# Tag Info

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 ...

7

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 ...

7

Note that $\beta$ is the coefficient of the portfolio regressed on the benchmark. That is \begin{align*} r_P = \alpha+\beta r_B + \varepsilon, \end{align*} where $\varepsilon$ is the residual. The standard deviation of the residual is called the residual risk. Specifically, \begin{align*} std(\varepsilon) &= \sqrt{var(r_P-\beta r_B-\alpha)}\\ &=\sqrt{...

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

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$$w_jCov(R_i,R_j) In the above notation, the sigmas are summing over i and j ... 5 I would not necessarily call it a failure. CAPM explained ~70% of returns (on average) so this may quite be one of the 30% that could not be explained (see link). However, an improved approach or extension of the CAPM would be the the Famma-French factor model which explains roughly 90% of returns (see link). Again, the Famma-French is an extension of CAPM ... 5 Because you have CAPM therefore the following holds:$$r_i = r_f + \beta_i (r_M - r_f) + \epsilon_i$$where r_i is the expected return of stock i, r_f is the risk free return and r_M is the expected market return, and \epsilon is an idiosyncratic return adjustment or an error. Now if you take the \text{Var}[\cdot] operator over the ... 5 Below you find some observations... In CAPM, we assume people are risk-averse and people get compensated for the systematic risk they suffer. The assumption that most people are risk-averse makes sense, but why are the rational investors also risk-averse? The "rational investors" prefer high (expected) returns and low volatity. In this sense, the ... 5 Active return: R-R_m i.e. your security (or portfolio) compared to the market portfolio. Used to judge performance before the CAPM was invented Excess return: R-R_f the security compared to the risk free rate, appears on the left hand side of the CAPM equation. Excess return on the market: R_m-R_f, appears on the right hand side In words the CAPM ... 5 Let us ignore the riskless rate for simplicity of the presentation. If you have (historical or simulated) return series r_i for the portfolio and r_i^M for the market, then the beta is the OLS regression beta:$$ \beta = cov(r_i,r_i^M)/var(r_i^M). $$Then if you write r_i = \alpha + \beta r_i^M + \epsilon_i on the other hand$$ \epsilon_i = r_i - ( \... 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 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 ... 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 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: States:... 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 ... 3$\sigma_p=\sqrt{\omega_a^2 \sigma_a^2+(1-\omega_a)^2 \sigma_b^2+2 \omega_a (1-\omega_a) \rho_{ab} \sigma_a \sigma_b}$with$\rho_{ab}=-1$the term under the square root simplifies to$(\omega_a \sigma_a-(1-\omega_a) \sigma_b)^2$which is equivalent to$(-\omega_a \sigma_a+(1-\omega_a) \sigma_b)^2$therefore$\sigma_p=\omega_a \sigma_a-(1-\omega_a) \...

2

In the CAPM, $E(r_i) = r_f$, if $\beta_i=0$ holds ALWAYS. BUT: recall the calculation of $\beta$: $\beta_i=\cfrac{\sigma_{iM}}{\sigma_M^2}$ and ${\sigma_{iM}}=\Sigma_jw_j\sigma_{ij}$ if your asset is uncorrelated with all other assets in the market, the last expression simplifies to: ${\sigma_{iM}}=\Sigma_jw_j\sigma_{ij}=w_i\sigma_i^2$ so for $\... 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$\Sigma_{residual}\equiv\Omega-\beta\sigma_{world}^{...

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

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

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

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

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 ...

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