Is there any proof of correlation between risk & return?

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Re: Fama-French correlation of Risk and Return

Postby Aptenodytes » Fri Sep 09, 2011 2:57 am

Taylor Larimore wrote:Bogleheads:

IFA has an excellent article with charts showing the strong correlation of Risk and Return in the Fama/French Three Factor Model.

Chart 8-15a shows Market Risk correlation; Chart 8-18 shows Size Risk correlation; and Chart 8-16a reveals the Value Risk correlation.

http://www.ifa.com/12steps/step8/step8page4.asp

I peek in on this thread out of morbid curiosity, and want to thank Taylor for taking the time to patiently point out the basic facts for the sake of people who might be new to this and vulnerable to being distracted by all this noise and confusion.

The last time I scanned the thread I was convinced this was just a giant trolling exercise. I don't think that any more, but it is no less pleasant to watch.
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Postby Verde » Fri Sep 09, 2011 6:03 pm

efalken wrote: 2) Per the ERP being 5%: I think it's more like 2%, which is near what Fama/French, or Ivo Welch, think (they are around 3%). Pablo Fernandez surveys 5% as the average. For average investors, however, trade too much, incurring inefficient tax losses, and invest procyclically so that annual average returns are greater than the returns to the average dollar invested. I think it's conservative to say your average retail investor would be better in a money market account.
3) Per data not reflecting expected returns, that argument is possible, but unlikely. The Ang, Hodrick et al (2008?) piece highlight the low-vol anomaly in 17 developed country markets, and others have found this in emerging markets. So, everywhere, since data have been collected, low beta/vol stocks outperform the average, and high beta/vol have underperformed. Averages are not expectations, but the larger the sample the more average should be close to expected value, and its the only data we have. Further, Sharpe and Amromin (2009) actually looked at surveys and found investors assumed risk and return were inversely correlated over time.


Ok, it appears I understood your arguments from the start. That is unfortunate, I hoped there was some meat in your 150 page thesis which I was just too stupid to grasp.

I think your attack against the historic ERP has some glaring errors:

Equity Premium Adjustments
1. Geometric vs. Arithmetic Averaging 2.0%
2. Survivorship Bias/Peso Problems 2.0% *
3. Taxes 3.0%
4. Adverse Market Timing 3.0%
5. Transaction Costs 3.0%
Sum 13.0%

On point 1. Using Robert Shiller’s S&P 500 data which he extended back to 1871, I calculate an ERP over Tbills up to 2010 of 6%. The geometric mean was 4.4%. Dimson, Marsh and Staunton report geomean ERP’s over bills for 17 countries for the 109 years 1900-2008 in a range between 2% and 6%, and for the world index over this period of 4.2%.

Point 2. The ERP is a forward looking concept. The historical ERP does not reflect the expected ERP priced by the market over this period. In the case of the successful US market it is reasonable to conclude that more risk was priced than that which actually showed up hence the higher than expected historic ERP. In the case of the disastrous Belgian market less risk was probably priced than that which showed up hence the 2% historic ERP. You think the market prices for a 2% ERP, and you say Fama& French think it is 3% (You don’t clarify if this is the mean or geomean). The fact is the market discounts stocks to provide an expected ERP, it is not risk neutral.

Points 3 and 5. You claim that taxes and costs reduce the ERP by 6% per annum. I am far from convinced by the evidence you present. Even if this number was remotely ballpark, smart investors only pay a fraction of this. The market does not price stupidity. If it did there would be a huge arbitrage opportunity for informed investors. The market correctly prices only for taxes and costs incurred by the smart money to eliminate this arbitrage opportunity. Any Boglehead will be able to tell you that this is only about 50bps.

Point 4. Your argument about adverse market timing costing the average investor 3% fails the basic arithmetic test. The average investor’s returns cannot be below average. Every incidence of bad market timing must be balanced out by good market timing taking the other side of the trade. The Dichev paper you cite was refuted in a 2008 paper ‘Dollar-Weighted Returns to Stock Investors: A New Look at the Evidence’ by Keswani and Stolin.
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Postby Verde » Fri Sep 09, 2011 6:05 pm

efalken wrote:
5) If you cross-tab beta after controlling for size, there is no positive return to beta. If you simply sort stocks into betas, but exclude the noninvestible little companies, returns peak at beta~1, then decline dramatically. So, I think any beta~return correlation is totally explained by little companies that can't be traded, more measurement error than anything real anyway (ergo the initial 12% small cap effect is now thought to be around 2-3%).


There is a basic problem with subdividing the market into various segments.
The market only prices systematic risk, it offers no discount for specific risks.
When you sort stocks by beta you expose these specific risks which are diversified away in the total market.
These specific risks overwhelm the market risk when viewed in isolation. Therefore you should not expect to find correlation when you sort by beta.
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verde follow up

Postby efalken » Sat Sep 10, 2011 5:04 pm

verde:

You appear to agree geometric vs. arithmetic average as an appropriate adjustment, so that's 2%.

You state "The ERP is a forward looking concept. The historical ERP does not reflect the expected ERP priced by the market over this period." That seems rather anti-empirical. 100 years and 17 countries is a lot of data. I imagine another lifetime of data would also be insufficient, so it seems non-falsifiable via this reasoning.

On taxes and transaction costs, you argue smart investors pay much less. I agree. But even smart investors could not avoid significant transaction costs until the 1970's, so all that data prior to that periods needs such an adjustment. Average returns to average investors are the data that supports or rejects asset pricing theory. Taxes too are generally ignored, but Blum and Gannon went through a pretty transparent and reasonable simulation and found the 3% adjustment.

You state "the market correctly prices only for taxes and costs incurred by the smart money to eliminate this arbitrage opportunity." Perhaps this is why the historical return on stocks since 1990 has been so meager, because prior to this, costs were significantly higher to even the smartest investors (low cost mutual funds were pretty insignificant prior to this time).

As per adverse timing, the argument that average investor’s returns cannot be below average" is not correct if you are equally weighting annual returns to periods when there's 2x as much money in the bad year returns, half as much in the good times. The return to the average dollar invested is a superior metric to the return to the average year, because it better describes what happened to the average investor. Surely with hindsight people would have lowered their exposures in 2000 and 2008, but they didn't.

Keswami and Stolin agree with Dichev's aggregate numbers for the US, then argue that various subsamples give a better estimate, though they present a 73-02 sample that one can't compare to Dichev's 65-02 subsample. They then show a different international dataset gives a different estimate. Both reduce Dichev's findings by about 1%. They then look at data after 2002, to 2007, and find this reduces his findings another 1% or so. But Keswami and Stolin's paper is much less thorough, 16 pages vs. 36, and for some reason not published in peer-reviewed literature even though it's exactly the type of paper that, if true, would be greatly appreciated by the major journals. After 3.5 very tumultuous years, I bet if their results held, they would have updated it, but instead, let it die. As academics there job is to publish, so I suspect this isn't for lack of interest, but rather, the data aren't agreeable. It would be useful to see this updated, nonetheless.

You state that "When you sort stocks by beta you expose these specific risks which are diversified away in the total market. These specific risks overwhelm the market risk when viewed in isolation. Therefore you should not expect to find correlation when you sort by beta.". I don't really understand the point unless you state what the risks are, which we could then look at to see if they generated a return premium. However, clearly beta, in the context of size and value, something that tells you anything about average returns, and presumably expected returns. Historical data is future data, but it's all we got.

A good theory identifies some sorting criteria to isolate higher risk, and presumably higher returning, portfolios. Most academics agree 'beta' won't do that. Now 'size' and 'value' are the candidates, but these have a very subtle relation to risk as opposed to characteristics of stocks that have done well. One might as well add high cash flow and low volatility as measures of risk simply because they have, historically, been associated with higher returns, but this seems a clear rationalization of risk to match the theory.
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Postby Verde » Sun Sep 11, 2011 4:15 pm

efalken wrote:You appear to agree geometric vs. arithmetic average as an appropriate adjustment, so that's 2%.


Yes, but when you say: “Per the ERP being 5%: I think it's more like 2%, which is near what Fama/French, or Ivo Welch, think (they are around 3%)”, do you refer to mean or geomean?.

efalken wrote: You state "The ERP is a forward looking concept. The historical ERP does not reflect the expected ERP priced by the market over this period." That seems rather anti-empirical. 100 years and 17 countries is a lot of data. I imagine another lifetime of data would also be insufficient, so it seems non-falsifiable via this reasoning.


If we assume normally distributed returns with an annualised SD of 20% and an annualised expected return of 3% the chances are about 62% that the actual outcome will fall outside the range 2-4% after 100 years. We basically have one 100 year independent data point (data from the 17 countries are not really independent) from an unstable distribution. We cannot infer with any confidence from the one 4% historic geomean ERP for the world index whether the market priced a 1% or 7% ERP. So I don’t think empirical research into the ERP tell us much about the forward looking ERP.
More importantly, if there actually was proof that the priced ERP = the realised ERP, it would imply that the market has psychic powers. The stock market’s purpose is not to predict the future. Its purpose is to move capital from those who have it to those who can use it productively, and it certainly does not need to predict the future to accomplish this.

efalken wrote: On taxes and transaction costs, you argue smart investors pay much less. I agree. But even smart investors could not avoid significant transaction costs until the 1970's, so all that data prior to that periods needs such an adjustment. Average returns to average investors are the data that supports or rejects asset pricing theory. Taxes too are generally ignored, but Blum and Gannon went through a pretty transparent and reasonable simulation and found the 3% adjustment.
You state "the market correctly prices only for taxes and costs incurred by the smart money to eliminate this arbitrage opportunity." Perhaps this is why the historical return on stocks since 1990 has been so meager, because prior to this, costs were significantly higher to even the smartest investors (low cost mutual funds were pretty insignificant prior to this time).


On taxes. You think it reasonable to assume that historically the average $ invested in the US stock market lost 3% per year to taxes. The Shiller data gives us a nominal geomean return for US stocks of 8%. If we assume your numbers are correct we have to deduct 6% from this to account for costs and adverse market timing. So investors would have reported a 2% return to the IRS which you believe was taxed at a rate of 150% (3%/2%).
The research you cite seems to ignore the fact that a huge proportion of invested funds is tax advantaged (pensions, insurers, foreign investors etc).

I would like to know if your argument that the stock market prices on a risk neutral basis applies to the cost of capital as well. Returns = cost of capital, so do you contend that companies expect that the cost of debt and equity capital are equal?

I don’t see why only returns to investors should count towards the ERP. Returns which end up in the pockets of other market participants (the middle men and helpers) is still a reward for the risk incurred by the investor – these participants have to either provide a useful service to investors, or find a way to extract gains from foolish investors.

Your argument about a correlation between lower costs and meagre recent returns would be persuasive if valuations went up as costs came down and stayed there. The fact that valuations have come down from their absurd levels counters this argument. The S&P500 real return 1990-2010 was 5.7%- not exactly meagre.

I believe a successful market has to return the initial dividend yield + the growth in dividends to long-term buy and hold investors less a bit of dilution. It also has to pay arbitrageurs and liquidity providers a fair wage for performing useful functions like tightening spreads and eliminating inefficiencies. Other investors are engaged in a zero sum game.

Those long term investors always incurred low costs, they only pay trading costs on their initial investments, when they reinvest divs and rebalance. This has always been an order of magnitude less than 3% per year. ETF’s and index funds has only benefited the small buy and hold investor which has always been a small proportion of the market.
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Postby Verde » Sun Sep 11, 2011 5:08 pm

efalken wrote:As per adverse timing, the argument that average investor’s returns cannot be below average" is not correct if you are equally weighting annual returns to periods when there's 2x as much money in the bad year returns, half as much in the good times. The return to the average dollar invested is a superior metric to the return to the average year, because it better describes what happened to the average investor. Surely with hindsight people would have lowered their exposures in 2000 and 2008, but they didn't.




I want to clarify the issue of time weighted vs. $ weighted returns to eliminate misunderstandings.

The quotes are from the Dichev paper:

To the extent that transactions between investors cause market cap to change, there can be no difference between time and $ weighted returns. Only to the extent that capital flows between companies and investors can there be such a difference.

These flows include straightforward items like dividends, stock repurchases, and stock issues but could also include more complicated items like contribution and distribution of non-cash assets, exercise of stock options, spin-offs and equity carve-outs, issuing stock in mergers and acquisitions, and listing and delisting of new stocks.



Distributions = MVt-1*(1 + rt) - MVt
where MV is market capitalization (typically calculated as number of shares*stock price) and rt is the total return for that period (including dividends). Positive amounts of Distributions indicate capital flows from the company to investors for that period, and negative amounts indicate flows from investors to the company.3 The intuition behind this calculation is that market value changes result from both real price appreciation and net investor capital flows. Thus, controlling for returns, market value changes identify the capital flows


So if the market drops 50%, and then recovers without there being net flows between companies and investors, the two metrics will be the same. Sure some idiots might sell low and experience a permanent loss, but another investor buys low, in aggregate there is no bad timing cost.

If new stock capital contributions and distributions are “random”, and earn the same average rate of returns as the rest of the stock capital, then there will be no difference between buy-and-hold
and dollar-weighted returns. However, buy-and-hold and dollar-weighted returns will differ if there are material correlations between the timing of capital contributions and distributions and past and future stock returns.


So, overall investors can incur a bad timing cost if companies consistently time these new capital contributions and distributions favourably.

These results imply that infusions of stock market capital tend to happen after superior past returns and before subsequent inferior returns; the converse applies for distributions of market capital.


But,

Finally, the findings of this study seem to offer some practical investment implications. One implication is that passive investment strategies are likely to do well because they avoid both transaction costs and the negative effects of timing. Another implication and an area of potential future inquiry is that one might be able to use aggregate timing signals to build superior investment strategies. Taken literally, the evidence in this study implies that a “contrarian” strategy based on taking investment position opposite to that implied by the aggregate market inflows and outflows is likely to be successful.


So once again only fools pay this cost. Therefore I see no reason why the market has to price for it.

A few years ago I tried to replicate Dichev’s work, but I only had about 20 years data and my year by year results were swinging wildly from a positive effect one year to a negative effect the next. I need to see a lot more research and endorsements from big names in finance before I would rely on Dichev’s results.
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Re: Is there any proof of correlation between risk & return?

Postby Verde » Thu Apr 26, 2012 10:14 am

Dr. Falkenstein published this entry on his blog:

http://falkenblog.blogspot.com/2012/04/ ... emium.html

He seems to have changed his mind from 'The equity risk premium is zero', to 'The equity risk premium is zero for the marginal investor'.

Relative risk theory only makes sense if the equity premium is zero, but then everything is consistent. While the equity risk premium is down to about 3.5% these days, and realistic standard errors are about 4%, no one thinks it is zero. I asserted it was zero previously because it seemed possible, and was really the only missing piece of the puzzle. Now I'm thinking I made an error. The equity risk premium is positive, close to the 3.5% most others believe. They key is that this return is not a scalar, but rather an array.


This blogpost is relevant to the discussion we had in this thread last year. I agree with almost everything in this blogpost and I think it supports Boglehead thinking rather than undermine it, as was the perception created by his views on the way markets price risk expressed in the paper quoted by the OP.

Another quote from the blogpost:
The efficient investor avoids active management, invests steady amounts over the business cycle, generates few taxable events, trades to minimize trade impact and the effect of a bid-ask spread. Your average investor trades inefficiently, and generates a lot of profit for a large and vibrant financial community
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Re: Is there any proof of correlation between risk & return?

Postby Joe S. » Thu Apr 26, 2012 10:54 am

Lbill wrote:The following quote is from a working paper by economist Eric Falkenstein entitled "Why Risk is not Related to Returns", 2007... ...The author makes a strong argument that, outside of a few special cases, there is no general evidence to support the commonly accepted wisdom that there is a positive correlation between investment risk and investment returns. Taking investment risk - for the most part - goes unrewarded, except when you're lucky.

You need to read the paper more closely. The author makes absolutely no argument that there is not a positive correlation between risk and reward between different investment classes. Rather he argues that there is not a positive correlation between risk and reward inside different investment classes.

In other words he does not argue against the well documented evidence that stocks have a higher risk and higher return than bonds. Rather, he argues that high risk stocks do not have a higher return than low risk stocks.

If you go to Table 3.6 (Page 133), he gives a list of asset classes that don't show a positive risk-return premium. Most of them are assets that Bogleheads would never dream of owning, like lottery tickets, movies, currencies, horse racing, futures, and sports betting.

He also admits that bonds are an exception to his rule, Bonds do show a risk-return premium on the short end of the yield curve.

Indices wrote:Sadly the risk/return relationship is one of the biggest myths on this forum.

His article does not advance your claim because he never compares high risk asset classes to low risk asset classes. He only compares assets inside the same asset class. He never studies whether stocks have a higher return than bonds. Stocks and bonds are what Bogleheads invest in.
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Re: Is there any proof of correlation between risk & return?

Postby protagonist » Fri Apr 27, 2012 12:21 am

This is a fascinating thread and I am only through the first page (sleep beckons).

My intuition (for what it is worth) is that a large percentage of people buy a particular investment not because of any innate ability to assess risk/reward ratios, but because they are TOLD it is a good idea to buy it....whether by their broker, or the article in USA today, or what they see on Fox News, or by their buddy at the gym who read the article in USA today. The pundit who sold the product may have presented it in terms of potential risk and reward, but the main motive for the majority of pundits is really only their own profit. People don't go to McDonald's in droves because McDonald's makes the best hamburger in town- they go there because they are the best advertisers. Ditto in many of our elections where large segments of the population are convinced to foolishly vote against their own self-interest. I would be skeptical of any hypothesis based on a premise that our DNA is finely tuned to assess highly complex risk-reward puzzles, as those that exist in the world of finance.
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Re: Risk and Return correlation.

Postby getRichSlower » Fri Apr 27, 2012 1:45 am

Taylor Larimore wrote:
Indices wrote:No there is no correlation. I am increasingly convinced that virtually all investments given enough time return about the rate of inflation possibly less. Sadly the risk/return relationship is one of the biggest myths on this forum.


Hi Indices:

The chart in the link below (Figure 1.2) in Mr. Bogle's book, Common Sense on Mutual Funds, shows the risk (jagged vs. smooth lines) and real return of stocks, bonds, and gold, plus the effect of inflation from 1802 to 2008.

Total Real Return on $10,000 initial Investment

You can see, the different investments do not "return the rate of inflation possibly less." The return of stocks and bonds and gold is vastly different over longer periods and all beat inflation.

The correlation of Risk and Return is not a "myth."

1) If that chart had taxes taken into account, then the values of stocks and bonds would be substantially lower.
2) Given that investors should only be rewarded for non diversifiable risk and mutual funds only came into existence in the US in the 1890, wouldn't the real returns to stock and bonds prior to say 1900 be useless for making predictions about the future?
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Re: Is there any proof of correlation between risk & ret

Postby tadamsmar » Fri Apr 27, 2012 7:40 am

GregLee wrote:
Lbill wrote:
For 40 years researchers have looked for corroborating evidence of risk and reward having a positive and a linear relationship ...

Why should the relationship be linear? Even if there is no positive linear relationship between risk and reward, that wouldn't show that there is no relationship between the two.


The theory implies it should be linear at least under some transform. That's what the reasoning behind the Sharpe Ratio implies.

The theory (and the assumption that many of us make) that risk and beta is correlated for stocks has little or no evidence in it's favor.

If you attribute high returns to a small risk factor or value risk factor, then you can rehabilitate the theory that risk and return are correlated.
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Re: Is there any proof of correlation between risk & ret

Postby GregLee » Fri Apr 27, 2012 10:03 am

tadamsmar wrote:The theory implies it should be linear at least under some transform. That's what the reasoning behind the Sharpe Ratio implies.

I don't understand that. I don't understand what relation "linear under some transform" could exclude, nor what "the theory" refers to. I've just read over Sharpe's 1994 article on the Sharpe ratio, and I can't find the reasoning about linearity that you're referring to -- could you please point it out?
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Re: Is there any proof of correlation between risk & return?

Postby umfundi » Fri Apr 27, 2012 3:09 pm

Lump me in with the morbidly curious crowd.

Perhaps a naive question: Isn't the Efficient Frontier a graph of risk vs. return?

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