140% Stocks (or why the Kelly criterion is cool)
 backpacker
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140% Stocks (or why the Kelly criterion is cool)
Common sense says there is no limit to how fast things can travel. Common sense says there is no limit to how fast a portfolio is likely to grow as long as you are willing to take on more risk. Common sense is wrong about both. Nothing can travel faster than the speed of light. No portfolio is likely to grow faster than a 140% stock portfolio. Why is that? Read on to find out.
Last edited by backpacker on Sun Apr 26, 2015 10:00 am, edited 1 time in total.
 backpacker
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Re: 140% Stocks (or why the Kelly criterion is cool)
Suppose you have $1,000. I offer you a bet on a fair coin. Heads you double your money. Tails I keep whatever you bet. You can make one bet a day for as long as you want. How much should you bet each day?
You could bet everything every chance you get. That would maximize your expected returns. It would not maximize your likely longterm returns. Eventually the coin will land tails and you will have no money.
There is an answer to how much you should bet if you want to maximize your longterm winnings. That answer is given by the Kelly criterion. What you do to find the optimal strategy is (a) tell the Kelly criterion what the odds of different outcomes are and (b) tell the Kelly criterion how much you gain or lose in each outcome. The formula then tells you how much to bet to maximize your longterm returns. For the coin flipping game, the answer is 25%. You should bet 25% of your "portfolio" on the game at any given time.
The Kelly criterion for stocks is a simple formula, assuming that returns are normally distributed. That formula is: The historic excess return of the S&P 500 is about 5.6%. The standard deviation about 20%. So the Kelly criterion says to put 140% of your portfolio in stocks to maximize your likely longterm returns.
If you're paying attention, you noticed that I just made some silly assumptions. I assumed that stock returns are normal. They're not. I assumed that historic S&P 500 returns represent the real odds of investing in stocks. They don't. So 140% isn't really optimal. What is? That's the million dollar question.
Why does this matter then? In debates about 100% stocks, someone always says: Why 100% stocks? Why not 105%? Or 110%? The assumption being that you can always trade more risk for higher likely returns in the long run. So there is no place to draw the line, to point at which an investor who wants higher returns in the longrun can draw the line. That's completely false. If we knew the real odds of investing in stocks (i.e. if we knew the real return distribution), we could use the Kelly criterion to find the portfolio with the highest likely longterm returns. That portfolio is the "speed limit". If you take on more or less risk than that portfolio, you will likely get lower returns in the longrun. Who knows what the speed limit is. It's probably lower than 140%. But I would bet it's not too far from 100% stocks.
(This post is largely a summary of what I learned from these great notes on investing and favorable bets.)
You could bet everything every chance you get. That would maximize your expected returns. It would not maximize your likely longterm returns. Eventually the coin will land tails and you will have no money.
There is an answer to how much you should bet if you want to maximize your longterm winnings. That answer is given by the Kelly criterion. What you do to find the optimal strategy is (a) tell the Kelly criterion what the odds of different outcomes are and (b) tell the Kelly criterion how much you gain or lose in each outcome. The formula then tells you how much to bet to maximize your longterm returns. For the coin flipping game, the answer is 25%. You should bet 25% of your "portfolio" on the game at any given time.
The Kelly criterion for stocks is a simple formula, assuming that returns are normally distributed. That formula is:
Code: Select all
(optimal percentage in stocks)=(average excess return of stocks over bonds)/(standard deviation of excess returns)^2
If you're paying attention, you noticed that I just made some silly assumptions. I assumed that stock returns are normal. They're not. I assumed that historic S&P 500 returns represent the real odds of investing in stocks. They don't. So 140% isn't really optimal. What is? That's the million dollar question.
Why does this matter then? In debates about 100% stocks, someone always says: Why 100% stocks? Why not 105%? Or 110%? The assumption being that you can always trade more risk for higher likely returns in the long run. So there is no place to draw the line, to point at which an investor who wants higher returns in the longrun can draw the line. That's completely false. If we knew the real odds of investing in stocks (i.e. if we knew the real return distribution), we could use the Kelly criterion to find the portfolio with the highest likely longterm returns. That portfolio is the "speed limit". If you take on more or less risk than that portfolio, you will likely get lower returns in the longrun. Who knows what the speed limit is. It's probably lower than 140%. But I would bet it's not too far from 100% stocks.
(This post is largely a summary of what I learned from these great notes on investing and favorable bets.)
 nisiprius
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Re: 140% Stocks
I've been trying to absorb William Poundstone's book, Fortune's Formula. It is a great book and not at all what I thought it would be (I thought it would be an adulatory puff piece making unverifiable claims of certain people getting rich by being geniuses with a formula). He touches on all the things that seems to get discussed here and explains them very clearly. I think I need to read it a second time, though.
Here's a question to argue about. In its original formulation, it seems perfectly clear the Kelly Criterion refers to a situation where a gambler has inside information about a situation and knows that the true odds are different from the offered odds. The "inside information" wouldn't necessarily be anything illegal, but it has to be real "asymmetrical information"some actual knowledge that is different from what "the market" (e.g. the parimutuel board) knows. It is the optimal betting strategy when you actually know something that the oddsmaker doesn't know.
In Kelly's original paper, the "situation which we have chosen here"the only situation consideredis:
OK, obviously this applies to the stock market if you have something resembling inside information. It is the optimum way, by some measure, to exploit a market inefficiencybut you have to have the market inefficiency. Don't you?
Does it or does it not actually apply to a buyandhold indexfund investor who is not even seeking an "edge?" Does it apply simply because the stock market has a positive expected return? Does it actually apply to an efficient market? Or does it only apply in a situation where we are convinced we know something the market doesn't know (and are right?)
In the stock market analogy, for an ordinary indexfund investor who is trying to decide on an asset allocation and considering using leverage, what is the analog to the "posted odds" and what is the analog to the "edge?"
Here's a question to argue about. In its original formulation, it seems perfectly clear the Kelly Criterion refers to a situation where a gambler has inside information about a situation and knows that the true odds are different from the offered odds. The "inside information" wouldn't necessarily be anything illegal, but it has to be real "asymmetrical information"some actual knowledge that is different from what "the market" (e.g. the parimutuel board) knows. It is the optimal betting strategy when you actually know something that the oddsmaker doesn't know.
In Kelly's original paper, the "situation which we have chosen here"the only situation consideredis:
He then goes on to consider the case of a "noisy channel," where the gambler has only an edge, not a certainty, and that's mostly what the article is about.The Gambler with a Private Wire
Let us consider a communication channel which is used to transmit the results of a chance situation before those results become common knowledge, so that a gambler may still place bets at the original odds. Consider first the case of a noiseless binary channel, which might beused, for example, to transmit the results of a series of baseball games between two equally matched teams...
OK, obviously this applies to the stock market if you have something resembling inside information. It is the optimum way, by some measure, to exploit a market inefficiencybut you have to have the market inefficiency. Don't you?
Does it or does it not actually apply to a buyandhold indexfund investor who is not even seeking an "edge?" Does it apply simply because the stock market has a positive expected return? Does it actually apply to an efficient market? Or does it only apply in a situation where we are convinced we know something the market doesn't know (and are right?)
In the stock market analogy, for an ordinary indexfund investor who is trying to decide on an asset allocation and considering using leverage, what is the analog to the "posted odds" and what is the analog to the "edge?"
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.
Re: 140% Stocks (or why the Kelly criterion is cool)
There are two issues with the Kelly criterion.
The more important issue is that it assumes a static portfolio. If you are adding to your portfolio while working, or withdrawing in retirement, your portfolio will not grow at the same rate as its investments.
The other issue is the discontinuity of interest rates; you cannot lend at the same rate you borrow. This is likely to make 100% stock optimal for a large range of investors, as the interest cost in going from 100% to 110% is higher than the lost interest in going from 90% to 100%. (You can still get the benefit of more than 100% stock by investing in riskier stocks; the stock of a company with a lot of debt might have a beta of 1.5, so that a portfolio of such companies would rise or fall by 150% as much as the stock market.)
The more important issue is that it assumes a static portfolio. If you are adding to your portfolio while working, or withdrawing in retirement, your portfolio will not grow at the same rate as its investments.
The other issue is the discontinuity of interest rates; you cannot lend at the same rate you borrow. This is likely to make 100% stock optimal for a large range of investors, as the interest cost in going from 100% to 110% is higher than the lost interest in going from 90% to 100%. (You can still get the benefit of more than 100% stock by investing in riskier stocks; the stock of a company with a lot of debt might have a beta of 1.5, so that a portfolio of such companies would rise or fall by 150% as much as the stock market.)

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Re: 140% Stocks (or why the Kelly criterion is cool)
I was under the impression that the optimal Kelly leverage was only 117% stocks. Here's from a post from a prior conversation.
nisiprius wrote:Thorp's article, Understanding the Kelly Criterion, says:1) Is this the definitive statement of exactly what it is that the Kelly strategy optimizes?So, given any admissible strategy, there is a fractional Kelly strategy with which has a growth rate that is no lower and a variance of the growth rate that is no higher.
2) Does he mean "variance of the growth rate" or "variance of log(1 + growth rate)?
3) Earlier, tadasmar cited a paper by Rotando and Thorpe, The Kelly Criterion and the Stock Market, that says 117% stocks, i.e. 17% leverage is the Kelly strategy for the stock market.
This might be true if your investing universe is only stocks. What happens if you can invest in other things like bonds? I believe the answer may be different.backpacker wrote:Common sense says there is no limit to how fast things can travel. Common sense says there is no limit to how fast a portfolio is likely to grow as long as you are willing to take on more risk. Common sense is wrong about both. Nothing can travel faster than the speed of light. No portfolio is likely to grow faster than a 140% stock portfolio. Why is that? Read on to find out.
Re: 140% Stocks (or why the Kelly criterion is cool)
.
Last edited by 555 on Sun Apr 26, 2015 10:59 am, edited 1 time in total.
Re: 140% Stocks (or why the Kelly criterion is cool)
...backpacker wrote:The Kelly criterion for stocks is a simple formula, assuming that returns are normally distributed.
Also note that the Kelly criterion only applies when you "go bust" upon hitting 0 net worth. This is not the case as margin requirements will raise that threshold, and noncallable debt can let you go negative for substantial amounts of time.
The Kelly criterion applies to repeated bets made on independent trials of a game with well known expected returns. Stock market returns are not well characterized or independent.
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Re: 140% Stocks (or why the Kelly criterion is cool)
By the way, I don't think it is a straw man to ask "why not 200% stocks," because of this article (which was cited by "market timer" when he began a course of investments that resulted in his losing $200,000 more than he had).
In short, two writers who are explicitly offering advice to young investors either do not know or do not agree that the Kelly criterion puts a sanity ceiling of 140% on stock investing.
In a later book, they suggest a strategy based on LEAPS, but not because of risk, but becauseto them, regrettably401(k) plans do not offer the ability to use margin. "In the not too distant future, courts may determine that failing to offer employees the option to diversify temporarally" (by not offering young 401(k) participants a means of using 200% leverage) will fail the legal test of fiduciary responsibility.You should be more than 100% in equities when you are young. An exposure of 200% to start would be a better idea. That's rightif you are young, you should be buying on margin. Pay down the debt as you age and then ease off to a 5050 stockandbond mix at the beginning of your retirement.
In short, two writers who are explicitly offering advice to young investors either do not know or do not agree that the Kelly criterion puts a sanity ceiling of 140% on stock investing.
Last edited by nisiprius on Sun Apr 26, 2015 11:31 am, edited 1 time in total.
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Re: 140% Stocks (or why the Kelly criterion is cool)
The last part is certainly not true. To get higher leverage in stocks, you use the Emini SP500 futures (ES). To get higher leverage in bonds, you use the various US Govt Bond Futures: 5yr, Note, 10+Yr, etc. The embedded leverage in those is very very reasonable. In fact, you can infer the appropriate interest rate from the basis of these instruments vs. the cash products.grabiner wrote:There are two issues with the Kelly criterion.
The more important issue is that it assumes a static portfolio. If you are adding to your portfolio while working, or withdrawing in retirement, your portfolio will not grow at the same rate as its investments.
The other issue is the discontinuity of interest rates; you cannot lend at the same rate you borrow. This is likely to make 100% stock optimal for a large range of investors, as the interest cost in going from 100% to 110% is higher than the lost interest in going from 90% to 100%. (You can still get the benefit of more than 100% stock by investing in riskier stocks; the stock of a company with a lot of debt might have a beta of 1.5, so that a portfolio of such companies would rise or fall by 150% as much as the stock market.)

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Re: 140% Stocks (or why the Kelly criterion is cool)
I believe that the authors interpretation of Kelly criterion might be different, than say, what the objective of an ordinary investor is.nisiprius wrote:By the way, I don't think it is a straw man to ask "why not 200% stocks," because of this article (which was cited by "market timer" when he began a course of investments that resulted in his losing $200,000 more than he had).In a later book, they suggest a strategy based on LEAPS, but not because of risk, but becauseto them, regrettably401(k) plans do not offer the ability to use margin. "In the not too distant future, courts may determine that failing to offer employees the option to diversify temporarily" (by not offering young 401(k) participants a means of using 200% leverage) will fail the legal test of fiduciary responsibility.You should be more than 100% in equities when you are young. An exposure of 200% to start would be a better idea. That's rightif you are young, you should be buying on margin. Pay down the debt as you age and then ease off to a 5050 stockandbond mix at the beginning of your retirement.
In short, two writers who are explicitly offering advice to young investors either do not know or do not agree that the Kelly criterion puts a sanity ceiling of 140% on stock investing.
My interpretation of where to take leverage (not necessarily of the Kelly criterion) is to take leverage in the asset with the best riskadjusted returns.
E.g., one of the best known trades in US treasuries is that by taking more leverage in the juiciest part of the yield curve, you can beat the riskadjusted returns of the highest yielding part of the yield curve.
In fact, if you use dollar risk as a limiting factor, the optimal portfolio will be something like:
50% stocks,
130% bonds.
(numbers not precise)
 ResearchMed
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Re: 140% Stocks (or why the Kelly criterion is cool)
As an aside, should the word "temporarily" be "temporally"?nisiprius wrote:By the way, I don't think it is a straw man to ask "why not 200% stocks," because of this article (which was cited by "market timer" when he began a course of investments that resulted in his losing $200,000 more than he had).In a later book, they suggest a strategy based on LEAPS, but not because of risk, but becauseto them, regrettably401(k) plans do not offer the ability to use margin. "In the not too distant future, courts may determine that failing to offer employees the option to diversify temporarily" (by not offering young 401(k) participants a means of using 200% leverage) will fail the legal test of fiduciary responsibility.You should be more than 100% in equities when you are young. An exposure of 200% to start would be a better idea. That's rightif you are young, you should be buying on margin. Pay down the debt as you age and then ease off to a 5050 stockandbond mix at the beginning of your retirement.
In short, two writers who are explicitly offering advice to young investors either do not know or do not agree that the Kelly criterion puts a sanity ceiling of 140% on stock investing.
What is the other book?
On first glance, doesn't this type of "strategy" ignore the risk, however small or large, of going totally belly up?
RM
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 backpacker
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Re: 140% Stocks
Cool! I'll have to dig up a copy.nisiprius wrote:I've been trying to absorb William Poundstone's book, Fortune's Formula. It is a great book and not at all what I thought it would be (I thought it would be an adulatory puff piece making unverifiable claims of certain people getting rich by being geniuses with a formula). He touches on all the things that seems to get discussed here and explains them very clearly. I think I need to read it a second time, though.
I think this is the idea. The odds are stacked against you in normal gambling. The bets are all unfavorable. Meaning that given public information, chances are that you will not make money. If you have inside information about a horse race, a bet that is unfavorable for everyone else becomes favorable for you. Chances are that you will make money even if that's not true for everyone else. Kelly appeals to inside information because he needs an example of a favorable betting game, one in which the chances are that you'll make money. The only way to get that with traditional gambling is to have inside information.nisiprius wrote: Here's a question to argue about. In its original formulation, it seems perfectly clear the Kelly Criterion refers to a situation where a gambler has inside information about a situation and knows that the true odds are different from the offered odds. The "inside information" wouldn't necessarily be anything illegal, but it has to be real "asymmetrical information"some actual knowledge that is different from what "the market" (e.g. the parimutuel board) knows. It is the optimal betting strategy when you actually know something that the oddsmaker doesn't know.
[...]
[O]bviously this applies to the stock market if you have something resembling inside information. It is the optimum way, by some measure, to exploit a market inefficiencybut you have to have the market inefficiency. Don't you? [...] Does it or does it not actually apply to a buyandhold indexfund investor who is not even seeking an "edge?"
Owning stocks instead of bonds is favorable bet that "occurs in nature". No inside information is required. Given public information, odds are that $100 in stocks will beat $100 in bonds over any given future time period. There's no guarantee, but buying stocks instead of bonds is not like putting your money in a slot machine. It's better than that. It's a favorable bet. Why owning stocks instead of bonds is a favorable bet is a puzzle. I'm not sure anyone really knows.
We Bogleheads think that active management is not a favorable bet. It's like paying someone to put your money in a slot machine. Given that, the Kelly criterion gives a clear answer about how much of your portfolio to put actively manage: 0%. This because the Kelly criterion tells us to avoid unfavorable bets completely.
Re: 140% Stocks
backpacker wrote:It's a favorable bet. Why owning stocks instead of bonds is a favorable bet is a puzzle. I'm not sure anyone really knows.
Re: 140% Stocks (or why the Kelly criterion is cool)
In theory, this absolutely does try to avoid that. When trying to maximize geometric mean, multiplying by zero is the last thing you want to do. In practice, this risk is really there. You don't know the "true" probability distribution of outcomes, so you need to err on the side of caution.ResearchMed wrote:On first glance, doesn't this type of "strategy" ignore the risk, however small or large, of going totally belly up?
Re: 140% Stocks
You are getting the wrong impression if you think you need inside information. You need a positive expected return on the bet, having inside information is just one example of how one might have a higher expected return. If you have a positive expected return, then you have an edge.nisiprius wrote:I've been trying to absorb William Poundstone's book, Fortune's Formula. It is a great book and not at all what I thought it would be (I thought it would be an adulatory puff piece making unverifiable claims of certain people getting rich by being geniuses with a formula). He touches on all the things that seems to get discussed here and explains them very clearly. I think I need to read it a second time, though.
Here's a question to argue about. In its original formulation, it seems perfectly clear the Kelly Criterion refers to a situation where a gambler has inside information about a situation and knows that the true odds are different from the offered odds. The "inside information" wouldn't necessarily be anything illegal, but it has to be real "asymmetrical information"some actual knowledge that is different from what "the market" (e.g. the parimutuel board) knows. It is the optimal betting strategy when you actually know something that the oddsmaker doesn't know.
In Kelly's original paper, the "situation which we have chosen here"the only situation consideredis:He then goes on to consider the case of a "noisy channel," where the gambler has only an edge, not a certainty, and that's mostly what the article is about.The Gambler with a Private Wire
Let us consider a communication channel which is used to transmit the results of a chance situation before those results become common knowledge, so that a gambler may still place bets at the original odds. Consider first the case of a noiseless binary channel, which might beused, for example, to transmit the results of a series of baseball games between two equally matched teams...
OK, obviously this applies to the stock market if you have something resembling inside information. It is the optimum way, by some measure, to exploit a market inefficiencybut you have to have the market inefficiency. Don't you?
Does it or does it not actually apply to a buyandhold indexfund investor who is not even seeking an "edge?" Does it apply simply because the stock market has a positive expected return? Does it actually apply to an efficient market? Or does it only apply in a situation where we are convinced we know something the market doesn't know (and are right?)
In the stock market analogy, for an ordinary indexfund investor who is trying to decide on an asset allocation and considering using leverage, what is the analog to the "posted odds" and what is the analog to the "edge?"
Last edited by tadamsmar on Sun Apr 26, 2015 11:41 am, edited 1 time in total.
 ResearchMed
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Re: 140% Stocks (or why the Kelly criterion is cool)
"In theory, theory and practice are the same. In practice, they are not."555 wrote:In theory, this absolutely does try to avoid that. When trying to maximize geometric mean, multiplying by zero is the last thing you want to do. In practice, this risk is really there. You don't know the "true" probability distribution of outcomes, so you need to err on the side of caution.ResearchMed wrote:On first glance, doesn't this type of "strategy" ignore the risk, however small or large, of going totally belly up?
attributed to Albert Einstein, Yogi Berra, and no doubt others.
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Re: 140% Stocks (or why the Kelly criterion is cool)
I could be wrong, but I believe one assumption of the "Mortgage your retirement" by Ayres and Nalebuff is that you are in the accumulation phase and are constantly infusing more money into your portfolio. That breaks the Kelly model where the only change is from the returns on your bets.
"Temporally" is the correct word. The idea is that you take higher risk investments early when you are young and less as you get older. You are spreading your risk over time, hence "temporally".ResearchMed wrote:As an aside, should the word "temporarily" be "temporally"?
What is the other book?
On first glance, doesn't this type of "strategy" ignore the risk, however small or large, of going totally belly up?
 backpacker
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Re: 140% Stocks (or why the Kelly criterion is cool)
I agree with both points. The cost of margin will tend to create a gravitational "pull" around 100% stocks.grabiner wrote:There are two issues with the Kelly criterion. The more important issue is that it assumes a static portfolio. [...] The other issue is the discontinuity of interest rates; you cannot lend at the same rate you borrow.
It depends on what numbers you use for the "historical" numbers. The paper you linked to uses a higher standard deviation than I used. Kelly hates risk, so will recommend less margin if you tell him that there is more risk.Rob Bertram wrote:I was under the impression that the optimal Kelly leverage was only 117% stocks.
I think that's right. The "portfolio" for A&N includes human capital. So 200% stocks isn't really 200% of your "portfolio" as they think of your portfolio. Tell Kelly that your portfolio is your cash plus the value of your human capital, and he too will tell you to leverage the heck out of your investments.Rob Bertram wrote:I could be wrong, but I believe one assumption of the "Mortgage your retirement" by Ayres and Nalebuff is that you are in the accumulation phase and are constantly infusing more money into your portfolio. That breaks the Kelly model where the only change is from the returns on your bets.
In the special cases it deals with, the Kelly criterion is equivalent to using expected utility with a logarithmic utility function. So the basic idea is easy to extend in principle to distributions failing these simple assumptions. The main point, as I see, is that investing comes with speed limits. There is always a point at which taking on more risk decreases the returns you are likely to get in the long run. Finding that speed limit is just harder in the real world than in simple idealized cases.691175002 wrote: The Kelly criterion applies to repeated bets made on independent trials of a game with well known expected returns. Stock market returns are not well characterized or independent.
Last edited by backpacker on Sun Apr 26, 2015 11:28 am, edited 1 time in total.
 backpacker
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Re: 140% Stocks
See "the equity risk puzzle".691175002 wrote:backpacker wrote:It's a favorable bet. Why owning stocks instead of bonds is a favorable bet is a puzzle. I'm not sure anyone really knows.
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Re: 140% Stocks (or why the Kelly criterion is cool)
Yes. Thanks. Fixed.ResearchMed wrote:As an aside, should the word "temporarily" be "temporally"?
"Lifecycle Investing," by Ayres and Nalebuff.What is the other book?
Ayres and Nalebuff say that in their backtesting it didn't seem to happen very often, and that when it does it's not a serious problem for a young investor because they have the time and wherewithal to pick themselves up, dust themselves off, and and start all over again. As "market timer" seems to have done.On first glance, doesn't this type of "strategy" ignore the risk, however small or large, of going totally belly up?
RM
Ayres and Nalebuff argue that the superior average outcomes in their simulations are well worth the small risk of financial ruin when young. You will have to ask "market timer" if he agrees.
Last edited by nisiprius on Sun Apr 26, 2015 11:46 am, edited 1 time in total.
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Re: 140% Stocks
We know that stocks have higher returns than bonds because of their higher risk. A debtholder has a senior claim on assets of a company and therefore should expect lower returns than an equityholder who will likely get nothing should the company fail.backpacker wrote:See "the equity risk puzzle".691175002 wrote:backpacker wrote:It's a favorable bet. Why owning stocks instead of bonds is a favorable bet is a puzzle. I'm not sure anyone really knows.
Owning stocks instead of bonds has not always been a favorable bet on a risk adjusted basis. Risk parity portfolios observe that you can leverage up a mostly bond portfolio to the same volatility as equity and produce superior returns
The equity risk puzzle only states that the equity risk premium is surprisingly high (when compared to the riskfree rate, not bonds in general). It is not generally accepted and its implications would be more subtle than "We don't know why owning stocks is better than bonds".
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Re: 140% Stocks (or why the Kelly criterion is cool)
The interesting thingPoundstone is very good on this pointthere has, all along, been continuing controversy about whether or not the Kelly criterion depends on a logarithmic utility function... or, in fact, any utility function. It's amazing to me but there just isn't any agreement on this point.backpacker wrote:...In the special cases it deals with, the Kelly criterion is equivalent to using expected utility with a logarithmic utility function...
The original Kelly paper says "The reason has nothing to do with the value function which he attached to his money, but merely with the fact that it is the logarithm which is additive in repeated bets and to which the law of large numbers applies." However, other peopleI don't have Poundstone's book at hand at the moment but I think Samuelson was one of theminsist that a logarithmic utility function is a smuggledin assumption somewhere.
Last edited by nisiprius on Sun Apr 26, 2015 11:50 am, edited 2 times in total.
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 ResearchMed
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Re: 140% Stocks (or why the Kelly criterion is cool)
If one is playing with fire, I mean margin, isn't there the chance of getting underwater, and sinking deeper and deeper?nisiprius wrote:Yes. Thanks. Fixed.ResearchMed wrote:As an aside, should the word "temporarily" be "temporally"?"Lifecycle Investing," by Ayres and Nalebuff.What is the other book?Ayres and Nalebuff say that in their backtesting it didn't seem to happen very often, and that when it does it's not a serious problem for a young investor as they have the time and wherewithal to pick themselves up, dust themselves off, and and start all over again. As "market timer" seems to have done.On first glance, doesn't this type of "strategy" ignore the risk, however small or large, of going totally belly up?
RM
Ayres and Nalebuff argue that the superior average outcomes in their simulations are well worth the small risk of financial ruin when young. You will have to ask "market timer" if he agrees.
I suppose the ultimate "dust oneself off" would then be "just" to declare bankruptcy?
Ethics aside (because this would sort of be a "planned" use of moral hazard), I suppose it would work if one had nothing else of value.
RM
This signature is a placebo. You are in the control group.

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Re: 140% Stocks (or why the Kelly criterion is cool)
I haven't read Kelly, I haven't read the notes, I haven't read Poundstone although I have the book on my shelf.
So, while very interesting, whether you or Kelly are correct ends up not being important for the great majority of investors. In fact talking about this false God, er, goal, of maximizing returns, tends to be distracting and ultimately destructive to nearly all investors  present company excluded, since Bogleheads are a special group and would never be distracted by something as prurient as simple greed.
(backpacker, I'm not saying you are advocating this line of thinking, or yourself being a Pied Piper of the investment world. It's rather widespread, though thankfully not ubiquitous, that the media and financial shills talk primarily about returns, and only occasionally, if at all, about risk.)
95% or 99% of investors should not be solving for this X. The X they should be solving for is, how do I minimize the odds of failing to meet my minimum needs for (retirement, college, whatever long termgoal).So the Kelly criterion says to put 140% of your portfolio in stocks to maximize your likely longterm returns.
So, while very interesting, whether you or Kelly are correct ends up not being important for the great majority of investors. In fact talking about this false God, er, goal, of maximizing returns, tends to be distracting and ultimately destructive to nearly all investors  present company excluded, since Bogleheads are a special group and would never be distracted by something as prurient as simple greed.
(backpacker, I'm not saying you are advocating this line of thinking, or yourself being a Pied Piper of the investment world. It's rather widespread, though thankfully not ubiquitous, that the media and financial shills talk primarily about returns, and only occasionally, if at all, about risk.)

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Re: 140% Stocks (or why the Kelly criterion is cool)
For those who haven't read the book "Lifecycle Investing" bay Ayres and Nalebuff, Market Timer is a case study in one of the chapters.nisiprius wrote:"Lifecycle Investing," by Ayres and Nalebuff.What is the other book?Ayres and Nalebuff say that in their backtesting it didn't seem to happen very often, and that when it does it's not a serious problem for a young investor because they have the time and wherewithal to pick themselves up, dust themselves off, and and start all over again. As "market timer" seems to have done.On first glance, doesn't this type of "strategy" ignore the risk, however small or large, of going totally belly up?
RM
Ayres and Nalebuff argue that the superior average outcomes in their simulations are well worth the small risk of financial ruin when young. You will have to ask "market timer" if he agrees.
Here's what Market timer said recently in a discussion on leverage:
market timer wrote:Yes, I agree with the rationale of Mortgage Your Retirement. In fact, I think OP's method of leveraging into a balanced fund makes more sense than a leveraged bet on equities, due to capturing the benefits of the term premium (positive expected return, negatively correlated with equities). Implementing this strategy is a challenge. There aren't any good investment vehicles that do the work for you, so it has to be DIY. The cheapest funding is actually not using margin, but options and futures.madbrain wrote:market timer,
So, do you still believe using margin to leverage longterm investments has merit ?
If so, what kind of strategy would have not just survived this 2008 period until today, but also offered a return greater than holding the overall market ?
And is it something you would ever recommend to someone who hasn't already "hit their number", unlike the OP ?
As for performance, it would not be hard to create ex post a balanced fund with leverage that survived 2008 and outperformed 100% stocks.
I'd recommend MYR to those in accumulation stage with stable careers. However, careers are sometimes shorter than expected. Mine lasted all of five years before I burned out. After reaching one's number, I'd be more inclined to transition to a LMP mindset.
Re: 140% Stocks (or why the Kelly criterion is cool)
Whether you can end up underwater depends on whether by "margin" you literally mean a margin loan. If so then yes, as market timer's thread illustrates.
A better way of owning 140% stocks is to make 40% of your portfolio LEAPs calls with a strike price of onehalf the current market price of the underlying security (SPY, say). Then the other 60% you just invest in equities. The value of the option can't go negative, so while a 2008 or 1929 scenario would be very hurtful, you wouldn't be underwater.
A better way of owning 140% stocks is to make 40% of your portfolio LEAPs calls with a strike price of onehalf the current market price of the underlying security (SPY, say). Then the other 60% you just invest in equities. The value of the option can't go negative, so while a 2008 or 1929 scenario would be very hurtful, you wouldn't be underwater.
Most of my posts assume no behavioral errors.
Re: 140% Stocks (or why the Kelly criterion is cool)
We are kind of abusing the applicability and assumptions here (to a very high degree), but if you look at the 140% number in the title and make some adjustments, you might end up close to 100% or possibly under.
Credit Suisse yearbook has US equity premium from 1900 being 4.5%. For the world, 3.3%. If we keep around a 20% standard deviation assumption (these days with correlations being as they are, including exUS may drive up standard deviations if you go global), that's implying something more like 82.5% stocks to 112.5% stocks.
Potentially no leverage or high beta needed!
Credit Suisse yearbook has US equity premium from 1900 being 4.5%. For the world, 3.3%. If we keep around a 20% standard deviation assumption (these days with correlations being as they are, including exUS may drive up standard deviations if you go global), that's implying something more like 82.5% stocks to 112.5% stocks.
Potentially no leverage or high beta needed!
Re: 140% Stocks (or why the Kelly criterion is cool)
d'Alemberts betting sequence suggests betting 1 unit initially and increasing the stake by 1 unit after each losing play, reducing by 1 unit after each winning play (minimum stake 1 unit).
I bet heads, 1 unit, lose
I bet heads, 2 units, lose
I bet heads, 3 units, win
In total outlayed 6 units in bets and received back 6 units won.
After 2 losing, 1 winning plays I'm at break even.
Applied to stocks, start with $500 stake out of $1000 bankroll, 1000 stock index level, and if share prices halve down to 500 then double up the stake by adding another $500. Stocks can't fall below zero so there's no next downleg. You just have to sit it out and wait until stocks recover back 33% (500 x 1.33 index level = 666 index level) at which point you're back to breakeven (despite the index being down 333 below the 1000 original start date level). If the index returns back to 1000 you are sitting on a $500 profit  and reduce the stake having had a winning play.
$500 invested at 1000 index level, index declined 50% to 500 at which point the shares held value was $250, another $500 added ($750 value at that 50% lower price), allin sit it out and you're back to breakeven at 666 index level, 50% up at 1000 index level.
Index down, down  and you're stuck at allin awaiting a rebound back to 33% below the original start date share price level in order to breakeven.
Index down, up and despite being at the same level as at the start you're sitting on profits
Index up and you're in profit  but not as by much as if you'd been fully invested from the start.
Versus 140% stock and if the index drops 71% you're wiped out because you were forced out of the market as you couldn't cover the margin call.
I bet heads, 1 unit, lose
I bet heads, 2 units, lose
I bet heads, 3 units, win
In total outlayed 6 units in bets and received back 6 units won.
After 2 losing, 1 winning plays I'm at break even.
Applied to stocks, start with $500 stake out of $1000 bankroll, 1000 stock index level, and if share prices halve down to 500 then double up the stake by adding another $500. Stocks can't fall below zero so there's no next downleg. You just have to sit it out and wait until stocks recover back 33% (500 x 1.33 index level = 666 index level) at which point you're back to breakeven (despite the index being down 333 below the 1000 original start date level). If the index returns back to 1000 you are sitting on a $500 profit  and reduce the stake having had a winning play.
$500 invested at 1000 index level, index declined 50% to 500 at which point the shares held value was $250, another $500 added ($750 value at that 50% lower price), allin sit it out and you're back to breakeven at 666 index level, 50% up at 1000 index level.
Index down, down  and you're stuck at allin awaiting a rebound back to 33% below the original start date share price level in order to breakeven.
Index down, up and despite being at the same level as at the start you're sitting on profits
Index up and you're in profit  but not as by much as if you'd been fully invested from the start.
Versus 140% stock and if the index drops 71% you're wiped out because you were forced out of the market as you couldn't cover the margin call.

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Re: 140% Stocks (or why the Kelly criterion is cool)
With just a margin account, the chances of going negative are low. Brokers will liquidate your position before they lose money. Going to zero is a definite possibility if over leveraged.ResearchMed wrote:If one is playing with fire, I mean margin, isn't there the chance of getting underwater, and sinking deeper and deeper?
I suppose the ultimate "dust oneself off" would then be "just" to declare bankruptcy?
Ethics aside (because this would sort of be a "planned" use of moral hazard), I suppose it would work if one had nothing else of value.
If I remember correctly Market Timer was taking cash advances on multiple credit cards and putting that money into the market. That is a way to get underwater.

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Re: 140% Stocks (or why the Kelly criterion is cool)
Why is it common sense that there is no limit to how fast things can travel when so many natural phenomena appear to approach a limit as some input variable x approaches some value a?backpacker wrote:Common sense says there is no limit to how fast things can travel. Common sense says there is no limit to how fast a portfolio is likely to grow as long as you are willing to take on more risk. Common sense is wrong about both. Nothing can travel faster than the speed of light. No portfolio is likely to grow faster than a 140% stock portfolio. Why is that? Read on to find out.
Re: 140% Stocks (or why the Kelly criterion is cool)
Kelly assumes an indefenite number of bets. At any point in the progression, you could see as much as a 90% decline in your bankroll. Now, if you can live forever, you can recover from such a drastically bad series of losing bets, and ultimately achieve the "optimal expected return." Unfortunately, we don't live forever, we start withdrawing funds for expenses, or we quit the game when down 90% for behavioral reasons. Kelly doesn't work for normal people. For institutions, endowment funds, and so on....maybe.
 backpacker
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Re: 140% Stocks (or why the Kelly criterion is cool)
You may be surprised, but I mostly agree.letsgobobby wrote:I haven't read Kelly, I haven't read the notes, I haven't read Poundstone although I have the book on my shelf.
95% or 99% of investors should not be solving for this X. The X they should be solving for is, how do I minimize the odds of failing to meet my minimum needs for (retirement, college, whatever long termgoal).So the Kelly criterion says to put 140% of your portfolio in stocks to maximize your likely longterm returns.
The Kelly criterion is the financial equivalent of the idealized plane. Kelly is guaranteed to beat any essentially different strategy over the "longrun" in the following sense: Given any favorable repeatable gamble (like investing in stocks instead of bonds), the odds of any essentially different strategy beating Kelly go to zero as the number of times the game is played goes to infinity. If an angel could invest in the stock market, Kelly would tell her how to do it.
We are not angels, but Kelly still has things to teach us. The main lesson is that there is a speed limit to investing. Even if you want to maximize your likely longterm returns, risk be damned, there is a point at which you should stop adding risk. This because it will lower your likely longterm returns.
Re: 140% Stocks (or why the Kelly criterion is cool)
For the higher frequency trading I used to do I used a log stochastic based approach. Identify a volatile target stock and set a high price above which you were prepared to be totally out of the stock and a low price at which you were allin and potentially sitting on a potentially prolonged buy and hold awaiting recovery position, or considered that point to be a stoploss  exit with a loss and move on to other choices. Those boundaries being defined by 2 standard deviations or charting (projection lines progressed across the prior peaks (troughs).backpacker wrote:the Kelly criterion is equivalent to using expected utility with a logarithmic utility function.
= ( log(current)  log(bottom) ) / log(top)  log(bottom) )
That depicted an appropriate amount of cash reserve to be holding at that current level. When a turning point was apparent (such as Point and Figure table based) revisit, recalculate and adjust the cash % amount (according to whether it was trade costs/spread etc. efficient to do so).
For example Dow 18,000 and happy to be all in at 12,000, all out at 22,000. log stochastic = ( log(18000)  log(12000) ) / ( log(22000)  log(12000) ) = 67% cash indicated (33% stock). Later if Dow = 16000 then 47.5% cash indicated.....etc.
Log scaling kept the volatility capture gains more linear across the entire spectrum of price levels.

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Re: 140% Stocks (or why the Kelly criterion is cool)
Any advisor who suggests options as a way to get leverage is out of their mind. They are immediately discredited on the basis of that alone.nisiprius wrote:By the way, I don't think it is a straw man to ask "why not 200% stocks," because of this article (which was cited by "market timer" when he began a course of investments that resulted in his losing $200,000 more than he had).In a later book, they suggest a strategy based on LEAPS, but not because of risk, but becauseto them, regrettably401(k) plans do not offer the ability to use margin. "In the not too distant future, courts may determine that failing to offer employees the option to diversify temporarally" (by not offering young 401(k) participants a means of using 200% leverage) will fail the legal test of fiduciary responsibility.You should be more than 100% in equities when you are young. An exposure of 200% to start would be a better idea. That's rightif you are young, you should be buying on margin. Pay down the debt as you age and then ease off to a 5050 stockandbond mix at the beginning of your retirement.
In short, two writers who are explicitly offering advice to young investors either do not know or do not agree that the Kelly criterion puts a sanity ceiling of 140% on stock investing.
Options are neither an accurate way of getting leverage, nor cheap. In fact, when you buy (or sell options) you are buying (or selling) something completely different: volatility, gamma, etc. Considering that selling volatility strategies generally have positive returns, any strategy that is persistently buying volatility has a huge drag. Not to mention that both the seller and the buyer of volatility pay to the market intermediaries.

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Re: 140% Stocks (or why the Kelly criterion is cool)
So let's say I have a portfolio X. It has a CAGR that is 1% less than the S&P 500 (5.6%  1% = 4.6%) and a standard deviation of 9%. Does that mean I should lever it 4.6%/(9%^2) = 568% to get the optimal return?backpacker wrote:The Kelly criterion for stocks is a simple formula, assuming that returns are normally distributed. That formula is:The historic excess return of the S&P 500 is about 5.6%. The standard deviation about 20%. So the Kelly criterion says to put 140% of your portfolio in stocks to maximize your likely longterm returns.Code: Select all
(optimal percentage in stocks)=(average excess return of stocks over bonds)/(standard deviation of excess returns)^2

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Re: 140% Stocks (or why the Kelly criterion is cool)
James T Kirk Solution (aka Gene Roddenberry):
Two are racing towards Earth; The Bugs (Orson Scott CardEnders Series, Robert HeinleinStarship Troopers The Movie) to destroy Earth and Ender and the The Federation Fleet to save Earth . Both using comparable but conventional FTL drives.
Who wins?
Neither.
Slartibartfast (Douglas Adams) wins using the Bistromathics Drive. Slartibartfast needs to get to Earth (on commission assignment from the Mice, who offers the Magnethians $1. Slartibartfast knows that $1 in hand is worth more than $1000 offered but worthless after Earth is destroyed ) to take Earth's measurements before the hostilities begin.
Two are racing towards Earth; The Bugs (Orson Scott CardEnders Series, Robert HeinleinStarship Troopers The Movie) to destroy Earth and Ender and the The Federation Fleet to save Earth . Both using comparable but conventional FTL drives.
Who wins?
Neither.
Slartibartfast (Douglas Adams) wins using the Bistromathics Drive. Slartibartfast needs to get to Earth (on commission assignment from the Mice, who offers the Magnethians $1. Slartibartfast knows that $1 in hand is worth more than $1000 offered but worthless after Earth is destroyed ) to take Earth's measurements before the hostilities begin.
Last edited by itstoomuch on Sun Apr 26, 2015 12:51 pm, edited 2 times in total.
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Re: 140% Stocks (or why the Kelly criterion is cool)
I challenge the original poster to provide a spreadsheet with his/her implementation of the 140% strategy using the Kelly Criterion.
1. Download the data from whatever source you want, e.g., yahoo finance,
2. Write a spreadsheet with a few columns that implements your interpretation of Kelly criterion,
3. Assume reasonably trading costs,
4. Post it back here.
I claim that the Kelly criterion is easy to understand in theory, but hard to implement in practice.
In fact, most trading ideas are hard to implement in practice. This coming from a professional trader.
1. Download the data from whatever source you want, e.g., yahoo finance,
2. Write a spreadsheet with a few columns that implements your interpretation of Kelly criterion,
3. Assume reasonably trading costs,
4. Post it back here.
I claim that the Kelly criterion is easy to understand in theory, but hard to implement in practice.
In fact, most trading ideas are hard to implement in practice. This coming from a professional trader.

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Re: 140% Stocks (or why the Kelly criterion is cool)
Furthermore, by reading the OPs post very carefully, I think his understanding of the Kelly criterion is completely wrong.
 backpacker
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Re: 140% Stocks (or why the Kelly criterion is cool)
I think that's right. But as already mentioned, Kelly makes lots of idealized assumptions. So its unclear to me how to translate that into an actual portfolio.Rob Bertram wrote:So let's say I have a portfolio X. It has a CAGR that is 1% less than the S&P 500 (5.6%  1% = 4.6%) and a standard deviation of 9%. Does that mean I should lever it 4.6%/(9%^2) = 568% to get the optimal return?backpacker wrote:The Kelly criterion for stocks is a simple formula, assuming that returns are normally distributed. That formula is:The historic excess return of the S&P 500 is about 5.6%. The standard deviation about 20%. So the Kelly criterion says to put 140% of your portfolio in stocks to maximize your likely longterm returns.Code: Select all
(optimal percentage in stocks)=(average excess return of stocks over bonds)/(standard deviation of excess returns)^2

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Re: 140% Stocks (or why the Kelly criterion is cool)
Almost, but the OPs formula is incorrect.Rob Bertram wrote:So let's say I have a portfolio X. It has a CAGR that is 1% less than the S&P 500 (5.6%  1% = 4.6%) and a standard deviation of 9%. Does that mean I should lever it 4.6%/(9%^2) = 568% to get the optimal return?backpacker wrote:The Kelly criterion for stocks is a simple formula, assuming that returns are normally distributed. That formula is:The historic excess return of the S&P 500 is about 5.6%. The standard deviation about 20%. So the Kelly criterion says to put 140% of your portfolio in stocks to maximize your likely longterm returns.Code: Select all
(optimal percentage in stocks)=(average excess return of stocks over bonds)/(standard deviation of excess returns)^2
Furthermore, using "BONDS" as the riskless asset is also incorrect.
Furthermore, the assumption that the riskless asset has a constant return is also incorrect.
The fact that returns are not normal is the least of OPs worry.
This analysis is so completely wrong, that in my opinion, posts like these should be heavily censored so as not to promote the dissemination of incorrect knowledge.
 backpacker
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Re: 140% Stocks (or why the Kelly criterion is cool)
I'm just reporting the calculation that starts on page 41 of the attached notes. It is obviously made using cartoonish assumptions. This is just a way of illustrating the main point that investing has a "speed limit". No one should be making delicate portfolio decisions based on this!TradingPlaces wrote:Almost, but the OPs formula is incorrect.Rob Bertram wrote:So let's say I have a portfolio X. It has a CAGR that is 1% less than the S&P 500 (5.6%  1% = 4.6%) and a standard deviation of 9%. Does that mean I should lever it 4.6%/(9%^2) = 568% to get the optimal return?backpacker wrote:The Kelly criterion for stocks is a simple formula, assuming that returns are normally distributed. That formula is:The historic excess return of the S&P 500 is about 5.6%. The standard deviation about 20%. So the Kelly criterion says to put 140% of your portfolio in stocks to maximize your likely longterm returns.Code: Select all
(optimal percentage in stocks)=(average excess return of stocks over bonds)/(standard deviation of excess returns)^2
Furthermore, using "BONDS" as the riskless asset is also incorrect.
Furthermore, the assumption that the riskless asset has a constant return is also incorrect.
The fact that returns are not normal is the least of OPs worry.
This analysis is so completely wrong, that in my opinion, posts like these should be heavily censored so as not to promote the dissemination of incorrect knowledge.
Last edited by backpacker on Sun Apr 26, 2015 1:11 pm, edited 1 time in total.

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Re: 140% Stocks (or why the Kelly criterion is cool)
For everyone who wants to understand the Kelly criterion CORRECTLY, I invite them to read the wikipedia article, which is:
(a) somewhat better written,
(b) reviewed,
(c) correct.
http://en.wikipedia.org/wiki/Kelly_criterion
Also, the explanation of the primary application of Kelly criterion to today's problem is found in the section called: "Application to the stock market".
However, the interpretation of that section needs to change as follows:
1. The riskfree rate should be the 3month treasury, and r, the riskless rate should be the rate of that,
2. The entire stock market should be considered 1 risky asset,
3. The second risky asset should be some intermediate bond index. For simplicity, we can take the constant maturity 10year bond, the constant maturity 5year bond, or your favorite Vanguard bond index.
To do the Kelly criterion correctly, one needs to compute the noncentral second moment of the risky assets, i.e., the covariance matrix, without removing the means, of {Stock, Bond} portfolio. It is well known that the offdiagonal entries in that matrix are weakly negative, i.e., that bondstock correlation is negative, at least historically.
And lastly, the expected return of stocks need to be obtained. That is the only input that is hard to get.
However, if you are able to input some reasonable value for that, then you will find out that outcome will surprize you. I think the optimal solution will be very close to being: extremely leveraged in bond risky asset, and a little allocation to the stock risky asset.
A shortcut interpretation of the Kelly criterion is to being extremely levered to the asset with a higher information ratio, and intermediate term bonds have had a higher information ratio over the last, oh, 3040 years.
And lastly, if your provide leverage limits, I think the solution you get will be something like: 120150% bonds, 3060% stocks.
Simply put, a 140% allocation to stocks is NOT optimal according to the Kelly criterion.
(a) somewhat better written,
(b) reviewed,
(c) correct.
http://en.wikipedia.org/wiki/Kelly_criterion
Also, the explanation of the primary application of Kelly criterion to today's problem is found in the section called: "Application to the stock market".
However, the interpretation of that section needs to change as follows:
1. The riskfree rate should be the 3month treasury, and r, the riskless rate should be the rate of that,
2. The entire stock market should be considered 1 risky asset,
3. The second risky asset should be some intermediate bond index. For simplicity, we can take the constant maturity 10year bond, the constant maturity 5year bond, or your favorite Vanguard bond index.
To do the Kelly criterion correctly, one needs to compute the noncentral second moment of the risky assets, i.e., the covariance matrix, without removing the means, of {Stock, Bond} portfolio. It is well known that the offdiagonal entries in that matrix are weakly negative, i.e., that bondstock correlation is negative, at least historically.
And lastly, the expected return of stocks need to be obtained. That is the only input that is hard to get.
However, if you are able to input some reasonable value for that, then you will find out that outcome will surprize you. I think the optimal solution will be very close to being: extremely leveraged in bond risky asset, and a little allocation to the stock risky asset.
A shortcut interpretation of the Kelly criterion is to being extremely levered to the asset with a higher information ratio, and intermediate term bonds have had a higher information ratio over the last, oh, 3040 years.
And lastly, if your provide leverage limits, I think the solution you get will be something like: 120150% bonds, 3060% stocks.
Simply put, a 140% allocation to stocks is NOT optimal according to the Kelly criterion.
 backpacker
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Re: 140% Stocks (or why the Kelly criterion is cool)
Shouldn't the riskfree rate be the investor's margin rate?TradingPlaces wrote:[T]the explanation of the primary application of Kelly criterion to today's problem is found in the section called: "Application to the stock market".
However, the interpretation of that section needs to change as follows:
1. The riskfree rate should be the 3month treasury, and r, the riskless rate should be the rate of that,
2. The entire stock market should be considered 1 risky asset,
3. The second risky asset should be some intermediate bond index. For simplicity, we can take the constant maturity 10year bond, the constant maturity 5year bond, or your favorite Vanguard bond index.

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Re: 140% Stocks (or why the Kelly criterion is cool)
In typical, secondrate academic fashion, the assumptions that MATTER less are brought into the foreground, while the realistic constraints of the presentday investor are simply overlooked.backpacker wrote: I'm just reporting the calculation that starts on page 41 of the attached notes. It is obviously made using cartoonish assumptions. This is just a way of illustrating the main point that investing has a "speed limit". No one should be making delicate portfolio decisions based on this!
Those Berkeley lecture notes provide the solution to the Kelly criterion using:
1. Cash (or the 3year treasury) as the riskless asset, and
2. SP500 as the risky asset.
However, by calling the first "BONDS", in a forum where bonds are generally interpreted to be the some intermediate bond index greatly muddies the waters. It appears that a typical investor should be 140% long stocks, and 40% short bonds, while a typical investor here is 60% long stocks, and 40% long bonds.
The fact that we are mixing CASH with BONDS is a fundamental flaw in this type of reasoning.
And second, the solution reached is extremely counterintuitive to a lot of conventional wisdom here.
The big issue I have is that the correct way of applying the Kelly criterion will give a solution that is OVERWEIGHT bonds, while the proposed "speed limit" solution is WAY, WAY overweight stocks, and reasonable solutions should not be so far from each other.
In fact, readers of both this post and the 200% leveraged crazy investor post might conclude that what was done wrong by the 200% leveraged guy was simply a mistake between being long 140% and 200%, while in fact, the mistake was fundamental. If you want to leverage, you should leveraged the HIGHER INFORMATION RATIO asset, not the LOWER INFORMATION RATIO asset.
That last point should follow from an intuitive understanding of the Kelly criterion.

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Re: 140% Stocks (or why the Kelly criterion is cool)
Short answer: YES and NO.backpacker wrote:Shouldn't the riskfree rate be the investor's margin rate?TradingPlaces wrote:[T]the explanation of the primary application of Kelly criterion to today's problem is found in the section called: "Application to the stock market".
However, the interpretation of that section needs to change as follows:
1. The riskfree rate should be the 3month treasury, and r, the riskless rate should be the rate of that,
2. The entire stock market should be considered 1 risky asset,
3. The second risky asset should be some intermediate bond index. For simplicity, we can take the constant maturity 10year bond, the constant maturity 5year bond, or your favorite Vanguard bond index.
Why complicate the problem further, while we are still struggling to explain and understand the simpler formulation.
Why should the investor's margin rate be the riskfree rate?
Where have you seen such a definition?
The riskfree rate should be the riskfree rate, i.e., the rate at which an investor can lend and receive their 100% principal, guaranteed, over a short horizon. I think he 3month treasury bill is the perfect riskfree rate.
Now, as far as whether a retail investor can BORROW at that rate: that is a complicated question. However, I firmly believe that by using:
(a) the Emini SP500 futures,
and
(b) any of the 5year, or 7/10year treasury futures on the CME, the implied borrow rate of a retail investor is as good as the 3month treasury rate.
So you see, when you borrow using the appropriate product, the borrowing cost of the investor BECOMES EQUAL to the risk free rate.
But the margin rate offered by the Charles "Thieves Incorporated" Schwab Brokerage is not the ideal borrowing rate.
To be sure, the 3month US treasury rate TODAY is roughly from 0 to 0.25%, and the implied borrow rate in the Emini and US Treasury futures is also firmly in that range.
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Re: 140% Stocks (or why the Kelly criterion is cool)
It wasn't nearly as fashionable to leverage bonds in 2007, when the Fed Funds rate was 5.25% and 30year Treasuries yielded roughly 5.0%. At the time, there was essentially no correlation between stocks and bonds. Now, of course, after the financial crisis, bonds are viewed as having a reliable negative correlation with stocks. I wonder what we will learn from the next crisis.TradingPlaces wrote:In fact, readers of both this post and the 200% leveraged crazy investor post might conclude that what was done wrong by the 200% leveraged guy was simply a mistake between being long 140% and 200%, while in fact, the mistake was fundamental. If you want to leverage, you should leveraged the HIGHER INFORMATION RATIO asset, not the LOWER INFORMATION RATIO asset.

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Re: 140% Stocks (or why the Kelly criterion is cool)
The CORRECT Kelly Criterion:
1. A Cash asset with riskfree rate of return, r, e.g., 3%, or whatever,
2. Multiple RISKY assets, with their full covariance matrix know. Assume two: (a) Stock market index, (b) Intermediate Bond Market Index,
 return vector of the risky asset set, in our case, 2dimensional: r_v = (r_s, r_b), where r_s is the EXPECTED return of the stock index, and r_b is the expected return of the bond index. E.g., r_v might be equal to (7.5%, 5.5%), or something like that,
 noncentral covariance matrix of the risk asset set, in our case, a 2x matrix S_v = ( (v_11, v_12), (v_21, v_22)), where index 1 refers to the stock market index, and index 2 refers to the bond market index. Square root of v_11 corresponds to the expected noncentral standard deviation of stock index returns, and square root of v_22 corresponds to the same for the bond index returns.
I think you can safely substitute the regular covariance matrix instead of what is required, because DAILY returns are much smaller than DAILY variance. However, when DAILY variance becomes close to DAILY returns, you would have to do the RIGHT thing.
3. Kelly criterion solution for the optimal bet for each RISKY asset:
u_v = (1 + r) * inverse (S_v) * ( r_v  r).
So:
(1+r) is a scalar, i.e., a 1dimensional entry,
S_v is 2x2, and inverse of (S_v) is also 2 x 2,
( r_v  r ) is 2 x 1,
Thus u_v is 2 x 2 x 2 x 1 = 2 x 1, i.e., a vector of the same dimensions as r_v.
The interpretation of u_v is:
u_v (index 1) > how much of ones total capital to bet on the stock index,
u_v (index 2) > how much of ones total capital to bet on the bond index.
Further, given limits to leverage, a more complicated form of the Kelly criterion needs to be solved involving limits to leverage.
Next, because the world is not a static place, this problem is dynamic, i.e., you need to have an iterative solution that solves for different values as they become available. For example, the risk free rate probably changes every year, every 2years, etc.
And lastly, you need some form of expected returns and covariance. Covariance is somewhat easy to come up with because covariances are stationary. Returns are a different story.
If you want to get an easy pass, then make this forwardlooking. I.e., go back in time to year T, and then as expected returns, use the NEXT 10years ACTUAL returns. This provides you with a RELATIVELY accurate estimate of overall future returns, HOWEVER, it is not SO EXACT as to outright cheat. If you use the 1year forward return as the expected return, you will be BLATANTLY cheating.
Also, the Kelly criterion is ABOUT TRADING. You will discover that you are TRADING a lot.
1. A Cash asset with riskfree rate of return, r, e.g., 3%, or whatever,
2. Multiple RISKY assets, with their full covariance matrix know. Assume two: (a) Stock market index, (b) Intermediate Bond Market Index,
 return vector of the risky asset set, in our case, 2dimensional: r_v = (r_s, r_b), where r_s is the EXPECTED return of the stock index, and r_b is the expected return of the bond index. E.g., r_v might be equal to (7.5%, 5.5%), or something like that,
 noncentral covariance matrix of the risk asset set, in our case, a 2x matrix S_v = ( (v_11, v_12), (v_21, v_22)), where index 1 refers to the stock market index, and index 2 refers to the bond market index. Square root of v_11 corresponds to the expected noncentral standard deviation of stock index returns, and square root of v_22 corresponds to the same for the bond index returns.
I think you can safely substitute the regular covariance matrix instead of what is required, because DAILY returns are much smaller than DAILY variance. However, when DAILY variance becomes close to DAILY returns, you would have to do the RIGHT thing.
3. Kelly criterion solution for the optimal bet for each RISKY asset:
u_v = (1 + r) * inverse (S_v) * ( r_v  r).
So:
(1+r) is a scalar, i.e., a 1dimensional entry,
S_v is 2x2, and inverse of (S_v) is also 2 x 2,
( r_v  r ) is 2 x 1,
Thus u_v is 2 x 2 x 2 x 1 = 2 x 1, i.e., a vector of the same dimensions as r_v.
The interpretation of u_v is:
u_v (index 1) > how much of ones total capital to bet on the stock index,
u_v (index 2) > how much of ones total capital to bet on the bond index.
Further, given limits to leverage, a more complicated form of the Kelly criterion needs to be solved involving limits to leverage.
Next, because the world is not a static place, this problem is dynamic, i.e., you need to have an iterative solution that solves for different values as they become available. For example, the risk free rate probably changes every year, every 2years, etc.
And lastly, you need some form of expected returns and covariance. Covariance is somewhat easy to come up with because covariances are stationary. Returns are a different story.
If you want to get an easy pass, then make this forwardlooking. I.e., go back in time to year T, and then as expected returns, use the NEXT 10years ACTUAL returns. This provides you with a RELATIVELY accurate estimate of overall future returns, HOWEVER, it is not SO EXACT as to outright cheat. If you use the 1year forward return as the expected return, you will be BLATANTLY cheating.
Also, the Kelly criterion is ABOUT TRADING. You will discover that you are TRADING a lot.

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Re: 140% Stocks (or why the Kelly criterion is cool)
That's why the Kelly Criterion suggest a dynamic trading problem.market timer wrote:It wasn't nearly as fashionable to leverage bonds in 2007, when the Fed Funds rate was 5.25% and 30year Treasuries yielded roughly 5.0%. At the time, there was essentially no correlation between stocks and bonds. Now, of course, after the financial crisis, bonds are viewed as having a reliable negative correlation with stocks. I wonder what we will learn from the next crisis.TradingPlaces wrote:In fact, readers of both this post and the 200% leveraged crazy investor post might conclude that what was done wrong by the 200% leveraged guy was simply a mistake between being long 140% and 200%, while in fact, the mistake was fundamental. If you want to leverage, you should leveraged the HIGHER INFORMATION RATIO asset, not the LOWER INFORMATION RATIO asset.
When the riskless rate can vary from 0% to 12%, you can not simply put the average rate of 3% for the past 40 years and think that you are doing a pretty good calculation.
If the riskfree rate is 5.25%, and the EXPECTED returns of the risky assets are LESS THAN THAT, then the Kelly criterion says you should BET ZERO on the risky asset.
Yet another reason why the 140% speed limit theory is discredited.
In fact, in 20062008 period, hoards of FatWallet finance users were borrowing from credit cards at 0% (arguably, a marketing ploy by credit card companies to offer belowmarket returns) and investing in the riskfree asset BECAUSE THAT WAS THE OPTIMAL THING TO DO! Let me tell you: most of them came out WAY ahead than our favorite guy who did the same with 200% leverage into stocks.
 nisiprius
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Re: 140% Stocks (or why the Kelly criterion is cool)
Is it really "well known?"TradingPlaces wrote:...It is well known that the offdiagonal entries in that matrix are weakly negative, i.e., that bondstock correlation is negative, at least historically...
Admittedly my trusty Ibbotson 2010 SBBI Classic Yearbook gets less and less trusty every year and I'm too cheap to buy a new one, but according to tables 62 and 64, p. 79, for 19262009, the correlations are as follows. "Large Company Stocks" is the S&P 500 and (supposedly comparable) predecessors. I'm going to bold "longterm government bonds" because that's typically what people cite for "bonds."
Inflationadjusted series (table 64)
Correlation of large company stocks with:
Longterm corporate bonds, +0.23
Longterm government bonds, +0.11
Intermediateterm government bonds, +0.08
Basic series (i.e. nominal) (table 62)
Correlation of large company stocks with:Longterm corporate bonds, +0.17
Longterm government bonds, +0.03
Intermediateterm government bonds, 0.01
I still have not figured out where the frequently claims of negative correlation actually come from. As market timer suggests, from recent short term data series that include 20082009, maybe.
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Re: 140% Stocks
This is extremely easy to construct:nisiprius wrote: Here's a question to argue about. In its original formulation, it seems perfectly clear the Kelly Criterion refers to a situation where a gambler has inside information about a situation and knows that the true odds are different from the offered odds. The "inside information" wouldn't necessarily be anything illegal, but it has to be real "asymmetrical information"some actual knowledge that is different from what "the market" (e.g. the parimutuel board) knows. It is the optimal betting strategy when you actually know something that the oddsmaker doesn't know.
1. World with 3 investable assets: (a) Cash, riskfree, aka 3month US Treasury, (b) the stock index, (c) the bond index,
2. If you want a noisy private channel, setup a 1ayear Kelly criterion problem, and for EXPECTED returns for the stock and bond index assets, take their 10year forward returns. Cheating? Yes, because it is forward looking. But that's your private channel, or "edge",
3. Do the Kelly criterion as a dynamic, 1ayear forward lookup, trading problem.
I think you should come out ahead.
Once you have this framework, try to use DIFFERENT ways of "predicting" the future returns. E.g.,: (a) past 10years returns, (b) past 40years returns, (c) Rick Ferry's predictions, (d) Shiller's predictions, (e) your neighbors kid's predictions, (f) have a monkey pick the returns.
I think this will be a good exercise in understanding how to apply the Kelly criterion.
Also, one thing to understand: even if the key parameters in the Kelly criterion do not change (EXPECTED returns and the covariance matrix), you WILL HAVE TO TRADE. Why? Because your bet ratios need to be KEPT CONSTANT relative to your portfolio size. I.e., if stocks rally 50%, and bonds go down 50%, you will have to SELL STOCKS and BUY BONDS. You decide how frequently you want to trade, but Kelly says: EVERY DAY!
Last edited by TradingPlaces on Sun Apr 26, 2015 2:06 pm, edited 1 time in total.