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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. ...


5

The two step Fama-Macbeth regression works as follows: First, run a cross sectional regression in each period. I believe that you want to estimate risk premia for each of the Fama and French factors. Therefore you run: $$r_{i,t} = \lambda_{t,MKT} \hat{\beta}_{i,MKT}+\lambda_{t,HML} \hat{\beta}_{i,HML}+\lambda_{t,SMB} \hat{\beta}_{i,SMB}+ \alpha_{i,t} \quad ...


5

Then for each month $t$, you run a cross-section regression: $r_{i,t} = \lambda_0 + \hat{\beta}_i {\lambda}_t + \alpha_{i,t}$ Where: $\hat{\beta}_i \equiv [\beta_{i, MktRf}, \beta_{i, SMB}, \beta_{i, HML}]'$, is a vector of the coefficients estimated on the first step. What you are looking for is to estimate the vector of $\hat{\lambda}_t \equiv [\...


4

First of all, it is not conceivable to do all that work by hand! You are crazy to have just thought it! Second, if you want to repeat your work with different datasets, I suggest you to use R, since, once you have written a script, you can use it all the times you want. But, there's a 'but': you cannot think we are going to write some code for you (you ...


4

You are trying to price an option through Monte Carlo simulations. Here is how it should work, assuming the Black-Scholes diffusion framework. Under the Black-Scholes model's assumptions, the value of a risky asset $S$ at the time $t=T$ is a random variable which reads $$ S_T = S_0 e^{\left(\mu-\frac{\sigma^2}{2}\right)T + \sigma \sqrt{T} Z}\tag{1}$$ with ...


3

Non overlapping periods would make for a far smaller sample


2

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.


2

It is true that FF always use December for their fundamental metrics as this is the end of the fiscal year for most companies. However, the annual reports of the companies are not directly available and so are fundamental data. Thus, the main reason for using June it to avoid look ahead bias.


2

Let's suppose $P$ is total annual deposits made continuously, then the change in value of total deposits $dV_t$ is (assuming no condition on additional deposits) $$dV_t= V_t r dt + P dt $$ where we assumed $r$ is constant. Solving above differential equation, we have: $$V_T = V_0 e^{rT} + \frac{P}{r} (e^{rT} -1)$$ Assuming $t_1$ is the time period at ...


1

The most common approach is to calculate returns as $r_t=\frac{p_t-p_{t-1}}{p_{t-1}}$ or $r_t=ln\left (\frac{p_t}{p_{t-1}}\right)$ using close prices at the end of the month.


1

APT assumes that idiosyncratic risk is zero on average: $E[e_i]=0$. The law of large numbers. From 1 and 2 it follows that as N increases, the weighted sum of idiosyncratic risks will converge to zero: $\lim\limits_{N\to\infty}\sum\limits_{i=1}^N e_p=\lim\limits_{N\to\infty}\sum\limits_{i=1}^N w_ie_i=0$ Strictly speaking some restrictions on the weights ...


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.


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

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

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 ...


1

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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