V5, p87, the two statements in the last paragraph : 1. Does it mean ? Yearly SD < monthly SD x 12^1/2 < weekly x 12^1/2 < daily SD < 250^1/2 ? SD : Standard Deviation, ^1/2 : Square Root 2. Does it mean ? The mean return used in the numerator of the calculated Sharpe ration is resulted from compounding 12 months returns while the Standard Deviation from a single month’s return (or the mean of 12 month returns) is used in the denominator ? Anyone can advise ?
- By lengthening the measurement interval - std deviation increases. Usually annual std dev > weekly std dev. But what is a way this number can be gamed? if you calculated a weekly std deviation and converted it into a annual std deviation using annual std dev calculated = weekly std dev * sqrt(52) (since there are 52 weeks in a year) the annual calculated std dev will be a smaller number. but numerator likewise would not be affected. It would be more or less the same number (r weekly * 52 would be approximately equal to r-annual.) but now since a lower annual std dev calculated is used - the sharpe ratio now would be a HIGHER number (since the denom. is lower). 2: Returns are compounded. So numerator = (1+rweek1)*(1+rweek2)*…(1+rweek52) - 1 std dev = monthly std dev*sqrt(12). (not compounded). return would most likely be equivalent to the annual return (or off by a very little bit). but std deviation calculated as above would be lower - so end result Sharpe Ratio would be higher.
CP, thank you very much for your response ! 1. What you meant are : Acutually (usually) : Yearly SD > monthly SD > weekly SD > daily SD if the Yearly SD is calculated as : monthly SD x 12^1/2 or weekly SD x 52^1/2 or daily SD x 250^1/2, the calculated Yearly SD will be lower than the actual yearly SD, i.e., the calculated yearly SD is an under-estimated one and the calculated Sharpe ratio will be HIGHER than the actual/real one. Am I right ? 2. In your example, I think the return in the numerator shall be : (1+r month1)* (1+r Month 2) * …(1+r month 12) -1 , since you use “monthly” SD in the denomintor. What is the monthly SD here ? The SD of a specific single month’s return ? And this will lead to higher Sharpe ratio (than real one) ? Since sqrt(12) = 3.464 and the annualized yearly return (compounded from monthly returns) is not necessarily 3.464x greater than the annualized yearly return, therefore, I think it is not necessarily that the calculated Sharpe ratio will be HIGHER than the actual/real one. i.e., there is a likelyhood that the the calculated Sharpe ratio will be LOWER than the actual/real one. But in any case, the calculated Sharpe ratio will be a distorted one. Am I correct ?
I come back to this issue because I still can’t get it. Anyone can help ?
AMA technical you may be but darn it you overthink everything …returns are linked geometically while standard deviation is a function of sqrt(T) “Simplify as much as possible, but no further.” my boi Bert
I couldn’t find a free edition of Richard Spurgin’s 2001 study outlining how Sharpe can be gamed by increasing the interval of measurement. I suspect it is as CP points out that you increase returns by the interval multiplier but divide by sqrt of interval multiplier , giving you increased sharpe. If you used Geometric compounding for the return , it would not be possible to game it quite as much , because Geometric Average ~ Arithmetic average - ( sigma/2) Even more funny to me is point #2: Compounding Monthly Returns but calculating the standard deviation from the not-compounded monthly returns. How exactly do you calculate standard deviation of COMPOUNDED monthly returns? I mean give me an example? If I give you 12 monthly returns , wouldn’t the standard deviation be the standard deviation of those 12 numbers? How would I compound these 12 , to get 12 other numbers to use? Help me here please!
- It may simply mean that SD(based on annual return) < SD(based on monthly return) and etc. It’s the “smoothing”, I think. 2) I also have a difficulty in understanding it. It is more a general problem than specific to Sharpe ratio. SD(based on monthly return) = SD_M x sqrt(12) R > Sum(12 monthly return)=Avg(monthly return)*12. It could be "
janakisri Wrote: ------------------------------------------------------- > I couldn’t find a free edition of Richard > Spurgin’s 2001 study outlining how Sharpe can be > gamed by increasing the interval of measurement. I > suspect it is as CP points out that you increase > returns by the interval multiplier but divide by > sqrt of interval multiplier , giving you increased > sharpe. If you used Geometric compounding for the > return , it would not be possible to game it quite > as much , because Geometric Average ~ Arithmetic > average - ( sigma/2) > > > Even more funny to me is point #2: > > Compounding Monthly Returns but calculating the > standard deviation from the not-compounded > monthly returns. > > How exactly do you calculate standard deviation of > COMPOUNDED monthly returns? I mean give me an > example? If I give you 12 monthly returns , > wouldn’t the standard deviation be the standard > deviation of those 12 numbers? How would I > compound these 12 , to get 12 other numbers to > use? Help me here please! Imagine having a return of 7% over two months period with standard deviation of 15% within the same period. Then you compound the return over one year to get 50% and then use the 15% standard deviation (without compounding, which should make the the standard deviation 37%) to measure risk adjusted return. You would have inflated the returns and kept the standard deviation based on a shorter investment period. This will skew the Sharpe ratio upwards.
- is a gaming. 2) is more like a problem of Sharpe Ratio itself. If not calculating that way, what else can we do?
pimpineasy Wrote: ------------------------------------------------------- > AMA technical you may be but darn it you overthink everything …returns > are linked geometically while standard deviation is a function of sqrt(T) > > “Simplify as much as possible, but no further.” > > my boi Bert This forum is open for discussions of any issue in the curriculum !
alta since you dont understand me i think u stand under me … never meant it as a dis or a dont discuss this topic …all i was pointing out is that AMA has a tendency to overthink things when a much simpler approach would be more elegant
Annualize St Dev (from daily return)=daily return*250(sqrt) Annualize St Dev (from monthly return)=monthly return*12(sqrt) So annual st dev computed using daily returns is higher than annual st dev computed using monthly returns. Going back to SR, using monthly returns will lower the st dev (denominator) so SR will be higher.
pimpineasy Wrote: ------------------------------------------------------- > alta > since you dont understand me i think u stand under me … > > never meant it as a dis or a dont discuss this topic …all i was pointing out is > that AMA has a tendency to overthink things when a much simpler approach would be more > elegant I don’t think he is overthinking, you can see here other candidates have same questions (including me). It good to raise question here and get clarification through discussions.
Maybe #2 is simply: Take numerator as compounded monthly returns ,then annualize: i.e. annual return = (compounded mthly returns)^(12/countofMonths) But take denominator as standard deviation of annual returns ( not compounded monthly returns) Because the monthly returns is most likely to have at least 1 +ve outsized return and 1 -ve outsized return , the monthly compounded standard return will likely be higher than a standard deviation of annualized returns which would have these bumps taken out. What I’m wondering is , how can they do these accounting tricks in the presence of GIPS compliance or past the eagle eyes of fund sponsors? Please note that this point is conceptually similar to point #5 which actually uses Total Return swaps to smooth monthly returns That MAY be within GIPS compliance but nevertheless is gaming the Sharpe because its only purpose is to fool investors into thinking your Sharpe is higher: point #5 is outlined in Spurgin’s ( free ) pdf : http://www.hedgefundprofiler.com/Documents/26.pdf But point #2 is plain cheating with numbers
whether you do annual return as a geometric mean of monthly returns or as an arithmetic mean of returns - you are not going to have returns going too far off from each other. however std dev of returns (denominator) would change. I think that is the point of this entire statement. – need to be consistent in the period used. – do not try to forecast a bigger period’s std. dev from a smaller period’s. (once you did that - you would have a lower std. deviation on the bigger period). – and then use that new lower std. deviation in the sharpe ratio - your sharpe ratio would be overstated.
Following are fundamental issues : In calculation of Sharpe Ratio, 1. Statement 1 Why ASD (Annualized Standard Deviation) of daily returns is generally higher than the weeky, which is, in turn, higher than the monthly ? I don’t have answer but I think statement 1 shall mean that ASD from monthly return (rather than ASD from weely return or daily return) which shall be lowest and shall be used in calculation of Sharpe Ratio (especially, for hedge funds, since monthly returns are reported). 2. Statement 2 What is the correct way to calculate the ARR (Annualized Rate of Return), given monthly or weekly or daily rate of retun ? I am sorry it seems I missed something because I don’t remember where this is stated formally in the curriculum. But it seems the “correct way” shall be : {[(1+r month1)* (1+r Month 2) * …(1+r month 12)]^1/12 -1} x12 when monthly rate of retun is given. Please refer to P.89~90 in this reading and EOC Q12B. In these 2 cases, the ARR calculated from : (1+r month1)* (1+r Month 2) * …1+r month 12) -1 are higher than those calculated by the “correct way” and this shall be a means to gaming. As for SD, I think basically no way to compound the SD from the monthly return and ASD = MSD x ^12 shall be used when monthly rate of retun is given. (MSD : Monthly SD) Any further response is appreciated !
- Statement 1 Why ASD (Annualized Standard Deviation) of daily returns is generally higher than the weeky, which is, in turn, higher than the monthly ? In a hypothetical case, the monthly return of last 10 years has been 0, but there was some daily fluctuation. Then, SD(monthly data)=0; so is SD(annualized from monthly data)=0. But, SD(daily data) > 0; so is SD(annualized from daily data) > 0. --Basically, I think this is more like a statistics Q, rather than a finance Q.
Correction to my previous message. I think statement 1 shall mean that ASD from DAILY (rather than ASD from weekly return or monthly return) which shall be HIGHEST and shall be used in calculation of Sharpe Ratio (especially, for hedge funds, since monthly returns are reported).