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16

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 ...


9

I would split the question into two sub-questions: Is market beta useful at all? Is market beta useful for high-frequency strategies that are fully hedged EOD? With regards to the first question, I would summarize the hundreds of papers on the subject as: yes, but not as much as it was initially believed. The reason being that multi-factor models are ...


9

It partly depends on the use case. If one is taking multiple strategies and assembling a portfolio that includes multiple different strategies and is mixing this with a heavy weighting to an equity index, then this might be a useful measure. Zero or negative beta does have meaning, in the same way that correlation has meaning. In the more traditional ...


6

Jennifer Bender of MSCI Barra has a paper from 2007 entitled: To Beta or Not to Beta: A Comparison of Historical Versus Fundamental Betas for Hedging Market Risk She deals specifically and exclusively with which method is superior for hedging long-only portfolios. Not surprisingly, she finds that Barra's approach is better. She tests long-only and ...


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.


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

The term 'rule of thumb' is ambiguous here. Because I don't think there are any rule of thumb, you just need to do the number crunching. However there are some stable characteristic through time linked to correlation. For instance it is a common fact that the hierarchy of correlation within different market is relatively stable. US equities are less ...


5

So my first answer was off base. For some reason I was thinking first moment (idiosyncratic returns), but he's looking for second moment (idiosyncratic volatility). There is a line of research on the returns to portfolios sorted on idiosyncratic volatility and I was hoping that there were descriptive statistics that said "fraction $\rho$ of stock/portfolio ...


5

I just want to give a qualitative assessment to your question: Volatility of a market is different than the volatility of a stock. Similarly like Copeland and Antikarov (2001) say that "...the volatility of a gold mine is different than the volatility of a gold..." If you want to quantitatively compute the percentage of a stock's volatility affected by ...


5

From a theoretical point of view (you mentioned beta, so assume we're in a CAPM world), you should hold the market portfolio (let's assume S&P500 index) and be long (or short) the risk-free asset to decrease (or increase) your return and risk. That is, if you'd like higher returns than the S&P500 offers and are willing to accept the risk, trade the ...


4

By William Bernstein, source: In June of 1992 academicians Eugene Fama and Kenneth French ("F/F") rocked the investing world with a study published in the Journal of Finance, innocuously entitled "The Cross-Section of Expected Stock Returns." The piece is the cognitive equivalent of an enormous hunk of marzipan cake which sits in your ...


4

Most portfolio managers look at the Sharpe ratio, or occasionally the Treynor ratio. In general, you want to maximize one of the these metrics, though there could be other issues that you haven't currently considered, like turnover or transaction costs associated with obtaining the portfolio.


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

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.


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)


3

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 ...


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

The empirical evidence shows that low to medium beta portfolios beat high beta portfolios on a risk adjusted basis. Search SSRN for "betting against beta." This flies completely in the face of CAPM but frankly CAPM is crap.


3

An Axioma research paper from August 2011, Using Multiple Risk Models for Superior Portfolio Management… A Practice Not Just For Quants, answers exactly your question, I believe. Note the graphs at the top of page 8. They compare their medium-horizon fundamental and statistical factor models from January 2008 to January 2009. At the start of the period, ...


3

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 ...


2

In response to your last question, "how do you use beta?" - I'd say try to as little as possible. Your use seems to be a bit out of the realm of what I'm used to, but whatever beta you get is so prone to observational error, that it may not be meaningful.


2

This blog post just came out: http://www.portfolioprobe.com/2011/02/08/4-and-a-half-myths-about-beta-in-finance/


2

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.


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

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

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 ...


2

Beta is calculated as Rstock = alpha + beta*Rindex. When you use slope in excel the first value is for the y's so you are doing it wrong, you should have slope(Stock returns, Index returns). While that is the formula you use above it is not the one in the excel, with the data you provide I get a beta of 0.96.


2

Beta calculation varies quite a bit as you've already noted. Using monthly or weekly closing prices is fairly common though; I don't know of anyone who uses daily prices. Yahoo gets its data from CapitalIQ so you may want to look over there. Good luck!



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