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Edge/Odds works when you lose all your money when you lose (like in typical bet as opposed to an investment)
For investments use P(win)/% bankroll you lose - P(loss)/%bankroll you win
- https://www.albertbridgecapital.com/drew-views/2019/5/23/woody-was-right
- https://www.lesswrong.com/posts/BZ6XaCwN4QGgH9CxF/the-kelly-criterion
- https://wizardofodds.com/gambling/kelly-criterion/
- https://moontowermeta.com/how-much-to-wager-when-you-have-edge-note-median-not-mean-outcomes/
- Merton Rule: https://share.getliner.com/5N8uL & https://elmfunds.com/gamblers-can-teach-buy-hold-crowd/
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Econompic twitter thread explainer https://twitter.com/EconomPic/status/1219129451230838786?s=20
A system where the time average converges to the ensemble average (our population mean) is known as an ergodic system. The system of gambles above is non-ergodic as the time average and the ensemble average diverge. And given we cannot individually experience the ensemble average, we should not be misled by it. The focus on ensemble averages, as is typically done in economics, can be misleading if the system is non-ergodic.
Losing wealth on a positive value bet
The first 100 periods of bets forces us to hold a counterintuitive idea in our minds. While the population as an aggregate experiences outcomes reflecting the positive expected value of the bet, the typical person does not. The increase in wealth across the aggregate population is only due to the extreme wealth of a few lucky people.
However, the picture over 1000 periods appears even more confusing. The positive expected value of the bet is nowhere to be seen. How could this be the case?
The answer to this lies in the distribution of bets. After 100 periods, one person had 70% of the wealth. We no longer have 10,000 equally weighted independent bets as we did in the first round. Instead, the path of the wealth of the population is largely subject to the outcome of the bets by this wealthy individual. As we have already shown, the wealth path for an individual almost surely leads to a compound 5% loss of wealth. That individual’s wealth is on borrowed time. The only way for someone to maintain their wealth would be to bet a smaller portion of their wealth, or to diversify their wealth across multiple bets.
- Marginal utility solution to the St. Petersburg Paradox applied to the "perpetuity paradox" https://elmfunds.com/perpetuity-paradox/
- Kelly, bet size, and the Haghani study which revealed that smart people will disproportionately construct betting strategies that are an order of magnitude worse than optimal (when you have an edge the optimal coruse of action is to bet in a way that correctly balances variance with risk of ruin while recognizing that you should increase bet size as bankroll increases)
- “suppose the gambler’s wife allowed him to bet one dollar each week but not to reinvest his winnings. He should then maximize his expectation (expected value of capital) on each bet.” 13 In other words, if you employ the first strategy, you should focus on average payout calculated with the arithmetic mean. In this case, the mean/variance approach is the way to go. In contrast, the Kelly Criterion assumes you parlay your bets, and says you should choose the opportunities with the highest geometric means."
- Geometric return is the product of N returns to the Nth root minus 1
- Substantial empirical evidence shows that stock price changes do not fall along a normal distribution. 15 Actual distributions contain many more small change observations and many more large moves than the simple distribution predicts. These tails play a meaningful role in shaping total returns for assets, and can be a cause of substantial financial pain for investors who do not anticipate them. As a result, mean and variance insufficiently express the distribution and mean/variance can at best crudely approximate market results. Notwithstanding this, practitioners assess risk and reward using a majority of analytical tools based on faulty mean/variance metrics. So the mean/variance approach has two major strikes against it. First, it doesn’t work for parlayed bets (even though most investors do reinvest). Second, it doesn’t consider the verity of nonnormal distributions. Yet most mainstream economists still argue that maximizing geometric returns is the wrong way to allocate capital. Why?
- Neoclassical Economic Objections to the Kelly Criterion One of the most vocal critics of geometric mean maximization happens to be one of the most well-known and well-regarded economists in the world: MIT’s Paul Samuelson. Poundstone notes that Samuelson likes to describe the Kelly Criterion as a fallacy.
- How do economists reconcile the apparently conflicting ideas that maximizing geometric mean will almost certainly result in higher wealth (theorem) with the notion that this approach is possibly inferior to other strategies (corollary)? Perhaps the clearest explanation of the mainstream economics case comes from Mark Rubinstein.17 First, he notes the geometric mean maximization strategy does not assure that you will end up with more wealth than other strategies. Since the approach is based on probability, there remains a very small chance an investor will do poorly. This low-probability, high-impact scenario may violate an individual’s utility function. Second, success of geometric mean maximization depends on investors staying in the market for the long run. If an investor needs access to the funds in the near-term, the benefits of compounding do not apply. Third, the system assumes the investment payoffs remain steady and the investment opportunities set is large enough to accommodate a rising asset base. Shifting investment payoffs undermine the system. Finally, Rubinstein invokes the macro-consistency test: to judge a strategy’s superiority, ask what would happen if everyone tried to follow it. His point is all investors cannot apply the geometric mean strategy successfully.
- One way to understand the difference of opinion is to distinguish between normative and positive arguments. Normative arguments stem from a view of how the world should be, while positive arguments reflect how things are and will likely be in the foreseeable future. Economists dismiss the strategy of maximizing geometric means based on a normative argument. Investors should have specific utility functions and act consistently with those functions. Since the small chance of a large loss will violate an individual’s utility function, geometric mean maximization is not right for everyone (Rubinstein’s first point). A positive argument is based on how people actually behave. Very few people take the time to quantify their utility functions, and those functions shift over time and with varying circumstances. In his famous Portfolio Selection, Markowitz advocates the geometric mean maximization approach. In spite of arguments by Jan Mossin (one of the founders of the capital asset pricing model) and Samuelson in the 1960s, Markowitz reconfirmed his endorsement of the geometric mean maximization strategy in the preface to his second edition published in 1970. Markowitz suggests utility-maximizing man “acts absurdly” over the long term.
- Why Many Money Managers Focus on Arithmetic Returns As we noted, geometric mean maximization requires an investor to be in the market over the long haul. If capital is free to come and go, however, as is the case with an open-end mutual fund, the portfolio manager may not have the luxury of thinking long-term. Even if geometric mean maximization is the best way to go, market realities may compel a short-term focus. The reasoning is straightforward: an open-end portfolio with poor short-run performance faces the very real prospect of losing assets. In turn, portfolio managers have a strong incentive to focus on the investment ideas they perceive will do well in the short term, even at the expense of ideas offering higher rates of return over the long term. Geometric mean maximization simply does not make sense for a portfolio manager in this short-term mindset.
- Poundstone highlights another important feature of the Kelly system: the returns are more volatile than other systems. While the Kelly system offers the highest probability of the most wealth after a long time, the path to the terminal wealth resembles a roller coaster. Another important lesson from prospect theory—and a departure from standard utility theory—is individuals are loss averse. Investors checking their portfolios frequently, especially volatile portfolios, are likely to suffer from myopic loss aversion. 24 The key point is that a Kelly system, which requires a long-term perspective to be effective, is inherently very difficult for investors to deal with psychologically. It is possible to reduce the strategy’s volatility by taking partial Kelly positions.
- Mean/variance is not the best way to think about maximizing long-term wealth if you are reinvesting your investment proceeds. If you face a one-time financial decision, you want to maximize your arithmetic mean. But with repeated favorable opportunities—either through time or diversification—chances are you will do better in the long term by maximizing geometric mean. Mean/variance may be deeply embedded in the investment industry’s lexicon, but it doesn’t do as good a job at building wealth as a Kelly-type system. • Applying the Kelly Criterion is hard psychologically. Assuming you do have an investment edge and a long-term horizon, applying the Kelly system is still hard because of loss aversion. Most investors face institutional and psychological constraints in applying a Kelly-type system.
Kelly's formula works in situations where you have returns that can be redeployed over a long period of time — that is, the wins from prior bets can be ploughed into future bets. If you were in a situation where you could bet only a fixed amount, or if you were in a situation where wins don't parlay forward to future bets, then standard means-variance analysis would work better.
- https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3472456 The Five Investor Camps that Try To Beat the Stock Market via K.C. Hamann
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- https://breakingthemarket.com/math-games/
- Single stock investing is super volatile; but indices have lowered the cost of owning a diversified portfolio thus lowering the vol.
- Repeated games of chance have very different odds of success than single games. The odds of a series of bets – specifically a series of products (multiplication)- are driven by, and trend toward, the GEOMETRIC average. Single bets, or a group of simultaneous bets -specifically a series of sums (addition)-, are driven by the ARITHMETIC average. The arithmetic average for the game is $1.05, as seen in game #1. The geometric average of game #3 is $0.949 per game ( √ {1.5*0.60} ). A loss of over 5 percent per play.
- M: if you can lose X on a coin flip you need to be able to make X/(1-X) when you win for the geometric mean to be zero
- Movement Capital: Compound returns matter more than average returns. This statement appreciates 'variance drain'
- An open question about optimal rebalancing frequency https://ofdollarsanddata.com/how-much-does-rebalancing-frequency-matter/
- Ergodicity https://share.getliner.com/LlAS1 Ole Peters https://www.nature.com/articles/s41567-019-0732-0
- rebalance is a mean reversion play. it's risk-reducing and pushes median return to mean return. It's also concave (see Hoffstein work) so can be thought of pairing well with trend or long vol which are convex https://twitter.com/KrisAbdelmessih/status/1251903372992475137?s=19 https://blog.thinknewfound.com/2020/02/ensembles-and-rebalancing/
- Different types of return; when dealing with wealth and repeated bets CAGR matters
- CAGR or Geometric return: (terminal wealth ^ 1/years)-1
- Arithmetic return = Total return / years
- Average annual return = simple average of your annual returns. These are used to compute vol
- Arithmetic return and avg annual return are not the same. Up 5% and then down 5% is negative arithmetic return but zero avg annual return
- Geometric return = r - .5 variance x time
- Optimizng for CAGR is a response to the St Petersburg Paradox. Since you have to bet your entire wealth on a volatile bet. Volatility matters and you can't simply seek to maximize expected return
- Diversify via barbell or Kelly https://taylorpearson.me/ergodicity/
Diversification is a free lunch because it allows you to "eat" geometric returns
- rebalancing increases a portfolio’s returns. The more often you rebalance the greater the benefit. With the 30 components of the Dow, increasing the rebalancing frequency increases both the portfolio’s arithmetic and geometric returns.
- look at Jegadeesh and Titman’s study again. Their methodology is actually an equal-weight rebalancing scheme, with the 3 month “holding period”, serving as a 3 month rebalancing period, and a 6 month rebalancing period, a 9 month rebalancing period, and finally a 12 month rebalancing period. The finding that “momentum” is strongest over the shorter period and fades as the holding period grows is not a finding about momentum. It’s exactly what you would expect from random behavior when adjusting portfolio rebalancing frequency. Yes the slope of the momentum curve is much higher, but momentum stocks are also much, much more volatile than dow components.
- First off, they should favor stocks that have the highest expected return. But they will also favor the stocks with the highest volatility. If stock returns are random, then by pure chance many of the recent top performers will have a lower expected return while displaying high volatility.
- Imagine two groups of 50 stocks. The first has an average return of 5% but volatility of 25%. The second has an average return of 10%, but a volatility of 15%. If you let the stocks randomly produce returns for a short period, and then select the 10 best stocks, is your sample more likely to come from the first group or the second?
- Because the first group is more volatile, it is more likely to have extreme losers and winners. Momentum is a gigantic volatility screen, more so than a “momentum” screen.
- The momentum screen will lean toward picking stocks with higher expected returns. But importantly it will also be filled with high volatility stocks even if they have average or poor returns.
- Momentum is said to "fade over time" but this is exactly what happens with random returns as "All random compounded returns start out producing returns equivalent to the asset’s arithmetic returns. But with every repetition, the returns will converge toward geometric return. A portfolio of stocks slows down this degradation of returns toward the geometric return, but it still happens."
- Note how a portfolio slows down the process of degradation vs single stocks
- We already know momentum screens select high volatility stocks. High volatility stocks will inherently have a large spread between their arithmetic and geometric returns.5 Therefore, the shape of the momentum return stream over time isn’t really an anomaly at all, but is expected.
- You don’t need to say the momentum effect works in the short run and then fades. It’s all explainable with randomness and geometric compounded returns. You don’t need stock “momentum” to explain the results of the study. The rules of the strategy alone create the illusion of momentum, even with random coin flips.
- Jesse Livermore got close: momentum failed to work in individual securities but worked in indexes.
- This turns out to be a hint as why momentum is "found everywhere". The act of rebalancing which is common to all the studies
- Technically, I’m not saying that randomness explains ALL of the momentum effect. It may. I’m saying randomness and rebalancing undoubtedly explain SOME of the findings of these papers. The process of selecting high volatility stocks and rebalancing them frequently produces most of “momentum’s” performance. If researchers compared their results to a random data set, they would see this.
Data Conclusions
2 Coin flip portfolio for 1000 weeks (19.2 yrs); uncorrelated coins.
The impact of volatility is stark; graph r/(1-r) https://www.derivative-calculator.net/
- Rebalancing pushes your CAGR closer to your arithmetic return. Need uncorrelated assets for this to work and positive expected returns
- Median returns are about 1/2 of expected mean return at around 15% vol if you rebalance!
- Median returns are about 90% of expected mean return at around 9% vol; rebalance is not important
- Rebalancing usually lowers your mean return when the vol is higher (intuition is it removes constrains the capital in the higher vol leg)
- https://moontowermeta.com/lesson-from-coin-flip-investing/
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| portfolio expected | 124% 1 | 124% | |
|---|---|---|---|
mean | 124% | 86% | |
median | 74% | 71% | |
A outperform % | 56.0% | ||
A average spread | 29% | ||
Daily Return Vol | 0.025 | 0 | |
Rebalance Period | 4 | ||
Edge | 0.1 | ||
weekly | Annual | ||
vol1 | 0.025 | 18.0% | |
vol2 | 0.000 | 0.0% | |
expected return1 | 0.00125 | 6.71% | |
expected return2 | 0 | 0.00% | |
edge of vol | 10% | - | |
0 | Expected Return | Annual Vol | Sharpe |
Portfolio | 3.36% | 9.0% | 0.37 |
portfolio expected | 249% | 249% | |
mean | 248% | 246% | |
median | 187% | 193% | |
A outperform % | 41.0% | ||
A average spread | -7% | ||
Daily Return Vol | 0.025 | 0.025 | |
Rebalance Period | 4 | ||
Edge | 0.1 | ||
weekly | Annual | ||
vol1 | 0.025 | 18.0% | |
vol2 | 0.025 | 18.0% | |
expected return1 | 0.00125 | 6.71% | |
expected return2 | 0.00125 | 6.71% | |
edge of vol | 10% | - | |
0 | Expected Return | Annual Vol | Sharpe |
Portfolio | 6.71% | 12.7% | 0.53 |
portfolio expected | 682% | 682% | |
mean | 689% | 545% | |
median | 250% | 322% | |
A outperform % | 42.0% | ||
A average spread | 147% | ||
Daily Return Vol | 0.025 | 0.05 | |
Rebalance Period | 4 | ||
Edge | 0.1 | ||
weekly | Annual | ||
vol1 | 0.025 | 18.0% | |
vol2 | 0.050 | 36.1% | |
expected return1 | 0.00125 | 6.71% | |
expected return2 | 0.0025 | 13.86% | |
edge of vol | 10% | - | |
Expected Return | Annual Vol | Sharpe | |
Portfolio | 10.29% | 20.2% | 0.51 |
portfolio | 140% | 140% | |
mean | 139% | 141% | |
median | 119% | 121% | |
A outperform % | 58.0% | ||
A average spread | 30% | ||
Daily Return Vol | 0.0175 | 0.0175 | |
Rebalance Period | 4 | ||
Edge | 0.1 | ||
weekly | Annual | ||
vol1 | 0.018 | 12.6% | |
vol2 | 0.018 | 12.6% | |
expected return1 | 0.000875 | 4.65% | |
expected return2 | 0.000875 | 4.65% | |
edge of vol | 10% | - | |
Expected Return | Annual Vol | Sharpe | |
Portfolio | 4.65% | 8.9% | 0.52 |
portfolio expected | 87982% | 87982% | |
mean | 15294% | 8044% | |
median | 53% | 231% | |
A outperform % | 45.0% | ||
A average spread | -2564% | ||
Daily Return Vol | 0.025 | 0.15 | |
Rebalance Period | 4 | ||
Edge | 0.1 | ||
weekly | Annual | ||
vol1 | 0.025 | 18.0% | |
vol2 | 0.150 | 108.2% | |
expected return1 | 0.00125 | 6.71% | |
expected return2 | 0.0075 | 47.48% | |
edge of vol | 10% | - | |
0 | Expected Return | Annual Vol | Sharpe |
Portfolio | 27.10% | 54.8% | 0.49 |
portfolio expected | 2061% | 2061% | |
mean | 1947% | 558% | |
median | 46% | 200% | |
A outperform % | 46.0% | ||
A average spread | 736% | ||
Daily Return Vol | 0.075 | 0 | |
Rebalance Period | 4 | ||
Edge | 0.1 | ||
weekly | Annual | ||
vol1 | 0.075 | 54.1% | |
vol2 | 0.000 | 0.0% | |
expected return1 | 0.00375 | 21.49% | |
expected return2 | 0 | 0.00% | |
edge of vol | 10% | - | |
0 | Expected Return | Annual Vol | Sharpe |
Portfolio | 10.74% | 27.0% | 0.40 |
portfolio expected | 557% | 557% | |
mean | 573% | 252% | |
median | 104% | 147% | |
A outperform % | 58.0% | ||
A average spread | 326% | ||
Daily Return Vol | 0 | 0.05 | |
Rebalance Period | 4 | ||
Edge | 0.1 | ||
weekly | Annual | ||
vol1 | 0.000 | 0.0% | |
vol2 | 0.050 | 36.1% | |
expected return1 | 0 | 0.00% | |
expected return2 | 0.0025 | 13.86% | |
edge of vol | 10% | - | |
0 | Expected Return | Annual Vol | Sharpe |
Portfolio | 6.93% | 18.0% | 0.38 |
portfolio expected | 319% | 319% | |
mean | 313% | 172% | |
median | 103% | 118% | |
A outperform % | 47.0% | ||
A average spread | 53% | ||
Daily Return Vol | 0.04 | 0 | |
Rebalance Period | 4 | ||
Edge | 0.1 | ||
weekly | Annual | ||
vol1 | 0.040 | 28.8% | |
vol2 | 0.000 | 0.0% | |
expected return1 | 0.002 | 10.95% | |
expected return2 | 0 | 0.00% | |
edge of vol | 10% | - | |
0 | Expected Return | Annual Vol | Sharpe |
Portfolio | 5.47% | 14.4% | 0.38 |