Monday, September 05, 2011

A Puzzle, a Solution, and Why It Cannot be Right. Maybe.

In the U.S. in the 20th century, stocks have consistently outperformed bonds. For an economist, this is puzzling. If you consistently get a better return buying stocks than buying bonds, why does anyone buy bonds? One would expect investment to shift out of bonds and into stocks until the return on both investments was equal.

There are several solutions that other people have offered to this puzzle. One is to argue that investors are risk averse and bonds have a more predictable yield; the rebuttal has been that, except over very short time periods, stocks almost always outperform bonds, so nobody investing for more than a few years can reduce his risk by buying bonds. Another is to claim that the U.S. market in the 20th century is a special case; investors in stocks just happened to be lucky, many times over.

I have a different solution, one that is elegant, coherent, and arguably cannot be right. More interesting still, the argument that shows it cannot be right also shows that there is no such thing as insider trading. 

I start with a bit of real world history:

When the Macintosh first came out, I was already familiar with the idea of a graphic interface and convinced that it was a better way of interacting with computers, having seen a video some years earlier on work at Xerox PARC. I observed that the rest of the world, or at least the people I could directly observe, had no idea what was going on; one colleague, told that I was buying a Mac, asked why I wasn't getting a PC Jr. instead, apparently in the belief that they were roughly comparable because of similar size. I concluded that if his response was typical, Apple stock was underpriced, so I bought some.  It turned out to be a correct decision. 

I was an insider for that transaction, not in the legal sense but in the  economic sense; I knew relevant things that most of the market did not know. Imagine a stock market in which every investor is in that sense an insider, each for a different tiny niche, a particular subset of investments with regard to which he has information that other investors do not have and cannot readily obtain. Like any inside trader, the investor can expect a better than market return when he invests on the basis of his inside information.

If expected return was all that mattered, each individual would invest only in the niche where he had specialized information. But individual investors are risk averse, so wish to diversify their investments. Having bought or sold stock in my niche to the point where any further bets would lose me as much in increased risk as they make me in expected return, additional investments will be outside of my niche. My investment income consists in part of a (say) 2% return on my capital, in part of an additional stream of income representing the rent on my specialized knowledge and obtained via a more than 2% return on capital invested in my niche.

It follows that the average return on investment, mine and everyone else's, is higher than the marginal return. My average return, pooling niche and non-niche investment, is something more than two percent. But my marginal return, what I would get if I invested another dollar, is only 2%, since I am already invested in my niche up to the limit imposed by risk aversion. The argument for equalizing returns on stocks and bonds is put in terms of average return—that, after all, is what we can observe. But the logic implies that what is equalized is marginal return. If bonds yield the same 2% return as stocks bought by an outsider—which, outside of my niche, I am—a 2% return on bonds is a sufficient reason not to shift capital out of them. 

We now have an explanation of how, in equilibrium, the average return on stocks can be consistently higher than the average return on bonds.

Unfortunately, the explanation cannot be right. The problem is that the outsider could choose some variant of the strategy sometimes described as throwing darts at the Wall Street Journal, buying stocks at random and so getting the market's average return. If he wishes to eliminate any random element, he could make his investment outside his niche by buying 1/100,000,000 of the stock of every firm on the  exchange,  guaranteeing himself the market's average return. He is  getting the average market return on his outside investments, more than the average market return on his inside investments. So is everyone else. 

Call the average market return R. I have just demonstrated that R equals a weighted average of Rout=R, the return on outside investments, and Rin>R, the turn on inside investments. Hence I have demonstrated that R>R, which is mathematically impossible.

I reached this point in the argument some years ago and eventually gave up, on the assumption that I must be making a mistake.  A day or two ago, it occurred to me that I had not only proved that my explanation of the stock/bond puzzle was wrong, I had also proved that insider trading, as normally imagined, does not exist.

To see why, apply the same argument to a market where only some investors are insiders. Anyone who wants can get the market return by investing in a random collection of stocks—or, to avoid any randomness, an equal fraction of every stock out there. Assuming, as economists routinely do, that investors are rational, all outsiders follow that strategy. Insiders get a return greater than the market return, outsiders get the market return. The market return is a weighted average of the return to insiders and outsiders. Hence R>R.

The only way out of this puzzle that I can see, whether for the general case of insider trading or my explanation of the stock/bond puzzle, is to assume that investors who are not insiders consistently follow a strategy that produces a lower return than another strategy readily available to them. When the insider buys or sells on the basis of his specialized knowledge there is some outsider willing to sell or buy, providing the other side of the transaction—despite knowing that doing so, in a market containing some insiders, is on average a losing game. That appears to contradict the assumption of rational actors, hence to be heretical from the standpoint of conventional economics.

Readers to whom all of this seems like confusing mumbo-jumbo are free to skip over it and wait for my next post, which will be on a different subject. Economist readers are invited to offer some solution to the puzzle that does not depend on investor irrationality.


At 3:06 PM, September 05, 2011, Blogger scottmatthew said...

"When the insider buys or sells on the basis of his specialized knowledge there is some outsider willing to sell or buy, providing the other side of the transaction—despite knowing that doing so, in a market containing some insiders, is on average a losing game."

The outsider only "loses" to the extent that he can expect a lower rate of return from his stock investing thatn an insider can expect. That my still be fine, if it is better rate of return than he believes is available for all other investments, which likely also have insiders to contend with.
I may buy Exxon for the dividend and hoped-for appreciation, and not worry that others with inside info can do even better. If I'm not an insider in any investable market, what is my alternative?
If I can't date the head cheerleader, does that mean I should die alone?

At 3:44 PM, September 05, 2011, Anonymous Anonymous said...

Bonds are a superior consumption hedge for periods relevant to mortals. If you were going to freeze yourself for 3000 years, you would put your money in stocks, not bonds because at very long horizons they are equally risky.

Regarding the insider trading puzzle: something you overlooked is that the relevant market includes private equity, which is not accessible to passive investors.

At 3:46 PM, September 05, 2011, Anonymous Robert Easton said...

How does this interact with the fact all index funds have tracking errors? i.e. an index fund investor does not get exactly the market rate of growth. As the relative values of companies change, the index fund investor changes his portfolio. But he's always slightly behind the news. Could this be where the extra return to "insiders" happens?

At 5:00 PM, September 05, 2011, Blogger Phil Birnbaum said...

You were not the only one looking at a Mac and noticing that it was better than the competition. By the time you did that, others had already noticed and bid up the price of Apple stock. You may not have made a premium on the deal, if that's the case.

Similarly, if it's insider information that the stock is too high, the price will quickly adjust down.

Now, suppose you buy an index fund today. You're not getting R. You're getting a little less than R. Why? Because there is a small chance that you are buying a stock at the exact time insiders are selling, and therefore paying too much. Therefore, your expected return on that purchase is R - delta.

Same when you eventually sell: you may be selling at exactly those few moments when insiders are buying. Again, you're losing a delta on the trade.

That doesn't solve the bond puzzle, but it does solve the insider puzzle, I think.

There could be something wrong with my logic, of course.

At 5:04 PM, September 05, 2011, Blogger Phil Birnbaum said...

As for the bond question ... couldn't it just be a matter of volatility? If, over the short term, many investors need to preserve the value of their capital (because they need to sell their investments shortly), they should be willing to accept a lower return in exchange for that security.

This can also apply to a 30-year bond. If you don't intend to keep it for 30 years, it might still makes sense to buy now and sell in 6 months. The fluctuation in price will be a lot smaller than for a stock portfolio.

Suppose I offer you 52:48 odds on a fair coin. Would you bet your house on it? Not for one toss. But if I offer you 1,000 tosses, betting 1/1000 of your house value on each toss, you'd take it.

For one toss, you might need 75:25 odds, or even higher, before you accepted.

Bonds could be the same thing. A lower expected value, but also with a lower standard deviation of outcome.

At 6:45 PM, September 05, 2011, Blogger Brandon Berg said...

If everyone buys all stocks in proportion to their market capitalization, then insider trading doesn't produce superior returns. Once the insider information is made public, it won't matter, because nobody is picking stocks based on this information.

For insider trading to work, you need stock-pickers to bid up the price of the stock once the information becomes public.

And when you have people picking stocks, there are going to be winners and losers. In the case of insider trading, the winner is the insider, because he's overweighted a high-performing stock in his portfolio, and also the first outsiders to get in immediately after the information becomes public. The loser is the person who sold the insider the stock, because he's underweighting a high-performing stock in his portfolio.

At 6:46 PM, September 05, 2011, Blogger John David Galt said...

I would expect that a lot of the demand (in the market sense) for bonds today is from institutions such as insurance companies and retirement funds, both private and governmental, because those buyers are at least partly restricted to investment choices traditionally considered safe -- both by the law and because investors (and policyholders) will likely sue them if their expectations aren't met.

I don't know of any comparable subset of investors who insist as a class on "less safe" investments.

At 6:54 PM, September 05, 2011, Blogger Brandon Berg said...

Also, it's not possible for an insider to overweight one stock in his portfolio without someone else underweighting it.

For example, consider a market with just you and me. We each own 50% of Acme's stock, and also 50% of Zenith's stock.

You get an advance look at Acme's quarterly earnings report and decide to buy some more. You can't, unless I sell it to you. And no matter how you get it, the fact remains that you'll have more of it than I will, and it will be more heavily represented in your portfolio than in mine.

At 7:42 PM, September 05, 2011, Blogger Robin Hanson said...

People who get an above market return from their info have to get that return at the expense of people on the other side of their trades. If uninformed traders just held their assets, and never traded them, informed traders could gain no added returns. Informed traders would just be have zero-sum duels with each other. The more than uninformed traders must trade to change the total amount of their investments, the more they will get below average return.

At 7:42 PM, September 05, 2011, Blogger Neolibertarian said...

In the U.S. in the 20th century, stocks have consistently outperformed bonds.

This must be the ivory tower in you speaking, as this statement is not at all true.

Bonds have outperformed stocks substantially over the last decade. A diversified mix of stocks, bonds, and hard assets will outperform a pure stock allocation both in terms of higher yield and in terms of lower variance.

I suppose this is the part where we are all reminded again that if you make false assumptions, the conclusions you arrive at are pure noise.

GIGO all the way.

At 8:18 PM, September 05, 2011, Anonymous Anonymous said...

An implicit assumption is that the insiders are beating the market based on their inside information -- my guess is that even insiders get it wrong as much as right for reasons independent of the 'nugget' of information they act upon.

At 8:33 PM, September 05, 2011, Blogger Milhouse said...

Neolibertarian, you may have noticed that the past decade is not part of the 20th century. If the past decade's performance does not match that of the 20th century, then either something important has changed, or it's just an aberration that will be corrected sooner or later.

At 8:51 PM, September 05, 2011, Anonymous Anonymous said...

> there is no such thing as insider trading

Yeah, right, sure. Google for index investment, noise trading - and recent insider trading lawsuits.

At 10:02 PM, September 05, 2011, Blogger Neolibertarian said...

@millhouse, I appreciate the nit picking, but the 20th century has two 10+ year stretches where bonds outperform bonds.

Bonds outperformed stocks in the 1930s and the 1970s. You need a 20 year window before stocks always outperform. And as noted above, a portfolio diversified across bonds, stocks, and hard assets outperforms stocks.

The idea that stocks are consistently better than bonds is the sort of naivite I might expect from a financial manager, not an "economist".

At 4:06 AM, September 06, 2011, Anonymous Anonymous said...

Some are "insiders" and can get above-average returns. Others are not insiders and consequently need to be satisfied with below-average returns (via index funds, let's say). Neither group would appear to be irrational. If I have no specialized knowledge, I have to be content with index-fund (below-average) returns, which still may be the best returns I can get, and hence are rational. (Average returns are somewhere between index-fund returns and insider-knowledge returns, and are unavailable to the "non-insider".) It sounds like everyone is rational here.

I think the error in Dr. Friedman's reasoning is where he says that the dart-throwing strategy will give you an average return. No! The average return is above the dart strategy. I think the confusion arises from the "average" and failing to consider what sets we are averaging over.

At 7:33 AM, September 06, 2011, Blogger Matt said...

Why not a real market in which a significant fraction of the investor population is partially (or "imperfectly" would perhaps be a better word) rational? That is, consciously choosing "rational" behavior when he is able to identify such, but not consistently able to do so correctly?

This being too complex a reality to model perfectly, we also have an imaginary market where every investor behaves "rationally", which we all agree to pretend is the reality because it predicts reality well enough most of the time, which no competing model has ever been able to do. This despite the fact that in the case of actual choices open to actual investors, the properly rational investment decision is frequently apparent only in hindsight.

Even a stopped clock is right twice a day, and it seems to me that Keynes was likewise right twice in his career:

1. "The market can stay irrational longer than you can stay liquid", and
2. "In the long run, we're all dead anyway".

In the "long run" much beloved of professional economists, market conditions do tend to revert to the mean of behavior predicted by the Efficient Market hypothesis and the Rational Investor hypothesis. But large deviations from that pattern are so common that we simply must accept the probability that these hypotheses are, while better than any others we've worked out in detail, nevertheless very imperfect proxies for reality on the ground.

If one accepts this, then your formulation no longer reduces to "R<R", meaning that while it still may be wrong, it is not manifestly absurd.

At 8:46 AM, September 06, 2011, Blogger Doc Merlin said...

The returns to the insiders are due in large part to timing. Outsiders don't have the advantage of timing.

lets say a stock is going to go up for a week then down for a week then back up for a week so it ends up roughly where it left off. An outsider is unlikely to earn any money from this transaction, while an insider is likely to earn money from it.

Remember, David, one can make money from any bet. One can make money by betting that the stock will go up, down, increase in volatility, decrease in volatility, heck you can even generate returns predicting that the price won't change. Therefor even if a market is going up at 6% every year, some individuals can be making 6% while others are making much more than 6%.

You basic problem comes from confusing returns with price increases, and by doing so assuming that everyone has a long-stocks strategy. This is simply not so. For example: right now most hedge funds are short the S&P 500 index.

At 9:10 AM, September 06, 2011, Anonymous Anonymous said...

I think the source of confusion is that you are assuming that the prices fully reveal information (or alternatively outsiders can fully observe insider actions).

Here is a simple example. Suppose an investment will be liquidated tomorrow and will be worth $100 or $200 with equal probability. You have no other investment options and you can either choose to invest and not invest. To make it simple, let’s say if you are an insider you know for sure that the investment will be worth $200.

If you are an insider your demand for this investment is your full wealth invested if $200 < Price of the investment.

If you are an outsider your demand for this investment is Expected value – Price / volatility of investment (you can get this result if you assume CARA utility). If the outsider is risk-neutral, it is simply fully invested if Expected investment return ($150) < Price.

Since the supply is fixed (that is fixed # of shares outstanding) the equilibrium price will determined with Demand = Supply. Assume that insiders and outsider have the same amount of wealth and outsiders make alpha % of the population. Eqbm price will solve: (1-alpha)*Indicator function (200 – Price) + alpha*(Expected value – price / volatility) = Supply. The eqbm price in this case will be < $200. You can then plug the eqbm price to find the respective returns of the insiders and outsiders.

Above we have assumed that outsiders completely ignore the actions of the insiders. If they observe the demand of the insiders then the eqbm price will be $200 and the returns to both insiders and outsiders will be 0. If they don’t observe the demand of the outsiders but just the price, then the price itself can partially reveal the demand of insiders (you can look at the early models of Grossman or Grossman and Stiglitz). If the price is fully revealing we are back to eqbm price = $200.

People usually add noise to make the price partially revealing to get interesting results. This is pretty standard finance stuff. As I mentioned you may want to look at the work of Grossman and Stiglitz who use this model, or Kyle who does something similar for market makers trading with insiders.


At 9:14 AM, September 06, 2011, Anonymous Anonymous said...

Btw, according to standard finance models, the % invested in stocks should not vary with investment horizon. They are not less risky in the long run. There are quite a few papers on this, but this quote from Samuelson sums it up nicely:

When a 35-year-old lost 82 percent of his pension portfolio between 1929 and 1932, do you think that it was fore-ordained in heaven that it would come back and fructify to +400 percent by his retirement at 65? How did the 1913 Tsarist executives fare in their retirement years on the Left Bank of Paris?


At 10:21 AM, September 06, 2011, Anonymous Simon said...

I'm not an economist, but I can't resist giving this a shot :-)

I think the key thing is this: Buying a random stock portfolio and holding it over a period of time will give you (on average) the average market return for the given period of time. But you still have to pick the time interval, and if you choose randomly, your average five-year investment will perform slightly below the average for a surrounding ten- or twenty-year interval.

To see this, consider a game with a single asset where the only decision is the timing of market entry and exit. The game has two players, Simon and David. In each round,
(1) the asset is owned by either Simon or David
(2) the asset yields a $1 profit or a $1 loss (with equal frequency) to its owner
(3) the owner decides if he wants to transfer the asset to the other player and writes down his decision
(4) the other player decides if he wants to receive the asset and writes down the decision
(5) if both players have written 'yes', the ownership changes in the subsequent round

Now, in this game, David has a (for me) unfortunate advantage: He knows the outcome ($1 profit or loss) of the next round and I don't. So he will keep (or offer to receive) the asset when it is about to make a profit and refuse (or offer to transfer) the asset when it is about to make a loss.

If I make my yes/no decisions by flipping a coin, David will earn money and I lose. How can I protect myself against David's superior knowledge (and his desire to pocket all my money)? The answer is to minimize the number of transfers, i.e., to always answer 'no'. If in the first round I own the asset, I'll hold on to it forever. If in the first round David owns it, I'll let him keep it forever. This optimal strategy leaves us both with an expected result of $0.

Similarly, in the stock market, there is a small expected loss associated with every timing decision the random investor makes in an environment of better-informed investors. So buying a portfolio and holding it for five years will yield a better return (on average) than holding it during five separate one-year periods. Holding it ten years will be even better - the expected return of the investment will approach the average market return as the length of the time interval increases. A portfolio bought at the beginning of time and sold at the end of time will match the market performance exactly.

The small expected loss associated with each randomly timed trade should explain how below-average performance is possible with rational investors and thus resolve the paradox. Or am I mistaken?

At 10:38 AM, September 06, 2011, Anonymous Anonymous said...

"Btw, according to standard finance models, the % invested in stocks should not vary with investment horizon. They are not less risky in the long run."

This only makes sense if you make the assumption that bonds are riskless. In reality, the riskiness of stocks and bonds converges in the (very) long run.

At 11:45 AM, September 06, 2011, Anonymous Max said...

The obvious other interpretation is that our actor buys some collection as above, same amount of each company, and sells it at the end of his "life" (whatever that means). There would hence be some, but very blotchy (e.g. sporadic temporally but dense upon occurence) action on the market. However, buying and selling these assets could only occur to "new" random buyers and insiders. Hence, the insiders would pay an informed price and the new random actors would pay whatever it was bid to, presumably by other random actors. This would create an unstable equilibrium whereby insiders would demand it until it exceeded it's true worth (only known to them) and random actors would demand it infinitely (unless we allow for some parameter regarding their "random buys," further complicating our model). As you can see, this illustrates the fact that this system involving never selling random (or "portfolio") buyers only functions if we initiate the system at some zero point, and then repeat this cyclically, a model which is impossible to realize (due, among other things, to non-synchronized life cycles).

At 11:48 AM, September 06, 2011, Anonymous Max said...

Sorry, the first part of my comment was lost in the shuffle. It was basically exploring the idea that people would randomly (according to some random variable they control) buy and sell stocks on the market, and showing that then insiders would be able to manipulate them at will and hence they'd perform below average. Then I explored never selling buyers and showed that this defeats the entire point of the stock market. I won't bother retyping these arguments as I'm sure you can reproduce them

At 5:53 PM, September 06, 2011, Blogger Curt C said...

This is a common problem for academic economists. Once you get into the real world of professionals, hedge fund managers, you would realize how worthless this assumption is. Combinations of stocks, bonds, and commodities perform even better over the long term. Sharpe ratio, or standard deviation of return is far more meaningful. Leverage as a variable for maximizing return also is more useful in low standard deviation products like bonds.

We can also throw in the moral hazard problem within banks. They would much rather risk immense amounts of money for a near guaranteed mediocre accounting profit than take a symmetric risk.

We can also add the period from 1981 onward as a time where long bonds outperformed stocks. Blindly following the past with out regard to fundamentals or price is similarly weak. There are far more successful models than to "just be long stocks."

Regardless, the efficient market hypothesis fails massively. I can give real world examples. Options pricing around near-term yrcw stock.

At 7:48 PM, September 06, 2011, Anonymous Tom Crispin said...

Anonymous from last night has it right: David's model assumes that insider knowledge has a positive value. If we assume that the average value of insider knowledge is zero, then the average rate of return R is just the average of insider trading.

In support:

a) insider knowledge that Betamax was a superior technology to VHS had a negative payoff - demonstrating that insider knowledge can be negative

b) before computers and widespread index funds, all trading was "insider": my stock charts are meaningful, yours are numeralogical gibberish.

At 11:58 AM, September 07, 2011, Anonymous Daublin said...

Parts of the argument depend on the investors in question being very small players, and others depend on the investors being very large players.

Much of the argument holds for small investors. However, one place it breaks down is the assumption that R is the weighted average of Rin and Rout. Instead, it's the weighted average of Rin, Rout, and RhalfThePlanet.

To account for RhalfThePlanet, you need to consider large investors, but then a different part of the argument has a problem. Large investors shift the stock price with their decisions. When the entire mass of insiders notices that some stock is underpriced, so they buy a lot of it, the price of that stock goes up. After a while, the price goes up enough that the insiders stop buying, but the outsiders carry on buying small quantities of it anyway. As a result, for any one stock, the insiders bought at a lower price than the outsiders.

Because insiders get the stock at a lower price, they get a better return on the very same stock. It is no longer true that Rout = R, because the outsiders bought at a worse than average price.

At 6:44 PM, September 07, 2011, Anonymous Anonymous said...

I'm not sure I understand the model here. If the outside rational investors just buy and hold, who trades with the insiders?

At 6:47 AM, September 08, 2011, Blogger neil craig said...

If "onsiders" are taking an excess of profits then the average profit from people following "what everybody knows" must be less and presumably close to what bonds make.

In theory if you fon't know anything you would thus go for the dartboard nethod and thus achieve the average but (A) us human beings feel nervous about that & (B) any stockbroker who peddled his tips on that basis would be unlikely to do well (though I understand that a few people use astrology or the bible which is a more respectable way of being random).

There is also the point that some stocks go bankrupt and you get nothing. It is little comfort when that happens to know that on average you would have outperformed the bond market.

At 9:21 AM, September 08, 2011, Anonymous Paul Birch said...

Others have pointed out how the insiders' gain comes from his better timing; even in a perfect market, outsiders can't see the corrected price until after the insiders have traded, so the insiders always have at least a one transaction advantage; that transaction is necessary to change the price.

The obvious exception to this is if outsiders use a proportional buy and hold strategy, when there are no further trades and everyone gets the average return in the form of dividends only.

What I'd like to point out is that the original argument also doesn't work to explain higher returns on stocks versus bonds, because insiders can pick better bonds too; the average return on bonds will be higher for insiders, just as for stocks and shares. The insider will either buy the bonds at a better price on the secondary bond market - or on first issue will make a better prediction of the likelihood of default and thus avoid the duds.

At 10:11 AM, September 08, 2011, Blogger Ari T said...

Btw. Bob Murhpy responded here.

Looking forward to a fruitful discussion.

At 5:41 PM, September 11, 2011, Blogger Vlad Tarko said...

Perhaps I'm missing something, but it seems to me there's an error with the math.

It should be

Rout = R - Rin

and not

Rout = R

If you take out of the average one of the best performing stocks you get a lower average. With this change you obtain R=R and not R>R.

To put it differently, if you buy one stock of X because you know it will be performing well, and buy one stock of all the others because you want to get the market rate, it's the same thing as just buying one stock of everything. Insider trading works because the insider relies only (or predominately) on Rin.

At 10:49 PM, September 13, 2011, Blogger Jon Leonard said...

One complication is that the various R aren't really scalar; they're drawn from some sort of distribution. It's not a simple matter of picking the investment with the higher expected return, because some amount of diversification is useful to hedge the possibility that returns are worse than expected.

This turns out to be an information theory problem, or at least the mathematics is identical (this application of information theory was first published in John Kelly's 1956 paper, "A New Interpretation of Information Rate.") If you know the expected return distribution of both stocks and bonds, then you can calculate what mix gives the best expected long term growth rate (with rebalancing). Unfortunately, knowing the actual distribution is impossible, and the answer is pretty sensitive to the likelihood and severity of negative-tail events (like the 1987 crash, for example.) Still, such methods are somewhat useful.

Coincidentally, the question of "inside" information turns out to be strongly related. The "inside" information is somewhat uncertain, and at least in theory, the excess return over a standard market portfolio should correspond to the information (in a mathematical sense) that the investor knows that the market does not.

My favorite book on this topic is William Poundstone's "Fortune's Formula", which is a very good popular treatment. I used Thomas and Cover's "Elements of Information Theory" to learn the math.

At 1:50 PM, September 17, 2011, Blogger Think a second time said...

Dr. Friedman, you have stumbled upon an incomplete version of an economic law I think I discovered a year ago:

"Return to risk will equal risk BUT ONLY IN SO FAR AS
divisibility is cost free and infinite."
I came to this when thinking about gambling. How can a casino make money? If you have for example, a raffle, and the prize is less than total ticket recites, a competitor can open start a competing raffle and ether offer a larger prize or charge less for each of the same number of tickets. This would continue until revenue=payouts...
EXCEPT that it costs something to operate a raffle. So, there is a cost in division.
What you said at the onset "he could make his investment outside his niche by buying 1/100,000,000 of the stock of every firm on the exchange, guaranteeing himself the market's average return." is untrue. Nether stocks nor the money used to buy them is infinitely divisible. Corporate management always tries to control the degree of divisibility of the companies' stock. They do so because this changes the kind of people who own the firm. If they want a larger number of share holders, each having a smaller stake, they do stock splitting. Some share holders will sell smaller peaces of the company than it was previously possible to sell. Or, if the management wants fewer owners of the company who each have a higher stake, the company will buy its' own stock.
The result is that you cannot buy 1/900th of a share in apple. You can buy into a hedge fund, but hedge fund managers very in skill and choosing a better than average one is as difficult as choosing a better than average oil company to invest in. And no hedge fund is larger enough to buy an equal share in all companies.
So, in the cases of a raffle or insurance or the stock market the risk one bears is different than the return he gets for it because it is not possible to infinitely divide the risk at zero cost.
An area in which it is possible to divide at almost zero cost almost infinitely is sports betting in Vegas. In a sports bar offers one team 2 to 5 odds and another offers that team 3 to 5 odds, you can make money at no risk by betting on the team in one bar and against it in another at the right ratio. Because bookies do this, the odds in every bar in Vegas are the same. This is only true because the minimum bet is low enough that they can afford to bet on both sides. If the minimum bet were raised significantly, the risk would be less divisible and we would observe a curious non-equilibrium in sports bars much like the one you noticed in stocks versus bonds.

By the way: two phenomenon are confused in the phrase "risk preference".
One is the degree one prefers making his situation better in more likely future possible worlds over making his situation better in less likely future possible worlds. In this, the man who buys insurance is no different than the man who buys a lotto ticket. But the man who operates a raffle or an insurance company is different than his patrons.
The other is if and to what extent one prefers to make himself better off in possible situations in which he is worse off or if he prefers to make himself better off in situations in which he is better off. In this, one who buys insurance is the opposite of one who buys a lotto ticket.
The law of diminishing marginal utility breaks down when two of a good can be used for something more than twice as good as what one of them can be used for. So, if what I can buy for $10 is worth more than twice as much to me than what I can buy for $5, it makes sense for me to bet $5 on a coin toss with 50-50 odds.

At 4:11 PM, September 18, 2011, Blogger Fearsome Pirate said...

The problem is that you've incorrectly defined outsiders. The outsiders get slightly less than the market return. Denote N number, R return, O outsiders, I insiders. Then the market return is

R(M) = (R(O)*N(O) + R(I)*N(I))/(N(O) + N(I))

The error comes because as N(O)/N(I) -> infinity, R(O) = R(M) in the limit...but only in the limit. For large values or N(O) and small values of N(I), they are very close, but not equal. They are close enough to approximate as equal for practical purposes, but that's not the same thing as being equal.

At 4:20 PM, September 18, 2011, Blogger Fearsome Pirate said...

Oh, and another problem with comparing stocks and bonds...they're different financial products. Stocks offer a higher average long term return, but I can't pay my electric bill with long-term returns. I can, however, pay it out of a consistent monthly stipend, something that's much easier to obtain with a fund full of bonds. Your risk of getting wiped out with bonds is much lower, and if you're depending on your investments for cash flow, or you need to make sure your assets stay above a certain point in order to avoid bankruptcy, they can be a better buy.

At 8:38 AM, December 05, 2011, Anonymous Anonymous said...

"Winner's curse" explains why the insiders win in the IPOs: if an outsider buys 1/10M of each company, in good IPOs he'll get less (due to overbooking) and he has to buy more afterwards from the market, at a higher price.

A similar problem happens whenever the outsider tries to trade something, e.g., add his share of the companies: maybe he gets poor shares more quickly than good ones.

It was recently in the news that some companies provide, at a price, a quick look at future stock purchase orders (e.g., "5000 X shares for $10.10") and thus allows the high frequency trading operators to buy 5000 shares at $10.09 some milliseconds before the actual order, thus profiting $50. This seems to be a minus-sum game.

Somebody claimed that an index fund has to buy more when stock prices change.

Why? If the fund owns 1/10M of each company, it only has to buy when the size of the fund changes, not when stock prices change.


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