The Investing Edge: The Win / Pay Odds Gap

Investing is too often looked at using a handful of academic models. Successful investing involves thinking about the investment process in as many different ways as possible. This article takes a look at investing using alternative views.

From Betting to Investing

Many of the ideas used by the investment community are adopted from the horse track and casino betting communities. Much of the failure that has dogged the investment community is due to rocket scientist PhDs misunderstanding the successful models of plebeian punters. The use of betting as an example is not an endorsement, just history.

To understand how the securities markets work you have to look no further than the horse bookies. Bookies take bets from the bettors. This is the first point that the public begins to misunderstand how betting, and therefore investing, works.

There are two potential misunderstandings:

Assuming that each horse has a uniform probability of winning, i.e. they are all just as likely to win.
Assuming that the bookie offers one to one payout odds, i.e. pays $1 for each $1 that is bet.

Grasping the significance of these statements is the key to successful investing. The bookie, equivalent to the investment bank or broker, will always make money. Always. They do this because they do not set payout odds depending on which horse they think will win, they set payout odds based on how people bet.

Understanding the Odds

To understand the rest of this article you need to understand some notation from probability. I’ll assume we’re at the racetrack.

Outcome odds, sometimes called winning odds and which I’ll simply call odds, are a way of representing the probability of winning. Odds of one to three, written 1:3, means that there is a 1/4 or 25% probability of winning. The arithmetic is that you take the first number and divide it by the sum of both numbers. Similarly odds of 1:9 means 1/10 or 10% chance of winning. In reality nobody actually knows the real odds, but they can quite often have a good idea of what they are.

Related are the payout odds. Payout odds are written the same way, 3:1 means that if you win you get $3 for each $1 you bet or invested, with the original dollar being returned. On the other hand, payout odds of 1:3 means that you win $1 for every $3 you bet. The relationship between outcome odds and payout odds is not that simple. Consider a two-horse race between Shadowfax, who has odds of 1:3 of winning, and Mr. Ed, who therefore has odds of 3:1 of winning. Many people would assume that the bookie should pay 3:1 on Shadowfax and 1:3 on Mr. Ed. This would quickly lead to the bookie going bankrupt.

Making a Market

The issue to realise is that it does not matter what the probabilities are for winning. What matters is how people bet, because if half the money bets that Shadowfax wins, then it would be a very bad idea to offer 3:1 payout odds on Shadowfax. If total bets are $100 in this scenario, then a bookie will pay out to the winners their $50 bet plus $150 in payoff, whilst the bookie will keep the $50 from the losing bets on Mr. Ed. In detail:

Betting income on Mr. Ed = $50
Betting income on Shadowfax = $50
Payout on winner Shadowfax
Return of original bet = -$50
Payout = 3 x -$50 = -$150
Total = -$200
Payout on loser Mr. Ed = $0
Net = $50 + $50 – $200 = -$100

This is a net loss. Not good.

What a bookie wants to do is to match his assets (cash from bets) with his contingent liabilities (payouts to winning bettors). This is no different from a bank matching its assets and liabilities, a function so vital that they have a committee of senior executives named ALCO to oversee this function that is managed by the treasury.

Using the previous example of half the money betting on each horse then the correct odds for the bookie to offer are 1:1 on each horse. That way, regardless of which horse wins, the bookie will not lose money. As another example, if 80% of the money bets on Mr. Ed and 20% on Shadowfax, then the bookies would offer payout odds of 1:4 on Mr. Ed. and 4:1 on Shadowfax. That way, whichever wins, the bookie loses no money.

Scenario if Shadowfax wins:

Betting income on Mr. Ed = $80
Betting income on Shadowfax = $20
Payout on winner Shadowfax
Return of original bet = -$20
Payout = 4 x -$20 = -$80
Total = -$100
Payout on loser Mr. Ed = $0
Net = $80 + $20 – $100 = 0

Scenario if Mr. Ed wins:

Betting income on Mr. Ed = $80
Betting income on Shadowfax = $20
Payout on winner Mr. Ed
Return of original bet = -$80
Payout = (1/4) x -$80 = -$20
Total = -$100
Payout on loser Shadowfax = $0
Net = $80 + $20 – $100 = 0

This is exactly how brokers, market makers and investment banks trade, although they shave the odds a bit to make money. An example of shaving the odds would be in the case of half the money is bet on each horse instead of offering 1:1 payout odds the bookie offers 2:1 on each horse. In such a case the bookie guarantees himself $25 in profit. Of course, the shaving is usually much smaller.

A financial services example. The treasury of a bank uses the currency markets. They get calls from other banks and corporate treasuries looking to buy and sell currency, such as US dollars (USD) for Japanese yen (JPY). The bank treasury will quote an exchange rate at which they are willing to sell USD, say 105 JPY per USD , and a rate at which they are willing to buy dollars, say 100 JPY per USD. They come up with these numbers not based on where they think exchange rates are going but solely based on where the current supply and demand for USD versus JPY is.

This methodology is also how the intelligent investor makes money.

The Winning / Payout Odds Gap

Here is the key to successful investing. It is not to maximise your return. In the horse betting example this would be equivalent to betting on the horse with the highest payout odds. Neither is it to minimise your risk. In the horse betting example this would be equivalent to betting on the horse with lowest probability of losing. The key is to pick the horse with the biggest gap between payoff odds and winning odds.

Let’s say Shadowfax has winning odds of 1:3, which is a 25% chance of winning, and Mr. Ed has winning odds of 3:1, which is a 75% chance of winning. If all bettors knew these odds then they would bet in a way that would lead to the bookie offering payout odds on Mr. Ed of 1:3, because if the bookie offered better odds, say 1:4, then bettors would increase their bets on Mr. Ed, forcing the bookie to bring down the pay out odds. Equivalently, if the bookie offered worse payout odds over the outcome odds, say 1:2, then bettors would reduce their bets forcing the bookie to increase the payout odds. I’m ignoring the shaving of odds here.

But what happens when not everyone is good at calculating the winning odds? Then they will bet in a manner that makes the bookie offer payout odds that are not in line with the winning odds. This means that the bettors who can understand that the payoff odds are much higher than they will have an edge, even if they do not know the actual winning odds.

The Investor’s Edge in Action

Now that we have introduced the idea of an winning / payout odds gap, let’s look at how it can be used. Consider two friends Amy and Bob. Bob learnt how to play poker believing that it will help him learn bluffing and betting strategies for a life in investing. Amy focussed on her academics, including probability and statistics.

Bob, now an experienced poker player, asks Amy if she would like to play this game. Amy, understanding the rules but never having played, accepts. At first blush things might look grim for Amy.

Poker usually requires that the players put up the same amount of money, called the stake. This is a mistake since Bob is a much better player. Amy, a neophyte poker player but an avid mathematician points out that it would only be fair to have odds involved, e.g. $1 from Amy equals $100 from Bob. This is simply setting payout odds.

Bob, who knows that he is at least 1,000 times better than Amy, accepts the 1:100 payout odds on him. This is his fatal mistake.

On the day of the game Amy seems to be in the bad position of having a 100:1 payoff but only a 1:1000 chance of winning. This is an improvement from the original 1:1 payoff with a 1:1000 outcome odds, but it is still a negative gap.

Here is the magic, the magic that is so difficult to develop, but when you have it, it seems to simple and you will win, not every time, not even the majority of the time, but enough times to trigger the payoff odds which compensates for your other losses and more. The magic is to now improve the winning odds. The target is to get it down to at least the payoff odds, i.e. from 1:1000 to 1:100 or better. But how?

In this scenario, it is to bet everything on every hand. You see, if you bet everything on a single hand then all skill goes out the window. Bluffing is useless. Calculating probabilities is useless. Counting cards is useless. The key is that it is useless for both players. In this case the winning odds become even , i.e. 1:1.

Most people would belittle forcing a 1:1 winning strategy. It is not sexy. The genius of Amy is not that she selected a strategy that gave her 1:1 odds of winning. The genius was that she understood that it was better than 1:1000 odds of winning. When you add the fact that she had negotiated 100:1 payoff odds, Amy has now transformed a massive negative gap to a large positive gap and, well, there you go. In effect, Amy is betting $1 for a 50% chance of winning $100!

Hedge Fund Trading: Losing Client Money

It is important to understand the potentially counter-intuitive lesson here: selecting a strategy of betting everything seems like the height of insanity, but it can make sense if it means altering your outcome odds to a point that creates a positive gap to your payout odds.

The second lesson is that moving payout odds is critically important. Normally, an investor cannot do this by negotiating, as Amy did. But they can do it using leverage, the easiest way to move payout odds. Remember you can leverage down as well as up and please make sure that you know what you are doing.

The startling result of applying this analysis to investing is that if you do not know how to invest then the best strategy in this case is to invest everything in a single trade. This is not advice, it is a logical conclusion.

This revelation helps explain the many proprietary traders and hedge funds that seem to lose spectacular amounts of money. They know full well that they have no idea how to invest and so they bet big. It is easier when it is other peoples’ money. If they get lucky then they will become rich and famous. If not, they go get another job or new clients.

The Risk of Ruin: Don’t Lose All Your Money

To recap, we have covered when to invest, when the outcome / payout odds are positive, and worked through a completely fictional example. The next step is examining how much to invest and this is related to the likelihood of losing all your money, frequently called the risk of ruin.

Consider the following example: two people, Bob and Amy, are similar in terms of retirement needs, say $100 each. They also have both received a monetary gift, Bob got $1 and Amy $2. Every day they each make an investment decision that at the end of the day will result in either a profit of $1 with probability 60% or a loss of $1 with probability 40%.

The question is: What is the probability that they will reach their target of $100, or conversely, that they will lose everything? The math is a little involved, but can be found on the internet under the name “Gambler’s Ruin.” The interesting result is that Bob has a 67% of losing all his money, or being ruined, and Amy has only a 44% chance of ruin.

Why are these results interesting? It shows that even when there is a good chance of profit (60%), that starting with too little of an initial investment has an extremely high probability of ruin. Incidentally, this is one of the reasons most start ups go bust, there is not enough cash kept in reserve, it is all poured into the business on day one.

What if Charlie joins this game, but he starts with $5? Well, his risk of ruin is a low 13%.

Investment Position Management and The Kelly Criterion

These ideas are mathematically complex, but they are critical to understanding why so many investment strategies seem to make sense but end up failing. Basically, even when a strategy can result in a profit, even a large one, the path to this profit can, and usually does, involve a number of losses. The investor who has not grasped this idea will lose their shirt as they invest far too large an amount given their balance sheets.

So the question becomes how much should a person invest? The most famous method to manage investment size, by far, is the Kelly Criterion. Again, the math is a little convoluted, but it basically formalises the earlier discussion on how successful investors make their investment decisions based the gap between the payout odds and the winning odds.

Continuing the example from earlier, consider a two-horse race whereby the first horse, Shadowfax, has 1:3 odds of winning and the second horse Mr. Ed has 3:1 win odds. If the payout odds offered on Shadowfax were 4:1 (as opposed to the 3:1 that the winning probabilities dictate) and therefore those on Mr. Ed are 1:4 then the discussion showed that the correct bet to make is on Shadowfax. What is left is to decide what percentage of your bankroll should be wagered.

Kelly looked at this problem and decided that the best way to tackle it is to consider a betting strategy that maximised the growth rate of winnings. After some mathematical black magic, the gap is basically defined as a ratio of the payoff odds to the outcome odds. From there Kelly derives the optimal amount to bet is a ratio of the odds gap divided by the payout odds. For the example above this comes out to 6.25% of the bankroll.

If you are confused think of it in these terms: Finding an investment edge is just the beginning of the battle. Developing strategies to determine how much to invest on each trade and how to avoid the risk of ruin even on a positive expected return are just as important. Unfortunately there is too much emphasis on the first issue and not enough understanding of the latter two issues.

The Kelly Criterion is useful in terms of understanding how to think about things but it is deeply flawed. As an example, if there is an investment that leads to a profit of $1 for each $1 invested with a probability of 95% and a loss of $1 for each $1 invested 5% of the time it looks like a great investment. The Kelly Criterion proscribes investing 90% of your cash each time. The problem with this is that if you lose on the first round, it will take 8 more investment rounds, all of which need to win, to make back your capital. Now imagine that these investments have a one year horizon. Not good.

The Impact of Volatility on Investment Returns

I introduced the Kelly Criterion as the most famous formula for answering the question of how much to invest and then subsequently showed it to be too volatile. Now I will delve into why and also took at an effective strategy.

The heart of the problem with the Kelly Criterion is the following behavior investment return behaviour: a 100% profit is reversed by a 50% loss and a 50% loss requires a 100% profit to make up for it. What this means is that if you make a -20% loss and then make +20% return you are still going to have a net loss of -4%.

The investment professionals will point out that they use the logarithm of returns. It is not necessary to understand what that means other than that this implies that investments are more likely to lose money than to make money. A simple example of why the logarithm of returns is an academic fiction considers two cases of a $100 investment either losing $50 or returning $50. In the case of the loss, the normal return percentage would be -50%. The log return would be -69%. On the other hand a $50 profit would be a normal return of +50% but a log return of only +41%.

If you would like to believe that losing $50 creates a different return percentage than earning a profit of $50, then you need read no further. This section is not for you.

So what does it mean to an investor if you need larger positive returns to offset smaller losses, e.g. +100% vs -50%? The first immediate issue is that if prices, and hence returns, move dramatically and/or often (this is called high volatility) then clearly a strategy based on pure luck for its profit and loss will result in a net loss. Returning +50% on a $100 initial investment followed by -50% will result in a net position of $75, or a -25% loss! Do it again, returning +50% then losing -50% results in a net position of $56!

This behaviour of total return to a symmetrical (equal up or down) increase in volatility is called being short gamma. You do not need to understand the mathematics behind this just the behaviour: short gamma is similar to selling insurance or selling options. You get paid an amount up front and every time something happens, you lose money.

Investment Lessons

Investment lesson number one is that, in simple terms, investing is not zero sum. An average investor, e.g. one who gets it right half the time, will not break even but will lose money. The more volatile the markets, the greater the amount he will lose even though he is right half the time.

Investment lesson number two, therefore, is that if you are investing in conventional securities such as equities or real estate, find something that is low volatility because you are naturally short gamma.

Investment lesson number two and a half is that the famous maxim “cut your losses short and let your winners run” should be re-examined in terms of the structural changes to the portfolio that it engenders, i.e. it increases the short gamma position. There is something to be said that the structural behaviour of an investment portfolio is just as important as position management.

Investment lesson number three is to deploy a strategy to mitigate this inherent challenge. A simple one is popularly known as the portfolio rebalancing strategy. In this, the investments are bought and sold to bring them back to the original percentages within the portfolio.

For example consider an investor who wants to invest half of his money in Apple and half in Google. After one year, Apple is up 10% and Google has doubled. The resulting weighting would be Apple 35% of the portfolio and Google 65%. The investor then sells shares in Google and buys shares in Apple to bring the portfolio back to 50% each.

This strategy became famous based on the idea that maintaining a constant mix of securities or assets made good sense. The real reason it does well is that it offsets the short gamma position described above. Buy selling more volatile stocks and reinvesting in lower volatility stocks this strategy mimics buying a put option and hence reverses the short gamma position.

For the option savvy reader, this is a delta one long gamma strategy. This is the best way to sell (not short) excess volatility in a long only market.

This article is not meant to be a primer on investing or give investment advice but to provide insight into an institutional investment process.

You can also find the article at: https://musings.al-binali.com/2017/09/investing-edge/