Regret Minimization

I did a mini research paper for Game Theory on the topic of Regret, it's pretty interesting when considered alongside Nash Equilibrium, and financial markets. While Equilibrium proves the existence of a single mixed strategy that guarantees a minimum payoff, this payoff could still be negative, and this would only serve as the maximum payoff if everybody else is achieving equilibrium. In real life games, players often deviate quite a bit from equilibrium.

Investments, trading

The whole buy & hold thing, if done with ETFs to avoid bankruptcy risk, is an equilibrium strategy where the player is guaranteed a return that roughly equals to the actual rate of inflation. This would be an optimal strategy, if everybody else is doing the same, including hedge funds and investment banks. Obviously, that is not so, resulting in the buy & hold folks considerable opportunity loss or "Regret". A change of strategy to minimize "Regret" then is needed to exploit the inefficiencies from other players deviating from equilibrium.


Here's a basic example of regret minimization from Wikipedia,

Minimax example

Suppose an investor has to choose between investing in stocks, bonds or the money market, and the total return depends on what happens to interest rates. The following table shows some possible returns:

Return	Interest rates rise	Static rates	Interest rates fall	Worst return
Stocks	−4	4	12	−4
Bonds	−2	3	8	−2
Money market	3	2	1	1
Best return	3	4	12

The crude minimax choice based on returns would be to invest in the money market, ensuring a return of at least 1. However, if interest rates fell then the regret associated with this choice would be large. This would be −11, which is the difference between the 1 received and the 12 which could have been received if the outturn had been known in advance. A mixed portfolio of about 11.1% in stocks and 88.9% in the money market would have ensured a return of at least 2.22; but, if interest rates fell, there would be a regret of about −9.78.

The regret table for this example, constructed by subtracting best returns from actual returns, is as follows:

Regret	Interest rates rise	Static rates	Interest rates fall	Worst regret
Stocks	−7	0	0	−7
Bonds	−5	−1	−4	−5
Money market	0	−2	−11	−11

Therefore, using a minimax choice based on regret, the best course would be to invest in bonds, ensuring a regret of no worse than −5. A mixed investment portfolio would do even better: 61.1% invested in stocks, and 38.9% in the money market would produce a regret no worse than about −4.28.

A Non-Random Walk Down Wall Street (Free PDF)

Awesome stuff, this is where Andrew W. Lo and A. Craig MacKinlay show analytically (i.e. statistical evidence) that stock prices are not random, but actually deterministic. The Princeton University Press has taken the trouble to make it freely available in PDF format.

(Click on the book cover for link)

Commercial traders short on NZD

Look at the latest Commitment of Traders

It looks like everyone's net long except Commercial ( typically Smart Money). However, they have covered some from the DX (USD Index) rally last week. Since the DX is negatively correlated with the stock markets, this probably also hints at further stock price drops.

Textbook Theories vs. Reality

Real world experience often brings to light academic fallacies, especially when it comes to the financial markets. Right off the bat, many business students here (Auckland) are taught that inefficiencies do not exist, or impossible to exploit; which completely disengages from reality.

Textbook theories vs. empirical reality

Just because an idea shows up in a textbook does not make it absolute truth. A widely known phenomenon, while Black-Scholes PDE presents an elegant option pricing formula, the “volatility smile” noticeably points out the formula’s inaccuracy and hence unreliability. Some may say “well, it’s all I got, better than nothing right?” No, a wrong solution is often WORSE than nothing.

Remember how the stochastic finance stock pricing model completely ignores credit risk and actual rate of inflation? Models like this have simply no practicality in actual trading.

Textbook contradictions (efficiency vs. inefficiency)

They do not even agree among each other. While some academics keep pushing Efficient Market Hypothesis, top business schools like Wharton use texts specifically on exploiting market inefficiencies like Understanding Arbitrage: An Intuitive Approach to Financial Analysis. What is up with that?!

Searching for truth

Knowledge requires actual experience, not just sometimes subjective ideas off some random textbook. It takes work, like everything else in life.

banks put 63 percent of their new cash into euros and yen

Noticed this at the NY Post,

"

Over the last three months, banks put 63 percent of their new cash into euros and yen -- not the greenbacks -- a nearly complete reversal of the dollar's onetime dominance for reserves, according to Barclays Capital. The dollar's share of new cash in the central banks was down to 37 percent -- compared with two-thirds a decade ago. 'Bernanke's other choice is to keep rates at zero, print even more money and sell more debt, but we'll see triple-digit inflation that could collapse the economy as we know it.'

"

The above article generalizes the immediate past, with no empirical evidence of likelihood for continued actions for the immediate future. And, the coming destruction of USD backed credit instruments still makes inflation/deflation immediate future uncertain.

Cointegration Requires Sound Economics

Because (Multi)cointegration could coincide with spurious regression, the relationships needs economic logic to hold validity. Blind number crunching then could lead to poorly designed relative value arbitrage strategies.

Cointegration/Multicointegration defined

From Wikipedia

“

Cointegration is an econometric property of time series variables. If two or more series are themselves non-stationary, but a linear combination of them is stationary, then the series are said to be cointegrated.

“

Example: 2 individual stock prices display varying probability distributions over time. When you subtract their daily returns, that difference in returns however does exhibit the same distribution at all periods, therefore becoming more “predictable”.

and…

“

Multicointegration extends the cointegration technique beyond two variables, and occasionally to variables integrated at different orders.

“

Issues with spurious correlation/regression

The non-Stationarity of individual security returns then makes random data set cointegration test unreliable. This partly explains how some incompetent banking wizards had picked tested numbers to make them “work”, backing unreliable risk assessments on credit instruments like CDOs, MBSs, and etc.

Back to fundamental economics

Number crunching alone is not enough, while those who acknowledged the crazy leverage and bad debt-packaging realized the oncoming crisis in 2007. When it comes down to it, understanding underlying economic relationships still rules them all.

Yield matters

When it comes to stocks, yield hints immediate future price moves via implied professional sentiment. Keep in mind that large, price affecting orders come from institutions and high value traders who make decisions largely by economic means, not historical price charts.

Yield vs. fixed interest

The relationship is simple. When dividend yield of a stock exceeds fixed interest instruments (bonds, savings accounts, and etc.), some large investors find the said stock more desirable due to a relatively higher return over time. So naturally when yield makes a historically significant high, large buying orders come and push the price up in the immediate future, and vice versa.

Of course if we take into account of credit risk (listed company going bankrupt) or direct market directional risk, it becomes a bit more complicated. Despite that, from personal experience and empirical evidence, not all institutional investors comprehend these issues. Remember how I mentioned that 80% of hedge fund employees did not understanding the technicalities of “hedging”?

How to apply this for an edge

Yield as a time series tends to remain stationary and has a negative correlation to the underlying stock price. Consider selling short when yield gets to a historical low, look at covering when it reverts to mean value; and vice versa for buying and selling.

The chart holds last two years of McDonald’s (MCD) along with its yield. Pretty easy right?

Pair Trading Explained

Market neutral, pair-trading strategies allow traders to reap profit off stock markets regardless of directional forecasting. It embraces simple principles, though the reward does not come easily.

Basic Tenet

Some institutional traders consider Pair-Trading a type of statistical arbitrage (like this one). It first takes two stocks or ETFs of very high positive correlation, i.e. they basically move in concert, and when their prices deviate you short the high, long the low, and close both positions for a profit when (and if) their prices converge.

The fundamental steps entail importing historical prices into statistical software, and then enter when they diverge by 2-3 standard deviations, closing positions if and when prices converge later on.

Main Risks

No adequate exit strategy if the pair never converges. Remember Long Term Capital Investments run by those Black-Scholes Nobel Prize winners? They lost a fortune betting on a pair of oil based stocks.

Too much time may lapse before convergence. Naturally if it takes several years for a pair to converge, the loss in opportunity cost may become quite significant.

Potential volatility may cause margin issues with open positions. To effectively manage this, potential reward becomes lowered, an uncomfortable trade off.

Liquidity issues still stand, especially with short positions becoming potentially squeezed. Some could argue that having a pair highly negatively correlated may solve this issue, yet while working with large volumes, liquidity and price impact matters will not go away.

Profitable?

Yes it can be, but not via simply entering and exiting based on guesses. It takes meticulous planning and active execution. It will not generate all winners, but as long as you can provide yourself a statistical edge, it will work.

Enhance Stock Investment Return via Option Selling

Long term investors could produce increased return (or lessen the same amount in losses) from selling call options. The concept of derivative trading may intimidating some learning investors, just read on and I will show you why this works and carries very-low risk.

Call-Option Basics

A Call-option gives you the right, not obligation, to purchase the underlying stock at the strike price; a Put-option does the opposite but right now we only take interest in Calls. Each option contract (according to CBOE regulation) carries 100 shares of the underlying stocks.

So, trading 1 contract of IBM Call-option gives you the right to purchase 100 shares of IBM at the strike price before the option contract expires (usually 3^rd Friday of expiration month). The value of traded option contracts is termed the “premium”, the price that option buyers pay, and option writers receive.

(The most renowned option exchange today operates at CBOE, Chicago Board of Option Exchange. They provide a brief and concise primer on options. Spend half an hour on it and you will understand all the fundamentals.)

Out-of-money Call-Options

The option “money-ness” describes the relationship between the strike price and underlying value. For Call-Options, this means a strike-price higher than the underlying price, e.g. holding an IBM call-option contract with strike at $110 while IBM sells at $105/share. The premium decreases to zero for out-of-money options at expiration.

Writing Out-Of-Money Call-Options

When you write a contract of Calls, you become obligated to sell 100 shares of the underlying stock at strike price IF assigned (this happens randomly, and infrequently according to statistics). This naturally requires tremendous risk management if you do not own shares of the underlying. However if you do hold sufficient underlying stock positions (i.e. covered), this tactic could prove very worthwhile.

Detailed Example

Let’s say you own 100 shares of QQQQ (closed at $44.28 on 4/11/08). You could consider selling a contract of Calls with strike price ≥$45 expiring in May. Last quoted Bid for a $45 Call stands at $1.16, and you decide to write 1 contract.

The following possibilities will ensue for the 3^rd Friday of May.

1. QQQQ lowers in value to say, $41/share, and you lose $328 on the ETF position and the options expire worthless. Due to premium gained on the written Calls; you reduce that loss by $116.

2. QQQQ remains at the same price of $44.28, and you made no return on the stock position, and the options expire worthless. With the premium earned, you make a risk-free profit of $116, roughly 2.6% of the underlying position.

3. QQQQ increases to say, $48/share, and you gain $372 on the ETF position, lose $300 on the written option position (IF assigned), and end with a net profit of $72. Yes in this case your profit could become lower; nevertheless it is still about “taking profit”, not a loss.

So, this scheme reduces risks and increases your probability of success (2 out of 3 net-profitable potential scenarios), and you can pull it off easily. Pretty good, huh!