How to properly annualize sharpe ratio.

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How to properly annualize sharpe ratio.

Post by Anon9001 »

The numerator part of sharpe ratio "Arithmetic mean of simple returns" does not make sense. Consider Investment having 50% return in one year and -50% return in next year. Arithmetic mean of these returns is 0% but actual return is -25%.This problem goes away if arithmetic mean is taken from log returns. Also multiplying the arithmetic mean by 12/252 to annualize it only makes sense for log returns. I heard conflicting advice on whether to use log/simple returns for sharpe ratio so I would like to know what is the proper way to get mean for simple returns and annualize the mean.
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Re: How to properly annualize sharpe ratio.

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Anon9001 wrote: Fri Jul 08, 2022 8:41 am The numerator part of sharpe ratio "Arithmetic mean of simple returns" does not make sense. Consider Investment having 50% return in one year and -50% return in next year. Arithmetic mean of these returns is 0% but actual return is -25%.
If there is an equal probability of returns being up 50% or down 50%, then the expected return is 0% and the median is -25%. Working through all the outcomes:

high-high: (1+.5)(1+.5) = 2.25. Gross return = 125%
high-low: (1+.5)(1-.5) = .75. Gross return = -25%
low-high: (1-.5)(1+.5) = .75. Gross return = -25%
low-low: (1-.5)(1-.5) = .25. Gross return = -75%

All four outcomes above are equal probability. Averaging across all four outcomes gives a gross return of 0%. The median is -25%. So the numerator of the Sharpe Ratio does capture the expected excess return. Just not the median excess return.
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Re: How to properly annualize sharpe ratio.

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I am confused- the question you are asking does not relate to the Sharpe Ratio. The units are “excess return per unit of risk”. There is no time dimension thus nothing to annualize. Geometric returns, while important, have nothing to do with the Sharpe formula.
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Re: How to properly annualize sharpe ratio.

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Ben Mathew wrote: Fri Jul 08, 2022 12:35 pm If there is an equal probability of returns being up 50% or down 50%, then the expected return is 0% and the median is -25%. Working through all the outcomes:

high-high: (1+.5)(1+.5) = 2.25. Gross return = 125%
high-low: (1+.5)(1-.5) = .75. Gross return = -25%
low-high: (1-.5)(1+.5) = .75. Gross return = -25%
low-low: (1-.5)(1-.5) = .25. Gross return = -75%

All four outcomes above are equal probability. Averaging across all four outcomes gives a gross return of 0%. The median is -25%. So the numerator of the Sharpe Ratio does capture the expected excess return. Just not the median excess return.
Thanks Ben for insights. So I can just compute sharpe ratio using simple returns and annualize the sharpe ratio by multiplying by square root of 12/252 and I dont need to use log returns to compute sharpe ratio?
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Re: How to properly annualize sharpe ratio.

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Anon9001 wrote: Fri Jul 08, 2022 12:43 pm
Ben Mathew wrote: Fri Jul 08, 2022 12:35 pm If there is an equal probability of returns being up 50% or down 50%, then the expected return is 0% and the median is -25%. Working through all the outcomes:

high-high: (1+.5)(1+.5) = 2.25. Gross return = 125%
high-low: (1+.5)(1-.5) = .75. Gross return = -25%
low-high: (1-.5)(1+.5) = .75. Gross return = -25%
low-low: (1-.5)(1-.5) = .25. Gross return = -75%

All four outcomes above are equal probability. Averaging across all four outcomes gives a gross return of 0%. The median is -25%. So the numerator of the Sharpe Ratio does capture the expected excess return. Just not the median excess return.
Thanks Ben for insights. So I can just compute sharpe ratio using simple returns and annualize the sharpe ratio by multiplying by square root of 12/252 and I dont need to use log returns to compute sharpe ratio?
I'm not sure why you are multiplying by 12/252 to annualize. What returns are you starting from? Are they not annual returns to begin with?
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Re: How to properly annualize sharpe ratio.

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Ben Mathew wrote: Fri Jul 08, 2022 12:49 pm I'm not sure why you are multiplying by 12/252 to annualize. What returns are you starting from? Are they not annual returns to begin with?
Daily/Monthly Returns depending on what asset I am comparing.
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Re: How to properly annualize sharpe ratio.

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alex_686 wrote: Fri Jul 08, 2022 12:41 pm I am confused- the question you are asking does not relate to the Sharpe Ratio. The units are “excess return per unit of risk”. There is no time dimension thus nothing to annualize. Geometric returns, while important, have nothing to do with the Sharpe formula.
If you are computing sharpe using monthly/daily returns it will be monthly/daily Sharpe Ratio. Assuming returns are I.I.D the general advice is to multiply sharpe ratio by square root of 12/252(this multiplies the numerator by 12/252 and demoniator by square root of 12/252) to get Annualized Sharpe Ratio. I figured out how to annualize standard deviation of simple returns correctly but I did not figure how to annualize the arithmetic mean of simple returns correctly.
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Re: How to properly annualize sharpe ratio.

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Anon9001 wrote: Fri Jul 08, 2022 1:30 pm
alex_686 wrote: Fri Jul 08, 2022 12:41 pm I am confused- the question you are asking does not relate to the Sharpe Ratio. The units are “excess return per unit of risk”. There is no time dimension thus nothing to annualize. Geometric returns, while important, have nothing to do with the Sharpe formula.
If you are computing sharpe using monthly/daily returns it will be monthly/daily Sharpe Ratio. Assuming returns are I.I.D the general advice is to multiply sharpe ratio by square root of 12/252(this multiplies the numerator by 12/252 and demoniator by square root of 12/252) to get Annualized Sharpe Ratio. I figured out how to annualize daily/monthly simple returns correctly but I did not figure how to annualize the arithmetic mean of simple returns correctly.
You don’t want to do this.

If your units are “volatile per month” and want to convert your units to “volatility per month” what you are saying is absolutely correct.

If you follow the math behind the Sharpe ratio you will see that the time units cancel out. Thus you can’t convert. There is no time dimension.

As a analogy we can say that that it is 3000 miles between New York and LA. You can’t convert that value to mph. The distance between the 2 places are fixed and do not charge if you do it by bike or by plane.
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Re: How to properly annualize sharpe ratio.

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Anon9001 wrote: Fri Jul 08, 2022 1:13 pm
Ben Mathew wrote: Fri Jul 08, 2022 12:49 pm I'm not sure why you are multiplying by 12/252 to annualize. What returns are you starting from? Are they not annual returns to begin with?
Daily/Monthly Returns depending on what asset I am comparing.
I see. So it's 12 or 252 depending on whether you're starting out with monthly returns or daily returns. I thought you were multiplying by 12 divided by 252.

Annualizing the Sharpe Ratio is not straightforward. I know how to convert the numerator (expected excess returns), but not the denominator (standard deviation of returns). Standard deviation of log returns is easy to convert across different units of time if you assume independent returns, but I'm not sure how to convert from standard deviation of log returns to standard deviation of returns.

To calculate a Sharpe ratio for annual returns, I would suggest first calculating annual returns in the raw data, and then calculating the annual Sharpe Ratio directly from that.

But in case it's useful, here's how to convert expected returns from monthly to annual, assuming independent returns:

1. Convert monthly returns to monthly log returns.
2. Calculate the expected monthly log return by averaging monthly log returns.
3. Expected annual log return = 12*expected monthly log return
4. Expected annual return = exp(expected annual log return + adjustment factor)-1. If annual log returns are normally distributed, then the adjustment factor is variance of annual log returns divided by 2.
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Re: How to properly annualize sharpe ratio.

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Ben Mathew wrote: Fri Jul 08, 2022 2:39 pm I see. So it's 12 or 252 depending on whether you're starting out with monthly returns or daily returns. I thought you were multiplying by 12 divided by 252.

Annualizing the Sharpe Ratio is not straightforward. I know how to convert the numerator (expected excess returns), but not the denominator (standard deviation of returns). Standard deviation of log returns is easy to convert across different units of time if you assume independent returns, but I'm not sure how to convert from standard deviation of log returns to standard deviation of returns.

To calculate a Sharpe ratio for annual returns, I would suggest first calculating annual returns in the raw data, and then calculating the annual Sharpe Ratio directly from that.

But in case it's useful, here's how to convert expected returns from monthly to annual, assuming independent returns:

1. Convert monthly returns to monthly log returns.
2. Calculate the expected monthly log return by averaging monthly log returns.
3. Expected annual log return = 12*expected monthly log return
4. Expected annual return = exp(expected annual log return + adjustment factor)-1. If annual log returns are normally distributed, then the adjustment factor is variance of annual log returns divided by 2.
Is this adjustment factor necessary? I am doing 5 Year Rolling Sharpe comparison between two assets and I prefer monthly/daily returns to get good sample size. If I were to take annual returns for assets whose returns are going back to 1979 I would have 8 samples to look at.
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Re: How to properly annualize sharpe ratio.

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Anon9001 wrote: Sat Jul 09, 2022 10:33 am
Ben Mathew wrote: Fri Jul 08, 2022 2:39 pm I see. So it's 12 or 252 depending on whether you're starting out with monthly returns or daily returns. I thought you were multiplying by 12 divided by 252.

Annualizing the Sharpe Ratio is not straightforward. I know how to convert the numerator (expected excess returns), but not the denominator (standard deviation of returns). Standard deviation of log returns is easy to convert across different units of time if you assume independent returns, but I'm not sure how to convert from standard deviation of log returns to standard deviation of returns.

To calculate a Sharpe ratio for annual returns, I would suggest first calculating annual returns in the raw data, and then calculating the annual Sharpe Ratio directly from that.

But in case it's useful, here's how to convert expected returns from monthly to annual, assuming independent returns:

1. Convert monthly returns to monthly log returns.
2. Calculate the expected monthly log return by averaging monthly log returns.
3. Expected annual log return = 12*expected monthly log return
4. Expected annual return = exp(expected annual log return + adjustment factor)-1. If annual log returns are normally distributed, then the adjustment factor is variance of annual log returns divided by 2.
Is this adjustment factor necessary? I am doing 5 Year Rolling Sharpe comparison between two assets and I prefer monthly/daily returns to get good sample size. If I were to take annual returns for assets whose returns are going back to 1979 I would have 8 samples to look at.
The adjustment factor will make a nontrivial difference. It's not that hard to calculate, so no reason to not include it. But you still have the problem of figuring out the denominator.

You don't have to take non-overlapping windows for the annual data. The years can overlap. I.e. the first one year period starts Jan 1 1979. The second one year period starts Jan 2 1979 etc. So you're not throwing away any data. There will be a lot of serial correlation in the annual data. But the mean and standarad deviation estimates calculated from serially correlated data will be fine if there is enough data.
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Re: How to properly annualize sharpe ratio.

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Ben Mathew wrote: Sat Jul 09, 2022 11:01 am You don't have to take non-overlapping windows for the annual data. The years can overlap. I.e. the first one year period starts Jan 1 1979. The second one year period starts Jan 2 1979 etc. So you're not throwing away any data. There will be a lot of serial correlation in the annual data. But the mean and standarad deviation estimates calculated from serially correlated data will be fine if there is enough data.
How much data (sample size) do you need to get reliable mean and standard deviation estimates if there is serial correlation?

Thanks for help.
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Re: How to properly annualize sharpe ratio.

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Anon9001 wrote: Sat Jul 09, 2022 11:07 am
Ben Mathew wrote: Sat Jul 09, 2022 11:01 am You don't have to take non-overlapping windows for the annual data. The years can overlap. I.e. the first one year period starts Jan 1 1979. The second one year period starts Jan 2 1979 etc. So you're not throwing away any data. There will be a lot of serial correlation in the annual data. But the mean and standarad deviation estimates calculated from serially correlated data will be fine if there is enough data.
How much data (sample size) do you need to get reliable mean and standard deviation estimates if there is serial correlation?

Thanks for help.
Estimates of the expected return will be unbiased even with serial correlation. So your numerator will be fine. Estimates of the standard deviation will be biased low. So your denominator will be biased low. This bias declines quickly as sample size increases. i.e. The bias declines more from n=10 to n=20, and than from n=20 to n=30. So I suspect you'll be okay. But if you want to formally calculate the size of the bias and correct for it, the following article may be a good place to start:

https://en.wikipedia.org/wiki/Unbiased_ ... rrelation)
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Re: How to properly annualize sharpe ratio.

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Ben Mathew wrote: Sat Jul 09, 2022 12:29 pm Estimates of the expected return will be unbiased even with serial correlation. So your numerator will be fine. Estimates of the standard deviation will be biased low. So your denominator will be biased low. This bias declines quickly as sample size increases. i.e. The bias declines more from n=10 to n=20, and than from n=20 to n=30. So I suspect you'll be okay. But if you want to formally calculate the size of the bias and correct for it, the following article may be a good place to start:

https://en.wikipedia.org/wiki/Unbiased_ ... rrelation)
Is this video accurate regarding how to scale volatility when there is autocorrelation?: https://m.youtube.com/watch?v=_z-08wZUfBc
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Re: How to properly annualize sharpe ratio.

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Anon9001 wrote: Sat Jul 09, 2022 12:48 pm
Ben Mathew wrote: Sat Jul 09, 2022 12:29 pm Estimates of the expected return will be unbiased even with serial correlation. So your numerator will be fine. Estimates of the standard deviation will be biased low. So your denominator will be biased low. This bias declines quickly as sample size increases. i.e. The bias declines more from n=10 to n=20, and than from n=20 to n=30. So I suspect you'll be okay. But if you want to formally calculate the size of the bias and correct for it, the following article may be a good place to start:

https://en.wikipedia.org/wiki/Unbiased_ ... rrelation)
Is this video accurate regarding how to scale volatility when there is autocorrelation?: https://m.youtube.com/watch?v=_z-08wZUfBc
I don't think so. He's assuming return over 5 day period = sum of the 5 daily returns. This will be true only for log returns, not for regular returns.
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Re: How to properly annualize sharpe ratio.

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Ben Mathew wrote: Sat Jul 09, 2022 2:22 pm I don't think so. He's assuming return over 5 day period = sum of the 5 daily returns. This will be true only for log returns, not for regular returns.
Thanks but for the rf part of sharpe ratio a simple annualization can be done right? Example If I have monthly rf I just multiply by 12 to get annualized rf.
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Re: How to properly annualize sharpe ratio.

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Anon9001 wrote: Sat Jul 09, 2022 2:47 pm
Ben Mathew wrote: Sat Jul 09, 2022 2:22 pm I don't think so. He's assuming return over 5 day period = sum of the 5 daily returns. This will be true only for log returns, not for regular returns.
Thanks but for the rf part of sharpe ratio a simple annualization can be done right? Example If I have monthly rf I just multiply by 12 to get annualized rf.
To convert from monthly to annual returns, you would need to use

annual return = (1+monthly return)^12-1

This would be true even for the risk free return.
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Re: How to properly annualize sharpe ratio.

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Ben Mathew wrote: Sat Jul 09, 2022 4:45 pm To convert from monthly to annual returns, you would need to use

annual return = (1+monthly return)^12-1

This would be true even for the risk free return.
I would like you to check if this Rolling 5 Year Sharpe calculation I done in Excel is correct. This comparison is between Vanguard High Yield Bond Yield Fund and Vanguard 500 Fund going back to 1985. Here is the sheet. I also attached images of my method of computing rolling 5 Year Sharpe in Excel.
Last edited by Anon9001 on Mon Jul 11, 2022 11:04 am, edited 2 times in total.
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Re: How to properly annualize sharpe ratio.

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Ben Mathew wrote: Sat Jul 09, 2022 4:45 pm To convert from monthly to annual returns, you would need to use

annual return = (1+monthly return)^12-1

This would be true even for the risk free return.
I made error earlier in the excel sheet. I reuploaded it. Please click the download link again.
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Re: How to properly annualize sharpe ratio.

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Anon9001 wrote: Sun Jul 10, 2022 3:13 pm
Ben Mathew wrote: Sat Jul 09, 2022 4:45 pm To convert from monthly to annual returns, you would need to use

annual return = (1+monthly return)^12-1

This would be true even for the risk free return.
I made error earlier in the excel sheet. I reuploaded it. Please click the download link again.
The formulas look okay. But I missed the fact earlier than you were using only 5 years worth of returns for each Sharpe ratio estimate. Standard deviation estimates may be unreliable for five year windows. Maybe 20 or 30 year windows will work better? I don't think you can read much into 5 year performances of two investments anyway, whether for expected excess return or for risk. But it may still be useful if one series is better than other most of the time, even if it isn't very meaningful at any one instant of time.
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Re: How to properly annualize sharpe ratio.

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Ben Mathew wrote: Sun Jul 10, 2022 3:39 pm The formulas look okay. But I missed the fact earlier than you were using only 5 years worth of returns for each Sharpe ratio estimate. Standard deviation estimates may be unreliable for five year windows. Maybe 20 or 30 year windows will work better? I don't think you can read much into 5 year performances of two investments anyway, whether for expected excess return or for risk. But it may still be useful if one series is better than other most of the time, even if it isn't very meaningful at any one instant of time.
I have devised alternative method to calculating rolling 5 year sharpes. For computing 5 Year Sharpe on Jan 1990 I take the average and standard deviation of annual returns occuring on Jan 1986, Jan 1987, Jan 1988, Jan 1989 and Jan 1990. For computing 5 Year Sharpe on Feb 1990 I take the average and standard deviation of annual returns occuring on, Feb 1986, Feb 1987, Feb 1988, Feb 1989 and Feb 1990. Is this method correct or should I stick with previous method? Link to Excel sheet which is having this new method here. Thanks.
Last edited by Anon9001 on Mon Jul 11, 2022 11:04 am, edited 1 time in total.
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Re: How to properly annualize sharpe ratio.

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Ben Mathew wrote: Sun Jul 10, 2022 3:39 pm The formulas look okay. But I missed the fact earlier than you were using only 5 years worth of returns for each Sharpe ratio estimate. Standard deviation estimates may be unreliable for five year windows. Maybe 20 or 30 year windows will work better? I don't think you can read much into 5 year performances of two investments anyway, whether for expected excess return or for risk. But it may still be useful if one series is better than other most of the time, even if it isn't very meaningful at any one instant of time.
In this new method the sample size for each 5 year sharpe ratio estimate is 5. Is this too low?
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Re: How to properly annualize sharpe ratio.

Post by alex_686 »

Ben Mathew wrote: Sun Jul 10, 2022 3:39 pm
Anon9001 wrote: Sun Jul 10, 2022 3:13 pm
Ben Mathew wrote: Sat Jul 09, 2022 4:45 pm To convert from monthly to annual returns, you would need to use

annual return = (1+monthly return)^12-1

This would be true even for the risk free return.
I made error earlier in the excel sheet. I reuploaded it. Please click the download link again.
The formulas look okay. But I missed the fact earlier than you were using only 5 years worth of returns for each Sharpe ratio estimate. Standard deviation estimates may be unreliable for five year windows. Maybe 20 or 30 year windows will work better? I don't think you can read much into 5 year performances of two investments anyway, whether for expected excess return or for risk. But it may still be useful if one series is better than other most of the time, even if it isn't very meaningful at any one instant of time.
5 years is way better than 20 years.

Sharpe makes the assumption that we are working with a normal distribution curve. The markets do not have a normal distribution curve. You can get away with this for periods were volatile is constant. Say 5 years. You are not going to get away with this foe longer periods.

Investment theory is one of those great fields that the more data you throw at it the worse the statically results will be.

Now, once again, please don’t annualize the standard deviation. It is bad math.
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Re: How to properly annualize sharpe ratio.

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Ben Mathew wrote: Sun Jul 10, 2022 3:39 pm The formulas look okay. But I missed the fact earlier than you were using only 5 years worth of returns for each Sharpe ratio estimate. Standard deviation estimates may be unreliable for five year windows. Maybe 20 or 30 year windows will work better? I don't think you can read much into 5 year performances of two investments anyway, whether for expected excess return or for risk. But it may still be useful if one series is better than other most of the time, even if it isn't very meaningful at any one instant of time.
Anon9001 wrote: Mon Jul 11, 2022 4:19 am I have devised alternative method to calculating rolling 5 year sharpes. For computing 5 Year Sharpe on Jan 1990 I take the average and standard deviation of annual returns occuring on Jan 1986, Jan 1987, Jan 1988, Jan 1989 and Jan 1990. For computing 5 Year Sharpe on Feb 1990 I take the average and standard deviation of annual returns occuring on, Feb 1986, Feb 1987, Feb 1988, Feb 1989 and Feb 1990. Is this method correct or should I stick with previous method? Link to Excel sheet which is having this new method here. Thanks.
Anon9001 wrote: Mon Jul 11, 2022 7:55 am In this new method the sample size for each 5 year sharpe ratio estimate is 5. Is this too low?
I forgot to include screenshots of this new method.
Image
Image

Also the earlier screenshots of the previous method were incorrect. They are screenshots of the prevoious method before I fixed the error I done. Here is the correct screenshots.
Image
Image

EDIT: Fixed some mistakes in the sheet and updated the screenshot to reflect this change.
Last edited by Anon9001 on Mon Jul 11, 2022 11:18 am, edited 1 time in total.
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Re: How to properly annualize sharpe ratio.

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Anon9001 wrote: Mon Jul 11, 2022 4:19 am
Ben Mathew wrote: Sun Jul 10, 2022 3:39 pm The formulas look okay. But I missed the fact earlier than you were using only 5 years worth of returns for each Sharpe ratio estimate. Standard deviation estimates may be unreliable for five year windows. Maybe 20 or 30 year windows will work better? I don't think you can read much into 5 year performances of two investments anyway, whether for expected excess return or for risk. But it may still be useful if one series is better than other most of the time, even if it isn't very meaningful at any one instant of time.
I have devised alternative method to calculating rolling 5 year sharpes. For computing 5 Year Sharpe on Jan 1990 I take the average and standard deviation of annual returns occuring on Jan 1986, Jan 1987, Jan 1988, Jan 1989 and Jan 1990. For computing 5 Year Sharpe on Feb 1990 I take the average and standard deviation of annual returns occuring on, Feb 1986, Feb 1987, Feb 1988, Feb 1989 and Feb 1990. Is this method correct or should I stick with previous method? Link to Excel sheet which is having this new method here. Thanks.
Both methods have the problem that you have only 5 independent draws of annual returns. That's just not enough to reliably estimate the standard deviation of a distribution.

For five year windows, it may be better to stick with daily or monthly returns and calculate the daily or monthly Sharpe ratios. You don't know how to annualize it, but is that necessary for your purposes? At least the daily or monthly Sharpe ratio will be more reliably estimated.
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Re: How to properly annualize sharpe ratio.

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I believe that instead of annualizing the sharpe ratio, each component of the sharpe ratio should be annualized. For instance, you take the annualized portfolio return, subtract the annualized RFR, and then divide by the annualized standard deviation.
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Re: How to properly annualize sharpe ratio.

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Ben Mathew wrote: Mon Jul 11, 2022 11:04 am Both methods have the problem that you have only 5 independent draws of annual returns. That's just not enough to reliably estimate the standard deviation of a distribution.

For five year windows, it may be better to stick with daily or monthly returns and calculate the daily or monthly Sharpe ratios. You don't know how to annualize it, but is that necessary for your purposes? At least the daily or monthly Sharpe ratio will be more reliably estimated.
The main reason for annualzing the sharpe ratio is for comparison sake right? Sharpe ratios which are computed using monthly returns cannot be compared against sharpe ratios which are computed using daily returns unless they are both annualized. If I don't care about that then I should be fine with not annnualizing the sharpe ratio right?
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Re: How to properly annualize sharpe ratio.

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Anon9001 wrote: Tue Jul 12, 2022 12:02 pm
Ben Mathew wrote: Mon Jul 11, 2022 11:04 am Both methods have the problem that you have only 5 independent draws of annual returns. That's just not enough to reliably estimate the standard deviation of a distribution.

For five year windows, it may be better to stick with daily or monthly returns and calculate the daily or monthly Sharpe ratios. You don't know how to annualize it, but is that necessary for your purposes? At least the daily or monthly Sharpe ratio will be more reliably estimated.
The main reason for annualzing the sharpe ratio is for comparison sake right? Sharpe ratios which are computed using monthly returns cannot be compared against sharpe ratios which are computed using daily returns unless they are both annualized. If I don't care about that then I should be fine with not annnualizing the sharpe ratio right?
Yes. If you compute daily or monthly Sharpe ratios for both series, it's an apples to apples comparison. You don't need to annualize to compare them.
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Re: How to properly annualize sharpe ratio.

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Ben Mathew wrote: Tue Jul 12, 2022 12:17 pm Yes. If you compute daily or monthly Sharpe ratios for both series, it's an apples to apples comparison. You don't need to annualize to compare them.
Okay that should be it. Thanks for the help Ben. I will not annualize the sharpe ratio from now on as I dont care about comparing sharpe ratios which are computed using daily returns to sharpe ratios which are computed using monthly returns. If I want to compare I will do like you said and use daily/monthly returns to compute both sharpe ratios. :sharebeer
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Re: How to properly annualize sharpe ratio.

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Ben Mathew wrote: Tue Jul 12, 2022 12:17 pm Yes. If you compute daily or monthly Sharpe ratios for both series, it's an apples to apples comparison. You don't need to annualize to compare them.
Unrelated to this topic I wonder how much population/sample size do you need to get accurate Kurtosis, Skewness estimates for daily/monthly/annual returns? Is it going to be higher than population/sample size you need to get for accurate standard deviation estimate as kurtosis and skewness is higher moment?
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Re: How to properly annualize sharpe ratio.

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Ben Mathew wrote: Tue Jul 12, 2022 12:17 pm Yes. If you compute daily or monthly Sharpe ratios for both series, it's an apples to apples comparison. You don't need to annualize to compare them.
Is this website correct? They are saying standard error of skewness/kurtosis can be computed by sqrt of 6/n for skewness or sqrt of 24/n for kurtosis. if the absolute value of skewness/kurtosis is higher than 2*standard error than the skewnees/kurtosis value is indicating returns distribution is not normal.
https://www.real-statistics.com/tests-n ... -kurtosis/
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Re: How to properly annualize sharpe ratio.

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^ I don't know the answers to these questions. Sorry.
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Re: How to properly annualize sharpe ratio.

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Ben Mathew wrote: Fri Jul 08, 2022 2:39 pm But in case it's useful, here's how to convert expected returns from monthly to annual, assuming independent returns:

1. Convert monthly returns to monthly log returns.
2. Calculate the expected monthly log return by averaging monthly log returns.
3. Expected annual log return = 12*expected monthly log return
4. Expected annual return = exp(expected annual log return + adjustment factor)-1. If annual log returns are normally distributed, then the adjustment factor is variance of annual log returns divided by 2.
Sorry for bothering you but I am wondering the averaging you are talking about in Step 2 is arithmetic mean right? Also in #4 I am assuming "exp" means exponent please clarify if I am wrong or right in my assumption.

What I am thinking now is I use this method to get correct annualized mean but for annualzing volatility I just multiply by square root of 12. I understand this multiplying by square root of 12 is only exact for log returns but I dont know of alternative method on how to annualize volatility. I will check later whether this annualized volatility is close to annual volatility by comparing them.
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Re: How to properly annualize sharpe ratio.

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Anon9001 wrote: Tue Aug 02, 2022 7:12 am
Ben Mathew wrote: Fri Jul 08, 2022 2:39 pm But in case it's useful, here's how to convert expected returns from monthly to annual, assuming independent returns:

1. Convert monthly returns to monthly log returns.
2. Calculate the expected monthly log return by averaging monthly log returns.
3. Expected annual log return = 12*expected monthly log return
4. Expected annual return = exp(expected annual log return + adjustment factor)-1. If annual log returns are normally distributed, then the adjustment factor is variance of annual log returns divided by 2.
Sorry for bothering you but I am wondering the averaging you are talking about in Step 2 is arithmetic mean right? Also in #4 I am assuming "exp" means exponent please clarify if I am wrong or right in my assumption.
That is correct.
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Re: How to properly annualize sharpe ratio.

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Ben Mathew wrote: Tue Aug 02, 2022 3:34 pm That is correct.
:D

Again sorry for annoying you but I am wondering is the adjustment factor you mentioned earlier going to be accurate if I only have 10 indepedent draws of annual returns (10 because I am doing 10 Year Rolling Sharpe)?
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Re: How to properly annualize sharpe ratio.

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Ben Mathew wrote: Fri Jul 08, 2022 2:39 pm Standard deviation of log returns is easy to convert across different units of time if you assume independent returns, but I'm not sure how to convert from standard deviation of log returns to standard deviation of returns.
Again sorry to bug you but I am wondering what are you meaning here. If I take this formula "exp(stdev of monthly log returns*sqrt(12))-1" do I get Annualized Geometric Standard Deviation?
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Re: How to properly annualize sharpe ratio.

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Anon9001 wrote: Wed Aug 17, 2022 2:44 pm
Ben Mathew wrote: Fri Jul 08, 2022 2:39 pm Standard deviation of log returns is easy to convert across different units of time if you assume independent returns, but I'm not sure how to convert from standard deviation of log returns to standard deviation of returns.
Again sorry to bug you but I am wondering what are you meaning here. If I take this formula "exp(stdev of monthly log returns*sqrt(12))-1" do I get Annualized Geometric Standard Deviation?
No. There is no such mathematical concept of geometric volatility. In theory volatility is constant so all you have to do is square over the time period. If you break that assumption - i.e. assume that volatility either scales up or down over time - then you can no longer use the Sharpe Ratio which does make trhat assumption.
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Re: How to properly annualize sharpe ratio.

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So let us step back a second and think about why this whole line of questioning is off.

First there is the hard math. There is just nothing in the math that suggests that this is even a remotely good idea. So to be fair where are you getting this idea from? For context I have taken formal mathematical courses in this subject and calculating these things were my day job for a while.

Second, let us try to use a little intuition on why this won't work. I am going to guess that this comes from thinking about Sequence of Return Risk or Volatility Drag.

Geometric Returns = Average Annual Returns - Volatility Drag

or

Geometric Returns - Average Returns - (Standard Deviation Squared /2)

So volatility drives the difference the Geometric and Average returns.

But doing anything "geometric" with volatility makes no sense. That would imply that the length of time would have some sort of impact on volatility. That a person holding the S&P 500 for 10 years would have a different volatility than a person who has only held it for a day.

Volatility assumes a standard bell curve. As you expand or contract time the size of the bell curve changes but not its shape and it is the shape we are concerned about.
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Re: How to properly annualize sharpe ratio.

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With due respect there is no such thing as volatility drag. This volatility drag is coming from thinking arithmetic mean is appropriate for simple returns when it isnt. Geomean of simple return is the return investor gets. I have recieved this GSD as suggestion from Vinevez in order to create Geometric Sharpe. It wasnt my idea but it makes sense tho. The traditional SD is calculated as follows:

1. You take the arithmetic mean of all data points.

2. You calculate variance of each data point by subtracting the arithmetic mean from each data point.

3. You square the variance of each data point.

4. You take sum of squared variance values.

5. You divide the sum of squared variance values by number of data points
in the data set less 1.

6. You take the square root of the quotient from Step 5.

So the traditional SD measures dispersion of data relative to arithmetic mean which is not suitable if we are using geometric mean in the numerator.
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