15

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) ...


9

If you want to learn more about price pressure, you should look after market impact of metaorders, which is a more adequate term. Because of the microstructure (i.e. the mix of orderbboks dynamics, trading rules, participants behaviours and habits, etc), the more you buy or sell, the more you influence the price an unfavorable way (for your trades). Just ...


6

You could use the two factor model of Schwartz-Smith. It's a very standard model in commodities, where you observe this kind of long term mean reversion (where "long-term" is here around a year). It's a mean reversion model where the long-term mean reversion is itself a brownian proccess. This way you can have the desided stochasticity in the short term, ...


6

Investor preferences for higher level moments are probably most easily explained by behavioral finance. Investors' tendency to overvalue out-sized positive and negative outcomes, such as gamblers' willingness to play negative expectancy casino games, is consistent with many of the intuitions underlying Prospect Theory. There are several possible behavioral ...


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

Edited Comments: Sharpe Ratio covers both future and historical time frames (as @Freddy points out). Referencing the "Geometric Return and Portoflio Analysis", for the historical calculation, you want to make as few assumptions as possible (in my opinion). Let $m_i \triangleq$ the monthly return for period $i$ and $r_t \triangleq$ annual return, for $i\...


4

Portfolio returns are analyzed to account for risk factors only to determine what the risk factor contributed to the returns, was it the underlying assets or the skill of the portfolio manager. Fama French model explains the returns in terms of principal component such SMB and HML besides the market related returns from CAPM. These links have more detais ...


4

The most common transformations you see for these three variables on credit desks is to compute "returns" on the credit variables. So, rather than taking the straight daily differences $\Delta s_t$ of swap spreads and $\Delta H_t$ of the high yield index (by which I assume here you mean on-the-run CDX HY), practitioners will transform to $\frac{\Delta s_t}{...


4

There are a few reasons to use factor models. Most importantly, stocks tend to move together. Stated alternately, the first principal component of the securities in a domestic market tends to explain a large share of the variance. If you're concerned with multiple securities (as in portfolio optimization), then you have to account for this or you will ...


4

One economic model you could look at is the Habit model of Campbell and Cochrane (1999). The basic idea is that as the consumption of the representative investor approaches the (appropriately defined) habit level of consumption the representative investors risk aversion spikes: this means discount rates increase dramatically and we see a big drop in stock ...


3

It depends on what makes more economic sense: If you are calculating CAGR for FX (which is traded effectively 24/7) strategy returns for instance, it would seem fair to use 365.25 calendar days. If you are calculating CAGR for internal reporting of trading strategy returns on a product with 5 market sessions per week, it would seem fair to use 252 calendar ...


3

Your intuition is not exactly right. To start with often the facts that small minus big or high minus low explain the cross-section of returns is called a puzzle. It is called a puzzle precisely because there is no unifying explanation for them. It is fairly agreed among academics that the Size effect is most likely a January effect, or probably it even ...


3

Evans and Schmitz (2015) might give an answer to your question if the Fama-French factors are indeed working or not. Value, size and momentum have a long history as stock price predictors, and similar indicators have been applied to stock indices in order to predict the performance of one national index against another. Published back tests of trading ...


3

The answer depends on what model you assume for the underlying. The situation, that the underlying can become negative also occures for interest rate spreads and even for interest rates. Here some people use absolute changes, that is $X_{i} - X_{i-1}$ instead of relative changes $\frac{X_{i} - X_{i-1}}{X_{i-1}}$ or (which is almost the same as relative ...


3

I think you need to exactly define which ratio you are talking about. For example the ex-post Sharpe ratio's components are all well known. You have your realized returns, risk free returns (or whatever other benchmark you define your excess returns against) and realized volatility of returns. For realized asset returns you should not use log returns but ...


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 ...


2

Another important reason for using risk-adjusted returns is to disentangle "skill" from "risk-taking". Think of a equation for a fund's performance like: $r_{i,t}-r_f=\alpha_i+\epsilon_{i,t}$ where $\alpha_i$ gives you the average excess return of fund $i$. Alpha is often interpreted as measure of a managers' skill in timing the market and selecting ...


2

You can compute daily gross returns and then simply multiply them: $R_w=\prod_i \left(1+R_i \right)-1$, where $R_i=P_{close,i}/P_{close,i-1}-1$


2

If you do step 1 and step 2 every day, then you indeed assume that you rebalance the strategy every day. If you want to assume differently, for example monthly, you need to first compound the returns for each asset separately during the whole month and then do a weighted sum of the compounded returns using the weights of each asset at the beginning of the ...


2

It is just partial answer to your question. The Fama and French three factor model can be written as: $$R_{it}=\beta_{im}R_{Mt}+ \beta_{iSMB}SMB_t+\beta_{iHML}HML_t + e_{it}$$ In this model the market index is supposed to capture systematic risk originating from macroeconomic factors. Whereas, SMB and HML are firm specific variables and are chosen ...


2

That's a quite interesting problem, a few thoughts on how to attack it: Calculate the correlation and beta between the benchmark and the fund. If the above imply a link between these two then proceed with the betas' comparison. Regarding the three approaches you mention, the one which subtracts the betas sounds mathematically-speaking wrong since beta is a ...


2

Although I agree with jd8 answer, practical implementation issues may arise. Here I suggest a parsimonious engineering solution relying on economic intuition of Habit model of Campbell and Cochrane (1999). 1 – Assume time varying mean and standard deviation in standard GBM dynamics. 2 – Use “drawdown” as an observable variable for measuring risk aversion. ...


2

I just wrote two papers on a related topic. Let us not use the log method right now as it was originally intended as an approximation from the time we used punch cards. You can, but we will come back to why you may not want to. If we begin with a simple AR(1) model $$x_{t+1}=\beta{x}_t+\varepsilon_{t+1},$$ then we know that we are buying assets with the ...


2

They are "essentially" the same thing. IPP (or excess IRR) is the excess return over the annualized benchmark such that the adjusted PME is 1: $$\text{PME} = 1 = \frac{\sum_{i=1}^n d_i \left(1 + \dfrac{b_{T_i, T_N}}{q} + \dfrac{r}{q}\right)^{q(T_N-T_i)} }{\sum_{j=1}^m c_i \left(1 + \dfrac{b_{T_j, T_N}}{q} + \dfrac{r}{q}\right)^{q(T_N-T_j)}}, $$ ...


2

No. I just published a paper on this. If return is defined as $$r_t=\frac{p_{t+1}q_{t+1}}{p_tq_t},$$ and since returns are not data while prices and volumes are, then it follows that the distribution of returns depends entirely upon the distribution of prices and the distribution of quantities. For example, in bankruptcy, $q_{t+1}=0$. As it is a very ...


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

In a nutshell, this is the "variance drag" problem. The mechanics of how you short something matter, and it's relevant to the discussion of levered/inverse ETFs that behave differently from classic/vanilla positions. Consider an XYZ future at 100. A day later it's 1% up, at 101. Two days later, it's up 1% again, at 102.1. If I go long, I make 2.1 profit. ...


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*} ...


1

If you are not interested in correlations, etc. I'd just make a column for each rebalance date, let's say you want to rebalance at the end of each year for 50 years, you'd have 50 columns, and a row for each number of simulations. Let's say 10,000 (each row is going to be the same, but rand() will make the result different for each. The first cell would be ...


1

You will need a 'pseudo' random number generator - most stats programming languages have them (Matlab, R, Python...). But GBM is defined with Normal increments $N(0,\sigma^{2}(T-t))$ so I dont think using Student's t distribution is a good idea, never seen it in any literature/applications. It is however used for instance in GARCH modelling.... Random ...


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