Evidence of Intent

General form for the return R:

Optimal conditions:

Specializing to previous case, where i runs from 1 to 2:

Notice that when , , so as disaster becomes ever more unlikely, you would bet a proportion of your wealth up to the loss ratio. This is a reflection of the Kelly criterion’s tendency to never allow anything to go to zero, under any circumstance.

We’ll consider why this is undesirable in some future post. For now, Let’s make most use of the formulation, and try to find good ways of summarizing win/loss ratios and frequencies to fit into the form above.

The terms for each outcome ()  sum to zero for the optimal . Since , These terms are either monotonically increasing or decreasing with depending on the sign of .

The question then is how one would represent the two groups of monotonically increasing and decreasing terms so has to figure out which they net out at. This will be covered in the next post.

Suppose you have a bet which loses money most of the time, but wins a massive amount now and then, how much money should you put on it?

Let’s say the 1 time you win, you win $a for each dollar you bet, and the N times you lose, you lose $b for each dollar you bet. By the Kelly criterion, the geometric average rate of gain if you bet of your wealth would be

Setting , you get

Suppose you are asked to flip a coin, and heads you win $3, and tails you lose $1—then , and therefore , i.e., you should bet 25% of your wealth.

If you have a lottery ticket that has a 1 out of 5,000 chance of winning $10,000 that costs $1, and you are only allowed to buy one number, then , and
and you should only bet 0.01% of your wealth at a time.

Conversely, if you were selling a lottery ticket that had 1 out of 10,000 chance of winning $5,000 that cost $1, , and and you should be trying to have about 0.01% of your wealth at stake.

Related: Do not play the lottery unless you are a millionaire

When I first found out about Case-Shiller futures, I was pretty excited at the possibility of buying a house, enjoying the cheap loan and tax benefits, and hedging out most of the risk.

Alas, the futures have no volume. As of today the open interest on the Feb 2010 New York Case-Shiller futures (NYMG10) is 2 (as in the first integer greater than 1).

It’s difficult to make markets when the underlying and the instrument differ so much in liquidity? Granted, SP500 futures are more liquid than the basket of stocks too, but that gap is bridgeable. What differentiates a bridgeable from an unbridgeable gap? What implications does this have for the existence of noise traders?

Money is a medium of exchange. When exchanging A for B, we prefer to exchange A for money and then money  for B.

Money is a store of value. On top of holding money temporarily while exchanging A for B, we also hold money when we haven’t determined what B we want yet. Holding money is preferable to holding A because A might be bothersome to store (e.g., a truckload of sand) or it might become less valuable with time (e.g., a truckload of apples).

The usefulness of money gives rise to a liquidity preference. Keynes enumerated three ways in which money is useful: as a buffer to smooth out short-term volatility (known unknowns) in income and expenditure, for use as emergency reserves (unknown unknowns), and for use in speculation, i.e., using knowledge of prices to buy assets at low prices and sell them at high prices. The first two forms of liquidity preference tend to grow with income, whilst the last is more affected by the interest rate and expectations of future interest rates.

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When the economy is in a state of equilibrium, each party holds a constant amount of money, and it is possible to think of the flow of money as consisting of many cases of multiparty barter, i.e., every dollar flows in a circle, with goods and services flowing in the opposite direction. This money flux is the GDP, and is a measure of economic activity.

Some goods are not directly consumed, and instead are used to produce other goods and services – these are called investment goods. The accumulation of investment goods increases production.

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When stimulating the economy by printing money, one dumps money into certain regions, and that money proceeds to flow outwards from the introduction points. Iff that flow results in the accumulation of investment goods, the economy is successfully stimulated.

Income from an asset, , depends on manager decision and random factor , so . Assume that managers are motivated by personal income , such that the action taken where is the manager’s utility function and is the incentive-compensation scheme.

The problem in incentive compensation (principle-agent theory) is that of finding , the incentive compensation scheme under which payoff to the owners is maximized. The inputs considered are the utility (including risk adversity) of the manager, , and the relationship of effort to outcome, .

By designing , you want to maximize the incentive (the responsiveness of compensation to managerial effort) while minimizing the actual payoff, all while sharing risk so that the manager is not crippled by the risk involved.

Net-net, it seems to suggest that we don’t want managers that are too rich, since the manager needs to have a significant proportion of his wealth vested in the company (to align risk interests and prevent moral hazard) at the same time the owners are trying not to pay him too much, and hence limit his ownership of the company. In fact, it seems to me that an equitable arrangement always results in the manager’s percentage ownership of the company growing with time, an effect only partially offset by the growth in asset value as a whole. After all, what good reason can a manager give for not wanting to own more of the company they have control of?

The possibility of separating ownership from management is made possible by knowledge of responsiveness of managerial effort to reward (motivation, ) and the responsiveness of asset income to managerial effort (). This knowledge cannot be guaranteed to always be of a form that makes the separation possible.

Bought Brooks Brothers shirts for the first time today. Very curious pricing structure — 1 shirt for $80, 3 shirts for $200, $60 off for going above $300.

The marginal prices of the 6 shirts I got were then $80, $80, $40, $80, $20, $40, making for an average price of $56.

There’s something awful about thinking that personal capabilities are strongly ordered – that for every two people with different capabilities, one of the two can do everything that the other can. This feeling is behind the aversion to taking money too seriously, because money is definitely strongly ordered. If money were everything, I’d either be better or worse than you, with no other possibility.

Thank god money isn’t everything then. More accurately, thank god prices do not completely represent my desire for things. If everyone had the same desires and productive abilities, then prices would indeed represent desires, but because of the different desires (demands) and productivities (supplies), there is much to gain from exchanging goods I produce with other people.

To the extent that people conform in their desire for scarce items, however, prices are inevitably accurate measures of personal demand, and it’s an endless rat race. To the extent to which prices of things are aligned with the degree what I want them, there is only one way to become happier – to make more money.

Drawing 5 cards out of a deck 100,000 times, I got

      1 SF
     23 4K
    110 FH
    220 FL
    412 ST
   2144 3K
   4706 2P
  42322 1P
  50062 HC

Drawing 7 cards out of a deck 100,000 times, I got

      5 SF
    181 4K
   2595 FH
   3016 FL
   4633 ST
   4907 3K
  17251 HC
  23565 2P
  43847 1P

One-pair becomes the most common hand!

See also: http://en.wikipedia.org/wiki/Poker_probability

Hat tip: CMoh

http://www.playwinningpoker.com/poker/betting/

In heads-up poker games, if both 22 and AK were to go all-in pre-flop, they would be evenly (very close to 50/50) matched. In real play, however, 22 loses to AK by a big margin. This is because the  betting and folding options are worth a lot more to AK than to 22, which is to say that the unknown flop, turn and river introduce big variances into the AK’s expected return.

In that sense, AKs are preferable because their riskiness boosts the value of options possessed.

Lesson: seek risk with respect to which you possess options!

The more I think about poker, the more apt of a metaphor for life it becomes. Roommate chris informed me that high level matches go to showdown about one in 200 times on the site he plays on. My first reaction was that of distaste, that there was a huge barrier to newcomers learning anything since they have precious little objective evidence to learn from. I then thought that maybe there were players out there with mentors, and that you would be outcompeted by rule-following rote-learners if you tried to learn the game from scratch.

Then I realized that I had been going about it the wrong way, and I had been too bent on my own definition of reality, which involved knowing what cards the other players have, or having some theory about them, in order to learn. That is, after all, the objective reality on which winning or losing the game was based on.

Or is it? This is really a complex question, the objectivity of a situation, when the underlying reality percolates up to observed reality in such a convoluted manner that you might as well make up a fresh theory for what happens on the surface. Which, I think might just be the case here, that poker (barring certain obtusities) depends on understanding the manners in which others seek patterns as much as, or even more than, the underlying mechanics of shuffling cards and objective odds.

The stock market is the same. The long time it takes for earnings to prove themselves surely makes the movement of prices analogous to a showdown only one-in-200 times, and yet is the ultimate reality on which every other movement is based. Reality is mixed here, neither objective nor arbitrary.

Why animal rights? Because we think of animals on a continuum with humans, and having animal rights forms a buffer for human rights. It is difficult for someone to grant animal rights without granting human rights, and so pushing for animal rights makes human rights more dependable. Sensible moves based on using the structure of the world. Fruitarians go too far though.