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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 ...

5

There are sufficiently different ways to calculate the Sharpe ratio that the best advice I can give is to do whatever your boss wants. Also, if it is for a paper or research document, just make clear you document your method. My approach is usually to calculate the highest frequency Sharpe ratio I can based on the data. The higher frequency choice is to get ...

4

As indicated by @AlexC and @amdopt, the formula is exact for log returns and approximate for discrete returns. Define the factor by which a price changes as $k$ so that price tomorrow $P_{t+1}$ is the price today times $k$ : $P_{t}*k$.Then the change in the price over a business year is $$\prod_{i \in [1, 252]}{k}$$ The log of the change is by properties of ...

3

I do not have access to the exact time-series of the MSCI world, but looking at the returns from the tracking ETF, since 2001 the average return is negative. Thus regardless of the risk-free you use you will get a negative sharpe ratio.

3

I can only repeat myself because your mentioned previously asked question is essentially identical: => I would say do not include non-trading days, do not include days with zero position, do not include days where the asset did not trade for whatever other reason. Here some reasons and pointers: Sharpe measures excess returns scaled by volatility. The ...

2

Here is an example calculation according to the formula by William F. Sharpe, 1994. The OP's method of annualising the variance (as used below), is also specified by the Committee of European Securities Regulators in this document, page 5, box 1. For this example, taking 24 months of returns of risk-free proxy (US 4-week T-bills) and an example stock, (and ...

2

It makes no difference whether you work with annualized numbers or not. If you work with monthly logarithmic returns $\{r_1,r_2,\cdots,r_{12}\}$ then the return for the year is $R=r_1+r_2+\cdots+r_{12}$. Assuming only that the returns are i.i.d and the standard deviation $\sigma$ exists, then the standard deviation of the annual return $R$ is $\sqrt {12} \... 2 Your second suggestion can be interpreted as an actual return, while the first one would be a hypothetical return, given you could keep making the same gains you did for those days over the entire 3 year period. Everything else depends on your assumptions. You can take (actual days)/365, often used is actual/360 or even actual/252 if you only consider the ... 2 Yes, it mostly makes sense. The process you are outlining would give you a VaR estimate using the assumption that the returns of the cryptos are Normally distributed, and have a zero drift value. I think those assumptions are a bit of a stretch for cryptos in practice. I would multiply the variance matrix of the daily changes by 365. 365 would be the best ... 1 If you make the assumption that your returns are iid normally distributed$R_i \sim \mathcal{N}(0, 1)$, then with the second definition, and using @ZRH provided formula... $$E[DD] = \sqrt{\int_{-\infty}^{c}x^2\frac{1}{\sqrt{2 \pi}}exp(-\frac{x^2}{2}) dx }$$ So expanding this out (integration by parts) you get; $$E[DD] = \sqrt{ \left [ - x \frac{1}{\sqrt{2 ... 1 For finding the returns during a calendar year (or other period) there are two methods: a) The IRR (internal rate of return) method requires the values of the portfolio at the end of year n-1 and end of year n, and the dates and amounts of any cash additions/withdrawals from the portfolio during the year. If you put these dates and amounts into an Excel ... 1 The annuity expression a_{4}^{(12)}is written as:$$a_{4}^{(12)}= \frac{1-(1+i)^{-4}}{i^{(12)}} = \frac{i}{i^{(12)}} a_4$$where, i is the effective annual rate of interest and i^{(12)} is nominal rate of interest convertible monthly, which is equal to$$i^{(12)}=12((1+i)^{1/12}-1)$$There is no closed formula to get the interest rate, you have to ... 1 What happened to the cash in the intervening periods? Was it in a mattress somewhere or in the money market in some form? The problem to me appears less you have these gaps but you only want to count these return periods. That would make anyone you are dealing with extremely nervous. Your capital exists in between these investments and the returns on ... 1 The industry standard is to display annualized returns on a yearly basis because it is requested by most of the authorities (SEC...) and you shouldn't mix years. So you should present your returns as: 2013 : X % 2014 : X % 2015 : X % Also you should use formula 2) because formula 1) is misleading : If you tell someone that you have a 8% of yearly ... 1 Let's assume that you are manager of a fund and you have made the aforementioned investments and gains (and only those). When I ask you what was the annualized return of your fund during 2013-2015, the answer most certainly is 3.228% and not 8.037%. 1 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_*$is the return over the$...

1

I'm currently also using daily returns which I want to annualize. This is my approach: For every month, I calculate the simple return using the formula: (end-of-month closing price / beginning-of-month closing price) - 1. I use the Excel formula somproduct(geomean(A1:A12+1)-1) to find the monthly compounded return. Finally, I annualize the result of step 2 ...

1

If you assume that your monthly returns are independent from each other, then the annualized variance of each series, and the covariance can be annualized. This assumption allows you to use V(x1+X2+...+x12) = V(x1) + V(x2) + ... + V(x12) where xi is the return for the month "i". Actually, for this to happen you only need a weaker assumption: that is that ...

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