# Tag Info

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

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

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 ...

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 ...

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

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

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

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

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.

4

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 ...

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

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

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

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

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 ...

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

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

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: ...

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

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

This is definitely not a Kalman filter's issue: if you replace this line of code args <- eapply(env = env, FUN = function(x){ClCl(x)}) with this one args <- eapply(env = env, FUN = function(x){ClCl(x)})[Symbols] eapply() will keep the order of the original Yahoo query from quantmod. You can check and you will see each $\beta_{t}$ matches about ...

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

If you assume that your monthly returns are independent from each other, then the annualized variance of each series, and the covariance can be annualized. This assumption allows you to use V(x1+X2+...+x12) = V(x1) + V(x2) + ... + V(x12) where xi is the return for the month "i". Actually, for this to happen you only need a weaker assumption: that is that ...

1

You still want to perform portfolio optimization. Put everything into one bucket, run 'global' portfolio optimization, build the portfolio. Even if you prefer Sharpe ratios, you should do that on the overall portfolio - not just on individual ones. Be careful of sharpe ratios for low risk, low return assets. Dividing one small number by another small number ...

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