Financial FallaciesIgor Rivin (Temple U/Brown U/Meteque Holdings)

Fallacy I

“A dollar is a dollar”

A dollar is a dollar?

A dollar is a dollar?

Depends to whom. The question was first analyzed by Daniel Bernoulli back in 1735, prompted by his cousin Nicholas Bernoulli’s St Peterburg game paradox.

St Petersburg Game

I offer you a game: I flip a fair coin. If it lands heads, I pay you $1. If it lands tails, we flip again, and if it lands heads, I pay you $2. If it lands tails, flip again, and if it lands heads, I pay you $4, and so on.

Now, how big an entry fee will you pay to play this game?

St Petersburg game

Easy to compute that the expectation is infinite!

Will you give me all your money?

St Petersburg Game

Empirically, the answer is a resounding NO! And it is unlikely that anyone will pay more than $10 or so to play.

Why? Bernoulli’s thesis is that if Peter has $100000, and Paul has $1MM, then $10000 is worth about the same to Peter as $100000 is to Paul.

A dollar is a dollar?

An example: a 10% per year return is viewed as a steady return. A $100000 per year return is strange.

Bernoulli’s example: insurance. If you insure your house for a bit more than the expectation of loss, good for you and the insurance company.

A dollar is a dollar

A bit of analysis leads to the conclusion that the UTILITY of N dollars is proportional to log N.

A dollar is a dollar

So, giving the guy money is good for society (that way be socialism…)

St Petersburg game

And of course, the logarithmic expectation of the St Petersburg game is finite.

Fallacy II

log (1+x) = x

prevalent in the finance community (because pension fund managers do not understand logs?)

log(1+x) = x

leads to complete confusion, not the least manifestation of which is the Sharpe ratio.

And its cousins, the CAPM and Markowitz portfolio optimization.

Sharpe Ratio

Defined as the mean of portfolio excess returns divided by the standard deviation of excess return.

Should be the mean of excess log returns divided by their variance.

(call this the KT-ratio, for Kelly-Thorp, more on this below)

Three confused Nobelists

Harry Markowitz

Paul Samuelson Bill Sharpe

And their intellectual offspring

Nassim Taleb

There are two methods to consider in a risky strategy:

The first is to know all parameters about the future, and engage in optimized portfolio

construction, a lunacy, unless one has a god-like knowledge of the future. Let’s call it

Markowitz-style.

What if you do know about logs?

Optimal betting strategies

Problem: you have an edge on the house (say, counting cards, or doing statistical analysis of stock returns). How much should you bet?

Optimal betting strategies

ANSWER: depends on what you want. If you only have one shot, you should bet all of your money (if you are really, really sure you have an edge).

Optimal betting strategies

But if you can play as long as you want, betting all your money every time is a terrible idea, since you will LOSE all your money quickly, even if you have an edge.

Optimal betting strategies

But, the answer has been figured out (at Bell Labs! — not as surprising as it seems, since the phone company made a lot of money from bookies back in the day).

Optimal betting strategies

The answer is “the Kelly Criterion”

The Kelly Criterion

The Kelly Criterion

Interesting properties: betting LESS than the Kelly criterion reduces your returns, but also decreases the volatility

Betting more ALSO decreases returns, while INCREASING the volatility

The Kelly Criterion($1000 initial bankroll, p=0.51, 2000

bets)($1000 initial bankroll, p=0.51, 2000

bets)

The half-Kelly ($1000 initial bankroll, p=0.51, 2000 bets)

The Kelly Criterion ($1000 initial bankroll, p=0.54, 2000 bets)

The half-Kelly ($1000 initial bankrol, p=0.54, 2000 bets)

Some Kellyists

Ed Thorp

Jim Simons

Ray Dalio

No Nobel prizes, and no regrets

Back to fallacies

Madoff Returns

Impossible! Or is it?

Take a look:

See any similarity?

The martingale strategy

You go to the casino, and bet $1 on red. If you win, you walk away.

If you lose, you bet $2 on red. If you win, you are up $1, and walk away.

If you lose, you bet $4, and so on.

The martingale strategy

It’s amazing! You win $1 with 100% probability.

Well, except for running out of money (in a casino, there are table limits, but in the stock market, there aren’t…)

The martingale strategy

So, we found a hedge fund — MG partners, which has $1000 under management to begin, and every day we run the martingale. What happens?

The martingale for fair coin toss (2000 bets, $1000 starting bankroll), betting $1

every time.

The martingale for fair coin toss, betting 0.05% every time ($1000 starting capital,

2000 bets)

The martingale with an edge ($1000 starting capital, p=0.54, $1 bet, 2000

bets)

Still looks a little variable…

The “standard trick” (or so people tell me…) is to found a few funds, and only tell people about the good one(s), just to be on the safe side.

As the amount of investment goes up, the risk of catastrophic failure goes down.

When things crash

The investors are screwed, but the manager walks away with the 2/20.

Of course, this is not nice to the investors, so…

We borrow money from the bank

At the end, the manager has done well, the investors have done well, the bank, umm…

Not so nice to the bank. But now there is a really simple solution.

Instead of $1000, use $10Bn

Then, the manager does (very) well.

The investors do quite well

The bank earns nice interest income

And when things blow up, the bank is too big to fail; the Fed bails it out with money it prints, so

Everybody wins!

Quantopian exchange

How many “trading systems” are the martingale in disguise? People like winning

every day…

Of course, once the fund gets big enough

Hire enough physics PhDs to have SOME alpha.

Then inertia is your friend.

(that was another fallacy):

You need to have good performance to raise money.

Another fallacy (on a smaller scale):

Ride your winners, dump your losers

Ride your winners, dump your losers

Infinitely many backtests show that mean reversion is quite noticeable in the market.

Which means that this is exactly the wrong thing to do.

Thank you for your attention!

Thanks to Quantopian for organizing!