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12

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

10

This is actually a rather involved question in some sense, and different interpretations exist. Going down the stochastic discount factor road, factor asset pricing models (eg. Fama-French 3 Factor, Carhart 4 Factor etc...) imply a stochastic discount factor that is linear in the factors. You can take a narrow, linear algebra based interpretation that the ...

9

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 ...

8

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 ...

8

Alphas from a time-series regression are error terms in the cross-sectional, linear relationship between expected returns and factor betas. If a factor model were correct those error terms (the alphas) would be zero. Discussion A carefully written version of a standard time-series regression of returns in excess of the risk free rate on market excess ...

8

To be consistent with the average daily returns that you specified, your first strategy would need to have a daily standard deviation of 31,749 USD and the second a standard deviation of 7,937 USD. How much weight you should assign to each strategy depends on your goal. You might want to maximize the daily profit, minimize the volatility, or maximize the ...

7

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 ...

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{...

7

Infinity is rather non-sensical. A better question perhaps is whether you can put some theoretical bounds on an asset's market beta. An asset's volatility bounds its market beta Let $R_i$ be the return of security $i$ and $R_m$ be the return of the market. Market beta would be given by: $$\beta_i = \frac{\operatorname{Cov}(R_i, R_m)}{\operatorname{Var}(... 6 You might want to read this: Size, Value, and Momentum in International Stock Returns by Fama and French (2011) Abstract: In the four regions (North America, Europe, Japan, and Asia Pacific) we examine, there are value premiums in average stock returns that, except for Japan, decrease with size. Except for Japan, there is return momentum everywhere,... 6 There is a rationale here and it has to do with the connection between diversification and the number of assets in a portfolio. Suppose we purchase an equal-weighted portfolio of n stocks. The variance of the return is then: \sigma_{p}^2 = \sum \sum w_i$$w_j$Cov($R_i$,$R_j$) In the above notation, the sigmas are summing over$i$and$j$... 6 Active return:$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$, appears on the right hand side In words the CAPM ... 6 If you really believed the CAPM's prediction that$\alpha=0$, then imposing$\alpha=0$in your estimation would indeed lead to your 2nd formula. The problems? The CAPM doesn't work so imposing a false restriction during estimation is problematic. More generally, taking factor models extremely seriously and imposing$\alpha=0$in estimation to gain ... 5 What you observe in your regression is not strange at all. The regression beta you estimated is$\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 ... 5 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 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 equation ... 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 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 $$\epsilon_i = r_i - ( \... 5 You're not going to get an analytic formula except in special cases of function \rho(x). And you're probably going to want \rho convex. If \rho is convex, the problem is a convex optimization problem and can be efficiently solved numerically. If \rho isn't convex, the optimization problem may be difficult to solve. If \rho(x) = |x| you basically ... 4 Beta as a measure of risk has serious drawbacks, particularly in emerging markets. You need to consider alternative risk metrics (cost-of-capital build-up method or volatility, for example), or if you do use beta consider what the market index refers to and the composition of that index. This paper actually happens to touch on beta estimation and uses ... 4 The CAPM states that the expected return of an asset i is related to the expected market return by$$\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)) $$In your formulation, the denominator ... 4 The risk-free rate is the y-intercept of the Security market line. If the risk free rate goes negative the y-intercept of the Security market line would simply be below the x-axis. So if the risk-free rate decreases the whole line shifts down. This just means people are willing to pay for safety. According to the formula for the SML: E(Ri) : expected return ... 4 \sigma_p=\sqrt{\omega_a^2 \sigma_a^2+(1-\omega_a)^2 \sigma_b^2+2 \omega_a (1-\omega_a) \rho_{ab} \sigma_a \sigma_b} with \rho_{ab}=-1 the term under the square root simplifies to (\omega_a \sigma_a-(1-\omega_a) \sigma_b)^2 which is equivalent to (-\omega_a \sigma_a+(1-\omega_a) \sigma_b)^2 therefore \sigma_p=\omega_a \sigma_a-(1-\omega_a) \... 4 As you know the equation that describes them is the same. The single index model is an empirical description of stock returns. You do some regressions using data and you come up with Alphas, Betas etc. That's all. It is useful for example in modeling risks of a bunch of stocks in a simple way. The CAPM is an economic theory that says that Alpha in the long ... 4 The solution provided can be derived using the CAPM. For asset A you have:$$R_A-R_f = \alpha_A +\beta_A(R_M-R_f)+\epsilon_A$$Similarly for asset B:$$R_B-R_f = \alpha_B +\beta_B(R_M-R_f)+\epsilon_B$$Calculate the covariance:$$\text{Cov}(R_A, R_B) = \text{Cov}(\beta_AR_M, \beta_BR_M)$$Here I have dispensed with all the constant terms, and also ... 4 The question above looks somewhat confused. Where's the error term? A recipe for a standard calculation It's customary to work with monthly returns. For each portfolio$i$, calculate monthly excess returns$R^x_{i,t} = R_{i,t} - R^f_t$where$R^f_t$denotes the 1-month risk free rate. Calculate or download the monthly excess return of the market$R^m_t - ...

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A linear relationship between expected returns and covariance with a risk factor is a necessary consequence of a linear asset pricing function In theory, a CAPM relationship can be derived when a pricing kernel $S$ is affine in the return of the market portfolio. Different sets of assumptions lead to this affine relationship. Be aware that the CAPM is an ...

4

In a word, yes. That's a correct and valid view to take but, as you'll always find in finance, it really depends on context and the question that you're trying to answer. This is the case in markets but more broadly in business and something that academically minded scientists/engineers struggle often understand and appreciate fully. This boils down to the ...

4

CAPM states that the expected return of any given asset should equal $ER_i=R_f+β_i (R_m-R_f)$, with α being the error term of the previous equation. Now, as α has an expected value of zero, then only way to achieve higher expected returns is taking on more β (given that $E[(R_m-R_f )]>0$). Every individual stock has some idiosyncratic risk in addition to ...

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 ...

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