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Why do we usually model returns and not prices?

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 ...
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16 votes
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Term structure of Equity returns

Intro: Duration-Based Asset Pricing Similar to bonds, we can define the duration of stock $i$ as $$ Dur_{i,t} = \sum_{s=1}^\infty s\cdot\frac{\mathbb{E}_t[CF_{i,t+s}]e^{-s r_{i,t}}}{P_{i,t}},$$ where $...
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13 votes

Should Sharpe ratio be computed using log returns or relative returns?

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 ...
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10 votes
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Definition of Return of A Long/short Portfolio

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 ...
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10 votes
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Reasons for negative autocorrelation

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 ...
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  • 1,267
9 votes

Should I use an arithmetic or a geometric calculation for the Sharpe Ratio?

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 ...
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8 votes
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How to annualise the volatility of non-iid returns?

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 ...
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8 votes

Why do we usually model returns and not prices?

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 ...
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8 votes

What is the difference between squared returns and variance?

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 ...
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7 votes
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Risk-adjusted returns ratio that does not reward high risk for negative returns

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 ...
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6 votes

How to annualise the volatility of non-iid returns?

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 ...
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6 votes
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Using central limit theorem to test whether population average return is the same, before and after the recession

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. ...
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6 votes
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If markets are efficient, why are most returns systematically high?

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 ...
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6 votes
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When modelling ARCH/GARCH effects, do we use excess returns?

GARCH models have little to do with the economics of the data generating process of the series you model, so both returns and excess returns (and log-returns, and inflation-adjusted ones, even ones ...
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6 votes
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CAPM - Expected vs. actual returns

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 ...
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6 votes
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Convert arithmetic returns to log returns

Transmuting one to the other is pretty straightforward without the underlying sequence of prices. To go from log to simple: $R = exp(r) - 1$ To go from simple to log: $r = log(R+1)$
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  • 3,605
6 votes
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Returns and logreturns differences

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 ...
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6 votes
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Is it always better to use the entire distribution of a financial returns series, not just $\mu$ and $\sigma$?

It depends. For example, if you're doing option pricing in the log normal world returns are completely described by the mean and standard deviation. If you add jumps, you would also need to ...
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  • 7,598
6 votes

Are cumulative returns stationary?

Hi: Even if returns were stationary ( which is probably dependent on the time series one is considering ), cumulative returns, where $n$ is not fixed ( as it in say a rolling sum with a fixed window ...
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  • 1,017
5 votes
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Geometric Returns values less than -100%

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}\...
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  • 5,291
5 votes

Fama-French Data from daily to monthly returns

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 ...
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  • 111
5 votes
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Calculating log-returns across multiple securities and time

In Python, simple geometric returns: ...
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5 votes
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Sharpe ratio: discrete or continuous returns?

For client reporting purposes, it is customary to use discrete returns. For backtesting, it pretty much make no difference.
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  • 10.9k
5 votes

Proof that linear returns aggregate across securities

I think you are simply confusing percentage weights and number of assets. In your definition the initial percentage weight of the $m$ assets in the portfolio are given by $w_i^{t - 1}$ and they sum ...
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5 votes
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Predict probability of returns: How does changing volatility affect the return pdf?

I have written an entire paper on this approach at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2828744 As to your specifics 1) "Volatility" as defined by variance does not exist, which is ...
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  • 4,046
5 votes
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average return Vs cumulative return interpretation

Consider these two simple portfolios: Portfolio 1 returns -10% in month 1 and 10% in month 2. Average arithmetic return is zero, and cumulative return is $(1-10\%)(1+10\%)=0.99$. Portfolio 2 returns -...
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5 votes

Have any other factor "styles" which explain equity returns been uncovered?

A wonderful recent paper that might be of interest is Feng, Giglio, and Xiu's "Taming the Factor Zoo." First, the paper lists nearly 100 "factors" that have been proposed from 1965 through 2016. The ...
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5 votes
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How to calculate necessary gain to compensate a loss in a financial transaction?

Let x represent the percent change-e.g. 2%, let k represent the number of decreases, and z the number of increases. Something like this? We want to find z such that: $\left(1-x\right)^k\left(1+x\...
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5 votes
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Normality or Log-Normality of Regular Returns

You're right but a GBM doesn't assume that percentage returns are normally distributed. It's about log-returns. If the log-return $r_t=\ln\left(\frac{S_{t+dt}}{S_t}\right)$ is normally distributed (...
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  • 13.8k
5 votes
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Calculating Dollar-Neutral Strategy Net Return

Yes, you can just do IGE - SPY if you assume the short finances the long. The Sharpe ratio will be the same whether or not you divide by 2.
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