# Tag Info

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This the "Joint Hypothesis Problem". Basically, any test for abnormal returns is also implicitly a test of the model you use to define "abnormal". If you see a significant and positive $\alpha$, that could either mean that you actually are generating excess risk-adjusted returns, or it could mean that your risk model is incomplete. This is basically what ...

3

Non overlapping periods would make for a far smaller sample

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Fitting Fama-French or Carhart is as simple as learning how to perform Bayesian regression. Pretty much every introductory book Bayesian estimation will cover this. There are analytic formulae under certain assumptions, but I would definitely try to learn the basics of MCMC and Gibbs sampling before trying this out in practice. Here are two papers. The book ...

3

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

Let $X$ be endowed with the following partial order: $y \geq x$ means that $\Bbb P(y\geq x) = 1$. The AOA condition in your case states that the pricing law $p$ is strictly inctreasing with respect to $\geq$, whereas LOP says that $p$ is a linear function. Neither if the two implies another one in general. For example, if $X = \Bbb R$ then $p(x) = x^3$ is a ...

2

They are actually exactly the same thing. CAPM say that expected risk premia are “explained" by the risk premium on the mean variance efficient (MVE) portfolio $$R^i_{t+1} - R^f = \delta (R^{MVE}_{t+1}-R^f) + \varepsilon_{t+1}$$ De facto, you are saying that the systematic risk is just the projection of risk premia on MVE risk premium, and OLS are exactly ...

2

The coefficients assuming they are statistically significant can be interpreted whether or not the underlying portfolio is efficient. The CAPM or FF4 simply tries to decompose a portfolio into a series of linear exposures + an intercept (alpha) which can be viewed as constant added value. In mathematical terms the regression is explaining how much of ...

2

if you have $p=0.5$ For example: $U(w)=ln(2w)$ why is that? relative risk aversion is given by $$RRA=\frac{-wU''(w)}{U('w)}=\frac{-w*(-1/4w^2)}{1/2w}=0.5$$ Now you can apply your formula. take for example: $x= 10000$ and $\pi=0.5=1-\pi.$ then expected utility is equal to $EU(x,w)=0.5*ln(2*(w+x))+0.5*ln(2*(w-x))=0.5ln(220000)+0.5ln(180000)$ you want to ...

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)$and pocket the ... 2 Unfortunately, I do not know the model you talk about. However, the law of one price is a direct implication of the no-arbitrage assumption, which is assumed in many models (if not all). I do agree that the law of one price should be stated as a theorem rather than a definition. Anyway. Consider the case in which two portfolios A and B have the same value ... 1 It is due to parameter variation. By overlapping portfolios they can better show that their results are not a one-off result that only works given this very specific set of inputs. Without testing with different parameters (stocks, timeframe, etc), results are liable to blow up given a different input. That is not to say using parameter variation always ... 1 Yes leverage amplifies the exposure of equity to systematic risks. Just consider the standard textbook formula (Modigliani-Miller):$\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$then consumption growth is has a unit root and is not stationary, and so will be the model. 1 The R code is correct. You could also use the I() operator. You can look here on page 53. The code then would be lm(stock~market+I(market^2)+I(market^3), data=example) EDIT: going more into detail: Doing the above you define regressors$market^2$and$market^3$. The coefficients will be calculated the usual way (covariance of response with the regressors ... 1 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 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 ... 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. 1 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 ... 1 The answer depends somewhat on the type of bond. An odd lot of corporate bonds may be different than municipal bonds or US Government Agencies. Generally speaking, an odd lot would be any trade under$100,000 in face value.

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Supply and demand... If you want an event that produce a change in the value of a currency, just look at the ruble. As Russia, gets more and more isolated and inflation spins out of control the ruble lose its value against other currencies.

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In small sample, there is no reason why $x' \epsilon$ will be 0. In fact, there is no real reason why $\epsilon$ should be independent. The fact that you are assuming a linear specification for the returns means you are to some extent making assumptions of linear regression. Justifying the errors being uncorrelated with the independent variables is justified ...

1

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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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 ...

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