant can be interpreted whether or not the underlying portfolio is efficient. The CAPM or FF4 simply tries to decompose a portfolio into a series of linear exposures + an intercept (alpha) which can be viewed as constant added value. In mathematical terms the regression is explaining how much of ...


2

WACC is the weighted average cost of capital therefore from the business's standpoint, they would want to have a lower WACC because it is an average of the % cost of capital. From an investor's standpoint: it can be mixed. For a bondholder, they would want WACC to be a bit high but not by too much. For example, a higher WACC may mean the company is paying a ...


2

They are actually exactly the same thing. CAPM say that expected risk premia are “explained" by the risk premium on the mean variance efficient (MVE) portfolio $$ R^i_{t+1} - R^f = \delta (R^{MVE}_{t+1}-R^f) + \varepsilon_{t+1} $$ De facto, you are saying that the systematic risk is just the projection of risk premia on MVE risk premium, and OLS are exactly ...


2

PerformanceAnalytics in R and PortfolioAnalytics in R Here is a tutorial from UW http://faculty.washington.edu/ezivot/econ424/portfolioFunctionsPowerPoint.pdf


2

Although Rf can be negative (but not too negative), Rm cannot be less than Rf as in your example. It is a non-equilibrium situation, no one would invest in risky securities if they have an expectation lower than risk-free securities. So Rm > Rf is a necessary assumption of the CAPM, whether rates are positive or negative. Also, algebra is algebra and the ...


2

The simple answer is no. You need historical data to backuo the implied correlation. A smart way to do it is to use Buss and Vilkov (2009) methodology. Denote the risk-neutral correlation between each pair of stocks: $\rho_{ij,t}^Q$. The presence of the correlation premia led Buss and Vilkov (2009) to estimate the risk-neutral correlation by making: ...


1

Error term The error term tells the difference between the theoretical and the observed values of the dependent variable. As such it is referred to the single observations. In your equation, as you say, $i$ stands for the $i$-th share, therefore the meaning of $\varepsilon_i$ is unclear (as it can't be the $i$-th observation) and the related equations ...



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