Long-run Sharpe Ratios in an ideal world

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TD2626
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Long-run Sharpe Ratios in an ideal world

Post by TD2626 » Thu Aug 24, 2017 3:05 pm

I have a question about theory that I've been pondering:

Assuming that the strong form of the efficient market hypothesis holds, would all diversified investments have identical Sharpe ratios (before expenses) if an investment has an infinite time horizon and no black swan events occur?

(I know these approximations aren't perfect - in reality markets are fairly efficient but not perfectly efficient and time horizons for many investors may be "long" but not infinitely so).

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Re: Long-run Sharpe Ratios in an ideal world

Post by SimpleGift » Thu Aug 24, 2017 6:57 pm

TD2626 wrote:
Thu Aug 24, 2017 3:05 pm
Assuming that the strong form of the efficient market hypothesis holds, would all diversified investments have identical Sharpe ratios (before expenses) if an investment has an infinite time horizon and no black swan events occur?
I've wondered about this question myself. The only empirical evidence I’ve seen is from the study discussed in this recent Forum thread, which reported global stock and bonds returns and standard deviations for the 145-year period from 1870-2015. These are the Sharpe Ratios I calculated (i.e., the ratio between average annual returns above the risk-free rate and the annual standard deviation of those returns):
  • Bonds……..…...….0.19
    Stocks………...…..0.27
They’re not that different over almost a century-and-a-half — but I'd be interested to hear the theoretical justification.
Last edited by SimpleGift on Thu Aug 24, 2017 7:44 pm, edited 1 time in total.
Cordially, Todd

bigred77
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Re: Long-run Sharpe Ratios in an ideal world

Post by bigred77 » Thu Aug 24, 2017 7:15 pm

I'm very interested to read this thread. I think it's an excellent question.

I would say no, even if markets were perfectly efficient sharp ratios would not be equal, because there are still other real world constraints.

Investors don't have free access to leverage at the risk free rate to get an efficient portfolio up to their desired risk level (or risk tolerance) which means to compensate they could crowd into "riskier assets" to bear more risk, even if that causes the returns from those assets to be less efficient than the lower risk assets.

Regulations can distort supply and demand (I'm thinking of rules that limit some entities to only hold investment grade fixed income).

Uncertainty. We can never perfectly predict the future. We only have our expectations. Sharp ratios measure "after the fact" results which can be distorted by black swan events. That doesn't mean the EMH has been violated. That just means the risk showed up.

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Re: Long-run Sharpe Ratios in an ideal world

Post by jbolden1517 » Thu Aug 24, 2017 7:34 pm

TD2626 wrote:
Thu Aug 24, 2017 3:05 pm
I have a question about theory that I've been pondering:

Assuming that the strong form of the efficient market hypothesis holds, would all diversified investments have identical Sharpe ratios (before expenses) if an investment has an infinite time horizon and no black swan events occur?

(I know these approximations aren't perfect - in reality markets are fairly efficient but not perfectly efficient and time horizons for many investors may be "long" but not infinitely so).
No not in general. Assuming strong EMH two assets with perfect correlation would always have identical Sharpe ratios. If there are two assets X and Y that don't perfectly correlate then they would get bid up or down in price relative to one another so that if m represents the market weight of X relative to Y: the asset (mX + Y) would have better risk adjusted return (Sharpe ratio) than either X or Y.

This is BTW one of the big problems with the Boglehead EMH. You can either have:
a) The price of a stock is the best (perfect) estimate of its individual future dividends discounted at the market rate
b) The price of a stock is the best (perfect) estimate of its individual future dividends discounted relative to the adjustment based on its covariances with all other stocks

but not both. Price gives you one degree of freedom. Only one equation at most can be satisfied.

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Re: Long-run Sharpe Ratios in an ideal world

Post by bigred77 » Thu Aug 24, 2017 7:49 pm

jbolden1517 wrote:
Thu Aug 24, 2017 7:34 pm
TD2626 wrote:
Thu Aug 24, 2017 3:05 pm
I have a question about theory that I've been pondering:

Assuming that the strong form of the efficient market hypothesis holds, would all diversified investments have identical Sharpe ratios (before expenses) if an investment has an infinite time horizon and no black swan events occur?

(I know these approximations aren't perfect - in reality markets are fairly efficient but not perfectly efficient and time horizons for many investors may be "long" but not infinitely so).
No not in general. Assuming strong EMH two assets with perfect correlation would always have identical Sharpe ratios. If there are two assets X and Y that don't perfectly correlate then they would get bid up or down in price relative to one another so that if m represents the market weight of X relative to Y: the asset (mX + Y) would have better risk adjusted return (Sharpe ratio) than either X or Y.

This is BTW one of the big problems with the Boglehead EMH. You can either have:
a) The price of a stock is the best (perfect) estimate of its individual future dividends discounted at the market rate
b) The price of a stock is the best (perfect) estimate of its individual future dividends discounted relative to the adjustment based on its covariances with all other stocks

but not both. Price gives you one degree of freedom. Only one equation at most can be satisfied.
I would like to propose C.) The price of a stock is the best estimate of its individual future dividends (and buybacks) discounted at a rate deemed appropriate by the consensus of investors to compensate them for their perceived risk of that individual stock.

Wouldn't different discount rates be used for different individual stocks. Wouldn't the market rate only be appropriate for stocks with an individual beta of a perfect 1.00?

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Re: Long-run Sharpe Ratios in an ideal world

Post by jbolden1517 » Thu Aug 24, 2017 8:06 pm

bigred77 wrote:
Thu Aug 24, 2017 7:49 pm
I would like to propose C.) The price of a stock is the best estimate of its individual future dividends (and buybacks) discounted at a rate deemed appropriate by the consensus of investors to compensate them for their perceived risk of that individual stock.

Wouldn't different discount rates be used for different individual stocks. Wouldn't the market rate only be appropriate for stocks with an individual beta of a perfect 1.00?
If you are having them discounted at beta then (c) is just a variant of (a). If you are proposing some other discounting then (c) could be an entirely different option than either (a) or (b). Which is fine, my point was just that strong-EMH for stocks and CAPM (strong EMH at the stock market level) produce different results.

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Re: Long-run Sharpe Ratios in an ideal world

Post by TD2626 » Thu Aug 24, 2017 10:43 pm

My opinion is that to first approximation, Sharpe ratios would be equal. The strong form of the EMH seems to require this.

But - is there a second order effect due to correlations? Do efficient markets take into account low correlations?

MPT says that one can improve portfolio Sharpe ratio by buying risky assets that have low correlations (say, a 60/40 stocks/bond portfolio). However, MPT is widely known. Wouldn't, over long periods, perfectly efficient markets take into account correlations?

If this were the case, assets with the lowest correlations to the market portfolio would have slightly lower Sharpe ratios on their own, but low correlations would make up for this by improving overall portfolio Sharpe ratio via MPT.

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Re: Long-run Sharpe Ratios in an ideal world

Post by afan » Fri Aug 25, 2017 4:25 am

There is a trivial reason the Sharpe ratios would not be identical: variance is not the only risk that is priced. Thus, two assets could have identical risk adjusted returns but different Sharpe ratios.

In the real world, in addition to the distortion caused by inability to borrow at the risk free rate, there are big difference in tax rates between unrealized capital gains, realized gains, and different forms of dividends. Theory ignores these effects.
We don't know how to beat the market on a risk-adjusted basis, and we don't know anyone that does know either | --Swedroe | We assume that markets are efficient, that prices are right | --Fama

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Re: Long-run Sharpe Ratios in an ideal world

Post by afan » Fri Aug 25, 2017 10:24 am

Also, keep in mind that stocks are priced prospectively, based on what people expect the risk adjusted returns to be. Sharpe ratios are calculated retrospectively, based on what the return and variances turned out to be. Thus, investments are priced by the market to have equivalent EXPECTED risk adjusted returns. But the forecast expected returns and variances are almost never exactly correct. It would be remarkable if they were.

So two investments with identical expected risk adjusted returns would be highly unlikely to have identical observed risk adjusted returns. Long run does not help here.
We don't know how to beat the market on a risk-adjusted basis, and we don't know anyone that does know either | --Swedroe | We assume that markets are efficient, that prices are right | --Fama

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Re: Long-run Sharpe Ratios in an ideal world

Post by rkhusky » Fri Aug 25, 2017 10:40 am

afan wrote:
Fri Aug 25, 2017 4:25 am
There is a trivial reason the Sharpe ratios would not be identical: variance is not the only risk that is priced. Thus, two assets could have identical risk adjusted returns but different Sharpe ratios.

In the real world, in addition to the distortion caused by inability to borrow at the risk free rate, there are big difference in tax rates between unrealized capital gains, realized gains, and different forms of dividends. Theory ignores these effects.
+1
Standard deviation is only an approximation for risk. And, even if it was the definition for risk, who's to say that it is related to price in the form of the Sharpe ratio (i.e. a plot of return vs. risk is a straight line)?

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Re: Long-run Sharpe Ratios in an ideal world

Post by Swelfie » Fri Aug 25, 2017 10:48 am

rkhusky wrote:
Fri Aug 25, 2017 10:40 am
afan wrote:
Fri Aug 25, 2017 4:25 am
There is a trivial reason the Sharpe ratios would not be identical: variance is not the only risk that is priced. Thus, two assets could have identical risk adjusted returns but different Sharpe ratios.

In the real world, in addition to the distortion caused by inability to borrow at the risk free rate, there are big difference in tax rates between unrealized capital gains, realized gains, and different forms of dividends. Theory ignores these effects.
+1
Standard deviation is only an approximation for risk. And, even if it was the definition for risk, who's to say that it is related to price in the form of the Sharpe ratio (i.e. a plot of return vs. risk is a straight line)?
Not to mention, not even taking into account other types of risk, the Sharpe ratio ONLY deals with the variance and modality of a standard distribution. It completely ignored the skewness and kertosis of returns as well as return distributions that are not standard. A perfectly efficient market would correlate their assets across all of these factors and would likely choose a portfolio quite different than what the Sharpe ratio would suggest.

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Re: Long-run Sharpe Ratios in an ideal world

Post by TD2626 » Fri Aug 25, 2017 9:57 pm

Swelfie wrote:
Fri Aug 25, 2017 10:48 am
rkhusky wrote:
Fri Aug 25, 2017 10:40 am
afan wrote:
Fri Aug 25, 2017 4:25 am
There is a trivial reason the Sharpe ratios would not be identical: variance is not the only risk that is priced. Thus, two assets could have identical risk adjusted returns but different Sharpe ratios.

In the real world, in addition to the distortion caused by inability to borrow at the risk free rate, there are big difference in tax rates between unrealized capital gains, realized gains, and different forms of dividends. Theory ignores these effects.
+1
Standard deviation is only an approximation for risk. And, even if it was the definition for risk, who's to say that it is related to price in the form of the Sharpe ratio (i.e. a plot of return vs. risk is a straight line)?
Not to mention, not even taking into account other types of risk, the Sharpe ratio ONLY deals with the variance and modality of a standard distribution. It completely ignored the skewness and kertosis of returns as well as return distributions that are not standard. A perfectly efficient market would correlate their assets across all of these factors and would likely choose a portfolio quite different than what the Sharpe ratio would suggest.
As I discussed towards the end of this thread (viewtopic.php?f=10&t=222695&p=3438116), I think that there are two primary effects that make returns not perfectly normally distributed:
1: Kurtosis risk/fat tail risk/black swan risk
2. Autocorrelation/mean reversion/serial negative correlation

I think that it may be the case that these effects, in theory, could partially cancel each other out over very long time periods. (I.e. after a fat tail risk event, things revert to the mean). If this is true, and the "Gaussian approximation" is a relatively decent one, then standard deviation is roughly equal to risk.

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Re: Long-run Sharpe Ratios in an ideal world

Post by TD2626 » Fri Aug 25, 2017 10:11 pm

Simplegift wrote:
Thu Aug 24, 2017 6:57 pm
TD2626 wrote:
Thu Aug 24, 2017 3:05 pm
Assuming that the strong form of the efficient market hypothesis holds, would all diversified investments have identical Sharpe ratios (before expenses) if an investment has an infinite time horizon and no black swan events occur?
I've wondered about this question myself. The only empirical evidence I’ve seen is from the study discussed in this recent Forum thread, which reported global stock and bonds returns and standard deviations for the 145-year period from 1870-2015. These are the Sharpe Ratios I calculated (i.e., the ratio between average annual returns above the risk-free rate and the annual standard deviation of those returns):
  • Bonds……..…...….0.19
    Stocks………...…..0.27
They’re not that different over almost a century-and-a-half — but I'd be interested to hear the theoretical justification.
I read through the tread you cited and looked through some empirical evidence myself - and it seems as though:
1. Sharpe ratios do seem to be suspiciously close to each other over long periods
2. There's probably not enough empirical evidence alone to conclusively prove anything one way or the other.

0.19 vs 0.27 is suspiciously similar - but it's not conclusive (it would be conclusive if it were, say, 0.2345 vs 0.2346).

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Re: Long-run Sharpe Ratios in an ideal world

Post by TD2626 » Fri Aug 25, 2017 10:30 pm

TD2626 wrote:
Thu Aug 24, 2017 10:43 pm
My opinion is that to first approximation, Sharpe ratios would be equal. The strong form of the EMH seems to require this.

But - is there a second order effect due to correlations? Do efficient markets take into account low correlations?

MPT says that one can improve portfolio Sharpe ratio by buying risky assets that have low correlations (say, a 60/40 stocks/bond portfolio). However, MPT is widely known. Wouldn't, over long periods, perfectly efficient markets take into account correlations?

If this were the case, assets with the lowest correlations to the market portfolio would have slightly lower Sharpe ratios on their own, but low correlations would make up for this by improving overall portfolio Sharpe ratio via MPT.
Expanding on the implications of this:

If this is the case, the assets with the highest Sharpe ratios by themselves would be those with characteristics of both stocks and bonds (as they have a high correlation to a market portfolio of 60/40 stocks and bonds).

These sorts of assets are supposedly "safer" stocks (dividend stocks/widows & orphans stocks/blue chips/utilities), "riskier" bonds (long-term and/or lower credit quality corporate and high-yield bonds), and hybrid securities (convertible bonds and preferred stocks)

Would this seem to suggest one could construct a theoretical "inverse Larry Portfolio" that has a slight tilt toward diversified funds that hold some of those aforementioned moderate-risk, moderate-return assets?

The Larry portfolio tilts towards low risk, low reward (short term bonds) and simultaneously tilts towards high risk, high reward (small-value stock).
One could envision an opposite portfolio that instead tilts towards intermediate risk, intermediate return (dividend stocks/long-term corporate bonds). If both portfolios (LP and inverse-LP) are constructed to have the same expected risk, which would have the higher return given the conditions described in my original post in this thread?

(Note: This "inverse Larry portfolio" is based on a lot of questionable assumptions and approximations and is for theoretical debate only. Also, it is worth noting that it isn't supported by historical data.)

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Re: Long-run Sharpe Ratios in an ideal world

Post by asset_chaos » Fri Aug 25, 2017 10:37 pm

At the risk of being roundly criticized for ignoring correlations, stock prices and other things, there is a mean field type of argument one can make that only requires there be a relationship between the diversified portfolio return minus the risk-free return, call it R, and the riskiness of the diversified portfolio, call it r, defined in whatever way you like, but presumably defined in some kind of consistent and measurable way. Since the original question asked about an ideal world, I shall idealize. If you can plot a curve of R versus r, then by definition the Sharpe ratio is the ratio of the coordinates of any point along that curve, and it's fairly easy to show by taking the derivative of Sharpe ratio with respect to r that these two statements are equivalent:

1) if the Sharpe ratio does not change with risk level, then
2) the return and risk (R versus r) plot along a straight line with the Sharpe ratio as its slope. And vice versa.

So plot R versus r obtained over a long time and for whatever set of diversified portfolios you care about and see what it does. If the curve is a straight line, then all risky diversified portfolios have the same Sharpe ratio. I don't know what it does. Actually, I have Bernstein's Intelligent Asset Allocator within reach, and his table 2.1 gives return and standard deviation for several kinds of US stocks and bonds over the time 1926 to 1998. And the plot of R versus r is not a straight line. It's a reasonable straight line at lower risk, but at higher risk the curve flattens out. Empirically over that time the Sharpe ratio was approximately a constant for lower risk and declined at higher risk. As a further check, there is IFA's data https://www.ifa.com/portfolios/risk-and-return/ for diversified portfolios that systematically vary the risk, and their data appears to cover 1928-2016. If I take the IFA graph at face value, those diversified portfolios are much closer to a straight line than Bernstein's asset class data; however, the IFA curve clearly still flattens at higher risk levels. The conclusion is the same, but diversified portfolios appear to be somewhat closer to converging to the same Sharpe ratio than asset classes.

A simple analysis and a bit of reasonably long term data support the conclusion that Sharpe ratios do not converge to identical values for all risk levels. There seem to be reasonably convergent Sharpe ratios at lower risk levels and a decline of Sharpe ratios at higher risk levels.
Regards, | | Guy

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Re: Long-run Sharpe Ratios in an ideal world

Post by afan » Sat Aug 26, 2017 9:16 am

You are still.assuming there is a single dimension of risk. The data say that at least variance, skewness and kurtosis are priced. That means there is not a single metric of risk adjusted returns. You need to account for, at least, three risk metrics. The Sharpe ratio does not account for this. It is not a matter of picking a different definition of risk, but recognizing that you have multiple risk measure. You cannot plot risk vs return on a two dimensional chart.
We don't know how to beat the market on a risk-adjusted basis, and we don't know anyone that does know either | --Swedroe | We assume that markets are efficient, that prices are right | --Fama

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Re: Long-run Sharpe Ratios in an ideal world

Post by acanthurus » Sat Aug 26, 2017 2:48 pm

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Last edited by acanthurus on Tue Oct 31, 2017 6:29 pm, edited 1 time in total.

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Re: Long-run Sharpe Ratios in an ideal world

Post by TD2626 » Sun Aug 27, 2017 1:17 pm

acanthurus wrote:
Sat Aug 26, 2017 2:48 pm
TD2626 wrote:
Fri Aug 25, 2017 9:57 pm
As I discussed towards the end of this thread (viewtopic.php?f=10&t=222695&p=3438116), I think that there are two primary effects that make returns not perfectly normally distributed:
1: Kurtosis risk/fat tail risk/black swan risk
2. Autocorrelation/mean reversion/serial negative correlation

I think that it may be the case that these effects, in theory, could partially cancel each other out over very long time periods. (I.e. after a fat tail risk event, things revert to the mean). If this is true, and the "Gaussian approximation" is a relatively decent one, then standard deviation is roughly equal to risk.
I apologize in advance if my comment is not quite up to the level of discussion going on, as you all seem quite knowledgeable, but the mention of kurtosis and market returns not being perfectly normally distributed caught my attention. For the overall market to exhibit a normal distribution without excess kurtosis over any time period, I would think the movements of every individual stock would need to be independent of all other stocks. If this independence holds, then by the central limit theorem the market in aggregate should exhibit normally distributed returns without any excess kurtosis. Once independent behavior breaks down, that's no longer guaranteed.

If one looks at historical kurtosis over short (<=1 year) rolling time periods, you see massive spikes around major market corrections and euphoria. These "running for the exits" periods of history would definitely cause independent behavior of stocks to break down. Indeed, the longer the rolling time period, the less excess kurtosis, with a 252-day kurtosis having a mean value of 3.8.

http://www.portfolioprobe.com/2012/02/1 ... s-history/

This may have been obvious to you guys, but I though some graphs might be nice, and it does seem to support your idea that over long time periods the gaussian approximation becomes more correct.
Thank you for the link, it was very helpful.

I was searching through the forum and found this old thread (viewtopic.php?f=10&t=28687), where forum member sgr000 did some remarkably thorough analysis. In one post on the first page, statistical tests run on annual returns showed that there wasn't enough evidence to show that returns aren't Gaussian. A later post on the second page showed that monthly returns suggested fat tail risk.

To quote the conclusion from that analysis:
sgr000 wrote:
Sat Dec 06, 2008 2:29 pm
Summary:
  • Using Shiller's yearly data, there is no convincing evidence of non-normality in the distribution of returns.
  • This may be an averaging effect, where fat-tailed daily returns give rise to less fat-tailed monthly returns which in turn give rise to very nearly normal yearly returns.
  • However, with N=136 data points in the yearly dataset, this means we can't depend on seeing events with probability much less than 1%, i.e, we can't look out beyond maybe 3 sigma. So who knows?
  • However, the monthly data over the same time period show dramatic evidence of non-normality, exhibiting both left and right fat tails (kurtosis > 3). I.e., "improbable" things are much more probable than you think.
  • The quantile-quantile plots both go much further out to the right than to the left (15 vs 10 sigma), which is strong indication of positive skewness as well. I.e., "good improbable" things are more probable than "bad improbable" things.
  • It is unclear what, if any, impact this has on a long-term, low-cost, broadly diversified, index-like portfolio.
In my opinion, over short time periods, there are fat tails - but over longer time periods, reversion to the mean could cancel this out. After a sudden, precipitous drop in asset prices, the market comes to its senses. This all happens within a single year. (This analysis involves a lot of approximations and may be incorrect; a black swan risk could cause assets to fall 100%, resulting in no chance of recovery).

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Re: Long-run Sharpe Ratios in an ideal world

Post by TD2626 » Sun Aug 27, 2017 1:32 pm

afan wrote:
Sat Aug 26, 2017 9:16 am
You are still.assuming there is a single dimension of risk. The data say that at least variance, skewness and kurtosis are priced. That means there is not a single metric of risk adjusted returns. You need to account for, at least, three risk metrics. The Sharpe ratio does not account for this. It is not a matter of picking a different definition of risk, but recognizing that you have multiple risk measure. You cannot plot risk vs return on a two dimensional chart.
It is entirely possible to plot risk vs return on a 2-D chart - but whether this is a reasonable approximation is up for debate. I feel it is reasonable to draw up an efficient frontier plot, but others may say that that is making too many approximations.

Is there another type of plot that could work better? A 3-dimensional chart? Or an alternative measure of risk (other than the Sharpe ratio)? I know that people often use other measures... but I think using standard deviation as risk and Sharpe ratio as risk-adjusted return is very reasonable. If we want to use Sorotino Ratio, Treynor ratio, etc, that could be done, though. And CVAR or downside risk instead of standard deviation can aldo be used.

If someone could produce a 3-d chart or a better measure of risk (that took into account skew and kurtosis, as well as an investors goals and loss aversion), that would be interesting.

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Re: Long-run Sharpe Ratios in an ideal world

Post by TD2626 » Sun Aug 27, 2017 2:01 pm

asset_chaos wrote:
Fri Aug 25, 2017 10:37 pm
At the risk of being roundly criticized for ignoring correlations, stock prices and other things, there is a mean field type of argument one can make that only requires there be a relationship between the diversified portfolio return minus the risk-free return, call it R, and the riskiness of the diversified portfolio, call it r, defined in whatever way you like, but presumably defined in some kind of consistent and measurable way. Since the original question asked about an ideal world, I shall idealize. If you can plot a curve of R versus r, then by definition the Sharpe ratio is the ratio of the coordinates of any point along that curve, and it's fairly easy to show by taking the derivative of Sharpe ratio with respect to r that these two statements are equivalent:

1) if the Sharpe ratio does not change with risk level, then
2) the return and risk (R versus r) plot along a straight line with the Sharpe ratio as its slope. And vice versa.

So plot R versus r obtained over a long time and for whatever set of diversified portfolios you care about and see what it does. If the curve is a straight line, then all risky diversified portfolios have the same Sharpe ratio. I don't know what it does. Actually, I have Bernstein's Intelligent Asset Allocator within reach, and his table 2.1 gives return and standard deviation for several kinds of US stocks and bonds over the time 1926 to 1998. And the plot of R versus r is not a straight line. It's a reasonable straight line at lower risk, but at higher risk the curve flattens out. Empirically over that time the Sharpe ratio was approximately a constant for lower risk and declined at higher risk. As a further check, there is IFA's data https://www.ifa.com/portfolios/risk-and-return/ for diversified portfolios that systematically vary the risk, and their data appears to cover 1928-2016. If I take the IFA graph at face value, those diversified portfolios are much closer to a straight line than Bernstein's asset class data; however, the IFA curve clearly still flattens at higher risk levels. The conclusion is the same, but diversified portfolios appear to be somewhat closer to converging to the same Sharpe ratio than asset classes.

A simple analysis and a bit of reasonably long term data support the conclusion that Sharpe ratios do not converge to identical values for all risk levels. There seem to be reasonably convergent Sharpe ratios at lower risk levels and a decline of Sharpe ratios at higher risk levels.
I like your idea of thinking of Sharpe ratio as the slope of the risk vs return plot. I think the issues that are seen may be a result of several things:

1. There could be an issue at risk=0. Sharpe ratio has risk in the denominator, and as you approach risk-free securities, you could have an issue. Standard deviations can't be negative - you can't have negative risk. So there may be an asymptote at x=0. Note, though, that in theory, the numerator should be zero at risk=0 (as with no risk, you would get the risk free rate), so there may not have to be an issue. You do have a 0 divided by 0 case here, though.

2. There could be an issue as risk gets large. Say you extend the chart out to some of the riskiest securities - for example, Emerging Market value. Say these securities (for illustrative purposes only) have a standard deviation of 35% and an expected return of 8%. There's a 0.1% chance (from http://www.calculator.net/probability-c ... &x=87&y=30) that returns are below -100% if using standard deviation = risk. Those who invest in broadly diversified funds (without a margin account) are in theory not at risk of loosing more than 100%, so this means that a part of the left tail is clipped off. Essentially, stock prices can't decline below $0.00, but the standard deviation/normal distribution doesn't take this into account. Thus, one could be seeing the Sharpe ratio declining for the riskiest securities.

3. This could be a figment of the short time span of data we have. We would need hundreds of years of far more accurate data to really test this theory empirically.

4. This could be a result of many investors aiming for "return at any cost". They want the best-returning investments and are willing to accept a low Sharpe ratio in exchange. If this is the case, though, I would think it would be a temporary, behavioral thing instead of a long-term, stable phenomenon.

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Re: Long-run Sharpe Ratios in an ideal world

Post by afan » Sun Aug 27, 2017 3:39 pm

TD2626 wrote:
Sun Aug 27, 2017 1:32 pm

It is entirely possible to plot risk vs return on a 2-D chart
By definition if you use a 2D plot you are only plotting two variables. This cannot work if there are more than 2 variables to plot.
TD2626 wrote:
Sun Aug 27, 2017 1:32 pm
- but whether this is a reasonable approximation is up for debate.
It is not a "debate", there are ample data showing that risks beyond variance are priced.
TD2626 wrote:
Sun Aug 27, 2017 1:32 pm
Is there another type of plot that could work better? A 3-dimensional chart?
At least 4 dimensions, whether you need more is an area of ongoing research.
TD2626 wrote:
Sun Aug 27, 2017 1:32 pm
I know that people often use other measures... but I think using standard deviation as risk and Sharpe ratio as risk-adjusted return is very reasonable. If we want to use Sorotino Ratio, Treynor ratio, etc, that could be done, though.
Those all use a single risk metric.
TD2626 wrote:
Sun Aug 27, 2017 1:32 pm
If someone could produce a 3-d chart or a better measure of risk (that took into account skew and kurtosis, as well as an investors goals and loss aversion), that would be interesting.
For an introduction to the problem with Sharpe ratios, see the paper "Sharpening Sharpe Ratios". Other good papers on the topic of multidimensional risk

"The Market Price of Skewness"
"Asset Pricing When Returns are Nonnormal"
"Effects of skewness and kurtosis on portfolio
rankings"
"On the Direction of Preference for Moments of Higher Order than the Variance"

And the references therein.

It is a fascinating body of literature. There is no question that elements of risk other than variance are priced in the market. The relative weights of the higher moments certainly has not been reliably determined. Since there is more than one dimension of risk, there is no reason that everyone has to have the same risk preferences. One person could care more about minimizing variance, while another found high skewness to be relatively more desirable. These two investors would rank a set of portfolios differently because they were looking for different rewards and seeking to avoid different types of risk.

In theory you could have a multidimensional plot of X variables with local maxima identified. There would probably be some portfolios that had so much risk on multiple measures or such low return that most people would find them to be inferior.

The only data on real world investor risk preferences comes from extracting it from the actual behavior of market prices.

Since very few people know their utility functions, if you showed them multiple portfolios with known mean, variance, skewness and kurtosis they would not know how to identify the portfolios that would make them the happiest.

Complicated, but really interesting. As you will see when you dive into the papers, these are just a handful of the work.

Most are available in near final form from SSRN.
We don't know how to beat the market on a risk-adjusted basis, and we don't know anyone that does know either | --Swedroe | We assume that markets are efficient, that prices are right | --Fama

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Re: Long-run Sharpe Ratios in an ideal world

Post by asset_chaos » Mon Aug 28, 2017 3:45 am

TD2626 wrote:
Sun Aug 27, 2017 2:01 pm
asset_chaos wrote:
Fri Aug 25, 2017 10:37 pm
At the risk of being roundly criticized for ignoring correlations, stock prices and other things, there is a mean field type of argument one can make that only requires there be a relationship between the diversified portfolio return minus the risk-free return, call it R, and the riskiness of the diversified portfolio, call it r, defined in whatever way you like, but presumably defined in some kind of consistent and measurable way. Since the original question asked about an ideal world, I shall idealize. If you can plot a curve of R versus r, then by definition the Sharpe ratio is the ratio of the coordinates of any point along that curve, and it's fairly easy to show by taking the derivative of Sharpe ratio with respect to r that these two statements are equivalent:

1) if the Sharpe ratio does not change with risk level, then
2) the return and risk (R versus r) plot along a straight line with the Sharpe ratio as its slope. And vice versa.

So plot R versus r obtained over a long time and for whatever set of diversified portfolios you care about and see what it does. If the curve is a straight line, then all risky diversified portfolios have the same Sharpe ratio. I don't know what it does. Actually, I have Bernstein's Intelligent Asset Allocator within reach, and his table 2.1 gives return and standard deviation for several kinds of US stocks and bonds over the time 1926 to 1998. And the plot of R versus r is not a straight line. It's a reasonable straight line at lower risk, but at higher risk the curve flattens out. Empirically over that time the Sharpe ratio was approximately a constant for lower risk and declined at higher risk. As a further check, there is IFA's data https://www.ifa.com/portfolios/risk-and-return/ for diversified portfolios that systematically vary the risk, and their data appears to cover 1928-2016. If I take the IFA graph at face value, those diversified portfolios are much closer to a straight line than Bernstein's asset class data; however, the IFA curve clearly still flattens at higher risk levels. The conclusion is the same, but diversified portfolios appear to be somewhat closer to converging to the same Sharpe ratio than asset classes.

A simple analysis and a bit of reasonably long term data support the conclusion that Sharpe ratios do not converge to identical values for all risk levels. There seem to be reasonably convergent Sharpe ratios at lower risk levels and a decline of Sharpe ratios at higher risk levels.
I like your idea of thinking of Sharpe ratio as the slope of the risk vs return plot. I think the issues that are seen may be a result of several things:
The Sharpe ratio is not the slope of that curve. The Sharpe ratio is the ratio of the coordinates at any point along the curve (y/x, not dy/dx). But the change in Sharpe ratio is related to the slope of that curve, and in the special case of the curve being a straight line with the same slope everywhere, then in that one special case the slope is the Sharpe ratio.
1. There could be an issue at risk=0. Sharpe ratio has risk in the denominator, and as you approach risk-free securities, you could have an issue. Standard deviations can't be negative - you can't have negative risk. So there may be an asymptote at x=0. Note, though, that in theory, the numerator should be zero at risk=0 (as with no risk, you would get the risk free rate), so there may not have to be an issue. You do have a 0 divided by 0 case here, though.
I think a singularity at zero risk is unreasonable, so in practice things are likely to go smoothly to zero, and data supports this.
2. There could be an issue as risk gets large. Say you extend the chart out to some of the riskiest securities - for example, Emerging Market value. Say these securities (for illustrative purposes only) have a standard deviation of 35% and an expected return of 8%. There's a 0.1% chance (from http://www.calculator.net/probability-c ... &x=87&y=30) that returns are below -100% if using standard deviation = risk. Those who invest in broadly diversified funds (without a margin account) are in theory not at risk of loosing more than 100%, so this means that a part of the left tail is clipped off. Essentially, stock prices can't decline below $0.00, but the standard deviation/normal distribution doesn't take this into account. Thus, one could be seeing the Sharpe ratio declining for the riskiest securities.
One sees the Sharpe ratio declining for the riskiest asset classes and diversified portfolios. Why that happens, I certainly don't know.
3. This could be a figment of the short time span of data we have. We would need hundreds of years of far more accurate data to really test this theory empirically.
If markets are not stable in the mean over a century or so, then it may be dubious to say anything about expected risk-adjusted return. Over several centuries I would worry that markets and economies would change structurally so much that comparisons from one era to another are unlikely to be useful. Over centuries the only investing constant I'd be reasonably sure of is the enduring nature of greed and fear of humans.
4. This could be a result of many investors aiming for "return at any cost". They want the best-returning investments and are willing to accept a low Sharpe ratio in exchange. If this is the case, though, I would think it would be a temporary, behavioral thing instead of a long-term, stable phenomenon.
I don't know.
Regards, | | Guy

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Re: Long-run Sharpe Ratios in an ideal world

Post by TD2626 » Tue Aug 29, 2017 12:31 pm

afan wrote:
Sun Aug 27, 2017 3:39 pm
TD2626 wrote:
Sun Aug 27, 2017 1:32 pm

It is entirely possible to plot risk vs return on a 2-D chart
By definition if you use a 2D plot you are only plotting two variables. This cannot work if there are more than 2 variables to plot.
TD2626 wrote:
Sun Aug 27, 2017 1:32 pm
- but whether this is a reasonable approximation is up for debate.
It is not a "debate", there are ample data showing that risks beyond variance are priced.
TD2626 wrote:
Sun Aug 27, 2017 1:32 pm
Is there another type of plot that could work better? A 3-dimensional chart?
At least 4 dimensions, whether you need more is an area of ongoing research.
TD2626 wrote:
Sun Aug 27, 2017 1:32 pm
I know that people often use other measures... but I think using standard deviation as risk and Sharpe ratio as risk-adjusted return is very reasonable. If we want to use Sorotino Ratio, Treynor ratio, etc, that could be done, though.
Those all use a single risk metric.
TD2626 wrote:
Sun Aug 27, 2017 1:32 pm
If someone could produce a 3-d chart or a better measure of risk (that took into account skew and kurtosis, as well as an investors goals and loss aversion), that would be interesting.
For an introduction to the problem with Sharpe ratios, see the paper "Sharpening Sharpe Ratios". Other good papers on the topic of multidimensional risk

"The Market Price of Skewness"
"Asset Pricing When Returns are Nonnormal"
"Effects of skewness and kurtosis on portfolio
rankings"
"On the Direction of Preference for Moments of Higher Order than the Variance"

And the references therein.

It is a fascinating body of literature. There is no question that elements of risk other than variance are priced in the market. The relative weights of the higher moments certainly has not been reliably determined. Since there is more than one dimension of risk, there is no reason that everyone has to have the same risk preferences. One person could care more about minimizing variance, while another found high skewness to be relatively more desirable. These two investors would rank a set of portfolios differently because they were looking for different rewards and seeking to avoid different types of risk.

In theory you could have a multidimensional plot of X variables with local maxima identified. There would probably be some portfolios that had so much risk on multiple measures or such low return that most people would find them to be inferior.

The only data on real world investor risk preferences comes from extracting it from the actual behavior of market prices.

Since very few people know their utility functions, if you showed them multiple portfolios with known mean, variance, skewness and kurtosis they would not know how to identify the portfolios that would make them the happiest.

Complicated, but really interesting. As you will see when you dive into the papers, these are just a handful of the work.

Most are available in near final form from SSRN.
Thank you for your response and paper reading suggestions. I read through many of these papers and found some interesting results. For example:

1. There are some proposals to create measures of "adjusted Sharpe ratio" that take into account skew and kurtosis
2. Investors appear to want positive odd central moments and low even central moments. Thus, investors want high & positive means and skews and low variance and kurtosis.

As I mentioned earlier in another thread (viewtopic.php?t=218859) it may be the case that since stock funds appear to have negative skew and individual stocks appear to have positive skew, a portfolio that has both stock funds and individual stocks may have zero skew. I think this characterizes my situation.

If high kurtosis is reduced over long time periods through reversion to the mean, then my portfolio may also have zero kurtosis over the long run. Thus, the Gaussian approximation appears reasonable to me over long time periods in my situation.

I think that risk measures other than Sharpe ratios are not ideal in my situation. You can measure downside risk - but this is only relevant for someone who is, for example, saving for a $X house in T years and wants to know the % chance they will have less than X in T years. I (and most investors, in my opinion) am focused on long-term return vs risk in general, with no single number X below which a purchase would be unobtainable and the outcome would have a sudden jump from good to bad.

It is true that it's unfortunate that standard deviation penalizes upside risk just as much as downside risk - but unless one has a $X below which success is unobtainable and above which no more money is needed, standard deviation seems to be the best measure of risk to me.

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Re: Long-run Sharpe Ratios in an ideal world

Post by TD2626 » Tue Aug 29, 2017 12:58 pm

asset_chaos wrote:
Mon Aug 28, 2017 3:45 am
TD2626 wrote:
Sun Aug 27, 2017 2:01 pm
asset_chaos wrote:
Fri Aug 25, 2017 10:37 pm
At the risk of being roundly criticized for ignoring correlations, stock prices and other things, there is a mean field type of argument one can make that only requires there be a relationship between the diversified portfolio return minus the risk-free return, call it R, and the riskiness of the diversified portfolio, call it r, defined in whatever way you like, but presumably defined in some kind of consistent and measurable way. Since the original question asked about an ideal world, I shall idealize. If you can plot a curve of R versus r, then by definition the Sharpe ratio is the ratio of the coordinates of any point along that curve, and it's fairly easy to show by taking the derivative of Sharpe ratio with respect to r that these two statements are equivalent:

1) if the Sharpe ratio does not change with risk level, then
2) the return and risk (R versus r) plot along a straight line with the Sharpe ratio as its slope. And vice versa.

So plot R versus r obtained over a long time and for whatever set of diversified portfolios you care about and see what it does. If the curve is a straight line, then all risky diversified portfolios have the same Sharpe ratio. I don't know what it does. Actually, I have Bernstein's Intelligent Asset Allocator within reach, and his table 2.1 gives return and standard deviation for several kinds of US stocks and bonds over the time 1926 to 1998. And the plot of R versus r is not a straight line. It's a reasonable straight line at lower risk, but at higher risk the curve flattens out. Empirically over that time the Sharpe ratio was approximately a constant for lower risk and declined at higher risk. As a further check, there is IFA's data https://www.ifa.com/portfolios/risk-and-return/ for diversified portfolios that systematically vary the risk, and their data appears to cover 1928-2016. If I take the IFA graph at face value, those diversified portfolios are much closer to a straight line than Bernstein's asset class data; however, the IFA curve clearly still flattens at higher risk levels. The conclusion is the same, but diversified portfolios appear to be somewhat closer to converging to the same Sharpe ratio than asset classes.

A simple analysis and a bit of reasonably long term data support the conclusion that Sharpe ratios do not converge to identical values for all risk levels. There seem to be reasonably convergent Sharpe ratios at lower risk levels and a decline of Sharpe ratios at higher risk levels.
I like your idea of thinking of Sharpe ratio as the slope of the risk vs return plot. I think the issues that are seen may be a result of several things:
The Sharpe ratio is not the slope of that curve. The Sharpe ratio is the ratio of the coordinates at any point along the curve (y/x, not dy/dx). But the change in Sharpe ratio is related to the slope of that curve, and in the special case of the curve being a straight line with the same slope everywhere, then in that one special case the slope is the Sharpe ratio.
1. There could be an issue at risk=0. Sharpe ratio has risk in the denominator, and as you approach risk-free securities, you could have an issue. Standard deviations can't be negative - you can't have negative risk. So there may be an asymptote at x=0. Note, though, that in theory, the numerator should be zero at risk=0 (as with no risk, you would get the risk free rate), so there may not have to be an issue. You do have a 0 divided by 0 case here, though.
I think a singularity at zero risk is unreasonable, so in practice things are likely to go smoothly to zero, and data supports this.
2. There could be an issue as risk gets large. Say you extend the chart out to some of the riskiest securities - for example, Emerging Market value. Say these securities (for illustrative purposes only) have a standard deviation of 35% and an expected return of 8%. There's a 0.1% chance (from http://www.calculator.net/probability-c ... &x=87&y=30) that returns are below -100% if using standard deviation = risk. Those who invest in broadly diversified funds (without a margin account) are in theory not at risk of loosing more than 100%, so this means that a part of the left tail is clipped off. Essentially, stock prices can't decline below $0.00, but the standard deviation/normal distribution doesn't take this into account. Thus, one could be seeing the Sharpe ratio declining for the riskiest securities.
One sees the Sharpe ratio declining for the riskiest asset classes and diversified portfolios. Why that happens, I certainly don't know.
3. This could be a figment of the short time span of data we have. We would need hundreds of years of far more accurate data to really test this theory empirically.
If markets are not stable in the mean over a century or so, then it may be dubious to say anything about expected risk-adjusted return. Over several centuries I would worry that markets and economies would change structurally so much that comparisons from one era to another are unlikely to be useful. Over centuries the only investing constant I'd be reasonably sure of is the enduring nature of greed and fear of humans.
4. This could be a result of many investors aiming for "return at any cost". They want the best-returning investments and are willing to accept a low Sharpe ratio in exchange. If this is the case, though, I would think it would be a temporary, behavioral thing instead of a long-term, stable phenomenon.
I don't know.

Thank you for your thorough response. Good point about the derivative (dy/dx) not equaling the y/x -based Sharpe ratio unless you have a straight line. Also, I agree with your point that the infinite-time horizon" approximation may not be realistic, since markets could have massive structural changes.

It's worthwhile to recall assumptions and approximations used often:

-The "infinite-time-horizon" approximation - short-term effects or the effects of a single business cycle are smoothed out because the investor is investing over a large number of business cycles. Probably reasonable if the investor's time horizon is 50+ years. Definitely dubious if the time horizon is less than 30 years.

-The "standard deviation equals risk" approximation - defining risk to equal standard deviation can lead to issues, but is probably reasonable in most cases.

-The "no behavioral errors" approximation - assuming that the investor is an rational, emotionless robot acting only in his or her own financial best interest at all times, making no behavioral errors and taking no emotional considerations into account. Based on the number of papers about kurtosis I've been reading lately, this rough approximation probably applies decently well to me - but it is a poor assumption for most investors, particularly those with no interest in or experience with investing.

-The "no black swan" assumption - assuming that over the time period considered, markets continue to function normally. By "normally", I mean "in their usual fashion" - and I also mean "normal" in the other sense of the word (Gaussian distribution). Events such as revolutions, nuclear war, asteroid impacts, and so forth obviously could cause investments to not perform as predicted. Thus, it is reasonable to say that "Assuming markets continue to operate, you have a 98% chance of meeting your investment goal by year 30 - but over the next 30 years, there is an X% chance of a black swan event like nuclear war, so in reality you actually have a (98-X) percent chance of meeting your goal.

-Other assumptions are also made. These are too numerous to list. There are many unknown unknowns.

I think that the benefits of making these approximations outweigh the issues that the approximations cause. However, I do doubt whether doing too much based on the approximations holding is a good idea.

It seems as though the only answers that theory (with the approximations mentioned) can provide are either that: A) One should have a very simple, globally cap weighted portfolio based on, say, Vanguard Total World as well as Total domestic and international bond funds; or B) One should have a very complex portfolio with large number of tilts and small allocations in a quest for perfection. It's hard to know whether A or B are correct.

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Re: Long-run Sharpe Ratios in an ideal world

Post by siamond » Tue Sep 05, 2017 5:06 pm

I have a slight suspicion that the OP's question is somewhat circular.

Small-Cap-Value (US) is the typical example that comes to mind where the Sharpe ratio proved better over time than Total-Market. But then Small-Cap-Value is a big thorn on the side of strong EMH. So... SCV might not be a good counter-example if we assume Strong EMH. But then SCV is the real-world, and strong EMH seems to be little more than a fantasy (*). So... we're closed in a loop here! :wink:

Anyhoo, in the real world, risk is much more than volatility, and investors are driven by many more factors than volatility and return, which easily explains why Sharpe ratios end up squarely different for fairly long data series. I stopped believing that the Sharpe ratio was carrying a lot of useful semantics a few years ago. It's one mildly interesting metric, not the center of the financial universe.

(*) not my words, although I totally agree. I just read a classic "The New Finance" from Prof. Robert Haugen, who railed mercilessly at "The Fantasy" (the efficient market hypothesis) by lining up an entire booklet full of value/growth studies... He must have been quite a colorful character.

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Re: Long-run Sharpe Ratios in an ideal world

Post by SimpleGift » Thu Sep 07, 2017 3:50 pm

TD2626 wrote:
Thu Aug 24, 2017 3:05 pm
Assuming that the strong form of the efficient market hypothesis holds, would all diversified investments have identical Sharpe ratios (before expenses) if an investment has an infinite time horizon and no black swan events occur?
siamond wrote:
Tue Sep 05, 2017 5:06 pm
Anyhoo, in the real world, risk is much more than volatility, and investors are driven by many more factors than volatility and return, which easily explains why Sharpe ratios end up squarely different for fairly long data series.
Not knowing much about the theoretical basis for similar Sharpe Ratios among asset classes over long time periods, I couldn’t help but empirically check whether the Sharpe Ratios for stocks and bonds have been similar for the 14 countries over the one century of time for which we have good data.

The table below shows the Sharpe Ratios for stocks and bonds over the 101-year period from 1900-2000, using the Dimson, Marsh and Staunton data for stocks, and using the Jorda, Schularick and Taylor database for bonds:
  • Image
Two observations:
  • 1) The Sharpe Ratios for stocks and bonds, averaged over the 14 countries and a century of time, aren't all that different.

    2) The Sharpe Ratios for stocks appear fairly consistent between countries over the time period — while the risk-return relationship for bonds varies wildly by country, perhaps due to disparate episodes of hyperinflation and deflation during the century.
Any thoughts?
Cordially, Todd

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Re: Long-run Sharpe Ratios in an ideal world

Post by siamond » Thu Sep 07, 2017 4:16 pm

Simplegift wrote:
Thu Sep 07, 2017 3:50 pm
Not knowing much about the theoretical basis for similar Sharpe Ratios among asset classes over long time periods, I couldn’t help but empirically check whether the Sharpe Ratios for stocks and bonds have been similar for the 14 countries over the one century of time for which we have good data.
That's an interesting idea! Did you do the math in local currency or USD? Did you use the risk-free rate from each country, or the US one?

Also, did you find a way to get the dividends series from the database (I didn't), or did you apply the math to price series? If the latter, I am not too sure this is entirely meaningful.
Simplegift wrote:
Thu Sep 07, 2017 3:50 pm
1) The Sharpe Ratios for stocks and bonds, averaged over the 14 countries and a century of time, aren't all that different.
Huh? Not that different? Stocks varying between 0.18 and 0.49... Bonds varying between -0.57 and 0.79... That is A LOT of variation for Sharpe ratios (even for stocks)... :shock:
Simplegift wrote:
Thu Sep 07, 2017 3:50 pm
2) The Sharpe Ratios for stocks appear fairly consistent between countries over the time period — while the risk-return relationship for bonds varies wildly by country, perhaps due to disparate episodes of hyperinflation and deflation during the century.
As usual for bonds, the pattern just varies wildly depending on the interest rates, hence (long) time periods. But yes, the outcome is quite eye-popping. Note that a Sharpe ratio doesn't change much if the math is done in nominal or real terms (since the inflation gets divided by itself).

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Re: Long-run Sharpe Ratios in an ideal world

Post by SimpleGift » Thu Sep 07, 2017 4:48 pm

siamond wrote:
Thu Sep 07, 2017 4:16 pm
Simplegift wrote:
Thu Sep 07, 2017 3:50 pm
Not knowing much about the theoretical basis for similar Sharpe Ratios among asset classes over long time periods, I couldn’t help but empirically check whether the Sharpe Ratios for stocks and bonds have been similar for the 14 countries over the one century of time for which we have good data.
That's an interesting idea! Did you do the math in local currency or USD? Did you use the risk-free rate from each country, or the US one?

All local currency; the risk free rate was from each country.

Also, did you find a way to get the dividends series from the database (I didn't), or did you apply the math to price series? If the latter, I am not too sure this is entirely meaningful.

This analysis used equity premium and standard deviation data for each country from DMS (their Triumph of the Optimists book); the short and long-term bond data for each country was from the JST database.
Simplegift wrote:
Thu Sep 07, 2017 3:50 pm
1) The Sharpe Ratios for stocks and bonds, averaged over the 14 countries and a century of time, aren't all that different.
Huh? Not that different? Stocks varying between 0.18 and 0.49... Bonds varying between -0.57 and 0.79... That is A LOT of variation for Sharpe ratios (even for stocks)... :shock:

I was just observing that the average Sharpe Ratios for all countries (0.33) wasn't too different from the all-country, average ratio for bonds (0.29). You're right that the variation of Sharpe Ratios between countries was very wide — but much less so for stocks than for bonds. This made me wonder if there is a more "universal" risk-return relationship for stocks? Perhaps their returns are just not as impacted by hyperinflation and deep deflation as bond returns are.
Simplegift wrote:
Thu Sep 07, 2017 3:50 pm
2) The Sharpe Ratios for stocks appear fairly consistent between countries over the time period — while the risk-return relationship for bonds varies wildly by country, perhaps due to disparate episodes of hyperinflation and deflation during the century.
As usual for bonds, the pattern just varies wildly depending on the interest rates, hence (long) time periods. But yes, the outcome is quite eye-popping. Note that a Sharpe ratio doesn't change much if the math is done in nominal or real terms (since the inflation gets divided by itself).

The difference between using nominal and real data for the Sharpe Ratio calculations had me puzzled, and am glad to learn that it doesn't make much difference. In this case, the nominal data was easier to find and analyze. Appreciate your comments, siamond, thanks.
Cordially, Todd

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Re: Long-run Sharpe Ratios in an ideal world

Post by siamond » Thu Sep 07, 2017 5:23 pm

Simplegift wrote:
Thu Sep 07, 2017 4:48 pm
I was just observing that the average Sharpe Ratios for all countries (0.33) wasn't too different from the all-country, average ratio for bonds (0.29).
Ooohhh... All right, got you. Thanks for the clarifications. Yes, that is intriguing (kind of amusing in a way), although the variations being so large between countries, I have to wonder if this isn't a statistical fluke.
Simplegift wrote:
Thu Sep 07, 2017 4:48 pm
You're right that the variation of Sharpe Ratios between countries was very wide — but much less so for stocks than for bonds. This made me wonder if there is a more "universal" risk-return relationship for stocks? Perhaps their returns are just not as impacted by hyperinflation and deep deflation as bond returns are.
Maybe I am subject to confirmation bias, but this just reinforces my view that when you look at bonds over diverse long time periods or geographical markets (instead of the past 30/40 years of US history), it just goes all over the place. Kind of a big thorn in the common wisdom that "bonds are for safety" if you ask me...

Now bonds do remain very useful as a low-correlation diversifier, and Sharpe ratios are handy to evaluate such portfolio effect. It shouldn't be difficult to expand your analysis at the portfolio level, actually. Say compare Asset Allocations of 80/20, 60/40 and 40/60 for each country. Ah wait... you don't have the annual returns for stocks, so the portfolios wouldn't be rebalanced every year if you just combine the aggregate returns at the end of the time period. Sigh. Let me try to bug the Jorda, Schularick and Taylor fellows one more time...

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Re: Long-run Sharpe Ratios in an ideal world

Post by TD2626 » Fri Sep 08, 2017 6:33 pm

Simplegift wrote:
Thu Sep 07, 2017 3:50 pm
TD2626 wrote:
Thu Aug 24, 2017 3:05 pm
Assuming that the strong form of the efficient market hypothesis holds, would all diversified investments have identical Sharpe ratios (before expenses) if an investment has an infinite time horizon and no black swan events occur?
siamond wrote:
Tue Sep 05, 2017 5:06 pm
Anyhoo, in the real world, risk is much more than volatility, and investors are driven by many more factors than volatility and return, which easily explains why Sharpe ratios end up squarely different for fairly long data series.
Not knowing much about the theoretical basis for similar Sharpe Ratios among asset classes over long time periods, I couldn’t help but empirically check whether the Sharpe Ratios for stocks and bonds have been similar for the 14 countries over the one century of time for which we have good data.

The table below shows the Sharpe Ratios for stocks and bonds over the 101-year period from 1900-2000, using the Dimson, Marsh and Staunton data for stocks, and using the Jorda, Schularick and Taylor database for bonds:
  • Image
Two observations:
  • 1) The Sharpe Ratios for stocks and bonds, averaged over the 14 countries and a century of time, aren't all that different.

    2) The Sharpe Ratios for stocks appear fairly consistent between countries over the time period — while the risk-return relationship for bonds varies wildly by country, perhaps due to disparate episodes of hyperinflation and deflation during the century.
Any thoughts?
First, nice empirical analysis. That sort of data is the kind of thing that is needed for this thread.

It does seem like Sharpe ratios appear to converge. (Of course, there's a lot of noise to this convergence, but that appears to be happening).

I tried searching through some academic literature to see if there was any theory to support this. I didn't find much - but someone who does a more thorough search may be able to locate something.

It does, though, intuitively seem reasonable to claim that "It's human nature to, over the long run, demand the same constant unit return per unit risk". However, this is simply conjecture backed by relatively shaky empirical evidence - nothing conclusive.

Maybe this is something that can't be proven. Maybe it's like P/E ratios - many people appear to think they converge to a natural long-run average. (Others suggest that that average may change - for example, this commentator seems to be suggest "25 is a new normal" -
https://seekingalpha.com/article/405655 ... t-new-norm). Note: I don't necessarily agree with this, and I tend to dislike the speculative reporting typical of SeekingAlpha)

I don't believe that investors should use P/E ratios - I think that they should just stay the course in their chosen AA. However, many attempt to use P/E as an indicator of overvaluation or undervaluation, or a predictor of reversion to the mean. Maybe that crowd would want to do an analysis of whether Sharpe ratios above the long-run average would correlate with a later reversion to the mean.

Anyway, unless someone can find solid theoretical proof, I feel that the idea that "Sharpe ratios have a natural, constant average over the long run across different asset classes and time periods" is something (like the idea that P/E ratios have a long run average) that can't be proven but could possibly instead be at best a rule of thumb. It may not even be true, though.

If the hypothesis was true, would there be substantial issues with MPT and correlations? If investors found a low-correlation thing that allowed their combined (say, 60/40) portfolio to have a Sharpe ratio above the market's, wouldn't investors rushing to take advantage of the low correlation depress bid up asset prices (or change correlations) until it became impossible to reliably get more unit return per unit risk than the market portfolio?

Is this related to Sharpe's idea that the global market portfolio has the highest Sharpe ratio? Would the implication be that that is a good portfolio to consider? (The market portfolio I am referring to is one which holds all assets at global cap weight; it is described here: viewtopic.php?f=10&t=227219&newpost=3525034; additional theory is discussed here: https://www.forbes.com/sites/phildemuth ... 01b4dd70d1)
Last edited by TD2626 on Sat Sep 09, 2017 2:30 pm, edited 1 time in total.

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Re: Long-run Sharpe Ratios in an ideal world

Post by siamond » Fri Sep 08, 2017 9:17 pm

TD2626 wrote:
Fri Sep 08, 2017 6:33 pm
It does seem like Sharpe ratios appear to converge. (Of course, there's a lot of noise to this convergence, but that appears to be happening).
Could you please elaborate of what makes you think that? It seems to me that those numbers are very diverse, and not converging in any way (besides the overall averages of stocks and bonds being somewhat similar).

PS for Simplegift: any way to break down those between 1900-1950 and 1951-2000? If there is some form of convergence over time, then we should notice it. Plus it's always interesting to separate the post-WW-II numbers from the older numbers.

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Re: Long-run Sharpe Ratios in an ideal world

Post by TD2626 » Sat Sep 09, 2017 12:50 pm

siamond wrote:
Fri Sep 08, 2017 9:17 pm
TD2626 wrote:
Fri Sep 08, 2017 6:33 pm
It does seem like Sharpe ratios appear to converge. (Of course, there's a lot of noise to this convergence, but that appears to be happening).
Could you please elaborate of what makes you think that? It seems to me that those numbers are very diverse, and not converging in any way (besides the overall averages of stocks and bonds being somewhat similar).
I examined more closely the table above that was posted by Simplegift. For stocks, the average Sharpe ratio was 0.33 and that number has a standard deviation of 0.08. The standard deviation is a small fraction of the mean. This would suggest that if another (out of sample) country's stock returns were listed, it would have a Sharpe ratio within +/- 0.08 of 0.33, 68% of the time (1 standard deviation); within +/- 0.16 of 0.33, 95% of the time (2 standard deviations), etc. (assuming a normal distribution).

The bond column is harder.
siamond wrote:
Tue Sep 05, 2017 5:06 pm
The table below shows the Sharpe Ratios for stocks and bonds over the 101-year period from 1900-2000, using the Dimson, Marsh and Staunton data for stocks, and using the Jorda, Schularick and Taylor database for bonds:
  • Image
Two observations:
  • 1) The Sharpe Ratios for stocks and bonds, averaged over the 14 countries and a century of time, aren't all that different.

    2) The Sharpe Ratios for stocks appear fairly consistent between countries over the time period — while the risk-return relationship for bonds varies wildly by country, perhaps due to disparate episodes of hyperinflation and deflation during the century.
As you can see from the bond column, the standard deviation is far higher than the average. I am wondering if this is due to changes in inflation. The table uses nominal returns - would it be better to use real returns? Also, currency effects could be coming into play. Do you know what currency was used, Simplegift? (Local currency vs USD returns could be different).

I looked to see if there was any correlation between a country's stock return and that same country's bond return. I plotted the table data columns (stocks vs bonds) against each other and the correlation is quite low. (R^2 of 7E-5 if I did the math right). Not sure what the absence of a correlation signifies or if it is relevant to this analysis - possibly it signifies that asset Sharpe ratios are independent each other.

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Re: Long-run Sharpe Ratios in an ideal world

Post by SimpleGift » Sat Sep 09, 2017 1:38 pm

TD2626 wrote:
Sat Sep 09, 2017 12:50 pm
As you can see from the bond column, the standard deviation is far higher than the average. I am wondering if this is due to changes in inflation. The table uses nominal returns - would it be better to use real returns? Also, currency effects could be coming into play. Do you know what currency was used, Simplegift? (Local currency vs USD returns could be different).
All the returns are in local currency. Real returns would have been nice, but nominal is what we had to work with.

I agree that the large variation in bond Sharpe Ratios between countries is likely due to periods of hyperinflation and deep deflation, which were quite different for each country. Stocks returns were perhaps not as greatly impacted by these large price swings, resulting in more consistent Sharpe Ratios between countries over the century. But I’m getting into speculation here.
Cordially, Todd

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Re: Long-run Sharpe Ratios in an ideal world

Post by TD2626 » Sat Sep 09, 2017 1:40 pm

I think that it may be helpful to specify the (unproven) hypothesis being considered and the variables used.

Please let me know if you see the hypothesis any other way.

As I see it, the hypothesis in two parts:

Hypothesis 1: "Averaged over centuries, due to constants of human behavioral and emotional aversion to risk, a reasonable measure of unit return per unit risk should converge to a similar value for all investment assets regardless of whether they are low, medium, or high risk".

Hypothesis 2: "The Sharpe Ratio is the correct reasonable measure to use"

If Hypothesis 1 is incorrect, then Hypothesis 2 is irrelevant. However, it may be the case that Hypothesis 1 is correct but Hypothesis 2 is incorrect and we should be using some other risk-return measure that takes into account skew and kurtosis (say, the Adjusted Sharpe Ratio proposed by Pezier and discussed in "How Sharp is the Sharpe ratio" by Bacon - the formula is given here: https://rdrr.io/github/cloudcello/Perfo ... Ratio.html; this was discussed some earlier in the thread) However, I feel that Sharpe Ratio is a reasonable measure for a rough approximation and the adjustments aren't needed.




There are two variables in this unproven hypothesis - time and risk.

The first is time. The time dimension says that over time (centuries), one asset's Sharpe ratio will have a stable long-run mean value that is reverted to. Such a single asset may be, say, large-cap stocks, or short-term treasuries, or small-value stocks.

The second dimension is risk. Assuming risk and return go hand in hand, (such that if one identifies a given risk level then the expected return can be automatically calculated), there is a risk spectrum. The risk spectrum goes from the least risky assets (something like ultra-short term high quality bonds) to the most risky (something like EM small-value).

The spectrum goes something like this (this is an approximation, and the exact order is of course subject to debate):

Low risk, Low return ------------------ <> -------------- <> ------------------- <> ---------------------- High risk , High return
"Risk-free" assets/Short term bonds / Inter-Term Bonds / Long Term Bonds / High Yield Bonds / Dividend Stocks / Large-Blend Stocks / Small Cap Stocks / Small Value / EM Stocks

This is essentially the kind of "risk potential" spectrum that Vanguard uses to rate it's funds by risk. (For example, this fund is shown as a 3: https://personal.vanguard.com/us/funds/ ... undId=1947).

If markets are efficient, shouldn't where you are on this risk potential spectrum not affect how much return per unit risk you get over a very long time horizon? If this hypothesis holds, then would the Sharpe ratio that you get (the unit excess return per unit risk) should be the same for each location on the spectrum? Of course, this doesn't take correlations into account - and the implications of this idea for portfolio design are tantalizing but not fully clear.

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Re: Long-run Sharpe Ratios in an ideal world

Post by siamond » Sat Sep 09, 2017 5:29 pm

Well, the Sharpe ratio is rather bizarre from a purely mechanical standpoint. First, yes, there is an implicit assumption of a normal distribution (definitely NOT the case). Next, who in their right mind would complain about an asset class going UP? Finally, small and short-lived variations do NOT matter whatsoever, you don't even notice them, or it barely registers... The only things that is a real risk are big/sustained drops. This being said, surprisingly enough, with all those mechanical deficiencies, in my experience, better designed metrics (e.g. Sortino ratio, Ulcer Index, etc) provide very similar outcomes than the good-old Sharpe ratio. And it's easy to compute. So... for all its flaws, it actually works better than one might expect.

Now, as to its semantics, this is where it gets downright silly imho. Volatility is emotionally disturbing for sure, but putting it on a big pedestal, claiming that this is THE definition of risk, and furthermore, putting on an equal footing with returns (by dividing one by the other) is just complete and utter non-sense imho, returns matter MUCH MORE than volatility. Sure, such ratio is convenient mathematically speaking, but that doesn't make it meaningful. Personally, I like to look at volatility on its own, because it does convey useful information, but that's just part of a wide spectrum of interesting metrics, and I would never optimize any decision based on one single ratio. And I don't think many professional investors make decisions based on it, far from it (I'm not saying they are right or not, but fact they don't). So you can guess that hypothesis #2 doesn't go well with me! LOL.

As to hypothesis #1, granted, the std-dev of Stock Sharpe ratios may be somewhat low in the OP's eye, but in my eye, this is still a very large spectrum and I can't equate that to any form of convergence. I actually fail to see why such convergence would occur. First, markets are NOT (strongly) efficient, as anybody having gone through the Internet and Financial crises (and previous ones) should have realized. Next, the forces driving market returns are extremely complicated (proof point? nobody figured them out! closest approximation coming from chaos theory, of all things!) and there is just no way that a simplistic little ratio like that would capture such powerful semantics.

Sure, volatility matters. But let's not put it on a huge pedestal, let's keep a much broader perspective. To come back to the thread's title, the real world is VERY messy. Just a fact of life.

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Re: Long-run Sharpe Ratios in an ideal world

Post by TD2626 » Sun Sep 10, 2017 3:40 pm

Simplegift wrote:
Sat Sep 09, 2017 1:38 pm
TD2626 wrote:
Sat Sep 09, 2017 12:50 pm
As you can see from the bond column, the standard deviation is far higher than the average. I am wondering if this is due to changes in inflation. The table uses nominal returns - would it be better to use real returns? Also, currency effects could be coming into play. Do you know what currency was used, Simplegift? (Local currency vs USD returns could be different).
All the returns are in local currency. Real returns would have been nice, but nominal is what we had to work with.

I agree that the large variation in bond Sharpe Ratios between countries is likely due to periods of hyperinflation and deep deflation, which were quite different for each country. Stocks returns were perhaps not as greatly impacted by these large price swings, resulting in more consistent Sharpe Ratios between countries over the century. But I’m getting into speculation here.
Stocks might not have been as affected by inflation because companies own physical assets (real estate, plant & equipment, inventory, commodities, etc) and these assets rise with rising inflation - thus the value of the company (in theory) rises with inflation. (Of course, this isn't a steady rise, and there's no contractual promise of inflation protection like with TIPS, but over the long run at least some inflation protection should be there. With nominal bonds, you're fully exposed to inflation risk.

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Re: Long-run Sharpe Ratios in an ideal world

Post by TD2626 » Sun Sep 10, 2017 4:49 pm

siamond wrote:
Sat Sep 09, 2017 5:29 pm
Well, the Sharpe ratio is rather bizarre from a purely mechanical standpoint. First, yes, there is an implicit assumption of a normal distribution (definitely NOT the case). Next, who in their right mind would complain about an asset class going UP? Finally, small and short-lived variations do NOT matter whatsoever, you don't even notice them, or it barely registers... The only things that is a real risk are big/sustained drops. This being said, surprisingly enough, with all those mechanical deficiencies, in my experience, better designed metrics (e.g. Sortino ratio, Ulcer Index, etc) provide very similar outcomes than the good-old Sharpe ratio. And it's easy to compute. So... for all its flaws, it actually works better than one might expect.

Now, as to its semantics, this is where it gets downright silly imho. Volatility is emotionally disturbing for sure, but putting it on a big pedestal, claiming that this is THE definition of risk, and furthermore, putting on an equal footing with returns (by dividing one by the other) is just complete and utter non-sense imho, returns matter MUCH MORE than volatility. Sure, such ratio is convenient mathematically speaking, but that doesn't make it meaningful. Personally, I like to look at volatility on its own, because it does convey useful information, but that's just part of a wide spectrum of interesting metrics, and I would never optimize any decision based on one single ratio. And I don't think many professional investors make decisions based on it, far from it (I'm not saying they are right or not, but fact they don't). So you can guess that hypothesis #2 doesn't go well with me! LOL.

As to hypothesis #1, granted, the std-dev of Stock Sharpe ratios may be somewhat low in the OP's eye, but in my eye, this is still a very large spectrum and I can't equate that to any form of convergence. I actually fail to see why such convergence would occur. First, markets are NOT (strongly) efficient, as anybody having gone through the Internet and Financial crises (and previous ones) should have realized. Next, the forces driving market returns are extremely complicated (proof point? nobody figured them out! closest approximation coming from chaos theory, of all things!) and there is just no way that a simplistic little ratio like that would capture such powerful semantics.

Sure, volatility matters. But let's not put it on a huge pedestal, let's keep a much broader perspective. To come back to the thread's title, the real world is VERY messy. Just a fact of life.
Thanks for your detailed response.

A few comments:

1. Yes, I agree that the fact that standard deviation penalizes upswings is hard to get over. I think it's theoretically justified to use the standard deviation because other metrics require a target rate of return (like the Sorotino ratio). Unless one has a specific target rate that one needs to achieve in order to meet goal X, then it's hard to pick a specific target. The Sharpe ratio is very simple yet very powerful.

2. Your comment of "returns matter much more than volatility" got me thinking. Other than correlations, there are four primary measures that I generally look at (mean, variance, skew, and excess kurtosis). In some of the papers Afan recommended earlier in this thread, it was shown that investors seem to want positive odd central moments and low even central moments. (Thus, investors want high & positive means and skews and low variance and kurtosis.). Could it be the case that lower central moments are more important to investors than higher ones? If this is the case, how much more important is mean than variance? How much more important is variance than skewedness? Maybe it's a square-root law - that instead of comparing mean and variance directly, one could compare mean to the square root of the variance. The standard deviation, though, is the square root of variance. The Sharpe ratio compares mean and standard deviation, not mean and variance. (I possibly am mistaken in the math here, let me know if I overlooked something).

3. The point where you said "I would never optimize any decision based on one single ratio" - I agree. Although finding the Sharpe-optimal portfolio through Modern Portfolio Theory and Markowitz optimization is a common approach, I think that one shouldn't just blindly go with whatever the computer says - one must also consider whether the model output is reasonable.

4. You mention you use other metrics in addition to risk and return. I of course look at things like skewedness, kurtosis, correlations, tax efficiency, tracking error, turnover, and so forth when evaluating funds. What sorts of things do you look at - are there other measures that you think are more important than, or on equal footing with, risk and return? I see your point that return is more important that risk and that they shouldn't be put on equally high pedestals.

5. I certainty agree that the real world is quite messy, and that it's hard to translate between theory and real-world. I think that markets are reasonably efficient over long periods (years). Over nanosecond timescales (faster than the fastest computer / algorithmic trader can react to news) markets aren't efficient. Over the long run, periods of overpricing and under-pricing cancel out, making markets efficient on average over many years. Of course, even long-run investors have to make trades at specific points in time, and if markets aren't giving a fair price then, that would be an issue.

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Re: Long-run Sharpe Ratios in an ideal world

Post by siamond » Sun Sep 10, 2017 9:41 pm

TD2626 wrote:
Sun Sep 10, 2017 4:49 pm
4. You mention you use other metrics in addition to risk and return. I of course look at things like skewedness, kurtosis, correlations, tax efficiency, tracking error, turnover, and so forth when evaluating funds. What sorts of things do you look at - are there other measures that you think are more important than, or on equal footing with, risk and return?
Well, my point was that risk is multi-faceted, and NOT equal to volatility (as captured by std-deviation). Now to answer your question, I try to look at things that seem meaningful by their impact to my life, on a day-to-day basis as well as over a full retirement period, and do that at the portfolio level (not per individual asset class). I do check standard-deviation (volatility), but I find drawdowns to be much more significant. And I am not speaking of the single worst drawdown (which is a single data point in history after all), but the various drawdowns on a trajectory of accumulation or distribution (retirement). A very cool metric is the aptly named "Ulcer Index", which you can find defined here (and implemented in the Simba spreadsheet). Between std-deviation and the Ulcer Index, I think this covers pretty well the emotional side of things (which certainly matters, but is not necessarily consequential).

What has the greatest impact on day-to-day life is purchasing power, notably during retirement. Consequently, I look at returns and portfolio balances and drawdowns in nominal terms AND in real terms. The nominal view is mostly emotional (and subject to anchoring), the real view is, well, real. And when you start playing with bonds and history, it's quite illuminating to see the impact of inflation on real returns and drawdowns (nope, bonds are NOT safe). My top-level concern is to maintain (and maybe expand) solid purchasing power for the rest of my life (and my spouse, and provide some level of bequest to my children and grandchildren), and to properly benefit from upsides as well as manage downsides as well as possible. This is not about the worst case scenario. This is about being adaptive to whatever will happen. Thinking LONG-TERM on such issues is crucial. Overloading on bonds can be great to manage emotions in the short term, but catastrophic for your mid/long-term purchasing power (add LTC to the equation just to make it more challenging) - this is where returns are so much more crucial, to keep an engine of growth in your portfolio. The SSR/PSR metrics that you can find in the Simba spreadsheet are informative for such purpose (combined with a data table to run numerous cycles), but the point is more to simulate the outcome of variable withdrawal methods, and unfortunately there is no single metric to summarize findings (I never truly warmed up to utility functions), but ranking outcomes in terms of percentiles proves very useful. And again, look at rosy scenarios as well as painful ones.

And finally, yes, tax efficiency counts all the way, and is another thing where long-term planning is crucial (e.g. Roth conversion strategies and so on).

Yeah, I know, sounds complicated, and it is. I'm sorry, I don't have a small neat set of equations to give you. Life is messy... Still, at the end, it is MOSTLY about purchasing power, and then we're quite far from the std-deviation of one's portfolio... Even if I would never underestimate the impact of emotions along the way.

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Re: Long-run Sharpe Ratios in an ideal world

Post by TD2626 » Mon Sep 11, 2017 12:48 am

Yes, it's very hard to get a single-number handle on risk, despite a large number of risk metrics available. The Ulcer Index is interesting as it largely excuses the short-term blips that many long-term investors (who infrequently check brokerage statements) may not even notice - and it also doesn't label as "risk" the chance of an unexpected gain. I agree that the "emotional side" of investing, and behavioral finance, is in many situations the elephant in the room. While I try to be the rational investor of the kind classic economic theory "assumes" everyone is, few people actually sit down and do the math that that entails.

Thanks for mentioning purchasing power/inflation. I knew I left something out. An asset's expected inflation performance is one of the most important characteristics of that asset. Nominal bonds, TIPS, stocks, REITs, commodities, and so forth all have different and well known expected responses to inflation. "What would this asset do in the event of inflation" is an important characteristic of any asset. (Though, one could argue that inflation is simply "inflation risk" in which case we're left with fewer variables). The nominal vs real point is also helpful. I try to do things in real if possible because one can't eat nominal... but often times nominal is easier to work with mathematically when doing certain calculations.

I think that the point of long-term, inflation adjusted total return being the primary thing to consider is a good one. If long-term real expected return is low or negative, then the investment can be hard to justify. Making sure that one stays the course is important as well. Pouring over long-term historical data and backtesting scenarios/results can really help educate one on the wide breadth of what is possible - and how beneficial it is to stay the course. The Simba spreadsheet is a great resource in this regard. Presumably, an investor who knows investing history would be better able to choose an allocation and stick too it.

It is unfortunate that it is hard to come up with a small set of equations that govern everything in investing. So much in investing is simple (in an elegant, brilliant way - like how Total Stock buys the same percentage stake in every company). In reality, though, life is complicated and at some point oversimplifying risks missing the bigger picture.

Oh, and of course savings and withdrawal rates are more important than questions of AA and far more important than questions of "do I tilt". Despite this, people spend more time discussing minor sub-allocations than overall top-line allocations, and more time discussing overall allocations than savings rate. That's all well and good if one wants to optimize the portfolio for a given savings rate, but people should know how big of an effect this has.

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Re: Long-run Sharpe Ratios in an ideal world

Post by siamond » Mon Sep 11, 2017 11:54 am

Yup, we're in sync. I think many people overestimate the importance of short-term risks and underestimate the importance of mid/long-term risks. I owe a big debt of gratitude to Dr. Bernstein, who articulated such framework so clearly in his wonderful 'Deep Risk' booklet, one of the few 'must read' book about personal investments. This was a big eye-opener for me.

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Re: Long-run Sharpe Ratios in an ideal world

Post by SimpleGift » Mon Sep 11, 2017 12:10 pm

TD2626 wrote:
Sat Sep 09, 2017 12:50 pm
As you can see from the bond column, the standard deviation is far higher than the average. I am wondering if this is due to changes in inflation. The table uses nominal returns - would it be better to use real returns?
For what it’s worth, in some weekend reading, I ran across real, inflation-adjusted Sharpe Ratios for U.S. stocks and bonds for the 1926-2017 period (table below). Once again, the ratios for equites are a bit higher than those for bonds — though corporate bonds, with their degree of equity risk, end up somewhere in between.
Not sure what to make of this. These Sharpe ratios just don’t seem similar enough to support the OP premise, that "if the strong form of the efficient market hypothesis holds, all diversified investments have identical Sharpe ratios.”
Cordially, Todd

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Re: Long-run Sharpe Ratios in an ideal world

Post by AlohaJoe » Tue Sep 12, 2017 12:52 am

TD2626 wrote:
Mon Sep 11, 2017 12:48 am
Yes, it's very hard to get a single-number handle on risk, despite a large number of risk metrics available.
To expand on this point, Andrew Ang's book Asset Management (which is my overall favorite book on investing) has one of my favorite discussions of risk and the many forms it can come in. It in the section on "Realistic Utility Functions"
In economics, there are many utility functions that realistically describe how people behave. In the asset management industry, unfortunately, only one utility model dominates -- by a long shot, it's the restrictive mean-variance utility model. It would be nice if we had a commercial optimizer where one could toggle between various utility functions, especially those incorporating downside risk aversion. It would be even better if an application could map a series of bad times, and how the risk of these bad times is perceived by an investor, to different classes of utility functions. Sadly, there are no such asset allocation applications that I know of at the time of writing that can do this. And yet all of the economic theory and optimization techniques are already published.
He then goes on to give examples of these other definitions of "bad times". I found it interesting because few of them are discussed on Bogleheads. Some of the interesting ones he discusses...

"Safety First"

Is the return greater than a predetermined level? If you have a liability (mortgage, children's college tuition, etc), this is a great definition of risk. If the return is greater than the level...then you are safe. If the return is less than this level, then disaster occurs.

"Habit Utility"

A "bad time" depends not just on wealth outcomes but also the investor's environment. It isn't just wealth that matters, it is wealth relative to a reference point. In most cases this means "to support the standard of living that I am used to". But it also handles that the standard of living can change over time. I personally think this is how most people planning for retirement think -- which means they should be using the math related to this utility function and not things like Sharpe ratios.

"Catching Up With The Jones"

Bad times are relative to other investors. This one is rarely discussed on Bogleheads -- and perhaps among Bogleheads it isn't that important -- but I think there are many investors for whom this is a risk. They don't want to have worse returns than their brother-in-law. They don't want to have worse returns than their best friend from high school. They don't want to have worse returns than the guys they play poker/golf/whatever with.

An example of this in the Bogleheads context might be the large number of Bogleheads who added REITs or Commodities a decade or so ago because that's what all the other Bogleheads seemed to be doing....

"Uncertainty Aversion"

With this agents define risk as "ambiguity". For instance, the returns of US Treasuries have low uncertainty. Chinese stocks have high uncertainty. (Whether they have high variance or not is beside the point.)

I think this "uncertainty aversion" plays a large role in how many Bogleheads create their asset allocations and judge their portfolios.

In particular, "uncertainty aversion" actually leads to much lower equity holdings portfolios than plain old "risk aversion" would suggest.

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Re: Long-run Sharpe Ratios in an ideal world

Post by jmk » Tue Sep 12, 2017 10:32 pm

EMH equalizing sharpe ratio assumes all that matters to the market is getting the most return per volatility. While that matters, others things matter too. So the fact the different sharpe ratios have persisted so long implies there are other trade offs at work.

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Re: Long-run Sharpe Ratios in an ideal world

Post by TD2626 » Wed Sep 13, 2017 12:13 am

Good points. I think that some portfolios that seem to work well in theory (i.e. they have a higher Sharpe ratio) are instead just exploiting differences between risk as measured by standard deviation and risk as priced by market participants. I guess the only way to have a true answer to this would be to wait a few hundred years to get better data - we've gone over most of the available data but there's just not enough centuries of data available to test this empirically. It does seem to me that if markets are efficient, then securities would need to be priced fairly based on their risk-return tradeoff, so measures of the risk-return tradeoff should be the same for all securities - but unfortunately it doesn't seem like it can be proven. (Besides, it relies on the strong-EMH, which is a imperfect approximation).

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