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19

Just to be painfully clear, it only seems to make sense to consider the logarithm of returns, i.e. $X=\log (1+\frac r{100})$ for a simple return of $r\%$ in an arbitrary period because this is what sums when returns are temporally aggregated. A basic property of cumulants is that cumulants of all orders are additive under convolution, for which a proof can ...


14

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 ...


13

In addition to John's answer and just to make things clear: The arithmetic mean is given by $$\mu_a = \frac{1}{n} \sum_{i=1}^n x_i$$ The geometric mean is given by $$\mu_g = \sqrt[n]{\prod_{i=1}^n (1+x_i)} -1$$ And we have that $$\mu_g \leq \mu_a$$ So not only would the geometric sharp ratio be taking into account the "actual" return of the ...


11

I'm not sure it makes sense to think of one as more correct than another. However, they do have significant differences. It may help to distinguish between ex-post evaluation of a strategy and ex-ante prediction of what the strategy's performance will be. For simplicity, let's assume the log returns of the strategy are approximately i.i.d. univariate ...


11

Concerning adjusted price series: Free yourself from terminology and definitions, as you can clearly see, Yahoo Finance got it wrong on the stock split you linked to (and as chrisaycock correctly pointed out). You need to focus on the problem not the term people use to describe the problem: You need to adjust time series for the stock split, period. So, it ...


10

I think this is a no-brainer. Only log-returns make sense. The average return can only be computed by averaging the sum of individual log returns. Taking the average of standard (relative) returns does not give you an average of the individual returns. Consider a simple case where the value of an investment alternates between 100 and 50 an odd number of ...


10

The initial investment is the capital in the account used to support the portfolio, not the cost of the assets in the portfolio. For example, when you sell a stock or bond short, your account doesn't actually accrue any cash. Instead you start receiving a regular cash flow. There isn't necessarily a difference between these quantities in a long-only ...


10

Looking at transaction prices, they would occur at the market bid if the active part is a seller, and at the ask if the active part is a buyer. With a random flow of sellers and buyers, the price will bounce between the bid and ask prices, creating a negative autocorrelation in returns. This penomenon is known as the bid-ask bounce, and has been discussed ...


9

I think this is a no-brainer. Only log-returns make sense. The average return can only be computed by averaging the sum of individual log returns. Taking the average of standard (relative) returns does not give you an average of the individual returns. Consider a simple case where the value of an investment alternates between 100 and 50 an odd number of ...


9

Hmm, this table looks wrong. Here's what it should look like. After the most recent corporate action, the Close and Adjusted Close should be the same; only prices from before the most recent action should have a different Adjusted Close. Here's another example. I think Yahoo just has the wrong information. If you wanted to derive your own adjustments for ...


8

So those are cumulative pnl figures and you are interested in the percent changes in pnl from one data point to the next? Don't use log returns, simply generate the percent changes through r(t)/r(t-1)-1. 4.3922/5.2735-1 = -16.71% (in your example time series I made the assumption that the time series is in ascending order. Given your description of the ...


8

We actually managed to come up with the answer to this question ourselves but wanted to share the answer since it might be relevant to others as well. The calculation depends on what method is used to calculate the cost. There is the FIFO, LIFO and the average cost method, see: http://www.accounting-basics-for-students.com/fifo-method.html If FIFO or LIFO ...


8

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.


8

Usually the formula for the sample variance of a stock is given by: \begin{equation} Var(R_{i}) = E (R_t - E(R_t))^2 \end{equation} If you are using daily data to compute the variance then the second term: $E(R_t) \approx 0$, therefore you can drop it from the computation. Which yields: \begin{equation} Var(R_{i}) \approx E (R_t)^2 \end{equation} ...


7

There are many variants proposed; some useful, some not so much. As an investor, the most important thing is to compare the exact same ratio, calculated in the exact same way, for each prospect. As the prospect/fund the most important thing is to be clear about the statistic you are reporting so your investors make well informed decisions. So let's start ...


7

An easy way to perform what you need is do it this way: if your data are daily then : > prices <- data$cl > log_returns <- diff(log(prices), lag=1) would provide you with daily log returns, if you change the $lag=1$ to $lag=5$ then you will get weekly moving log returns.


7

Whether its possible? Absolutely. However, you should probably keep in mind a couple points: * Many people claim a lot while proving very little to none. This is fine if the issue is a small-talk conversation. Believe it or not, no harm done. However, this is about money, and from my experience I cannot stress enough how important it is to do a very ...


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

You cannot use the clt to test something, it is a theorem about convergence. You can only use a statistical test to test something which basis is in many cases the clt. In this case you could e.g. use a so called t-test. In R you would e.g. type: t.test(data.Rb,data.Ra) to test whether the difference in the means is significant.


6

Yes, you are correct on both terms - it doesn't make much sense, and there exists a well-cited solution by C. Israelsen: "A refinement to the Sharpe ratio and information ratio." Journal of Asset Management 5.6 (2005): 423-427. The adjustment he gives is to define $$SR_{adj} = \frac{r}{\sigma^{\frac{r}{abs(r)}}},$$ which solves the ranking problem during ...


6

What you describe is known as the Equity Premium Puzzle - and it really is, as the name says, a real enigma: "The equity premium puzzle (EPP) is a phenomenon that describes the anomalously higher historical real returns of stocks over government bonds." Source: https://www.investopedia.com/terms/e/epp.asp#ixzz5HlCdHS2Z A good first introduction can be ...


6

Based on your comments on other answers, i would like to provide you a summary on the difference of the CAPM-Alpha and Jensen's-Alpha. CAPM The CAPM is an economic model for asset pricing. It states that the equation $$E[r_i - r_f] = \beta_i E[r_m- r_f]$$ holds for any asset $i$. $r_i$ denotes the return of asset $i$, $r_f$ the risk-free rate of interest, $...


5

In my opinion you have two choices: You calculate annual returns from the daily returns that you have - I guess it is clear how. Subsequently you calculate your statistics on these $11$ data points. When I look at your comment above, this could be what you want to achieve. Then you have the ex-post statistics on your data. The drawback is that $11$ data ...


5

You will struggle to put a number on the potential returns of high-frequency trading (HFT) and I think it wouldn't make any sense anyway if you don't take into consideration its risk and its leverage. Achieving 100% return with low volatility seems highly improbable; so ask the trader in question his Sharpe ratio to start with and compare it with yours. ...


5

The study you cited seems to be exaggerating slightly. 1) "An interesting fact of returns is that all of the stock returns since 1993 are from overnight returns" -> This is simply factually incorrect. Why don't you pick the S&P 500 names, you calculate the log returns taking into account price changes from the open to the close, then you do the same ...


5

Some of the used heavy-tail distributions are: Log-Cauchy and Log-Gamma Lévy Burr and Weibull Mixed normal Here two papers that cover some of them and others: http://ect-pigorsch.mee.uni-bonn.de/data/research/papers/Financial_Economics,_Fat-tailed_Distributions.pdf http://www.rff.org/RFF/Documents/RFF-DP-11-19-REV.pdf


5

You are calculating the geometric mean as if these are arithmetic returns. If you let $$L_{t}\equiv \frac{P_{t}}{P_{t-1}}-1$$ and $$C_{t}\equiv log(P_{t})-log(P_{t-1})$$ then $$L_{t}=exp\left(C_{t}\right)-1$$ Thus, to calculate the geometric return on log returns, you would recognize that $$\prod\left(1+L_{t}\right)=exp\left(\sum C_{t}\right)$$ The ...


5

You're compounding correctly but the discrepancy is not just because of rounding. SMB and HML are formed as averages of 6 and 4 different portfolios, respectively. As French's website explains, this results from cutting all stocks into 2x3 SizexBook portfolios. French compounds each of these portfolios to the proper horizon (eg monthly) and then averages ...


5

For client reporting purposes, it is customary to use discrete returns. For backtesting, it pretty much make no difference.


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