of return distributions. AnnStdDev (r 1, ..., r n) = StdDev (r 1, ..., r n) *. of Quarterly ROR) X SQRT (4) Note: Multiplying monthly Standard Deviation by the SQRT (12) is an industry standard method of approximating annualized Standard Deviations of Monthly Returns. Here is where we annualize the result. introduces a bias. (Obviously, neither P1 or P2 are normally distributed. ) The point about “comparing like with like” is what I am curious about, as there really is no relationship between a composite’s 3-year annualized return and its 3-year annualized standard deviation. 4 quarters objective is to understand why the standard deviation of a sample mean has a square root of n in the denominator. mathematically invalid procedure. I am not familiar with the notion of taking the number of observations into consideration, and don’t necessarily think it’s “the best way.” I do not know where Carl got this from; would have to review this part of his book to see if he cites something or if it’s his own creation. CFA Institute, Kaplan Best wishes, Assuming a Weiner process governs stock prices, variance is proportional to time. Perhaps I’m missing something. Just don’t try to compare that figure to the 36-month annualized returns! Dave. Variance also measures the amount of variation or dispersion of a set of data values from the mean. We cannot lose sight of the fact that standard deviation, within the context of GIPS compliance, serves two purposes: Let’s consider what I propose as answers to the above questions: The annualized standard deviation, like the non-annualized, presents a measure of volatility. If you want to transform it to annual volatility, you multiply it by the square root of the number of trading days per year. But, is it worth the effort to do something else? Dev. The units of Sharpe ratio are 'per square root time', that is, if you measure the mean and standard deviation based on trading days, the units are 'per square root (trading) day'. Sharpe ratios or estimates of them for arbitrary trailing periods are commonly used. The challenge that our Performance Measurement Think Tank member brought up was the same as I did in my article: can we in any way look at the distribution of returns for the 36-month period and relate them to the annualized standard deviation, as we do with dispersion, and the answer is “no.” But a bigger question: would we want to? 5 Year Annualized Standard Deviation. for calculating the annualized volatility measure rather than to opt for an expedient but The Spaulding Group. I've got a daily returns from 01.01 till 28.10 (or 10.28 for US standards) I would like to know how to annualize my standard deviation. method and presents two alternative measures of return volatility in which multiplying by This is why having the 3-year annualized return along with the 36-month standard deviation is desirable, since it makes this return to risk estimate even less “rough”. Analytics help us understand how the site is used, and which pages are the most popular. To present this volatility in annualized terms, we simply need to multiply our daily standard deviation by the square root of 252. Multiplying by the Square Root of Twelve to calculate annual standard deviation. annualized standard deviation. Using √12 for monthly or √4 for quarter has been done for decades, I believe. 3) Volatility is the measure that connects geometric average returns to arithmetic average returns. if you are annualizing monthly returns, you would multiply by square root of 12 since there are 12 months in one year. Why do we annualised risk is a good question. Annualized Standard Deviation. Learn more in our, What’s Wrong with Multiplying by the Square Root of The author illustrates the bias introduced by using this approach rather than the correct Winter CORRELATIONS FTSE100 SSE STOXX50 SP500 FTSE100 1 SSE 0.296528609 1 STOXX50 0.930235794 0.296123 3 1 SP500 0.704737525 0.250767 … Annualized standard deviation: Why? It’s just the number of observations in the annual period. Forcing consistency has benefits, no doubt; but with no explanatory power, there’s something lacking. Hopefully, not days, as they’re TOO NOISY. But, perhaps we can. You have multiplied by √12 .. In finance, volatility (usually denoted by σ) is the degree of variation of a trading price series over time, usually measured by the standard deviation of logarithmic returns.. What is your view? Here, 252 is the number of trading days in a year. It argues that the relationship between time and volatility, as measured by the standard deviation, increases with the “square root of time”. Nitin Let me try and give you an intuitive, though partial, explanation. So you would scale a Sharpe Ratio by multiplying by t/√t = √t, where t is the frequency you are annualizing from. The area is most undoubted worthy of some academic (or near-academic) research, to demonstrate this and to identify the appropriate methodology. 52 weeks of monthly returns rather than a sum of monthly returns. Dave. Paul, “flaky” may, in deed, be an appropriate term for this method. JAN options expire in 22 days, that would indicate that standard deviation … formula that uses monthly standard deviation and monthly average return to calculate Technically to do it all we have to assume that the returns are independent of each other – actually we know they are not so the calculation itself (multiplying by the square root of periodicity) is not valid. That is fine if all the potential client is doing is comparing risk to a benchmark, but not sufficient if the potential client wants to get a rough idea of the return to risk trade-off that is characteristic of the portfolio. rather than level returns because annual logarithmic return is the sum of its monthly Is annualised σ a valid measure in this situation? Thanks for your comments. Suppose you have a stock which you know is varying up or down by 12% per year. But is there really anything to be gained from comparing them? Thanks! For example, to get to 'per root … deviation of monthly returns by the square root of 12 to get annualized standard deviation Formula. Let me try and give you an intuitive, though partial, explanation. Again, I am not aware of any. We’re using cookies, but you can turn them off in Privacy Settings. Granted, there are some (e.g., Paul Kaplan of Morningstar) who soundly dismiss this approach, as it only applies to an arithmetic, not geometric, series. In extreme situations you might go over 100% in ex post as well. D. But how can you equate say 24 observations in a month with 12 observations in a year as per GIPS by just multiplying both by SQRT 12? “Of course, he added, if you are using weekly returns you have to multiply by the square root of 52 and if you are using monthly data you should multiply by the square root of 12. (The first equality is due to independence, the second is due to identical distributions.) I’m not sure how seriously I take someone with a nom de plume of “Whacko,Jacko,” but I will trust that the person behind it has at least some knowledge in this area; and no doubt, you are correct. Contrast this with what we do with risk, where we’re measuring standard deviation of 36 monthly returns. If you are using daily data: Compute the daily returns of the asset, Compute the standard deviation of these returns, Multiply the standard deviation by the square root of 260 (because there are about 260 business days in a year). The annual return for P1 is 12.7 while the annual return for P2 is 11.0. Thus, multiplying the standard deviation of monthly returns by the square root of 12 to get annualized standard deviation cannot be correct. Copyright 2018-2019. Ultimately, the best case would be to have the non-annualized standard deviation for a statistically significant number of annual returns rather than monthly. Historic volatility measures a time series of past market prices. David, 250 is a ‘sort of’ accepted standard for the number of business days in a year. This assumption has been shown to be inaccurate and therefore introduces error into the number. series with a standard deviation of 6%. Standard deviation, a commonly used measure of return volatility in annualized terms, is To demonstrate the extent of bias in the annual measure of standard deviation obtained by To "scale" the daily standard deviation to a monthly standard deviation, we multiply it not by 20 but by the square root of 20. CFA Institute does not endorse, promote or warrant the accuracy or quality of The Spaulding Group, Inc. GIPS® is a registered trademark owned by CFA Institute. Expect to see you in Boston! What meaning does this provide? The author suggests Fundamentals of Investment Performance Measurement, Performance Measurement for the Non-Performance Professional, PERFORMANCE MEASUREMENT FOR ASSET OWNERS AND CONSULTANTS, Past Articles of The Journal of Performance Measurement. Portfolio managers, performance analysts, and investment consultants commonly use standard alternative measure of return volatility involves estimating the logarithmic monthly And so, I’ve done that above. be annualized by multiplying by the square root of 12 without introducing any bias. It’s a very well established market standard – we all do it – but to repeat technically we have to assume returns are independent and we know they are not – so we shouldn’t really, Thanks, Carl. Applying Einstein's formula for annualized standard deviation to monthly return numbers At the risk of saying the obvious, if we expressed everything is variance terms, and we want to convert from monthly to annual, we would simply multiply by 12. Because an annual logarithmic return is This area needs a bit of clarification of terms and calculations, both Ex-Post and Ex-Ante. Suppose you have a stock which you know is varying up or down by 12% per year. Whacko (I agree their name lacks instant credibility) is correct in their logic for why the numbers are multiplied by the square root of 12. The Annualized Standard Deviation is the standard deviation multiplied by the square root of the number of periods in one year. This is discussed in your textbook as part of your supplementary readings. The author presents two alternative measures of return volatility whose monthly values can I think not. Daily volatility = √(∑ (P av – P i ) 2 / n) Step 7: Next, the annualized volatility formula is calculated by multiplying the daily volatility by the square root of 252. I agree with Carl, too, on the his points. I realize I am putting aside the non-normal distribution of returns because standard deviation is still the most widely used measure and I have not yet seen a viable, better alternative. Can we make any similar assessment using the annualized standard deviation? And I recall someone suggesting that firms should also display their 36-month annualized return along with it. The annualization factor is the square root of however many periods exist during a year. Read the Privacy Policy to learn how this information is used. first alternative measure is to sum monthly logarithmic return relatives (i.e., returns plus To be consistently wrong is not a good thing. A graph of direct versus estimated logarithmic standard deviation shows less than P This site uses functional cookies and external scripts to improve your experience. The composite’s non-annualized standard deviation, like the annualized, is lower, so we interpret this to mean that less risk was taken. If you want a mathematical proof the guys above did a great job in little space. I am exploring Paul’s argument in greater depth, and may report on it in a future post, newsletter, and/or article. I wish that there were a way to provide those over economically significant time periods rather than trailing time periods, but I haven’t thought or heard of a good way to identify those significant time periods and have everyone agree with them or have a pre-defined way of identifying them. There is no relation between the annualized standard deviation and the annualized return. Not sure this application does, either. But how does one do that with standard deviation? This assumes there are 252 trading days in a given year. where r 1, ..., r n is a return series, i.e., a sequence of returns for n time periods. Carl is also correct that there is an assumption of no serial correlation in the returns if you convert monthly to annual. The 36 months in GIPS as I see it can be treated as √250/36 or √250/375. Right. You can then annualise σ or VaR (makes no difference which) by * t ^(1/2). This now gives a whopping VaR of $52,019. Formula: (Std. Volume 43 The next chart compares those two lines to the theoretical result which takes the annualized standard deviation of the S&P 500 daily returns from 1950 to 2014 and divides it by the square root of time. What’s Wrong with Multiplying by the Square Root of However, that long of a track record would exclude many products. All fine and roughly comparable to an historical VaR calculation. FTSE100 SSE STOXX50 SP500 volatility 0.020023365 0.013795 8 0.0220276 1 0.0241014 9 The correlations are provided below. Standard deviation is associated with a normal distribution; we typically require at least 30 values in our distribution to have any statistical significance, so the 36 monthly returns meet and exceed this level. A plot of monthly average return versus the Calculating “annualized” standard deviation from monthly returns and the different month lengths. While the standard deviation scales with the square root of time, this is not the case for the variance. Mark Kritzman from State Street quantified what he referred to as interval error at a recent conference that I attended (https://northinfo.com/documents/738.pdf). We square the difference of the x's from the mean because the Euclidean distance proportional to the square root of the degrees of freedom (number of x's, in a population measure) is the best measure of dispersion. for 1,824 Canadian open-end funds for the 60-month period from November 2007 to October Further discussion, perhaps in person, or perhaps over dinner, would be worthwhile! (Stock price) x (Annualized Implied Volatility) x (Square Root of [days to expiration / 365]) = 1 standard deviation. When provided, the annualized standard deviation it is provided along with calendar year returns (so annual returns) for all managers. What does it mean? I’m not sure: it’s probably worth some discussion. the square root of 12 is appropriate to annualize the monthly measure. That is, when the x's have zero mean $\mu = 0$: I did a post some time ago about a vendor we encountered who annualizes rates of return using trade days: I came up with 10 reasons why this made no sense. asymmetrical nature of return distributions. What conclusion could we draw? I tried to address this by saying that unlike dispersion, where the distribution of returns relative to its mean has some value, volatility is quite different. ±1% difference between the two values for 96% of the funds, which validates the If we then convert this to a standard deviation, we would take the square root of the variance. Please chime in! This means that the standard deviation of 12 months of returns is smaller than the annualized standard deviation of 12 months of returns. “That’s simply an annualized standard deviation. I would very much like to see other views on this. Yes, we can argue that it’s flawed, for one reason or another. Comparing the annualized standard deviation values with their respective non-annualized, do you have any different interpretation? However, I learned that when you annualize monthly stock returns, you multiply the average monthly stock return by 12 to get the yearly stock return, and to get from the volatility (standard deviation) of the monthly stock return to a yearly stock return volatility you would have to multiply by the square root … What meaning do you draw from them? Once again, you need to consider they ‘why’ of providing standard deviation/variance (which has it’s roots in the sum of squared errors (SSE)). And while Bill Sharpe used non-annualized values in his eponymously named risk-adjusted measure, it is quite common to employ annualized values, and so, the annualized standard deviation would be plugged into the denominator. If I say that the average male height is 5.5 feet in some country and you say it is 66 inches, we are both saying the same thing. I forgot to mention that I do recognize that many would not believe that using the 36 month annualized standard deviation and the annual returns to get a rough idea of return to risk profile is a valid measure of return to risk, and I agree. it is important for asset managers to encourage the use of mathematically sound procedures Parametric VaR 95% would be 1.645*2%=3.29% or $3,250 for a $100,000 position. Multiply the standard deviation by the square root of 260 (because there are about 260 business days in a year). Thanks, and thanks for sharing the paper for Mark (I’ll review it when I return home from Vienna); we may reach out to see if he’d like to speak at PMAR next year. 01 Jan Standard deviation is the square root of variance, or the square root of the average squared deviation from the mean (see Calculating Variance and Standard Deviation in 4 Easy Steps ). Multiplying a series of monthly standard deviations by the square root of 12 (i.e., the square root of time) is quite standard. standard deviation obtained from multiplying the monthly measure by the square root of 12 If a non-annualized standard deviation of 36 monthly returns is provided, we have the standard deviation scaled to a one month return rather than scaled to an annual return. To be consistent with the scale for returns and to be consistent across firms, it makes sense to annualize standard deviations. Privacy Settings, CFA Institute Journal Review 1. In fact, it's more like: (Annual Standard Deviation)/Square-root-of-10 = 20.2/SQRT(10) = 6.4% >Aaah. 1) Annualization is a way of standardizing on a measure to make comparisons easier. An project worthy of someone’s (es’) time. I can’t address everything right now, but will at least touch on a bit of it. constituents, thus making multiplication by the square root of 12 appropriate. 12 months Dev. 2) Please define what test for significance you are using for saying that less than 30 observations are not significant. Is there an intuitive explanation for why … With annual returns N=5 We then calculated the Standard Deviation of those returns and multiply that by the Square Root of N Years. Depending on weekends and public holidays, this number will vary between 250 and 260. Yet we all do it – and to the extend we all do it consistently it’s probably OK – at least we are comparing like with like. Therefore, to some extent, volatility and standard deviation are the same, but… Why Volatility Is Not the Same as Standard Deviation. Ask Question ... Browse other questions tagged standard-deviation or ask your own question. Joshi. The standard deviation of this data set equals the daily volatility, which is 4.18%. Annualizing 7% yields 24.2%. I guess we do it because we tend to use annualised returns and therefore it makes sense to use annualised risk, Carl, Mean = 0 Standard Deviation = 1 Standard Deviation (N) = Annualized Standard Deviation/ sqrt (252/N) Where N is the N th day of the simulation. In principle, this rule only applies to the normal case, i.e. Note: recall that we are measuring the dispersion of annual returns within the context of GIPS’s dispersion; we aren’t annualizing a monthly standard deviation: the standard deviation is of annualized returns. That is because the standard deviation is defined as the square root of the variance. To summarize, Monthly Sharpe Ratios are annualized by multiplying by √12 I have spoken to others since and multiplying by SQRT12 has become a sort of industry standard. Vol. I think the key question remains: can we draw any different conclusions by comparing the composite and benchmark’s annualized standard deviations as we do with their non-annualized? The Annualized Monthly Standard Deviation is an approximation of the annual standard deviation. Consider the following: How do you interpret the annualized standard deviations? Given that it is only a linear transformation, you would not expect to draw any conclusions different than what would have been drawn from the comparison portfolio to benchmark monthly standard deviations. How does one compare them? Otherwise, you are agreeing to our use of cookies. I appreciate your rather detailed response. cannot be correct. If you continue to browse the site, it indicates you accept our use of cookies. Let’s say we have 5 years of returns as in the question posted above. 17 I am seeking to confirm that I have correctly calculated Tobin's formula for determining annualized standard deviation based on a series of monthly returns. What for? Of, perhaps one might suggest we compare it against the most recent one year period’s return. No. The bias from this approach is a function of the average monthly return as well as the standard deviation. Using an online standard deviation calculator or Excel function =STDEV (), you can find that the standard deviation of the data set is 1.58%. I think the comparison is solely between the composite’s and benchmark’s 3-year standard deviation, and whether that number is annualized or not, the comparison will be the same: that is, they will maintain their relative size differences (this is, I believe, a mathematical certainty). Most investment firms, for example, consistently use TWRR to calculate sub-portfolio return; however, in my view, as well as that of a growing number of more enlightened folks, IRR (MWRR) should be used. obtained by multiplying the standard deviation of monthly returns by the square root of 12. Twelve deviation of monthly returns is to multiply it by the square root of 12. The current Implied Volatility is 31.6%. of Quarterly ROR) X SQRT (4) Note: Multiplying monthly Standard Deviation by the SQRT (12) is an industry standard method of approximating annualized Standard Deviations of Monthly Returns. Take for example AAPL that is trading at $323.62 this morning. This speaks to your point about Mathematicians and their arguments, though I think statisticians are probably more appropriate critics. Since volatility is proportional to the square root of time, we next convert the annualized standard deviation of 40 into a weekly volatility by dividing it via the square root of time. Both have an average return of 1% per month. If you then said that the standard deviation was 6 inches and I said it was .5 feet, again we would be saying the same thing but both be internally inconsistent in our measurements. Since variance is an additive function, it is a simple transformation. (This is one reason why most risk attribution will look at contribution to tracking variance as compared to contribution to tracking error.) Twelve, Ethics for the Investment Management Profession, Code of Ethics and Standards of Professional Conduct, What’s Wrong with Multiplying by the Square Root of That it ’ s makeup take up, I probably would not have tried to understand the “ ”! Volatility, which are necessary for basic site functionality like keeping you logged in, are always.!, annualized standard deviation can be multiplied by the square root of N the! The benchmark, we can argue that it ’ s flawed, one... Or annualized standard deviation why square root benefits, no doubt ; but with no explanatory power, there ’ s simply an annualized deviation! And even though returns are not usually normally distributed, they ’ re enough. Did a great job in little space basis in GIPS for doing and... Th day of the average monthly return because of the variance of the simulation are trying to provide a to. The logarithmic monthly standard deviation of this data set equals the monthly deviation!, 252 is the N th day of the distribution might go over 100 % in post! Same time everything right now, but I applaud that annualized standard deviation why square root decision was to! Deviation you multiple the standard deviation can not be correct would be less right... This speaks to your point about mathematicians and their arguments, though I statisticians... I believe implied volatility looks forward in time, being derived from the mean value multiple standard! Extreme biases at extreme average returns to arithmetic average returns to arithmetic average returns reflect the asymmetrical nature return. A market-traded derivative ( in particular, an option ) how to re-express ratio... Some academic ( or near-academic ) research, to demonstrate this and the 3rd edition 2012 GIPS provides! The daily volatility or standard deviation = … annualized standard deviation, we ’ re enough! Touch on a bit annualized standard deviation why square root it without the article difference is directly related to the mean standard... And roughly comparable to an historical VaR calculation 30 observations are not significant the., if standard deviation values with their respective non-annualized, do you have any different interpretation % ie than. Is now 130 % ie more than your position someone ’ s say we have 5 of! Would take the square root of time like so: > so volatility. Of annualized standard deviation of return equals the monthly standard deviation just published monthly... Deviation can be multiplied by the square root of ( 12 ) ) is one. Like so: > so the volatility would be worthwhile disagree that there is no point to annualizing the deviation. Has a lower value than the benchmark, we would take the square root of 12 to obtain annualized! Explanatory power, there ’ s the point in annualizing it in context... When compared to contribution to tracking error. to re-express Sharpe ratio by multiplying by the square root the... Get annualized standard deviation of daily returns were 2 % =3.29 % or $ 3,250 for $. Following: how do you have any different interpretation so, I believe +/- one standard deviation, but applaud! A loss greater than 100 % in ex post as well therefore introduces error into the number of days... Good thing no difference which ) by * t ^ ( 1/2 ) basis... At the same time now gives a whopping VaR of $ 52,019..., r N =... The same time to do something else loss greater than 100 % would be less,?. Reason or another but what if it ’ s something we ’ re measuring standard deviation not. That potential clients do return times the square root of 252 ( * SQRT ( )... To time is 7 % … standard deviations proof the guys above did a great job in little.. A $ 100,000 position and then annualize the result can be multiplied by the square root of ( 12.. Contrast this with what we do with risk, too, at PMAR 2018 average returns difference )! It makes sense to annualize standard deviations a decision was made to consistency! Also display their 36-month annualized returns ( over 10 years ) look like so: > the. There ’ s return different units or another logic behind this is dead flaky appropriate methodology questions at the time! ) ) is just one example Browse the site is used you convert to! And 260 for arbitrary trailing periods are commonly used much like to see other views on to! ( often just called volatility ) the N th day of the variance published our monthly newsletter a! Gips handbook provides no examples I can ’ t address everything right now, am! In annualized terms as a measure of return volatility ( N ) = annualized standard SQRT. √T, where t is the square root of 252 very much like see... There is an additive function, it is something that potential clients do your experience ’ m sure. Use standard deviation is proportional to time there an intuitive explanation for why … that is at! In different units and therefore introduces error into the number of periods in one year period ’ something... Have any different interpretation % =3.29 % or $ 3,250 for a statistically significant number business! What ’ s probably worth some discussion * SQRT ( 252/N ) where is. Necessary for basic site functionality like keeping you logged in, are always.... ( so annual returns so both comparisons could be made take up, I suspected it be... 3Rd edition 2012 GIPS handbook provides no examples I can see use of cookies monthly... $ 100,000 position be less, right? ) P2 are normally distributed and independent from one another like:! Vary between 250 and 260 risk was taken that the standard deviation this... Vary between 250 and 260 multiple the standard deviation ( N ).! The amount of variation or dispersion of a sample mean has a square root of N the... “ flaky ” may, in deed, be an appropriate term for this method 2! At the same time volatility, which is 15.87 you logged in, are always enabled distributions... Get annualized standard deviation scales with the scale for returns and the different month lengths scaled standard.... Sake ; the annualization factor is the frequency you are correct, in order to get an standard... Other views on this to get annualized standard deviation display their 36-month annualized returns into the of! Day of the average monthly return because of the average monthly return because of the average monthly return as as. Against the most popular standard deviations why the standard deviation is defined as the standard curve! Would be 1.645 * 2 %, the variance no serial correlation in returns! Annual return is a ‘ sort of industry standard that less than 30 observations are not significant derived the... Return volatility involves estimating the logarithmic monthly standard deviation scales with the scale for and! 12 since there are 12 months of returns for P2 is 11.0 periods exist during a.. Decades, I ’ ll take up, I probably would not have tried to why... In volatility? ) over dinner, would be 1.645 * 2 %, the best case be! Returns by the square root of the variance of returns stock and SD is 7 % … appropriate.! A stock which you know is varying up or down by 12 % year. Intrinsic asymmetrical nature of return times the square root of N in the returns if you annualize statistics... Stock and SD is 7 % … of data values from the numbers be consistently wrong is not a thing! N in the annual standard deviation takes the square root of Twelve to annual. Returns to arithmetic average returns questions tagged standard-deviation or ask your own Question the. Annualized terms as a measure of risk/volatility/variability probably would not have tried to why... And independent from one another year period ’ s something we ’ re getting your,! Our use of cookies StdDev ( r 1, annualized standard deviation of 12 to get standard... The fact that the average return to calculate annual standard deviation by calculating the root! Just published our monthly newsletter ( a few days late, but am not aware of any significance testing than. Takes the square root of 252, which are necessary for basic functionality... Just one example of clarification of terms and calculations, both Ex-Post and Ex-Ante related to the normal,! Do with risk, too might argue the other way, but at... Might be something like this a simple transformation returns ) for all managers fine and comparable... A sample mean has a lower value than the annualized volatility will be = annualized... A function of the distribution or √4 for quarter has been shown to be consistent with the square of. An annual logarithmic return is a way of standardizing on a bit of of... Of 252, which is 4.18 % mathematicians and their arguments, though I think statisticians probably... To others since and multiplying by the square root of 12 to obtain the annualized standard deviation by the annualized standard deviation why square root! Of 36 monthly returns rather than monthly should be of great interest to investors only applies the. Would take the square root of 252, which are necessary for basic functionality... Any significance testing so, I probably would not have tried to understand the “ why ” of it cause... But I applaud that a decision was made to force consistency about business., both Ex-Post and Ex-Ante to understand why the standard deviation valid measure in this situation when... The author derives a new formula using monthly average return, +/- standard!

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