Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

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pezblanco
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Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by pezblanco » Tue Jan 09, 2018 6:31 pm

A short synopsis of the usual Kelly optimization problem is as follows. Sorry for the terseness but this is probably longer than most people are going to read. Go to last sentence to see the TLDR version. :D

1) Suppose we play a game with a independent identically distributed random sequence of outcomes (such as betting on a sequence of biased coin flips)

2) We have initial capital X_0. Our capital after the nth outcome is X_n.

3) Our betting strategy is to bet a fixed fraction of our capital on each outcome i.e the bet on outcome i, denoted as B_i = f X_{i-1} where f
can vary between zero and any positive number. If f>1, then we are leveraged and have to borrow capital in order to make the bet.

4) If the sequence of outcomes is a favorable game (i.e. E[X_n] > E{X_{n-1}]), then it is true that our capital increases exponentially with time. The rate after n outcomes is given by:

G_n(f) = (1/n)log(X_n/X_0) with expectation g_n(f) = E[G_n(f)] and limiting value g(f) = lim_{n -> infinity} g_n(f)
(I'm not going to discuss when/whether any expectations or limits exist ... lot of references on this if you're interested)

5) In some sense g(f) is average rate that our capital is expected to grow as we play the game. You can think of g(f) = log(1+r) where r is the yearly rate. The Kelly criterion proposes to choose f so that g(f) is maximized. I.e. we choose our bet size so that our expected rate of capital growth is maximized.

E.0 Thorp and others have tried to apply the Kelly criterion to various applications. Thorp famously applied it to bet sizing in blackjack when the player knows that he/she is getting a favorable bet (usually this knowledge comes about through card-counting). Thorp also applied this to the stock market ... In an example calculation, he approximated stock returns as truncated (2 sigma) Gaussian ... adjusted the mean and variance to coincide with empirical stock data, and applied the Kelly criterion. He found the optimal f to be 1.17, i.e. 17% leveraged stock position was optimal under his assumed truncated normal assumption.

If the assumed distributions are "nice", the usual formula for g(f) is

g(f) = E[log(1+fS)]

Where S is the return of the market over the riskless rate. One should note that the reason he used a truncated model, is that this formula doesn't exist for probability distributions with unbounded negative values .... such as the normal model.

I decided to do a similar study as the example given in Thorpe's paper except, I thought I would do the following:

a) I used 50 years of empirical data from the DFA matrix book from 1967 --- 2016 or REAL yearly returns for S&P500, Long Term Government Bonds (LTGB), and Short Term (One month) T-Bill (STTB) rates. I took these points to be my empirical distribution that I use to calculate g(f).

b) For the data presented, I assumed that our bet on each year is to have a portfolio of proportion f stock and (1-f) proportion LTGB. If f > 1, we borrow money at the STTB rate + 1 percent.

With these assumptions, we can compute g(f) based upon this last 50 years of data:

Image

So, we see that the optimal f is something like 1.52. So based on the last 50 years of market data, the best strategy for growing your porfolio under the Kelly criterion is to be leveraged out around 50%.

I then did a Monte Carlo simulation for results to be expected over a 30 year investment horizon and computed the distribution of realized rates for values of f = .6, 1, 2. The spread (variance) of the results increase markedly with increasing f. For shorter investment periods the spread is of course going to be even greater.

Image

I'm sorry for the poor quality, but if you look closely you can see the three distributions overlaid on one another. The more spread out the distributions also have slightly higher means.

I tried a variety of variants on the calculations .... things move around a little but not substantially
aa) I tried the cost of borrowing to be STTB + i, where i=0,1,1.5, 2, 3, 4 ... You have to get up to around 4 before you can get the optimal f = .8
bb) I tried assuming that LTGB and S&P500 returns were independent ... that made almost no change in the graphs
cc) I tried using just 25 years of past data and just 10 years of past data .... things move around a little but not qualitatively.

TLDR VERSION: Using empirical data from the markets for the last 50 years, the Kelly criterion of maximizing the growth rate of your portfolio says that you should be leveraged about 50% to stocks (i.e. 150% of your portfolio should be stocks).
Last edited by pezblanco on Thu Jan 11, 2018 2:33 pm, edited 2 times in total.

Thesaints
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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by Thesaints » Tue Jan 09, 2018 6:39 pm

You are correct in principle, but real cost of leveraging is to be accounted for, as well as the outcome of a -50% spike (
NASDAQ 2000, for instance) and the possibility of remaining leveraged.
I think your model assumes infinite credit on the investor's part.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by H-Town » Tue Jan 09, 2018 6:48 pm

I didn't verify the numbers but I won't go this route to maximize my portfolio growth rate. I'd rather maximize my saving rate and increase it over time. To me, that's the surest way to maximize my portfolio's potential growth.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by Thesaints » Tue Jan 09, 2018 6:50 pm

Once your portfolio grows savings rate may soon become irrelevant.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by KlangFool » Tue Jan 09, 2018 7:01 pm

OP,

Some smart economists try to run a hedge fund with their ideas. The end result is the followings:

https://en.wikipedia.org/wiki/Long-Term ... Management

KlangFool

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by pezblanco » Tue Jan 09, 2018 7:14 pm

KlangFool wrote:
Tue Jan 09, 2018 7:01 pm
OP,

Some smart economists try to run a hedge fund with their ideas. The end result is the followings:

https://en.wikipedia.org/wiki/Long-Term ... Management

KlangFool
Ha ha ha! But seriously, this is not a post advocating leveraging. Quite the opposite. Look at the first graph's vertical axis. Those numbers are the long term growth rates for various porfolios. A 60/40 portfolio has a growth rate of around 5.2% ... 100/0 has a rate of approximately 5.7%. Leveraged 50% has a rate of around 6.3%. And then look at the variance graphs! Why in the world WOULD you leverage given this data???
Thesaints wrote:
Tue Jan 09, 2018 6:39 pm
You are correct in principle, but real cost of leveraging is to be accounted for, as well as the outcome of a -50% spike (
NASDAQ 2000, for instance) and the possibility of remaining leveraged.
I think your model assumes infinite credit on the investor's part.
Well, yes I suppose. Though a -50% spike on a 50% leveraged portfolio would not wipe you out, right?

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by Blueskies123 » Tue Jan 09, 2018 7:27 pm

Yes, this is what blew up long term capital Management 20 years and cost US taxpayers 3.5 billion dollars.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by Thesaints » Tue Jan 09, 2018 9:17 pm

pezblanco wrote:
Tue Jan 09, 2018 7:14 pm
Well, yes I suppose. Though a -50% spike on a 50% leveraged portfolio would not wipe you out, right?
That depends on how fast the drop is. If it is gradual enough that you manage to keep your margin at 50%, then you only lose 75%.
If it is much faster, loss could be even higher.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by bogglehead125 » Tue Jan 09, 2018 10:50 pm

Blueskies123 wrote:
Tue Jan 09, 2018 7:27 pm
Yes, this is what blew up long term capital Management 20 years and cost US taxpayers 3.5 billion dollars.
"In September 1998, a group of 14 banks and brokerage firms invested $3.6 billion in LTCM to prevent the hedge fund’s imminent collapse. The arrangement was facilitated by the Federal Reserve, though the Fed did not lend any of its own funds." (https://www.federalreservehistory.org/e ... ar_failure)

Even more: "Under this new leadership, LTCM sold most of its remaining positions and returned the last of the group’s $3.6 billion investment by the end of 1999 (US General Accounting Office 2000)."

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by comeinvest » Thu Jan 11, 2018 4:31 am

Very interesting analysis!
I have questions about the second graph:
(1) "Monte Carlo simulation for results to be expected over a 30 year investment horizon" - How exactly did you do that simulation? There are only 21 different 30 year periods within your empirical data set of 50 years. Maybe I don't understand your model.
(2) The graph shows some negative return outcomes over 30 years for all 3 leverage ratios (f = .6, 1, 2). However, to my knowledge, there is no 30-year period with negative returns in the history of the S&P500, let alone for a 60/40 equities/cash portfolio. Please correct me if I'm wrong.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by comeinvest » Thu Jan 11, 2018 4:41 am

A question about the first graph: Are the numbers on the vertical axis nominal or return annual returns?
The number for cash (3.x%) seems to be the nominal historical cash returns, while the number for 100% equities is below the actual historical real return for U.S. equities of 6.x%.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by comeinvest » Thu Jan 11, 2018 4:47 am

pezblanco wrote:
Tue Jan 09, 2018 7:14 pm
100/0 has a rate of approximately 5.7%. Leveraged 50% has a rate of around 6.3%. And then look at the variance graphs! Why in the world WOULD you leverage given this data???
Depends how you view it. You are studying 30 year time frames. Over 30 years, a 5.7% annual return results in terminal wealth of 5.3 times the original capital, while a 6.3% annual return results in 6.3 times the original capital. That means a ca. 20% difference in terminal consumable wealth, or ca. 100% of the original capital.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by bgf » Thu Jan 11, 2018 8:40 am

something about this seems wrong... i dont think thorp would advise this.

he understood the Black-Scholes model for valuing options and used it effectively in practice by delta hedging with warrants. he did this before Black-Scholes were famous for their derivation, i think. probably most importantly, as others here have brought up LTCM, Thorp understood precisely where his model would fail. he was familiar with mandelbrot's work and believed he was correct that the normal distribution did not accurately reflect stock returns on the tails... instead, stock returns were 'wildly random.' and that if the model was going to fail it was going to fail during these less frequent and more extreme events. he said that he used the bachelier-osborne model anyway, but accounted for mandelbrot's work by providing himself a margin of safety on his trades. this worked wonders and his fund earned 20+% annual returns until he shut it down. he literally 'beat the market.'

i was always under the impression that he used the kelly criterion as he used it in card counting, he used it to determine the size of the 'bets' he made on individual warrant trades... and he took into account the assumptions required by his model. perhaps, i assumed wrong.

all that to say, i doubt he would agree that a static 1.5x leveraged position in the sp500 is the ideal investing strategy. it just doesn't jive. there is no margin of safety, and, even worse, the risk of total loss is absolutely untenable with that amount of leverage and no hedging. the number one rule for any money manager is don't lose money...
“TE OCCIDERE POSSUNT SED TE EDERE NON POSSUNT NEFAS EST"

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by pezblanco » Thu Jan 11, 2018 9:51 am

comeinvest wrote:
Thu Jan 11, 2018 4:31 am
Very interesting analysis!
I have questions about the second graph:
(1) "Monte Carlo simulation for results to be expected over a 30 year investment horizon" - How exactly did you do that simulation? There are only 21 different 30 year periods within your empirical data set of 50 years. Maybe I don't understand your model.
(2) The graph shows some negative return outcomes over 30 years for all 3 leverage ratios (f = .6, 1, 2). However, to my knowledge, there is no 30-year period with negative returns in the history of the S&P500, let alone for a 60/40 equities/cash portfolio. Please correct me if I'm wrong.
Thank you for your interest! I was thinking that I had directed this post to the wrong audience.

To get a 30 year horizon, we sample (with replacement) from the empirical distribution of the yearly data 30 times. So, for example, ONE of the possible 30 year sequences would be to get 30 years of the 2008 data point with its real 37% loss 30 years in a row! Of course the odds of that happening are very small ...
comeinvest wrote:
Thu Jan 11, 2018 4:41 am
A question about the first graph: Are the numbers on the vertical axis nominal or return annual returns?
The number for cash (3.x%) seems to be the nominal historical cash returns, while the number for 100% equities is below the actual historical real return for U.S. equities of 6.x%.
All numbers and data used and reported are REAL not nominal. According to the DFA Matrix Book, the real return on the S&P500 over the period 1967-2016 was 5.9 (5.86 to two decimal points).

[Edit added:] The numbers plotted are really the logarithm of one plus the rate. log (1 + r) is for small r very close to r. My bad though .... I'll add a warning to the original post. To transfer the numbers on the graph you just need to exponentiate and substract 1 to get the true rate. Anyway, the number you see for the 100% stock is log(1 + .01 times 5.86) = 5.69
comeinvest wrote:
Thu Jan 11, 2018 4:47 am
pezblanco wrote:
Tue Jan 09, 2018 7:14 pm
100/0 has a rate of approximately 5.7%. Leveraged 50% has a rate of around 6.3%. And then look at the variance graphs! Why in the world WOULD you leverage given this data???
Depends how you view it. You are studying 30 year time frames. Over 30 years, a 5.7% annual return results in terminal wealth of 5.3 times the original capital, while a 6.3% annual return results in 6.3 times the original capital. That means a ca. 20% difference in terminal consumable wealth, or ca. 100% of the original capital.
Yes ... but the spread on those possible values means that you're probably not going to be getting the "average" .... there is an uncomfortably large probability that you could do worse than not leveraging at all.
Last edited by pezblanco on Thu Jan 11, 2018 2:41 pm, edited 2 times in total.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by pezblanco » Thu Jan 11, 2018 10:00 am

bgf wrote:
Thu Jan 11, 2018 8:40 am
something about this seems wrong... i dont think thorp would advise this.

he understood the Black-Scholes model for valuing options and used it effectively in practice by delta hedging with warrants. he did this before Black-Scholes were famous for their derivation, i think. probably most importantly, as others here have brought up LTCM, Thorp understood precisely where his model would fail. he was familiar with mandelbrot's work and believed he was correct that the normal distribution did not accurately reflect stock returns on the tails... instead, stock returns were 'wildly random.' and that if the model was going to fail it was going to fail during these less frequent and more extreme events. he said that he used the bachelier-osborne model anyway, but accounted for mandelbrot's work by providing himself a margin of safety on his trades. this worked wonders and his fund earned 20+% annual returns until he shut it down. he literally 'beat the market.'

i was always under the impression that he used the kelly criterion as he used it in card counting, he used it to determine the size of the 'bets' he made on individual warrant trades... and he took into account the assumptions required by his model. perhaps, i assumed wrong.

all that to say, i doubt he would agree that a static 1.5x leveraged position in the sp500 is the ideal investing strategy. it just doesn't jive. there is no margin of safety, and, even worse, the risk of total loss is absolutely untenable with that amount of leverage and no hedging. the number one rule for any money manager is don't lose money...
Yes, Thorp seems to be a proponent of "fractional Kelly strategies" ... adding another layer of conservatism to the Kelly Criterion.

The use of heavy tailed distributions (like those you might see come in from fractal models) or even infinitely negative light tailed such as normals are outside the realm of the "usual" Kelly framework. Simply put, the needed expectations for quantities like E[G_n(f)] don't exist. I don't claim to know what all has been done in extending the theory in this field. Thorp is very careful in the papers I've skimmed by him to keep his comments and analysis to distributions with finite support. Of course in the little example I did, empirical distributions are always finite and so I didn't run into any problems.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by bgf » Thu Jan 11, 2018 10:44 am

pezblanco wrote:
Thu Jan 11, 2018 10:00 am
bgf wrote:
Thu Jan 11, 2018 8:40 am
something about this seems wrong... i dont think thorp would advise this.

he understood the Black-Scholes model for valuing options and used it effectively in practice by delta hedging with warrants. he did this before Black-Scholes were famous for their derivation, i think. probably most importantly, as others here have brought up LTCM, Thorp understood precisely where his model would fail. he was familiar with mandelbrot's work and believed he was correct that the normal distribution did not accurately reflect stock returns on the tails... instead, stock returns were 'wildly random.' and that if the model was going to fail it was going to fail during these less frequent and more extreme events. he said that he used the bachelier-osborne model anyway, but accounted for mandelbrot's work by providing himself a margin of safety on his trades. this worked wonders and his fund earned 20+% annual returns until he shut it down. he literally 'beat the market.'

i was always under the impression that he used the kelly criterion as he used it in card counting, he used it to determine the size of the 'bets' he made on individual warrant trades... and he took into account the assumptions required by his model. perhaps, i assumed wrong.

all that to say, i doubt he would agree that a static 1.5x leveraged position in the sp500 is the ideal investing strategy. it just doesn't jive. there is no margin of safety, and, even worse, the risk of total loss is absolutely untenable with that amount of leverage and no hedging. the number one rule for any money manager is don't lose money...
Yes, Thorp seems to be a proponent of "fractional Kelly strategies" ... adding another layer of conservatism to the Kelly Criterion.

The use of heavy tailed distributions (like those you might see come in from fractal models) or even infinitely negative light tailed such as normals are outside the realm of the "usual" Kelly framework. Simply put, the needed expectations for quantities like E[G_n(f)] don't exist. I don't claim to know what all has been done in extending the theory in this field. Thorp is very careful in the papers I've skimmed by him to keep his comments and analysis to distributions with finite support. Of course in the little example I did, empirical distributions are always finite and so I didn't run into any problems.
i want to learn more about the Kelly criterion, and I am glad you started this thread. if you have not already, Thorp's new book "A Man for All Markets" and Weatherall's "The Physics of Wall Street" are both good. I would recommend them if you are interested in this topic, as well as the history of other methods used in financial markets.

I have Fortune's Formula but have not yet read it.
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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by ThrustVectoring » Thu Jan 11, 2018 1:50 pm

My general heuristic for these sorts of models is that they tend to be overconfident, so for safety's sake you probably want to drop like 20% of the position size they recommend. Far better to be under-levered than over, and the slope of increasing gains flattens out near the point recommended by the Kelley Criterion. So for current market expectations, that means something like 1.2x levered instead of 1.5x.

For homeowners, especially given the generous terms from mortgages, repeatedly refinancing to 80% LTV to get more money in the market is likely enough up until it's time to start buying bonds ~10 years from retirement. Especially with the protection from calling in the loan, and if you're investing in retirement accounts, protection from creditors if the price of your house crashes wiping out your non-retirement position. Single-action states also likely have a bunch of creditor protection too, since they'll just take your house and can't go after your other assets.

Non-homeowners likely want something like 20% leverage. Call options (S&P 500 LEAPs, deep in-the-money) are likely better than margin loans, though I haven't done an in-depth analysis.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by FireProof » Thu Jan 11, 2018 2:02 pm

For a real world situation of continuous contributions over a working lifetime, the case for leverage is actually much clearer (although it starts high and tapers off quickly), because of the outsized impact of poor portfolio performance in the end (when it it is much larger). By leveraging at the beginning, you allow temporal diversification, which is far better than any diversification of type.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by pezblanco » Thu Jan 11, 2018 2:26 pm

ThrustVectoring wrote:
Thu Jan 11, 2018 1:50 pm
My general heuristic for these sorts of models is that they tend to be overconfident, so for safety's sake you probably want to drop like 20% of the position size they recommend. Far better to be under-levered than over, and the slope of increasing gains flattens out near the point recommended by the Kelley Criterion. So for current market expectations, that means something like 1.2x levered instead of 1.5x.

For homeowners, especially given the generous terms from mortgages, repeatedly refinancing to 80% LTV to get more money in the market is likely enough up until it's time to start buying bonds ~10 years from retirement. Especially with the protection from calling in the loan, and if you're investing in retirement accounts, protection from creditors if the price of your house crashes wiping out your non-retirement position. Single-action states also likely have a bunch of creditor protection too, since they'll just take your house and can't go after your other assets.

Non-homeowners likely want something like 20% leverage. Call options (S&P 500 LEAPs, deep in-the-money) are likely better than margin loans, though I haven't done an in-depth analysis.
Yes ... these less than 1.5 would be "fractional Kelly" strategies. I'm not sure I would be willing to bet my house on a decision based on 50 data points. :D

I wanted to look at this strategy from the point of view of a "simple" implementation. Right now you can borrow money at Interactive Brokers for around 2.5%. That seemed to fit in with the general rule of thumb of one month T-bill + 1%. The strategy is not super sensitive to borrow rates ... i.e there are very little differences to the curves or optimal points if you assume T-bill + 1.5% say.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by pezblanco » Thu Jan 11, 2018 2:27 pm

FireProof wrote:
Thu Jan 11, 2018 2:02 pm
For a real world situation of continuous contributions over a working lifetime, the case for leverage is actually much clearer (although it starts high and tapers off quickly), because of the outsized impact of poor portfolio performance in the end (when it it is much larger). By leveraging at the beginning, you allow temporal diversification, which is far better than any diversification of type.
That makes sense to me ...

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by ThrustVectoring » Thu Jan 11, 2018 3:00 pm

pezblanco wrote:
Thu Jan 11, 2018 2:27 pm
FireProof wrote:
Thu Jan 11, 2018 2:02 pm
For a real world situation of continuous contributions over a working lifetime, the case for leverage is actually much clearer (although it starts high and tapers off quickly), because of the outsized impact of poor portfolio performance in the end (when it it is much larger). By leveraging at the beginning, you allow temporal diversification, which is far better than any diversification of type.
That makes sense to me ...
I mean, it makes sense, but you still have to be sensible about it. See: viewtopic.php?t=5934

This is why I suggested leverage through high LTV on a mortgage in a single-action state. It's non-callable from the lender, and has an attractive strategic default option for the borrower.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by comeinvest » Fri Jan 12, 2018 3:19 am

ThrustVectoring wrote:
Thu Jan 11, 2018 3:00 pm
... This is why I suggested leverage through high LTV on a mortgage in a single-action state. It's non-callable from the lender, and has an attractive strategic default option for the borrower.
... but this safety comes at a price (like with everything else in finance): Even for folks with top credit scores, the equity risk premium of equity markets with respect to the rate of a 30-year mortgage is much smaller, than with respect to T-Bill or other short-term lending rates. On another note, in many real estate markets, the expected returns (based on cap rates) are currently only infinitesimal higher, arguably even lower than the mortgage rates (all numbers adjusted for inflation / expected growth). This means the potential out-performance from leverage is often slim, if not non-existing.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by gordoni2 » Fri Jan 12, 2018 3:33 am

You could also solve the problem analytically (see Wikipedia - Kelly criterion - Applications to the stock market):

Kelly's stock fraction = (u - r) / sigma^2 = (0.0575 - 0.0180) / 0.1544^2 = 166% stocks

where:

u = geometric Brownian motion drift = log(m / sqrt(1 + (vol / m)^2)) = 0.0575
sigma = geometric Brownian motion volatility = sqrt(log(1 + (vol / m)^2)) = 0.1544
m = 1 + arithmetic annual stock market real return = 1.0719
vol = annual stock market volatility = 0.1665
r = real return of risk free asset = 0.0080 + b = 0.0180
b = cost to borrow = 0.01

The trouble with the Kelly criterion is there is no good reason for choosing it. Why seek to maximize geometric mean wealth?

The Kelly criterion is a special case of Merton's portfolio problem when the coefficient of relative risk aversion is 1. Divide the stock fraction you get using the Kelly criterion by the coefficient of relative risk aversion (gamma) and you get the optimal asset allocation for Merton's portfolio problem:

Merton's stock fraction = (u - r) / (sigma^2 . gamma)

Economists seem divided, but a reasonable estimate of the coefficient of relative risk aversion might be around 2. Thus an asset allocation around 80% might be more reasonable for your scenario.

A final note, Merton's portfolio problem assumes all assets are tradable. If you have non-tradable assets such as Social Security, pensions, income annuities, or human capital, your asset allocation should probably be larger than predicted using Merton's formula, possibly a lot larger.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by comeinvest » Fri Jan 12, 2018 3:47 am

pezblanco wrote:
Thu Jan 11, 2018 9:51 am
... To get a 30 year horizon, we sample (with replacement) from the empirical distribution of the yearly data 30 times. So, for example, ONE of the possible 30 year sequences would be to get 30 years of the 2008 data point with its real 37% loss 30 years in a row! Of course the odds of that happening are very small ...
... But this means you are ignoring the temporal memory effect of the real world probability distribution. (You probably know that.) Because in the real world, a loss over 30 years never happened in the history of the U.S. stock market (not even over 20 years if my memory serves me right), it would appear that the optimal leverage ratio in the real world would be higher, as your model is too negative. I understand that your memory-less model would account for the possibility of scenarios that never happened in the past, but could happen in the future (e.g. 30 years of -37% in a row), thus adding a margin of safety. On the other hand, there seems to be evidence of "reversion to the mean" in financial markets, which I believe also has some common-sense explanation - with 30 years of -37%, or even milder scenarios like say 10 years of -15%, either the yields of securities would be sky high (for whatever reason), or the real economy would have shrunk to a fraction of its current size. I think in either case, with the underlying real world disaster (nuclear war?), you would have bigger problems than your portfolio...
The question is do we really need to introduce margins of safety accommodating scenarios that never happened in 100+ years including times of world war and all the other known historic disasters?

On another note, I don't have time to search for it right now, but there were several different studies discussed in this forum of optimal leverage ratios based on real historic data over certain time frames. The optimal leverage ratio was usually found to be between 1.5 and 2, depending on investors' borrowing rates, the specific leverage strategy, and other assumptions. Some of those studies also simulated margin calls (or lack thereof) depending on the leverage strategy and ratios. Warren Buffet's leverage ratio is estimated at 1.6 over the past 30 years. Some studies say this was the primary driver of Buffet's out-performance.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by pezblanco » Fri Jan 12, 2018 10:18 am

gordoni2 wrote:
Fri Jan 12, 2018 3:33 am
You could also solve the problem analytically (see Wikipedia - Kelly criterion - Applications to the stock market):

Kelly's stock fraction = (u - r) / sigma^2 = (0.0575 - 0.0180) / 0.1544^2 = 166% stocks

where:

u = geometric Brownian motion drift = log(m / sqrt(1 + (vol / m)^2)) = 0.0575
sigma = geometric Brownian motion volatility = sqrt(log(1 + (vol / m)^2)) = 0.1544
m = 1 + arithmetic annual stock market real return = 1.0719
vol = annual stock market volatility = 0.1665
r = real return of risk free asset = 0.0080 + b = 0.0180
b = cost to borrow = 0.01

The trouble with the Kelly criterion is there is no good reason for choosing it. Why seek to maximize geometric mean wealth?

The Kelly criterion is a special case of Merton's portfolio problem when the coefficient of relative risk aversion is 1. Divide the stock fraction you get using the Kelly criterion by the coefficient of relative risk aversion (gamma) and you get the optimal asset allocation for Merton's portfolio problem:

Merton's stock fraction = (u - r) / (sigma^2 . gamma)

Economists seem divided, but a reasonable estimate of the coefficient of relative risk aversion might be around 2. Thus an asset allocation around 80% might be more reasonable for your scenario.

A final note, Merton's portfolio problem assumes all assets are tradable. If you have non-tradable assets such as Social Security, pensions, income annuities, or human capital, your asset allocation should probably be larger than predicted using Merton's formula, possibly a lot larger.
I think there is a lot of discussion about the inadequacy of geometric BM models in financial modeling ... (see e.g. Mandelbrot, Taleb etc). One of the main things they have going for them is the ability to do a closed form analysis.

It is usually argued (please, I'm not an economist) that the wealth utility function for people isn't linear and might be closer to logarithmic Thorp makes an argument along these lines for the Kelly criterion as does Breiman. My first course in mathematical probability was out of Breiman so I'm going with him. :D

Thank you for the Merton's references. I assume the analytical result you give is for another geometric BM model?

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by pezblanco » Fri Jan 12, 2018 10:25 am

comeinvest wrote:
Fri Jan 12, 2018 3:47 am
pezblanco wrote:
Thu Jan 11, 2018 9:51 am
... To get a 30 year horizon, we sample (with replacement) from the empirical distribution of the yearly data 30 times. So, for example, ONE of the possible 30 year sequences would be to get 30 years of the 2008 data point with its real 37% loss 30 years in a row! Of course the odds of that happening are very small ...
... But this means you are ignoring the temporal memory effect of the real world probability distribution. (You probably know that.) Because in the real world, a loss over 30 years never happened in the history of the U.S. stock market (not even over 20 years if my memory serves me right), it would appear that the optimal leverage ratio in the real world would be higher, as your model is too negative. I understand that your memory-less model would account for the possibility of scenarios that never happened in the past, but could happen in the future (e.g. 30 years of -37% in a row), thus adding a margin of safety. On the other hand, there seems to be evidence of "reversion to the mean" in financial markets, which I believe also has some common-sense explanation - with 30 years of -37%, or even milder scenarios like say 10 years of -15%, either the yields of securities would be sky high (for whatever reason), or the real economy would have shrunk to a fraction of its current size. I think in either case, with the underlying real world disaster (nuclear war?), you would have bigger problems than your portfolio...
The question is do we really need to introduce margins of safety accommodating scenarios that never happened in 100+ years including times of world war and all the other known historic disasters?

On another note, I don't have time to search for it right now, but there were several different studies discussed in this forum of optimal leverage ratios based on real historic data over certain time frames. The optimal leverage ratio was usually found to be between 1.5 and 2, depending on investors' borrowing rates, the specific leverage strategy, and other assumptions. Some of those studies also simulated margin calls (or lack thereof) depending on the leverage strategy and ratios. Warren Buffet's leverage ratio is estimated at 1.6 over the past 30 years. Some studies say this was the primary driver of Buffet's out-performance.
Yes, I'm in broad agreement with most of your points. The fundamental problem is that of data. If we insist upon using empirical data with a possible temporal component, then at most we have 20 highly correlated data points over the last 50 years to simulate a 30 year investment horizon. I thought also, that by assuming independence of return data would probably be erring on the side of conservative.

I think that it is very interesting how all these approaches point to optimal values of leverage in the 150 to 200% range!

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by tadamsmar » Fri Jan 12, 2018 10:56 am

This paper:

https://sites.math.washington.edu/~morr ... nikhil.pdf

calcualated:

f∗ =(.058−.029)/.2162 ≈ .62

.058 = historical market real return
.2162 = historical market standard deviation
.029 = treasury (risk free) rate.

Implies to invest 62% in stocks (SP500)

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by pezblanco » Fri Jan 12, 2018 11:37 am

tadamsmar wrote:
Fri Jan 12, 2018 10:56 am
This paper:

https://sites.math.washington.edu/~morr ... nikhil.pdf

calcualated:

f∗ =(.058−.029)/.2162^2 ≈ .62

.058 = historical market real return
.2162 = historical market standard deviation
.029 = treasury (risk free) rate.

Implies to invest 62% in stocks (SP500)
Yes ... Kelly does a similar analysis (he calls it the log-normal diffusion model). I don't know why the numbers for this author differ so much from the others. I reason I think we might quibble about his risk free rate ... the geometric mean of the one-month T-bill real return for the last 50 years is more like .0077 .... that alone bumps the value up to over 1.

One advantage of the empirical data approach is that the risk free rate is also obtained from empirical data and is also random and tied to the actual returns from the market in stocks and long term bonds. These other approaches have to assume constant deterministic borrow rates.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by tadamsmar » Sat Jan 13, 2018 11:08 am

pezblanco wrote:
Fri Jan 12, 2018 11:37 am
tadamsmar wrote:
Fri Jan 12, 2018 10:56 am
This paper:

https://sites.math.washington.edu/~morr ... nikhil.pdf

calcualated:

f∗ =(.058−.029)/.2162^2 ≈ .62

.058 = historical market real return
.2162 = historical market standard deviation
.029 = treasury (risk free) rate.

Implies to invest 62% in stocks (SP500)
Yes ... Kelly does a similar analysis (he calls it the log-normal diffusion model). I don't know why the numbers for this author differ so much from the others. I reason I think we might quibble about his risk free rate ... the geometric mean of the one-month T-bill real return for the last 50 years is more like .0077 .... that alone bumps the value up to over 1.

One advantage of the empirical data approach is that the risk free rate is also obtained from empirical data and is also random and tied to the actual returns from the market in stocks and long term bonds. These other approaches have to assume constant deterministic borrow rates.
Opps, I looked at my source and it appears he use the treasury BOND rate. .029 seems way to big for a true risk free real rate.

That's probably the difference.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by tadamsmar » Sat Jan 13, 2018 11:37 am

Here's an analysis by Thorp himself that supports 117% investment in the S&p 500:

http://finance.martinsewell.com/money-m ... rp1992.pdf

That's assuming that you can borrow at the T-Bill rate.

Note that half-Kelly would be 58.5% in equities.

Given that the "win probabilities" in the stock market are not that persistently reliable compared with the analysis of a gambling game, fractional Kelly is good idea.

With fractional Kelly, the Kelly Criterion is roughly in line with Boglehead investing parameters.

PS: Even the sound math of Blackjack was a bit rosy given the realities of casino gambling. The casino bosses would send in a card mechanic to deal to Thorp. A card mechanic is a dealer/magician who is expert at slight of hand. Thorp often could not detect the cheating even if he had a watcher with him (he often used a watcher so he could just concentrate on the counts). He could only detect that his edge had evaporated over time. He took to leaving if there was a dealer change between shifts.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by pezblanco » Sat Jan 13, 2018 3:12 pm

tadamsmar wrote:
Sat Jan 13, 2018 11:37 am
Here's an analysis by Thorp himself that supports 117% investment in the S&p 500:

http://finance.martinsewell.com/money-m ... rp1992.pdf

That's assuming that you can borrow at the T-Bill rate.

Note that half-Kelly would be 58.5% in equities.

Given that the "win probabilities" in the stock market are not that persistently reliable compared with the analysis of a gambling game, fractional Kelly is good idea.

With fractional Kelly, the Kelly Criterion is roughly in line with Boglehead investing parameters.

PS: Even the sound math of Blackjack was a bit rosy given the realities of casino gambling. The casino bosses would send in a card mechanic to deal to Thorp. A card mechanic is a dealer/magician who is expert at slight of hand. Thorp often could not detect the cheating even if he had a watcher with him (he often used a watcher so he could just concentrate on the counts). He could only detect that his edge had evaporated over time. He took to leaving if there was a dealer change between shifts.
Yes, I mentioned/discussed this analysis in the original post ... you'll see that Thorp assumes a truncated normal as the return distribution over the T-bill rate. Thus His rates would be the rate OVER the T-bill rate .... anyway. The modeling approach (whether it be geometric BM or an assumed return distribution is great for getting a handle on what is going on. What I like about the empirical approach is that we can model directly the effects of realistic borrow rates and actual T-bill rates that historically existed along with the historical stock returns. We can model investing in LTGB when we are unleveraged and then model the effects of borrowing at the T-Bill rate + x%.

I didn't know that story about the casinos sending in "mechanics" to deal with him. I would have thought that the usual methodology was just to ban someone that exhibits too much skill.

I personally think that fractional Kelly is just a cop-out. I don't understand any mathematical reason to use it. You could argue that ANY allocation is a fractional Kelly ... you just need to choose your fraction!

Above, gordoni2 says:
gordoni2 wrote:
Fri Jan 12, 2018 3:33 am

The trouble with the Kelly criterion is there is no good reason for choosing it. Why seek to maximize geometric mean wealth?
and I think I fell down on the job a little there. With probability one, the capital at the nth step using the Kelly criterion divided by the capital at the nth step using any other criterion will go to infinity. This result doesn't depend upon utility functions or what have you. The policy of maximizing the E[X_n} in black jack or other games leads with probabiliy one to ruin. I think these are excellent reasons to choose the Kelly criterion regardless of economic utility considerations.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by gordoni2 » Sat Jan 13, 2018 10:37 pm

pezblanco wrote:
Sat Jan 13, 2018 3:12 pm
gordoni2 wrote:
Fri Jan 12, 2018 3:33 am

The trouble with the Kelly criterion is there is no good reason for choosing it. Why seek to maximize geometric mean wealth?
and I think I fell down on the job a little there. With probability one, the capital at the nth step using the Kelly criterion divided by the capital at the nth step using any other criterion will go to infinity. This result doesn't depend upon utility functions or what have you. [...] I think these are excellent reasons to choose the Kelly criterion regardless of economic utility considerations.
It should be that it is the shape of the distribution of wealth outcomes after some finite time that matters. To boil this distribution down to a single number of goodness you need to use a utility function. If your utility of wealth exhibits Constant Relative Risk Aversion (CRRA) with gamma = 1, i.e. logarithmic utility, the Kelly criterion will do just fine. If it is CRRA with gamma > 1 as seems more likely, the Kelly criterion will result in too high a probability of a small wealth outcomes. The Kelly criterion finds these small wealth outcomes are offset by much larger wealth outcomes, but because our utility of wealth is sub-logarithmic they are not.

Another problem with the Kelly criterion is it sees as the goal maximizing a particular function of terminal wealth. The purpose of investing should not be to maximize wealth but to maximize utility of consumption, as is the case with Merton's portfolio problem.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by bobcat2 » Sun Jan 14, 2018 1:24 pm

gordoni2 wrote:
Sat Jan 13, 2018 10:37 pm
Another problem with the Kelly criterion is it sees as the goal maximizing a particular function of terminal wealth. The purpose of investing should not be to maximize wealth but to maximize utility of consumption, as is the case with Merton's portfolio problem.
Or to put it in plainer terms, the purpose of investing should be to get you to a level of spending over time that you are happy with.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by pezblanco » Sun Jan 14, 2018 5:40 pm

gordoni2 wrote:
Sat Jan 13, 2018 10:37 pm
pezblanco wrote:
Sat Jan 13, 2018 3:12 pm
gordoni2 wrote:
Fri Jan 12, 2018 3:33 am

The trouble with the Kelly criterion is there is no good reason for choosing it. Why seek to maximize geometric mean wealth?
and I think I fell down on the job a little there. With probability one, the capital at the nth step using the Kelly criterion divided by the capital at the nth step using any other criterion will go to infinity. This result doesn't depend upon utility functions or what have you. [...] I think these are excellent reasons to choose the Kelly criterion regardless of economic utility considerations.
It should be that it is the shape of the distribution of wealth outcomes after some finite time that matters. To boil this distribution down to a single number of goodness you need to use a utility function. If your utility of wealth exhibits Constant Relative Risk Aversion (CRRA) with gamma = 1, i.e. logarithmic utility, the Kelly criterion will do just fine. If it is CRRA with gamma > 1 as seems more likely, the Kelly criterion will result in too high a probability of a small wealth outcomes. The Kelly criterion finds these small wealth outcomes are offset by much larger wealth outcomes, but because our utility of wealth is sub-logarithmic they are not.

Another problem with the Kelly criterion is it sees as the goal maximizing a particular function of terminal wealth. The purpose of investing should not be to maximize wealth but to maximize utility of consumption, as is the case with Merton's portfolio problem.

bobcat2 wrote:
Sun Jan 14, 2018 1:24 pm
gordoni2 wrote:
Sat Jan 13, 2018 10:37 pm
Another problem with the Kelly criterion is it sees as the goal maximizing a particular function of terminal wealth. The purpose of investing should not be to maximize wealth but to maximize utility of consumption, as is the case with Merton's portfolio problem.
Or to put it in plainer terms, the purpose of investing should be to get you to a level of spending over time that you are happy with.

BobK
Yeah, well .... I just don't see how utility functions solve anything. Everyone supposedly has their own private utility function ... which of course no one knows what it is.

I think the asymptotic almost-sure result of the Kelly criterion maximizing wealth is a much better result ... but then I'm not an economist.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by tadamsmar » Mon Jan 15, 2018 10:06 am

There are various examples that present problems for the Kelly Criterion.

Turns out most of us don't attempt to assure against every remote possibility that can wipe out our wealth. Those that try to do that seem a bit nutty.

Under the Kelly Criterion we are being reckless, the low probability event will eventually happen with probability = 1 and we will be wiped out.

Due to stuff like this, the Kelly Criterion is not generally considered to be normative ("You ought to use it") or descriptive ("People in general use it").

PS: Leo Szilard slept every night with his suitcase packed. Turned out to be good idea when he had to get out of Germany fast.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by pezblanco » Mon Jan 15, 2018 11:04 am

tadamsmar wrote:
Mon Jan 15, 2018 10:06 am
There are various examples that present problems for the Kelly Criterion.

Turns out most of us don't attempt to assure against every remote possibility that can wipe out our wealth. Those that try to do that seem a bit nutty.

Under the Kelly Criterion we are being reckless, the low probability event will eventually happen with probability = 1 and we will be wiped out.

Due to stuff like this, the Kelly Criterion is not generally considered to be normative ("You ought to use it") or descriptive ("People in general use it").

PS: Leo Szilard slept every night with his suitcase packed. Turned out to be good idea when he had to get out of Germany fast.
I'm not sure I'm understanding what you're saying, tadamsmar. Under the maximizing the mean wealth criterion, we end up with probability one of being wiped out. Under the Kelly criterion we ,with probability one, maximize our wealth against any other possible criterion (granted this is an asymptotic result) but with probability zero are we are ever wiped out!

I bow to your knowledge about whether it is normative or descriptive ....

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by bgf » Mon Jan 15, 2018 11:44 am

tadamsmar wrote:
Sat Jan 13, 2018 11:37 am
Here's an analysis by Thorp himself that supports 117% investment in the S&p 500:

http://finance.martinsewell.com/money-m ... rp1992.pdf

That's assuming that you can borrow at the T-Bill rate.

Note that half-Kelly would be 58.5% in equities.

Given that the "win probabilities" in the stock market are not that persistently reliable compared with the analysis of a gambling game, fractional Kelly is good idea.

With fractional Kelly, the Kelly Criterion is roughly in line with Boglehead investing parameters.

PS: Even the sound math of Blackjack was a bit rosy given the realities of casino gambling. The casino bosses would send in a card mechanic to deal to Thorp. A card mechanic is a dealer/magician who is expert at slight of hand. Thorp often could not detect the cheating even if he had a watcher with him (he often used a watcher so he could just concentrate on the counts). He could only detect that his edge had evaporated over time. He took to leaving if there was a dealer change between shifts.
thank you for sharing that. the caveats at the end seem the most important part however.
“TE OCCIDERE POSSUNT SED TE EDERE NON POSSUNT NEFAS EST"

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by tadamsmar » Mon Jan 15, 2018 2:36 pm

pezblanco wrote:
Mon Jan 15, 2018 11:04 am
tadamsmar wrote:
Mon Jan 15, 2018 10:06 am
There are various examples that present problems for the Kelly Criterion.

Turns out most of us don't attempt to assure against every remote possibility that can wipe out our wealth. Those that try to do that seem a bit nutty.

Under the Kelly Criterion we are being reckless, the low probability event will eventually happen with probability = 1 and we will be wiped out.

Due to stuff like this, the Kelly Criterion is not generally considered to be normative ("You ought to use it") or descriptive ("People in general use it").

PS: Leo Szilard slept every night with his suitcase packed. Turned out to be good idea when he had to get out of Germany fast.
I'm not sure I'm understanding what you're saying, tadamsmar.
I am saying that it's considered nutty to conform to the Kelly Criteria when we are betting against a very low probability event that would wipe us out. The Kelly calculation indicates that we will be wiped out in the long run, but we ignore this.

Another way to put it: The Kelly Criteria maximizes wealth in the long run. On reason we don't use it consistently (in all of it's possible applications) is because in the long run we are all dead.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by oldcomputerguy » Mon Jan 15, 2018 3:09 pm

Leveraged investing (investing on margin) was one of the greatest contributors to the severity and aftereffects of the market crash of 1929. No, thanks. I absolutely will not invest money I do not own.
It’s taken me a lot of years, but I’ve come around to this: If you’re dumb, surround yourself with smart people. And if you’re smart, surround yourself with smart people who disagree with you.

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by pezblanco » Mon Jan 15, 2018 5:42 pm

tadamsmar wrote:
Mon Jan 15, 2018 2:36 pm
pezblanco wrote:
Mon Jan 15, 2018 11:04 am
tadamsmar wrote:
Mon Jan 15, 2018 10:06 am
There are various examples that present problems for the Kelly Criterion.

Turns out most of us don't attempt to assure against every remote possibility that can wipe out our wealth. Those that try to do that seem a bit nutty.

Under the Kelly Criterion we are being reckless, the low probability event will eventually happen with probability = 1 and we will be wiped out.

Due to stuff like this, the Kelly Criterion is not generally considered to be normative ("You ought to use it") or descriptive ("People in general use it").

PS: Leo Szilard slept every night with his suitcase packed. Turned out to be good idea when he had to get out of Germany fast.
I'm not sure I'm understanding what you're saying, tadamsmar.
I am saying that it's considered nutty to conform to the Kelly Criteria when we are betting against a very low probability event that would wipe us out. The Kelly calculation indicates that we will be wiped out in the long run, but we ignore this.

Another way to put it: The Kelly Criteria maximizes wealth in the long run. On reason we don't use it consistently (in all of it's possible applications) is because in the long run we are all dead.
I don't know how I ended up being the champion of the Kelly criterion .... I don't leverage and probably never would ....

If you look at the expectation to be maximized: E[log(1+fS)] where S is the return over the riskless rate (note that I used a different formula that I derived since I wanted to look at the actual return with actual bond rates but anyway), you'll notice that there can be allocation of probabiiity to the range of where the argument of the log goes negative. So, the Kelly criterion NEVER lets you take a chance that would wipe you out, in the long run or in the short run.

That said, in my case, I'm using empirical stock market data ... in the last 50 years, the largest negative real return was -37%. In the universe of my simulations, as long as f is chosen so that, 1+f(-.37) > 0, we're OK. But, 10 years from now after we've implemented the Kelly criterion and we get a return of -80%, then yes, we could be wiped out .... the empirical data was inadequate to describe the reality of market returns. This is a fault of the data, not of the Kelly criterion.

... And yes .... in the long run, we are all dead .... you have to decide what sorts of tools and results are the best guides. I feel very comfortable with asymptotic arguments. I know enough to simulate what is going on for small time values and get a handle for when the asymptotics are expected to kick in. I personally get no insight whatsoever from people making "utility function" arguments. But, that's just me ....

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by tadamsmar » Tue Jan 16, 2018 2:51 pm

pezblanco wrote:
Mon Jan 15, 2018 5:42 pm
tadamsmar wrote:
Mon Jan 15, 2018 2:36 pm
pezblanco wrote:
Mon Jan 15, 2018 11:04 am
tadamsmar wrote:
Mon Jan 15, 2018 10:06 am
There are various examples that present problems for the Kelly Criterion.

Turns out most of us don't attempt to assure against every remote possibility that can wipe out our wealth. Those that try to do that seem a bit nutty.

Under the Kelly Criterion we are being reckless, the low probability event will eventually happen with probability = 1 and we will be wiped out.

Due to stuff like this, the Kelly Criterion is not generally considered to be normative ("You ought to use it") or descriptive ("People in general use it").

PS: Leo Szilard slept every night with his suitcase packed. Turned out to be good idea when he had to get out of Germany fast.
I'm not sure I'm understanding what you're saying, tadamsmar.
I am saying that it's considered nutty to conform to the Kelly Criteria when we are betting against a very low probability event that would wipe us out. The Kelly calculation indicates that we will be wiped out in the long run, but we ignore this.

Another way to put it: The Kelly Criteria maximizes wealth in the long run. On reason we don't use it consistently (in all of it's possible applications) is because in the long run we are all dead.
I don't know how I ended up being the champion of the Kelly criterion .... I don't leverage and probably never would ....

If you look at the expectation to be maximized: E[log(1+fS)] where S is the return over the riskless rate (note that I used a different formula that I derived since I wanted to look at the actual return with actual bond rates but anyway), you'll notice that there can be allocation of probabiiity to the range of where the argument of the log goes negative. So, the Kelly criterion NEVER lets you take a chance that would wipe you out, in the long run or in the short run.

That said, in my case, I'm using empirical stock market data ... in the last 50 years, the largest negative real return was -37%. In the universe of my simulations, as long as f is chosen so that, 1+f(-.37) > 0, we're OK. But, 10 years from now after we've implemented the Kelly criterion and we get a return of -80%, then yes, we could be wiped out .... the empirical data was inadequate to describe the reality of market returns. This is a fault of the data, not of the Kelly criterion.

... And yes .... in the long run, we are all dead .... you have to decide what sorts of tools and results are the best guides. I feel very comfortable with asymptotic arguments. I know enough to simulate what is going on for small time values and get a handle for when the asymptotics are expected to kick in. I personally get no insight whatsoever from people making "utility function" arguments. But, that's just me ....
You don't need to leverage to get to full Kelly. You can increase the risk of your portfolio to get you to full Kelly.

Are you at full Kelly? Are you practicing what you preach?

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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by pezblanco » Tue Jan 16, 2018 3:28 pm

tadamsmar wrote:
Tue Jan 16, 2018 2:51 pm
pezblanco wrote:
Mon Jan 15, 2018 5:42 pm

I don't know how I ended up being the champion of the Kelly criterion .... I don't leverage and probably never would ....

If you look at the expectation to be maximized: E[log(1+fS)] where S is the return over the riskless rate (note that I used a different formula that I derived since I wanted to look at the actual return with actual bond rates but anyway), you'll notice that there can be allocation of probabiiity to the range of where the argument of the log goes negative. So, the Kelly criterion NEVER lets you take a chance that would wipe you out, in the long run or in the short run.

That said, in my case, I'm using empirical stock market data ... in the last 50 years, the largest negative real return was -37%. In the universe of my simulations, as long as f is chosen so that, 1+f(-.37) > 0, we're OK. But, 10 years from now after we've implemented the Kelly criterion and we get a return of -80%, then yes, we could be wiped out .... the empirical data was inadequate to describe the reality of market returns. This is a fault of the data, not of the Kelly criterion.

... And yes .... in the long run, we are all dead .... you have to decide what sorts of tools and results are the best guides. I feel very comfortable with asymptotic arguments. I know enough to simulate what is going on for small time values and get a handle for when the asymptotics are expected to kick in. I personally get no insight whatsoever from people making "utility function" arguments. But, that's just me ....
You don't need to leverage to get to full Kelly. You can increase the risk of your portfolio to get you to full Kelly.

Are you at full Kelly? Are you practicing what you preach?
Well, I think that full Kelly computed from the last 50 years of empirical market data is f=1.5. So you are 50% leveraged with full Kelly. I don't use leverage as I've said many times ... and tadamsmar, I'm not preaching anything.

I don't plan to ever use leverage since I'm just not going to be in the market that long. I think these kinds of studies very much indicate that a young investor would not be crazy to think about leveraging out somewhere like 25% or so for the first 10 years of their investing life .....

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tadamsmar
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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by tadamsmar » Tue Jan 16, 2018 6:48 pm

pezblanco wrote:
Tue Jan 16, 2018 3:28 pm
tadamsmar wrote:
Tue Jan 16, 2018 2:51 pm
pezblanco wrote:
Mon Jan 15, 2018 5:42 pm

I don't know how I ended up being the champion of the Kelly criterion .... I don't leverage and probably never would ....

If you look at the expectation to be maximized: E[log(1+fS)] where S is the return over the riskless rate (note that I used a different formula that I derived since I wanted to look at the actual return with actual bond rates but anyway), you'll notice that there can be allocation of probabiiity to the range of where the argument of the log goes negative. So, the Kelly criterion NEVER lets you take a chance that would wipe you out, in the long run or in the short run.

That said, in my case, I'm using empirical stock market data ... in the last 50 years, the largest negative real return was -37%. In the universe of my simulations, as long as f is chosen so that, 1+f(-.37) > 0, we're OK. But, 10 years from now after we've implemented the Kelly criterion and we get a return of -80%, then yes, we could be wiped out .... the empirical data was inadequate to describe the reality of market returns. This is a fault of the data, not of the Kelly criterion.

... And yes .... in the long run, we are all dead .... you have to decide what sorts of tools and results are the best guides. I feel very comfortable with asymptotic arguments. I know enough to simulate what is going on for small time values and get a handle for when the asymptotics are expected to kick in. I personally get no insight whatsoever from people making "utility function" arguments. But, that's just me ....
You don't need to leverage to get to full Kelly. You can increase the risk of your portfolio to get you to full Kelly.

Are you at full Kelly? Are you practicing what you preach?
Well, I think that full Kelly computed from the last 50 years of empirical market data is f=1.5. So you are 50% leveraged with full Kelly. I don't use leverage as I've said many times ... and tadamsmar, I'm not preaching anything.

I don't plan to ever use leverage since I'm just not going to be in the market that long. I think these kinds of studies very much indicate that a young investor would not be crazy to think about leveraging out somewhere like 25% or so for the first 10 years of their investing life .....
Earlier you said you thought fractional Kelly was bunk, when I suggested it.

But now you say you are using it.

I am not questioning the mathematics. They are useful as a theoretical upper limit, for instance

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pezblanco
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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by pezblanco » Wed Jan 17, 2018 12:08 pm

tadamsmar wrote:
Tue Jan 16, 2018 6:48 pm
pezblanco wrote:
Tue Jan 16, 2018 3:28 pm

Well, I think that full Kelly computed from the last 50 years of empirical market data is f=1.5. So you are 50% leveraged with full Kelly. I don't use leverage as I've said many times ... and tadamsmar, I'm not preaching anything.

I don't plan to ever use leverage since I'm just not going to be in the market that long. I think these kinds of studies very much indicate that a young investor would not be crazy to think about leveraging out somewhere like 25% or so for the first 10 years of their investing life .....
Earlier you said you thought fractional Kelly was bunk, when I suggested it.

But now you say you are using it.

I am not questioning the mathematics. They are useful as a theoretical upper limit, for instance
I also said that any other option besides full Kelly must be fractional Kelly for some fraction. Calling it "fractional Kelly" just doesn't seem to be particularly fruitful. If someone came up with a good way to argue for the fraction, I would be very willing indeed to listen to them and be convinced.
I apologize to you if you felt that I was unfairly dismissing your arguments ... that wasn't my intention.

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tadamsmar
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Re: Maximizing Portfolio Growth Rate - Kelly Criterion - Empirical Market Data

Post by tadamsmar » Wed Jan 17, 2018 1:06 pm

There's some discussion of fractional Kelly here:

http://citeseerx.ist.psu.edu/viewdoc/do ... 1&type=pdf

The authors (including Thorp) discuss the relative advantages.

One application of fractional Kelly is when you don't trust you win probability estimate. You might think there is good chance it is too high, but you might be very confident that your error is less than 50%, so you confident that half-Kelly keeps out of the reckless region.

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