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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 ...

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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 ...

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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 ...

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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$$w_jCov(R_i,R_j) In the above notation, the sigmas are summing over i and j ... 5 I would not necessarily call it a failure. CAPM explained ~70% of returns (on average) so this may quite be one of the 30% that could not be explained (see link). However, an improved approach or extension of the CAPM would be the the Famma-French factor model which explains roughly 90% of returns (see link). Again, the Famma-French is an extension of CAPM ... 4 What you observe in your regression is not strange at all. The regression beta you estimated is \beta_i = \frac {\mathrm{cov}(r_i,r_m)}{\mathrm{var}(r_m)} where i represents the country/region (such as the USA or China) and m represents the "market" (which you take to be the ACWI). Since the USA is itself such a large component of the ACWI (about ... 4 The risk-free rate is the y-intercept of the Security market line. If the risk free rate goes negative the y-intercept of the Security market line would simply be below the x-axis. So if the risk-free rate decreases the whole line shifts down. This just means people are willing to pay for safety. According to the formula for the SML: E(Ri) : expected ... 4 In academics, Roll's critique of the CAPM is discussed a lot, for a start see Wikipedia page of Roll's critique. It is more of a principled "theoretical" critique of the CAPM than an empirical one. It says basically that the CAPM cannot be tested because every mean-variance efficient portfolio satisfies the CAPM the market portfolio is unobservable 3 This is in essence the idea behind Andrea Frazzini's paper 'Betting Against Beta'. There are various ETFs that aim to exploit the premium. In R, you can do just do a linear regression using the lm(Y~X) which includes an intercept or using lm(Y~X+0) which regresses without an intercept. Assuming you've saved the model in variable lm.r, then to get the ... 3 The CAPM states that the expected return of an asset i is related to the expected market return by$$\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 ... 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 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 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 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]$$In CAPM sense, there is no α . It only exists as idiosyncratic return. You would need lot more than one year of data to estimate the coefficients for regression. You can check p-values to see if the ... 2 I think there is not too much to say. At first glance it looks good if the manager loses 10\% if the whole market loses 30\%. But plugging the beta and the risk-free rate into the CAPM formula we see that we would have expected a loss of 2\% only. So the 10\% are much worse than expected. Note however that there are various reason's why CAPM just ... 2 The coefficients assuming they are statistically significant can be interpreted whether or not the underlying portfolio is efficient. The CAPM or FF4 simply tries to decompose a portfolio into a series of linear exposures + an intercept (alpha) which can be viewed as constant added value. In mathematical terms the regression is explaining how much of ... 2 WACC is the weighted average cost of capital therefore from the business's standpoint, they would want to have a lower WACC because it is an average of the % cost of capital. From an investor's standpoint: it can be mixed. For a bondholder, they would want WACC to be a bit high but not by too much. For example, a higher WACC may mean the company is paying a ... 2 They are actually exactly the same thing. CAPM say that expected risk premia are “explained" by the risk premium on the mean variance efficient (MVE) portfolio$$ R^i_{t+1} - R^f = \delta (R^{MVE}_{t+1}-R^f) + \varepsilon_{t+1} $$De facto, you are saying that the systematic risk is just the projection of risk premia on MVE risk premium, and OLS are exactly ... 2 PerformanceAnalytics in R and PortfolioAnalytics in R Here is a tutorial from UW http://faculty.washington.edu/ezivot/econ424/portfolioFunctionsPowerPoint.pdf 1 Error term The error term tells the difference between the theoretical and the observed values of the dependent variable. As such it is referred to the single observations. In your equation, as you say, i stands for the i-th share, therefore the meaning of \varepsilon_i is unclear (as it can't be the i-th observation) and the related equations ... 1 The answer is NO. It's mathematically incorrect. Simply look the correlation and covariance formulas. But here is a gedankenexperiment (thought experiment) that demonstrates that it's incorrect. Suppose, R1 = M. Then the claim Corr(M,R1) = Corr(M,R2) implies 1 = Corr(M,R2) for any R2, which is obviously wrong. 1 beta refers to the fact that on an average the stock has a degree of correlation with the movement of the index. the important thing is "on an average" because two different stocks may have the same beta but this average may have different weightages of different parts of that time period. so lets say that in the first part of the data, stock1 is not ... 1 Given that you're correctly measuring Alpha, the difference lies in the Beta exposures of the two managers. You may not be capturing certain tilts, which would show up in your error or incorrectly categorized as Beta. Consider the simple case where you have returns grouped into just technology vs. Energy for instance. R_p = B_0 + B_t$$R_m$ + ...

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Singer and Terhaar original paper can be found at this link. They do not provide an explanation about how to estimate this factor and just mention that both values provide a boundary. The CFA curriculum mentions that " For example, it has been observed that developed market bonds & equities are approx 80% integrated and 20% segmented.", however the ...

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Did the portfolio manager have the option of investing in emerging markets? If yes, use MSCI All-World. If the portfolio has holdings based in countries with "developed markets" yet has has emerging markets exposure to revenue/earnings, the convention is to use MSCI World.

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I have never seen such an adjustment. While monthly data are irregularly sampled in time (in every way...calendar days, trading days, seconds, etc), that irregularity is likely to be a smaller effect than your choice of data frequency (monthly, weekly, daily data). That said, your question is intriguing because in other fields they do have to deal with ...

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