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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 ...


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

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 $j$ ...


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 ...


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

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

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: ...


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

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

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 ...


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$ + ...


1

That is true. Utility would not be concave anymore under prospect theory (only for gains), but convex for losses, which is evidence against CAPM. CAPM is valid either : -if the utility function is quadratic (which is nonsense in terms of economic interpretation, and in general, Von Neumann- Morgenstern utility describes poorly reality and should be ...


1

The general problem of the investor is: $$ \max_{w\in[0,1]^n} U(\mu_p(w),\sigma_p(w))\quad s.t. \sum_{i=1}^n w_i=1$$ where $w$ being the portfolio weights, and $U$ utility function of portfolio risk $\sigma_p$ and return $\mu_p$. CAPM assumes investors with concave utility function $U=\mu_p-\frac{1}{2}\sigma_p^2$, from which then follows that all ...


1

I think the best strategy is to follow Ken French who posted all of the Fama-French factors on his website a while ago, including the risk-free rate. The latter is updated at the same frequency as the portolio returns, e.g. you can't use a 3-month - based rate if you work with monthly equity returns.


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

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 ...


1

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 ...


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

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.


1

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 ...


1

Compounding the monthly excess returns won't provide the annual excess return. You need to compute the difference between the annual return of the portfolio and the annual return of the benchmark. To illustrate this let's look at an example. Consider the following two situations: The benchmark performs well with a $2\%$ return each month; The benchmark ...


1

Your formula is adding where you should be multiplying, and you plugged your inputs into the wrong places (your levered Beta notably). In any case, the process for un-levering/re-levering the beta goes like so: Step 1: Find benchmark company/asset/project Beta. Step 2: Un-lever the benchmark Beta: Unlevered Beta = Levered Beta * (1 / ( 1 + (1 - ...


1

you get what should get. You can't prove that strategy long $X$ short $Y$ is market neutral: is strategy long EUR/USD short USD/CHF risk neutral? I wouldn't say that. It depends, on what? On relationship between these variables, so it is perfectly hedged only if dX=dY so your task is bad stated: it should be rather: what should be $b$ to assure that ...



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