What is the meaningful difference between two assets annualized sharpe ratios?

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What is the meaningful difference between two assets annualized sharpe ratios?

Post by Anon9001 »

Is it 0.1 (ie Asset A has Sharpe Ratio of 1 and Asset B is having Sharpe Ratio of 1.1)? 0.2(ie Asset A has Sharpe Ratio of 1 and Asset B is having Sharpe Ratio of 1.2)? I am comparing sharpe ratios of two different portfolios and wanting to know what sharpe ratio increase is meaningful?
Last edited by Anon9001 on Thu May 12, 2022 8:13 am, edited 1 time in total.
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

Post by rkhusky »

0.1 would be meaningful to me for two investments of the same class. Note that two investments could have identical Sharpe ratios, but be completely different (A: return of 0.1% and SD of 0.2%, B: return of 10% and SD of 20%)
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

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rkhusky wrote: Thu May 12, 2022 7:57 am 0.1 would be meaningful to me for two investments of the same class. Note that two investments could have identical Sharpe ratios, but be completely different (A: return of 0.1% and SD of 0.2%, B: return of 10% and SD of 20%)
So in that case if person has high risk appetite than they can choose B and if person has low risk appetite they can choose A as the risk adjusted returns are similar.
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

Post by nisiprius »

My rough, back-of-the-envelope, pragmatic way is this. We would like to believe that assets "have" a Sharpe ratio, that it is a fixed, predictable, persistent property. We would like to believe this about a lot of financial numbers. Mostly it isn't true. So my strategy for getting a "feeling" is to look at an assets with PortfolioVisualizer, split the time period in half, and see how different the two numbers are. Because if "the same" asset has two different Sharpe numbers, that tells us something about the accuracy, persistence, and predictability of the number.

For no particular reason, just to avoid mental math, I'll compare two calendar-year ten-year periods, to 2002-2011 to 2012-2021. And I'll do the exercise for one active and one passive fund

Sharpe ratios for 2002-2011 and 2012-2021

VTSAX, Vanguard Total Stock Market Index Fund: 0.20, 1.15
FCNTX, Fidelity Contrafund: 0.42, 1.24
VBTLX, Vanguard Total Bond Market Index Fund: 1.16, 0.71
PTTRX, PIMCO Total Return: 0.96, 0.92

Admittedly PTTRX looks stable, but the overwhelming impression is that Sharpe ratios aren't stable or persistent at all.

If the difference for a stock fund from one period to another can be like 0.8, then I'm not prepared to put too much weight on a difference of 0.2 between one stock fund and another.

If the difference for a bond fund from one period to another can be like 0.45, then I'm not prepared to put too much weight on a difference of 0.2 between one stock fund and another.

Of course I love to compare Sharpe ratios to prove, in a theoretical way, that some claim for superiority is dubious because it has higher return but lower Sharpe ratio, and that, therefore, you can get a higher return with vanilla assets if you adjust the allocation to equalize risks. But as a practical measure of superiority-inferiority it is just another unstable, fluctuating number with little persistence.
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

Post by rkhusky »

Anon9001 wrote: Thu May 12, 2022 8:17 am
rkhusky wrote: Thu May 12, 2022 7:57 am 0.1 would be meaningful to me for two investments of the same class. Note that two investments could have identical Sharpe ratios, but be completely different (A: return of 0.1% and SD of 0.2%, B: return of 10% and SD of 20%)
So in that case if person has high risk appetite than they can choose B and if person has low risk appetite they can choose A as the risk adjusted returns are similar.
Right. But you wouldn't know the difference if all you looked at was the Sharpe ratio. I would only use it for similar funds over the same time period, i.e. comparing two total market funds or two large value funds.
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

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nisiprius wrote: Thu May 12, 2022 8:57 am Of course I love to compare Sharpe ratios to prove, in a theoretical way, that some claim for superiority is dubious because it has higher return but lower Sharpe ratio, and that, therefore, you can get a higher return with vanilla assets if you adjust the allocation to equalize risks. But as a practical measure of superiority-inferiority it is just another unstable, fluctuating number with little persistence.
This is primarily according to old post from Vineviz due to the Sharpe Ratio Formula's Excess Portfolio Return varying heavily depending on time period but the standard deviation of the portfolio returns part of the formula stays near constant according to Vineviz. I constructed a Rolling 10 Year Annualized Sharpe Ratio (Based on Monthly Returns) to solve this problem so I am wondering if I am getting in 70% of the 10 year rolling time periods a 0.15 increase in Sharpe Ratio is that a meaningful difference? Link to Excel Sheet for Reference.
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

Post by SpaceTimeMorph »

I think this is going to be highly personalized but…

Practically, a sharpe ratio tells you the probability of returns exceeding a risk free rate. But it is done without the assumption of returns conforming to a specific model. We can, however, use a model to “convert” the sharpe ratio into a probability. For example, if we consider the volatility as the standard deviation of the return distribution then we can apply a model to convert this to probabilities.

If returns are normally distributed (big assumption!) then a sharpe of 1 would indicate an ~84% chance of an investment or asset class exceeding the risk free rate (~68% chance of returns being within 1sd of mean returns + 16% chance of exceeding 1sd on the high side). This is because the excess return is equal to the average amount of movement (vola). A sharpe of 2 with normally distributed returns would give you a probability of ~97.5% of a return in a given period exceeding the risk free rate. In this second case excess returns are twice what the average amount of movement in the stock is. And etc for higher sharpe ratios.

For smaller increments in sharpe ratio you’d have to use a cumulative distribution function to solve for the probability effect. If I have an equity portfolio with 10% return with 20% volatility and a 3.12% risk free rate, my sharpe ratio is 0.344. 1-CDF (assuming normal distribution) gives me a probability of ~63.5% (that is the probability of exceeding risk free rate of return). If I raise the sharpe ratio by 0.1 & hold volatility and risk free rate constant that would give me a return of 12% for a sharpe of 0.444. 1-CDF for the second sharpe is ~67.1%. So the real question in that case would be, “how much is a 3.6% change in probabilities worth to you?”

Excel formulas used for probabilities:
=1-norm.dist(.0312, .1, .2, TRUE) &
=1-norm.dist(.0312, .12, .2, TRUE)

Of course, you could reframe this as using a given interest rate on a loan you are trying to exceed or set the risk free rate in the numerator to 0, etc. The advantage of sharpe ratios is it gives you a visualization of risk influenced returns that is potentially model free (depends on the return and volatility calculations used).
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

Post by nisiprius »

Interestingly, while you were doing that, I was doing the same thing. I'm having trouble interpreting your spreadsheet, though. Maybe you can tell me if my results look more or less consistent with yours.

Image

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I used monthly returns for stocks or bonds and for bills; multiplied them by 12; subtracted them to get excess returns; calculated the population standard deviation of the monthly for stocks or bonds, multiplied by √12; calculated the Sharpe ratio as excess return / standard deviation.

I also found, but have not even looked at, a paper: Notes on the Sharpe Ratio, that seems to have a lot of relevant material on trying to do statistical tests.
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

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nisiprius wrote: Thu May 12, 2022 11:09 am Interestingly, while you were doing that, I was doing the same thing. I'm having trouble interpreting your spreadsheet, though. Maybe you can tell me if my results look more or less consistent with yours.

I also found, but have not even looked at, a paper: Notes on the Sharpe Ratio, that seems to have a lot of relevant material on trying to do statistical tests.
Yeah to get Annualized Sharpe Ratio according to PV you just have to multiply the Monthly Sharpe Ratio with Square Root of 12. The -0.38% is the RF I chosen. If that is where you are getting confused on.
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

Post by nisiprius »

I think you should probably be the one to do it... if you don't eventually I will... but why not pm Vineviz and see if he wants to comment?

My thoughts is that they're unstable and wander all over the place, but they are useful in calling out the very very very common practice of claiming "higher return" for an actively managed fund or ETF, or portfolio strategy, without accounting for the fact that it is taking higher risk.

As a general measure of fund quality, the Morningstar star ratings do adjust for risk in some way. Of course Morningstar itself acknowledges that the star ratings are not predictive or persistent, but at least they try to give a fairer measure of past performance than raw return.
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

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nisiprius wrote: Thu May 12, 2022 12:02 pm I think you should probably be the one to do it... if you don't eventually I will... but why not pm Vineviz and see if he wants to comment?

My thoughts is that they're unstable and wander all over the place, but they are useful in calling out the very very very common practice of claiming "higher return" for an actively managed fund or ETF, or portfolio strategy, without accounting for the fact that it is taking higher risk.

As a general measure of fund quality, the Morningstar star ratings do adjust for risk in some way. Of course Morningstar itself acknowledges that the star ratings are not predictive or persistent, but at least they try to give a fairer measure of past performance than raw return.
The Rolling Sharpe Ratios are already done. The only issue is I am having is knowing what difference between the two sharpe ratios is meaningful. As if I count Sharpe Ratio differences greater than 0.1 as meaningful then 50% Sensex 50% USA Portfolio is having 80% of the time superior risk adjusted returns than 100% Sensex Portfolio but if I count the Sharpe Ratio differences greater than 0.15 as meaningful then the amount of the time it is having superior risk adjusted returns compared to 100% Sensex Portfolio drops down to 70%. :?
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

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Anon9001 wrote: Thu May 12, 2022 12:18 pm The Rolling Sharpe Ratios are already done. The only issue is I am having is knowing what difference between the two sharpe ratios is meaningful.
I can't start a post about Sharpe ratios without first stating that I think Sharpe ratios are a terrible and useless metric, but I won't go into the details about why I think that for fear of derailing the thread.

But the issue in figuring out what a meaningful difference in Sharpe ratios boils down to something nisprius mentioned, with is the incredible amount of uncertainty involved in estimating the ratios themselves.

With 40 years of data, you only have four independent rolling 10-year Sharpe ratios to compare. And there's an incredible dispersion of rolling 10-year Sharpe ratios, which means a great deal of uncertainty.

Typical tests of statistical significance between two means would imply that you'd need a mean difference in Sharpe ratios of nearly 0.5 to suggest statistical significance (I'm estimating the variance by eye from the posted graph).

http://www.stat.yale.edu/Courses/1997-9 ... ancomp.htm

And that's assuming the means are stationary over time. For time series where that's highly doubtful (e.g. gold returns) or questionable (e.g. equity risk premium) this is a big assumption. Another big assumption is that you've properly matched the risk-free asset with the risky asset, which is also difficult.
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

Post by nisiprius »

vineviz wrote: Thu May 12, 2022 1:31 pm...I can't start a post about Sharpe ratios without first stating that I think Sharpe ratios are a terrible and useless metric, but I won't go into the details about why I think that for fear of derailing the thread...
But in my ignorant opinion, they are a better metric than the one which is commonly used: "return."

It would be interesting to have a survey of how 401(k) plan participants select the funds to contribute to. Hopefully most of them pick, or do not opt out of, the target-date option. Among the rest, I would guess that the commonest selection methods are a) contribute equally to all funds in the plan, and b) look at the table of 1-3-5 and 10-year return and pick the fund with the biggest number.
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

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nisiprius wrote: Thu May 12, 2022 1:44 pm
vineviz wrote: Thu May 12, 2022 1:31 pm...I can't start a post about Sharpe ratios without first stating that I think Sharpe ratios are a terrible and useless metric, but I won't go into the details about why I think that for fear of derailing the thread...
But in my ignorant opinion, they are a better metric than the one which is commonly used: "return."

It would be interesting to have a survey of how 401(k) plan participants select the funds to contribute to. Hopefully most of them pick, or do not opt out of, the target-date option. Among the rest, I would guess that the commonest selection methods are a) contribute equally to all funds in the plan, and b) look at the table of 1-3-5 and 10-year return and pick the fund with the biggest number.
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

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vineviz wrote: Thu May 12, 2022 1:31 pm With 40 years of data, you only have four independent rolling 10-year Sharpe ratios to compare. And there's an incredible dispersion of rolling 10-year Sharpe ratios, which means a great deal of uncertainty.

Typical tests of statistical significance between two means would imply that you'd need a mean difference in Sharpe ratios of nearly 0.5 to suggest statistical significance (I'm estimating the variance by eye from the posted graph).

http://www.stat.yale.edu/Courses/1997-9 ... ancomp.htm

And that's assuming the means are stationary over time. For time series where that's highly doubtful (e.g. gold returns) or questionable (e.g. equity risk premium) this is a big assumption. Another big assumption is that you've properly matched the risk-free asset with the risky asset, which is also difficult.
I am not using Annual Returns for the Rolling 10 Year Sharpe Ratios but Monthly Returns and then multiplying the Monthly Sharpe Ratio with the square root of 12 to get Annualized Sharpe Ratios so not sure why you are saying 4 Indepedent rolling ratios when in reality it is 393.
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

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Anon9001 wrote: Thu May 12, 2022 1:59 pm
vineviz wrote: Thu May 12, 2022 1:31 pm With 40 years of data, you only have four independent rolling 10-year Sharpe ratios to compare. And there's an incredible dispersion of rolling 10-year Sharpe ratios, which means a great deal of uncertainty.

Typical tests of statistical significance between two means would imply that you'd need a mean difference in Sharpe ratios of nearly 0.5 to suggest statistical significance (I'm estimating the variance by eye from the posted graph).

http://www.stat.yale.edu/Courses/1997-9 ... ancomp.htm

And that's assuming the means are stationary over time. For time series where that's highly doubtful (e.g. gold returns) or questionable (e.g. equity risk premium) this is a big assumption. Another big assumption is that you've properly matched the risk-free asset with the risky asset, which is also difficult.
I am not using Annual Returns for the Rolling 10 Year Sharpe Ratios but Monthly Returns and then multiplying the Monthly Sharpe Ratio with the square root of 12 to get Annualized Sharpe Ratios so not sure why you are saying 4 Indepedent rolling ratios when in reality it is 393.
If you have 40 years of annual data, there are only 4 fully independent 10 year periods in that sample.

Taking a 10 year average and shifting the period forward one month doesn't give you a completely independent observation from the first one, because 119 of the 120 months are overlapping. Most statistical tests assume the data are independent, and this is a challenge in the analysis of financial time series data..
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

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vineviz wrote: Thu May 12, 2022 2:33 pm If you have 40 years of annual data, there are only 4 fully independent 10 year periods in that sample.

Taking a 10 year average and shifting the period forward one month doesn't give you a completely independent observation from the first one, because 119 of the 120 months are overlapping. Most statistical tests assume the data are independent, and this is a challenge in the analysis of financial time series data..
Interesting so you are suggesting I shortern the rolling time period? To 5 Years from 10 Years?

Thanks.
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

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vineviz wrote: Thu May 12, 2022 2:33 pm
If you have 40 years of annual data, there are only 4 fully independent 10 year periods in that sample.

Taking a 10 year average and shifting the period forward one month doesn't give you a completely independent observation from the first one, because 119 of the 120 months are overlapping. Most statistical tests assume the data are independent, and this is a challenge in the analysis of financial time series data..
This might be a stupid question but I read the article you linked regarding flaws of Sharpe Ratio and it is recommending to use Actual Annual Standard Deviation instead of Annualized Standard Deviation due to overestimation of sharpe but I am wondering can the monthly return still be annualized and be divided by Annual Standard Deviation (After subtracting the Annualized Monthly Return from Annualized RF of course)?
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

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Anon9001 wrote: Thu May 12, 2022 2:43 pm Interesting so you are suggesting I shortern the rolling time period? To 5 Years from 10 Years?
It's unlikely to help much with the statistical tests. Yes, you'll have more independent samples. But because each sample is now noisier, due to the shorter averaging period, you're likely to get about the same significance.

In other words, doubling the number of observations but getting a 50% increase in sample standard deviation roughly puts you back where you started.
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

Post by nisiprius »

You don't even know that they are independent if they are non-overlapping. If they overlap, they certainly are not independent, but the reverse isn't necessarily true.

Just as a mental picture, I like this chart by Jim Otar. Don't take it too seriously. The neat little rectangles encourage you to perceive something that might not really be there.

Image

The point is that the stock market is chaotic. And, a lot of people feel--and I think there is actual math and academic papers to support this--that it is episodic. In old-fashioned market storytelling, you have a series of "markets" (as in bull market and bear market). Within each market, behavior is seemingly consistent and somewhat predictable. While it lasts, people count on the seemingly stable relationships... and build more and more sophisticated strategies that rely more and more on the belief that they will continue.

So you have relative consistency for a while, and then the old market ends and is superseded by a new market, whose behavior is initially not well known. Or the system slips from one metastable state to another. Or we start to see the effects of very-low-frequency, high amplitude noise.

The chart shows eight episodes in a period of 110 years, so they lasted for around 14 years each.

Naturally the next everyone asks is "can we predict how long they last and when it will turn the corner and what the next episode will look like?" My beliefs are no, no, and no.

Anyway, the point is a 110-year period doesn't give us 1,320 months to look at or 110 years to look at. It probably gives us about eight or so independent things to look at. We are looking at averages, standard deviations, correlation coefficients based on N = 8. Which gives us huge sampling error.

Within each period, there may what looks like a lot of variation when you're inside it, but there is enough overall similarity that the 14 years of one of those periods are closer to repeating the same number 14 times than 14 independent numbers.

A large number of arguments, e.g. US versus international, on inspection turn out to be based on a handful of these "episodes." There will be periods when the US outperforms international, and others when international outperforms. Basically a lot of arguments boils down to whether the seemingly-long time period included two good subperiods and one bad one, or two bad subperiods and one good one.
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

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vineviz wrote: Thu May 12, 2022 1:31 pm I can't start a post about Sharpe ratios without first stating that I think Sharpe ratios are a terrible and useless metric, but I won't go into the details about why I think that for fear of derailing the thread.
I am interested in what you think is a better way to measure risk adjusted returns if Sharpe Ratio is highly flawed.
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

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Anon9001 wrote: Fri May 13, 2022 7:10 am
vineviz wrote: Thu May 12, 2022 1:31 pm I can't start a post about Sharpe ratios without first stating that I think Sharpe ratios are a terrible and useless metric, but I won't go into the details about why I think that for fear of derailing the thread.
I am interested in what you think is a better way to measure risk adjusted returns if Sharpe Ratio is highly flawed.
If I had an intelligent answer to that question I'd be writing my Nobel Prize acceptance speech instead of this post.

I think "risk-adjusted" return is a useful concept in learning about finance. It's a hypothetical construct, though, and not something that I think CAN be directly measured.

I'd be okay using Sharpe in a portfolio context in limited circumstances: we've thoughtfully concluded that variance is a good proxy for risk, that variance is measured at the investor's actual time horizon, that the investor's risk-free rate is accurately specified, that limits on lending and/or leverage are properly accounted for, and that we account for the tremendous amount of parameter uncertainty. That's a lot of "ifs" though.
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

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vineviz wrote: Fri May 13, 2022 8:42 am If I had an intelligent answer to that question I'd be writing my Nobel Prize acceptance speech instead of this post.

I think "risk-adjusted" return is a useful concept in learning about finance. It's a hypothetical construct, though, and not something that I think CAN be directly measured.

I'd be okay using Sharpe in a portfolio context in limited circumstances: we've thoughtfully concluded that variance is a good proxy for risk, that variance is measured at the investor's actual time horizon, that the investor's risk-free rate is accurately specified, that limits on lending and/or leverage are properly accounted for, and that we account for the tremendous amount of parameter uncertainty. That's a lot of "ifs" though.
Thanks for clarifying that even though Sharpe Ratio is flawed there is no better measure of risk adjusted returns. So I have to ask does rolling sharpe ratios provide useful information? Should I use the sharpe ratio on full time period instead?
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

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Anon9001 wrote: Fri May 13, 2022 10:21 am Thanks for clarifying that even though Sharpe Ratio is flawed there is no better measure of risk adjusted returns.
Is that what I said? I don't think so. That's a bit like saying that ingesting ricin is safer than ingesting botulinum toxin, don't you think?

If I wanted to clarify my opinion about the Sharpe Ratio, I'd say something more akin to "the Sharpe Ratio is a terrible and misleading measure that should be confined to the dustbin of economic history".
Anon9001 wrote: Fri May 13, 2022 10:21 am I have to ask does rolling sharpe ratios provide useful information? Should I use the sharpe ratio on full time period instead?
The only reasonable answer to these questions is "don't use Sharpe Ratios at all".
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

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vineviz wrote: Fri May 13, 2022 10:37 am The only reasonable answer to these questions is "don't use Sharpe Ratios at all".
This is still better than using just raw Rolling Returns for comparing two assets right? I was using raw rolling returns before to compare assets.
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Anon9001
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

Post by Anon9001 »

This Research Affliates article is using p-value test to see if Sharpe Ratio Difference is meaningful. Could someone here tell me how this could be done in Excel?
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jg12345
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Re: What is the meaningful difference between two assets annualized sharpe ratios?

Post by jg12345 »

Hi OP,
1) The paper you link does not explain how they measure a p-value and under what assumptions for the underlying distribution. it's therefore impossible to tell you how to reproduce the p-value produced by the paper

The other point is that the explanation below the table is unfortunately suggesting that the author does not know much about what they are doing. A decent test to compare whether two distributions are similar is not measuring the p-value (of what hypothesis test?) and then compare the p-values. This tells me that there are little chances to even attempt to reproduce the results.

2) A p-value of a difference of proportions following a binomial distribution is my best guess of what they did, because it's simple for someone who doesn't have technical knowledge, and it's ok to do in excel (see https://courses.lumenlearning.com/intro ... oportions/). However, sharpe ratios do not follow binomial distributions, so even if they did that, it'd be wrong

3) If you want to compare two distributions of sharpe ratio for small caps vs large caps this might give you a better hint, requires knowledge, and R (which is free). https://stats.stackexchange.com/questio ... stribution

4) To complete, I have nothing to add to your original question that opened the post. other already have answered very well.
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