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7

A factor model has the form $$r_{j,t}=\sum_n \beta_{j,n} f_{n,t}+\epsilon_{j,t}$$ Where $r_{j,t}$ is the return of stock $j$ at time $t$, $\beta_{j,n}$ is the sensitivity (factor loading) of stock $j$ to factor $n$, $f_{n,t}$ is the return of factor $n$ at time $t$, and $\epsilon_{j,t}$ is the idiosyncratic non-factor return. One factor can be the constant. ...


3

Non overlapping periods would make for a far smaller sample


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In my experience HFT has to balance the reward of any strategy with risk. In the case of a news-based trading strategy, the risk can be enormous, which means the algo will need a very high expected profit in order to trade the news. After important news events, volatility skyrockets and persists for some time (sometimes even days). If the market were able ...


2

For non-normal asset price models you could look at the theory of Lévy-processes. If we assume that you work in the physical probability measure $P$ and that the random numbers that you have generated are daily log-returns, then you can do the following: Asset $i$ has starting price $S_0^i$ and for the future prices you can put $$ S_t^i = S_0^i ...


2

(1) To get an arbitrage, buy low and sell high. Consider the following strategy: at $k$, in the event $V_k(\Phi_1) < V_k(\Phi_2)$, buy $\Phi_1$ and sell $\Phi_2$ invest the difference at the risk free rate. At maturity, your portfolio is worth what the you put in the bank plus interest. Formally, if $V_t(\Phi_\alpha) = \sum_{i=0}^d \Phi^i_{\alpha,t} ...


2

Regarding (1). Assume for some time $k$, $\mathcal{V}_k(\Phi_1) > \mathcal{V}_k(\Phi_2)$ (w.l.o.g.) with full knowledge that these strategies have equal value at $T$ ,($\mathcal{V}_T(\Phi_1)=\mathcal{V}_T(\Phi_2)$). I claim that this situation admits arbitrage. I can sell $\mathcal{V}_k(\Phi_1)$ and buy $\mathcal{V}_k(\Phi_2)$ and pocket the ...


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Unfortunately, I do not know the model you talk about. However, the law of one price is a direct implication of the no-arbitrage assumption, which is assumed in many models (if not all). I do agree that the law of one price should be stated as a theorem rather than a definition. Anyway. Consider the case in which two portfolios A and B have the same value ...


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I would recommend "Active Portfolio Management" from Richard Grinold and Ronald Kahn. The book builds up most theories used in portfolio composition with much detail.


1

Using covariance to imply an inappropriate level of multicollinearity in a model can be very misleading, especially when the factors are measured in differing units or lack linear relationships. There will almost always be some level of collinearity in a multi-factor model (otherwise you run the risk of overfitting), especially one with a relatively small ...


1

I'll attempt an answer here --- but really, this is a relatively straightforward corporate finance question, and indeed, I'd argue this is effectively an accounting question. We have the accounting identity that, $$ Assets = Equity + Debt $$ At this point, I don't want to get into tax shields, financial distress costs and things of that sort (it's ...


1

It is due to parameter variation. By overlapping portfolios they can better show that their results are not a one-off result that only works given this very specific set of inputs. Without testing with different parameters (stocks, timeframe, etc), results are liable to blow up given a different input. That is not to say using parameter variation always ...


1

Yes leverage amplifies the exposure of equity to systematic risks. Just consider the standard textbook formula (Modigliani-Miller): $\beta_e = \beta_a \times (1+\frac{D(1-\tau)}{V})$ where $\beta_e$ is the sensitivity of the stock to systematic risk, $\tau$ is the tax-rate and $D/V$ is the leverage ratio. So beta (i.e. the exposure to systematic risk) ...


1

Well this is not my area of expertise but I have come across this sort of work before in Time Series Analysis/ Financial Econometrics. I don't know how much detail you want but from my understanding the author has written the two equation in State Space Form. I believe it is fairly common to write ARCH and GARCH models in this fashion. There are a lot of ...


1

After some careful thought, the answer is trivially simple, actually. If $\phi_1=0$ then consumption growth is iid. If If $\phi_1=1$ then consumption growth is has a unit root and is not stationary, and so will be the model.


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The R code is correct. You could also use the I() operator. You can look here on page 53. The code then would be lm(stock~market+I(market^2)+I(market^3), data=example) EDIT: going more into detail: Doing the above you define regressors $market^2$ and $market^3$. The coefficients will be calculated the usual way (covariance of response with the regressors ...


1

As a simple example: if stock A went up a lot in 2014 and also went up a lot in 2015 it could be: (a) that Stock A is a high Beta stock and the market was up in both years. This is the cross sectional property of expected returns. Some stocks, in this case high beta stocks go up more than others when the market goes up. (b) Somehow the fact that Stock A went ...


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Continuous time has a so-called elegance, but it is rarely correct. Most Q-measure people rarely care about correctness anyway, since they usually don't root their models in statistics. With no goodness of fit measures, continuous time models are elegant theory. In general, we also see that most ex-ante hedges are rarely good, ex-post. They have large ...


1

A unique state price vector does not have to exist for there to be no arbitrage. It sounds like the state price vector in question has infinitely many solutions. Try to reduce the price matrix to row echelon form and show that at least one state price vector exists.


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I just comment your second point, because in the definition i now of the LOP the state price vector (martingale measure) is involved. assuming the LOP holds then: state price vector is unique <=> the market is complete. to proof "=>" look at the trinomial model, show that the model is not complete by trying to find a hedge for a call, afterwards ...


1

The direct answer to your question on the choice of m is, "It depends." Your choice of m is dependent on the convention used by the source of your discount rate. Either may be appropriate. If you are actually looking to estimate a "fair" value, then the following will be relevant: A market yield(-to-maturity) approach assumes coupon reinvestment at that ...


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To many statistical questions you can get frequentist and Bayesian answers which actually coincide. Such a subject is covariance matrix shrinkage and Bayesian regression. Have a look at the article "Honey, I Shrunk the Sample Covariance Matrix" from Lediot and Wolf. They introduce a transformation of the covariance matrix, so that the diagonals become more ...


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I suggest you to read: Lancaster, Tony. An introduction to modern Bayesian econometrics. Oxford: Blackwell, 2004. I studied that to learn Bayesian Regression Model; the book is very clear and well-done and it is a good reference, IMHO, for who is a newbie, but, at the same time, is interested to the topic. Hope this helps.


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Singer and Terhaar original paper can be found at this link. They do not provide an explanation about how to estimate this factor and just mention that both values provide a boundary. The CFA curriculum mentions that " For example, it has been observed that developed market bonds & equities are approx 80% integrated and 20% segmented.", however the ...


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This is a very good observation that I wrote about in my undergrad studies. I also believed that markets were efficient but not precise. I used the example a few years back regarding a tweet (roughly after the Boston bombings). The tweet was regarding terrorist attacks in which markets fell sharply and then recouping all the gains as news later indicated ...



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