# Tag Info

26

Yes, the weights of the first eigenvector of a covariance matrix represent the market factor and also the largest source of systematic risk (variation of returns). Why PCA? Well, PCA simply identifies the eigenvector that maximally explains the variance of the system. It turns out that this is the "market factor" - i.e. the tendency of securities to rise ...

12

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

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From Yahoo! Finance Help The Beta used is Beta of Equity. Beta is the monthly price change of a particular company relative to the monthly price change of the S&P500. The time period for Beta is 3 years (36 months) when available. Source: https://help.yahoo.com/kb/finance/SLN2347.html?impressions=true (+Stock Price History)

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Proof: Recall that $$\beta_{i} = \frac{\mathrm{Cov}(r_{i},r_{m})}{\mathrm{Var}(r_{m})}.$$ Now, the returns on unlevered and levered equity are given by $$r_{U} = \frac{\mathrm{EBIT}(1-\tau) - \mathrm{CAPEX} + \mathrm{Depreciation}}{E_{U}}$$ $$r_{L} = \frac{\mathrm{EBIT}(1-\tau) - \mathrm{CAPEX} + \mathrm{Depreciation} + \mathrm{Net\ Debt} - \mathrm{... 7 Infinity is rather non-sensical. A better question perhaps is whether you can put some theoretical bounds on an asset's market beta. An asset's volatility bounds its market beta Let R_i be the return of security i and R_m be the return of the market. Market beta would be given by:$$ \beta_i = \frac{\operatorname{Cov}(R_i, R_m)}{\operatorname{Var}(...

6

I assume you're using returns to compute beta, not the prices. And yes, remove the "jumps", though this should happen automatically since you're looking only at intraday returns. One final piece of advice: you'll get more meaningful results if you smooth the returns via a moving average.

6

There are more ways to approach this but the method I propose should work reasonably well in practice, especially if you increase the number of assets you hold. Calculate the beta of the stocks you're holding with respect to an index Buy $N_f$ (sell when $N_f$ is negative) future contracts on that index $N_f$ can be calculated as $$N_f = \frac{\beta_T - \... 5 Have you checked out the vingette for DLM by Petris? Incidentally, Petris also has an R-book on the DLM package which includes estimation of beta as an example. 5 Let's first restate the formula of the beta of a portfolio P relative to a benchmark B:$$\beta_P=\frac{Cov(r_P,r_B)}{Var(r_B)} As chrisaycock said in his comment, the key thing to understand is that the beta is a statistical measure computed relative to a benchmark. Hence, I believe that the real question you should be asking is: Which benchmark ... 5 This is definitely a valid (and possibly viable) strategy. I think that your constraint of zero costs is a red herring and serves no useful purpose beside forcing you to take lopsided bets in the direction of the cheaper option. I would try instead to build a portfolio that has zero vega (hedged against overall moves in market-wide implied volatility) and ... 5 beta_A = correlation_A_Index * (stdd_A / stdd_Index ) The difference you see is due to correlation. The correlation between A and the index is lower than B and the index, and that's why you're seeing a lower beta. The moral of the story is that risk is subjective, and in fact you need to understand how your portfolio is correlated with these stocks in ... 5 Yahoo Finance calculates beta from monthly prices over a time of three years. The S&P500 is used as the benchmark You need 37 monthly prices (so you can get 36 returns) on the first trading day of each month. The final price should be on the first trading day of the previous month. The first price should be on the first trading day of the month 36 ... 5 What you're describing sounds like the reverse of a Fama-Macbeth regression. The original Fama-Macbeth approach estimated rolling time series regressions to get CAPM betas and then doing a cross-sectional regression to estimate the overall sensitivity of returns to beta. If I were to write down what the model looks like, I think you're talking about ... 5 This is a very good question. It can be argued that risk parity is one example of a smart beta strategy. Yet it is important to understand that both are coming from two different directions: risk parity is basically a form of risk management (in the sense of risk-adjustment) because its basic approach lies in diversification - like the alternative methods ... 5 I slightly disagree with Alex’s comment. The CAPM does not read as \begin{align*} r_{i,t} = r_{f,t}+ \beta_{i,t} (r_{m,t}-r_f) + \varepsilon_{i,t}. \end{align*} There is an important difference between the single index model (aka market model) (SIM) which reads as \begin{align*} r_{i,t} = \alpha_{i,t} + \beta_{i,t}(r_{m,t}-r_{f,t}) + \varepsilon_{i,t} \end{... 4 I did not look at the data, but recall that beta is a parameter in the following equation: r_A = \alpha + \beta r_B + \epsilon $$relating two returns (random variables, samples) r_A and r_B. To calculate beta you peform$$ \beta = \frac{cov(r_A,r_B)}{var(r_B)}. $$Thus if assets A and B exchange roles, then only the denominator changes. In your ... 4 I've started thinking about this, too. My gedanken conclusion turned out to be too simple once I found what I was after: http://www.investment-and-finance.net/derivatives/o/option-beta.html, which I've confirmed in Black & Scholes (1973) p10 (eq 15). In short:$$ \beta_{\text{option}} = \frac{S\cdot\Delta}{O}{\beta_S} $$where S is the underlying ... 4 Imagine a scenario where a beta neutral portfolio comprised being long one very high beta stock and short many low beta stocks. Such a portfolio clearly has extreme concentration of risk. Additionally imposing a 'dollar neutral' constraint, would help to spread the weights more evenly over all the stocks. A further observation is that measuring true 'beta'... 4 You need returns for 36 months, in particular data from 37 months. Yahoo also uses unadjusted closing prices for the reference index as far as i know. The data from 8/1/2015 got to be an error, I checked multiply data sources and found no similarities. After interpolating that point i got a beta of 0.48. 4 In a word, yes. That's a correct and valid view to take but, as you'll always find in finance, it really depends on context and the question that you're trying to answer. This is the case in markets but more broadly in business and something that academically minded scientists/engineers struggle often understand and appreciate fully. This boils down to the ... 4 To give you an idea of industry standards for funds (although not hedge-fund specific), Morningstar and Trustnet both use monthly returns and annualize their data. See, for an example plucked at random, https://www.trustnet.com/factsheets/o/gnol/aberdeen-asia-pacific--japan-equity-i-acc. Monthly returns remain the standard because some funds only publish ... 3 Suppose you have$$X\equiv\left(x_{1},\: x_{2}\right) $$where x_{1} are the daily log returns of the security and x_{2} are the daily log returns of the market. Assume further that X is iid multivariate normal$$X\sim N\left(\mu,\Sigma\right) $$People frequently calculate beta as$$\beta_{1,2}\equiv\frac{\Sigma_{1,2}}{\Sigma_{2,2}}  If you ...

3

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 ...

3

Not sure what the question is. As John points out: the method is linear regression. For the data you could look at Kenneth French's wegpage for US stocks. In the wikipedia article you find the links to factors for other countries (UK, Germny, Switzerland) - though I have not checked these links. Note however that the Fama-French model works better for ...

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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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You could sell a high realized volatility against a low implied all day and bust out in a month. Doing this as a inter-stock spread isn't going to make it much of a better trade. If you want to take advantage of realized vol vs. implied vol you need a model that describes the relationship between the two.

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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:...

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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.

3

I think one should look at the problem from two different angles to get an answer to this. Firstly, you can look (as you said you did) look at $\hat{\epsilon}$ in terms of a disturbance like you said, meaning the returns $R_{it}$ are depending linearly on the $R_{mt}$ - the market or factor returns. Then you can figure there is some regression involved an ...

3

In addition to the above I can suggest: ignore data point if returns are more than a certain threshold (2 s.d.) calculate at different sampling intervals and choose most stable beta with the best significance (certain longer intervals "smooth out" small to mid size jumps)

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