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6

Basically, prices usually have a unit root, while returns can be assumed to be stationary. This is also called order of integration, a unit root means integrated of order 1, I(1), while stationary is order 0, I(0). Time series that are stationary have a lot of convenient properties for analysis. When a time series is non-stationary, then that means the ...


4

Perhaps overly simplistic and repeating the pt above, but when doing statistics, ideally we want to compare like with like. Returns can be comparable with each other. Prices on the other hand always depend on the previous price.


4

Surely, there is; search for aggregational gaussianity in Google Scholar or ScienceDirect. In fact, 5 minutes returns are leptokurtic and fat-tailed; then as you increase timeframe, returns become more and more normal. Yearly data is almost normal, if you have enough points.


4

Based on Ito's isometry, \begin{align*} E_t (r^2_{t+1}) &= E_t \bigg(\int_t^{t+1} \sigma_s dW_s \int_t^{t+1} \sigma_s dW_s\bigg)\\ &= E_t \bigg(\int_t^{t+1} \sigma_{\tau}^2 \,d\tau\bigg) \\ &= E_t\bigg(\int_0^1 \sigma_{\tau+t}^2 \,d\tau\bigg) \\ &=\int_0^1 E_t\big(\sigma_{\tau+t}^2\big) \,d\tau. \end{align*} The identity \begin{align*} E_t ...


4

If you wanted to see the following (price $S_t$, log return $r$, simple return $R$) then $$ r = \log(S_{t+1}) - \log(S_t) = \log(S_{t+1}/S_{t}), $$ and $$ R = S_{t+1}/S_{t}-1, $$ thus $$ R = \exp(r)-1 $$ and $$r = \log(1+R).$$ Was this the question?


4

No there is no way since the calculated internal rate of return $r$ is by definition defined as: $0 = \sum_{i=0}^{I} \frac{C_{i}}{(1+r)^{i}} $ You need to know the entire cash flow distribution and its timing if you want to compute the Pooled IRR. One advantage of IRR is that it takes the irregular timings of cash flows into account, logically its ...


3

Just a bit of illustration added to @John's answer. Look at log prices $\log(P_t)$, assume that you know $P_0$ then $$ \log(P_t) = \log(P_0) + r_1 + \cdots r_t $$ where $r_i = \log(P_i)-\log(P_{i-1})$ are the log returns. By modelling the log-returns (which as already said take values on the whole real line which is a nice property for modelling) we model ...


3

Perhaps this works : http://en.wikipedia.org/wiki/Geometric_standard_deviation In particular, see under "Derivation"


3

My main reference will be "Dan Xu, Christian Beck - Transition from lognormal to chi-square superstatistics for financial time series" Non-equilibrium statistical mechanics (more specifically, superstatistics) gives some ideas of explaining the relation between time frame and its distribution: "...to regard the time series as a superposition of local ...


3

SMB is controlling for small stocks. Small and thinly traded are not equivalent. For instance, for most of its history, Berkshire Hathaway was a large stock, but thinly traded (b/c of its high price). There are a number of ways to handle liquidity risk. If you're looking to supplement a Fama-French regression, Pastor and Stambaugh (2003) uses order flow ...


3

The dummy function is always used to construct non-linear models. In your model, it is interpreted that the announcements have an non-linear effect on the return. So it is incorrect to say it is a linear regression problem, it should be called as a non-linear regression problem. In total, it means the announcements have asymmetric effects in explaining the ...


3

Exact solution: Assume we agree that for $y_1:=IRR(CF1)$, $y_2:=IRR(CF2)$, $y:=IRR(CF1+CF2)$, the following equations hold by definition: $$-1000+\frac{100}{1+y_1}+\frac{100}{(1+y_1)^2}+\frac{1100}{(1+y_1)^3}=0$$ $$-200+\frac{20}{1+y_2}+\frac{30}{(1+y_2)^2}+\frac{1}{(1+y_2)^3}=0$$ $$-1200+\frac{120}{1+y}+\frac{130}{(1+y)^2}+\frac{1001}{(1+y)^3}=0$$ These ...


2

You are doing it right. The differences are rounding issues and can be safely ignored for any practical purpose.


2

In practice, for heavily traded assets (above 60% quantile of average daily dollar volume), individual asset return is pretty scalable across different time frame by a factor of $\sqrt{T}$. However, for covariance among different assets, moving between different time frame is not linearly scalable (although it should be in math). This is known as "Epps ...


2

The simple answer is that when you calculate the value weighted return at time $t$ all you really need is the return during time $t$ and the market-capitalization weight as of $t-1$. You can filter the securities to remove the missing ones (and others that you may remove for other reasons, e.g. too small or price too low), calculate weights based on the ...


2

When position = 1, then you are long the S&P ETF. When position is -1, your portfolio consist of a short position of -1 S&P ETF. You will therefore have a vector like $Pos = (1,1,1,1,1,-1,-1,-1,-1,1,1,1,-1,-1,-1, \ldots)$, that will give you the evolution of your portfolio. Your returns are then the daily returns on the S&P multiplied by your ...


2

Yes, that can be really sophisticated even using such nice tools as pandas. But the basic idea is to find position enters & exits to derive cashflow. Here is my code to derive all that stuff from generated signals (in my backtester signals are fractions of 2 stocks in portfolio for each moment). I hope I've found all bugs here, but no warranties. ...


2

The correct formula is to compute multi period gross returns as products of single period gross returns. Conceptually it is equivalent to calculating the return on a self-financing portfolio initially made of 1 unit of stock, with each cash dividend reinvested in more stocks at the ex-dividend price.


2

To answer you correctly we'd need to see the exact inputs of your regression... and I doubt you can mix easily linear and binary variables like that. If the market return is 1% at time $t$ do you have $R_{m,t} = 0.01$ or $R_{m,t} = 1$. Same question for $R_t$ Assuming both are using the "0.01" convention, then a move of $1\% = 0.01$ results in a move of ...


2

Is this for one firm only? Is there positive and negative announcements (ie do the abnormal returns differ in sign)? As per Binder (1998): $$R_{it}=\alpha _{i} + \beta _{i}R_{mt} + \gamma _{i}D_{i} + u_{it} $$ where the coefficient $\gamma _{i}$ is the abnormal return for security $i$ during period $t$. If the events tend to affect the security prices both ...


2

Apply your trading strategy to history. Convert account equity to USD by applying historical USD/JPY rates. Calculate VAR/returns as usual. Remember, VAR calculated in such a way will underestimate the impact of extreme events: i.e. 95% VAR will return you minimum of what you can lose on 5% trading days.


2

Your opinion is correct. There is simply more information about risk-reward encoded in the Sharpe ratio than cumulative returns. The other thing that's important to know is that whatever ratio you choose is simply a social construct or conventional benchmark that people use to compare between each other. The ratio is only useful insofar as other people are ...


2

As far as your second model concerned: Abnormal returns for good news is $\beta_4$ The t-value of 3 tells it is significantly different from 0 The model does not account for effect of bad news so the effect of bad news will mostly be found in spikes in residuals around time of bad news releases. $\beta_0$ is return when all other factors in the model ...


2

Computing returns is one of the first things you learn when you start studying finance but I believe it's one the trickiest one once you get to complicated cases. The source you mentioned seems actually very good to me and it already takes into account different approaches and different subtleties like dividend payment. But this is in fact only the top of ...


2

First I thought about voting to close this question as it deals with Matlab synthax a lot. I ignore the Matlab stuff. You have 5-minutes data. So an estmator of volatility over any sample of size $N$ (e.g. 100) will be an estimator of the vol of your 5-min returns. Usually volatility is quotes as "per annum" or "pa". This means that using the square root of ...


1

There's no industry standard for calculating returns on derivative contracts. The reason is that derivative contract assets are different from what I'll call real assets. Examples of real assets are an ounce of gold, or an equity. Derivatives require you to invest, or allocate some amount of cash (i.e. margin), to accommodate losses. But this allocation is ...


1

$\sqrt{12}$ annualizes monthly deviations. But I don't understand why you measure tracking error with stdev. It should be $$ ATE = \sqrt{\frac{12}{36}\sum_{i=1}^{36}(r_{b,i}-r_{t,i})^2}$$ where $r_{b,i}$ is benchmark return for month $i$ and $r_{t,i}$ is tracking portfolio return for same period. So you shouldn't substract average error inside square.


1

I would definitely want to mark to market, instead of reporting realized + open p&l. Because the time ordering doesn't really affect the p&l, just the breakdown between realized and open. Which I don't think matters to most people. To do this you need a market price, to mark your open position at. In an intraday context you could use the last traded ...


1

This paper states that heteroskedasticity is a stylized fact in daily as well as intra-day returns: https://statistik.econ.kit.edu/download/doc_secure1/HandbookITandFinan.pdf


1

If high frequency returns are iid and the mean and variance are finite and vthe variance is greater than zero then the Central Limit theorem holds Then, regardless of the distribution of the high returns, when aggregated over time the aggregated returns will tend in distribution to a Normal distribution. The Lindeberg-Lévy-Feller version of the Central Limit ...



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