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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.

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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 ...

5

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 ...

4

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 ...

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.

3

Are you sure the return for two years is 0.7214? It should be 0.3422 per year if you are using 31/12/2011, and 0.3416 if you are using 01/01/2012 as the end date. Assuming the last number (because it makes for two full years, therefore easier to calculate), yes, there is a formula to derive it from the return of the individual years. It's the geometric ...

3

The PerformanceAnalytics library reflects several years worth of development by Brian Peterson and Peter Carl, as well as multiple collaborators. It is fairly widely used, tested and debugged. Basic software engineering practices suggest that you should strive to re-use it if possible. Options for that include accessing a remote R instance via RServe ...

3

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 ...

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 ...

2

2) you only take trading days for your analysis because taking in account days on which no price changes took place would shift results in a wrong direction. For exmple, you mostly take 250 trading days p.a. 3) Your time interval up to 2007 is okay and excludes the financial crisis, which is a non-normal circumstance. Therefore, your time interval can be ...

2

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 ...

2

I'm going to separate your question in two. The key thing you're asking is that how does Return.rebalancing treat your different frequencied and number of asset return and weight objects. Data munging: It subsets the first ncol(weight) columns of R (as ncol(edhec) > ncol(weights) ncol R is now 11. Checks if the first date in R is less than the first date ...

2

The coefficients assuming they are statistically significant can be interpreted whether or not the underlying portfolio is efficient. The CAPM or FF4 simply tries to decompose a portfolio into a series of linear exposures + an intercept (alpha) which can be viewed as constant added value. In mathematical terms the regression is explaining how much of ...

2

There are many ways answering this, here is one: We assume the asset price at $t=T$, $S_T = S_{T-1} \times (S_T / S_{T-1})$. Assuming continuous compounding, we can write, $S_T = S_{T-1} \times \exp(R_{T-1})$. Working the same way for the previous period, we get $S_{T} = S_{T-2} \times \exp(R_{T-1}+R_T)$. Working all the way back to the initial value of ...

2

You can't really combine the assets' log returns. You should calculate percentage returns for the three assets. Then at each time step, the portfolio's total return is: $r(i) = 0.5 \times \text{asset1_return}(i) + 0.25 \times \text{asset2_return}(i) + 0.25 \times \text{asset3_return}(i)$ Once you've calculated the time series of the portfolio's returns, ...

2

Unfortunately I don't think it's possible to compute returns purely based on yields... There are a few options: If you're on the buy side, you can easily get access to Barclay, Citi, or BofA's bond indices. These are very high quality datasets for studying historical bond returns. If you have Bloomberg, they've started providing bond indices as well. They ...

2

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 ...

1

Using a t-test should be ok because even when the underlying distribution is not normal you have a large enough sample size which justifies the assumption that the distribution of the sample means should be approximately normal due to the Central Limit Theorem.

1

If log returns have a symmetric distribution, prices will have a positively skewed distribution, since exponentiating induces positive skew.

1

Use your total wealth allocated to the trades as denominator. Total wealth allocated would include all collateral. In this way you (or your broker) make sure that the denominator is always positive. Presumably this would also reflect what you really want to track. The only problem that remains is what amount of your wealth needs to be allocated. But this is ...

1

Both approaches can be useful. For stocks, sorting into quantiles is popular because it's easy to understand and explain it's a simple matter to build factor portfolios and track or backtest their performance, while the translation from expected returns to a portfolio is a bit more involved more robust than a single-stock regression, because it is less ...

1

Given that other corporate events are reasonably modelled through regression models (compare The Detection of Earnings Manipulation I would try for using an regression approach. I believe a more recent and related paper has been published but I don't seem to find it at this time. Edit: and now I did - Earnings Manipulation and Expected Returns That said, ...

1

I found out that the upper time series is the result of a call > tail(Return.rebalancing(edhec,weights)) portfolio.returns 2009-03-31 0.005082048 2009-04-30 0.022982981 2009-05-31 0.037432398 2009-06-30 0.011107189 2009-07-31 0.025580507 2009-08-31 0.017983519 (by optical comparison. ;-) ) A glance ...

1

There are a lot of ways of doing this and what a good way of doing this will be driven by your needs as well. Criteria such as whether the method needs to be (in)sensitive to outliers and whether or not your groups need to be of the same size will influence this. One way to do this would be sorting the volatilities and group them: in groups of equal size ...

1

I don't think there is a correct answer to this question. If you're trying to study short-term correlations (e.g., to construct short-term trading signals), then 1-month or 3-month rolling correlation of daily returns is a feasible option. These short-term stock/bond correlations are quite unstable though. On the other hand, if you're studying long term ...

1

You can use both standards, but when you apply or compare this rate the standards must be equal, and it should be noted which convention you used. Note that 300/365 yeardays would in percentage be equal to 205/250 tradingdays, so its really just a convention that would make no difference in actual time.

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You can weight the returns and use them in calculations as shown below. From this site:- http://disc.sci.gsfc.nasa.gov/giovanni/additional/users-manual/G3_operation_time_series_stats.shtml The weighted mean is and the weighted standard deviation is So, making up some annualised returns for time spans, 1 yr, 3 yrs, 5 yrs & 10 yrs: r = {0.01, ...

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In your question you do not provide any reference. I believe that we are in front of two possibilities: annualized linear returns and Compound Annual Growth Rate (CAGR). If compounding is not mentioned, I would assume annualized linear returns. $n$-years Annualized linear returns $n$ = number of years $n * r_A = r_*$, where $r_*$ is the return over the ...

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Long story short, thanks to Dirk Eddelbuettel's suggestion I played a bit with rredis and indeed it offers quite a number interesting solutions. However, I still decided to start to write my own performance analytics library (albeit obviously smaller and more specific to my use case) in combination with an established Math/Stats library because I need more ...

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No, it's not. First, what you ought to be regressing are returns, not prices. Second, by interpolating you're underestimating the variance of the asset price in the interval between index price observations. Through your choice of interpolation method, you're essentially picking an arbitrary price in the middle. What you ought to be doing is maximum ...

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