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Arithmetic returns allow for easier cross-sectional aggregation and log returns allow for easier time-aggregation. The reason people use log returns (for equities) is that they are approximately invariant and hence easier to work with in estimating distributions. Meucci does better justice in describing invariance here. The basic idea (again, for equities) ...


12

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 ...


7

The correct answer has some intuition though it doesn't generalize to continuous time very easily: Think about the paper below like this: $Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y)$ The generalization is slightly hard because the dynamics of $\mu$ and $\sigma^2$ could be dependent for arbitrary returns. You can use a GMM estimator to derive the asymptotic ...


6

The answer is that it depends. In addition to the Lo paper above, there are a number of excellent references that go into depth about annualizing or time scaling non-i.i.d. returns, one of which is Roger Kauffman, "Long-Term Risk Management", 2005 which can be found at http://www.rogerkaufmann.ch/all-Budapest.pdf. There are some well known cases where the ...


6

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.


5

An example of non-overlapping one month returns: the return in January, the return in February, the return in March, etc. An example of overlapping 30 day returns: the return from January 1 to January 30, the return from January 2 to January 31, the return from January 3 to February 1, the return from January 4 to February 2, and so on. There are far fewer ...


5

The Black-Scholes-Merton (1973) model implies that the prices of the underlying asset at maturity $S_T$ are log-normally distributed $$ln(S_T)\sim N\big[ln(S_0)+(\mu-\frac{\sigma^2}{2})T,\;\sigma^2T\big]$$ so that the logarithmic returns to maturity $ln(\frac{S_T}{S_0})$ are normally distributed $$ln(\frac{S_T}{S_0})\sim N\big[(\mu-\frac{\sigma^2}{2})T, \;\...


5

Let $R$ denote the arithmetic return and $r$ the log returns. $$R=\frac{V_f-V_i}{V_i} \textrm { and } r=\ln\left(\frac{V_f}{V_i}\right)$$ Arithmetic and log returns are connected as: $$R=\frac{V_f-V_i}{V_i} =\frac{V_f}{V_i}-1$$ Hence, $R+1=\frac{V_f}{V_i}$. Taking log on both sides. $$\ln\left(\frac{V_f}{V_i}\right)=\ln(R+1) \textrm{ and } r=\ln(R+1)$$


4

In Python, simple geometric returns: import numpy as np import pandas as pd sp500 = pd.io.data.DataReader('^GSPC', 'yahoo')['Close'] simple_ret = sp500.pct_change() (1+simple_ret).cumprod()[-1] -1 0.74751768460019963 Log-returns: log_ret = np.log(1+simple_ret) np.exp(log_ret.cumsum()[-1]) -1 0.74751768460020074 In ...


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

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 ...


4

Actually, neither of your two results are quite correct. As explained in the Details for the Return.calculate function, most of the PerformanceAnalytics functions use discrete returns, not log returns. To get the correct results, you will have to convert your data from log returns to simple returns. Compare the charts from the following: charts....


4

The above relation really only approximately. If you consider arithmetic retunrs then it is exact. For the approximation you just need to look at the Taylor series of the exponential: $$ e^x = 1 + x + \text{ terms of higher order}. $$ These terms of higher order ($x^2$ and $x^3$) become small if $x$ is much smaller than one - which holds true for returns. ...


4

Large? ? The relationship between normal and log returns is $$(normal return) = exp(log return)-1$$ Therefore log-returns can be from $-\infty$ to $+\infty$ while normal ones can only be between $-1$ and $+\infty$.


4

There is no possibility to convert any two of your mentioned variables into the remaining one. For the compound and arithmetic return you can derive an inequality, but that's the best you can do. The definitions for your statements are: $$r_{\mathrm{compound}}= \prod_{t=0}^{n}{\left( 1+r_t \right)}$$ $$r_{\mathrm{arithmetic}}=\frac{1}{n} \sum_{t=0}^n{r_t}$...


3

Well, it wasn't easy because you didn't mentioned how your data is formatted. I create my own data.frame() basing on data you provided. You can skip this part if your data.frame is ready. Here's code I used to create a dataframe: > #given dates > dates=c("2000-1-3","2000-1-4","2000-1-5","2000-1-6","2000-1-7","2000-1-10","2000-1-11") > #formating ...


3

It looks like 1 and 2 are different portfolios of companies. 1 is a portfolio of dual-listed companies, and 2 is a portfolio of everything in the "market". Once you have constructed these these portfolios, let's say you put the returns for every time step into a vector, call it r, then the average return would be mean(r). You need some clarification as ...


3

Create a new price series that has a value for every minute, e.g. by carrying the last observation forward. Then compute returns from this new price series. (There are simpler approaches for this particular case, but I'd prefer the one outlined above as it is conceptually clear.) A sketch in R. (Disclosure: I am the maintainer of packages PMwR, from which ...


3

As you pointed out, not necessarily: I know that Fama-French have developed their 3 factor model using only simple returns That's because of a very common misconception: and, after all, the dependent variable in a regression needs to be iid. In fact, the dependent variable does not have to be normally distributed or i.i.d. for that matter. That ...


3

BS assumes prices NOT returns are log-normally distributed. Why making that assumption? 1.log-normal is not perfect but OK to fit potential prices distribution. 2.The nature of log-normal distribution will force the left tail to be above zero. 3. There are definitely distributions work better than log-normal in terms of fitting stock price data, but that ...


2

The high serial correlation you are getting in the first case is a spurious correlation. The correct way to do it is with returns. The price series has a unit root. You need to take diff(log(prices))) in order to have a stationary time series, on which you can then estimate autocorrelations, auto regressive coefficients, etc. properly. This was shown by ...


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

After editing my queston several times I decided to write an answer. In setting (B) we have the Ito correction term. As Gordon mentions this makes the expected value vanish. In setting (A) we introduce a positive drift of the size $\sigma^2 \Delta t/2$ even if we stay in discrete time. It is there. The following code illustrates this in R. The setting mu=...


2

The result is: $ e^{(-230%)} - 1 = -89% $


2

The document you're looking for is MSCI Index Calculation Methodology. At a very high level, MSCI indices are market-cap weighted. A crude aggregation would require that you weight the component returns by their market cap as a percentage of total market cap. For example, US currently has a 59% weighting in the index (where 59% = US market cap / world market ...


2

To my knowledge, adding a minus 1 does not transform a log normal variable into an normal distributed variable. The only thing that I can think of which make sense is the log normal represents a price ratio $\frac{P_t}{P_0}$ (for instance if the price process is a geometric browninan motion). In this case adding -1 does transform price ratio into arithmetic ...


2

This looks confused? I don't understand what you're saying in the second paragraph... Comment 1: "Best" forecast depends on what you mean by "best." Let $Y$ be a random variable and $\mathcal{F}$ be the information set. The "best" forecast depends on what the loss function is. If you're minimizing the expectation of squared loss: \begin{equation} \begin{...


2

In the colloquial sense of the word "justified," it is not justified. I will describe why it is justified mathematically and under what circumstances and in what case it is not justified. Let me begin with the simplest of equations $$\tilde{w}=R\bar{w}+\epsilon,\epsilon\sim\mathcal{N}(0,\sigma^2).$$ Let us assume that this equation is an element of our ...


2

OK, this need have nothing to do with any single sample of data. It's an inherent difference between the behaviour of linear vs logarithmic numbers. Which is what make up your respective simple and log returns, and associated averages. Imagine I offered you a bet in which you put a pound on the table, I put two down, we flipped a fair coin, and winner ...


1

All of the DMS returns are adjusting for dividends. Hence dividends are accounted in the sample. Moreover, DMS have also accounted for inflation. Hence, the real total net equity index return, now and hereafter, $e_{t}$, may be mathematically defined as \begin{equation*} e_{t}=\frac{1+\frac{P_{t}+D_{t}}{P_{t-1}}}{1+\pi_{t}}-1 \end{equation*} ...


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