22 March, 2022
Investing; Decision-making

If you haven't read my article on expected value, I recommend starting with that. Also, h/t to Nicholas Yoder for the charts used here.

The ideas in this article have been game-changing for me. Whilst they're not rocket science, I wouldn't be learning this bleary-eyed.

Let's begin...

The basic model of expected value is that you should take bets with positive EV.

But this misleading. It misses important nuance.

We must also consider how much money we have ('bankroll') and how many bets we get to make, amongst other things.

In other words, investments with the highest EV don't necessarily deserve the largest allocations of capital.

This seems counterintuitive, so let's consider some examples.

## Example 1: Positive EV bet with 99% chance of loss.

Suppose you can invest \$1k for a 1% chance of winning \$110K.

This gives an expected value of \$100.

Even though this is positive EV, you wouldn't allocate much money if you only had a single bet.

Why?

Because 99% of the time you still lose money!

The lesson?

Avoid taking positive EV bets with low probability of success if you only get to bet once.

Let's dive into the second example where stuff really starts to heat up.

## Example 2: 50/50 odds with 6% gain vs 5% loss

This time, there’s a 50/50 chance of gain or loss, with a gain of 6% and a loss of only 5%.

Let's consider you have \$100 of initial capital so you get +\$6 for a win and -\$5 for a loss, giving an EV of +\$1.

You can choose to make your bet as large or small as you like (i.e. use leverage) up to the possibility of total loss. In other words, you can bet up to 20x of \$100.

While the odds are in your favour, how much of your initial capital should you bet?

The trick is to balance the competing forces of betting more to achieve greater profit and betting less to limit the chance of going bankrupt.

Let's look at the two extremes.

If you bet 20x your capital, you stand to make \$120 if you win. But you go bankrupt on your first bet if you lose. Whilst this strategy maximises profit, 50% of the time you go bankrupt straight away.

On the other hand, if you bet 0.2x your capital, your returns are lower but even if you lose, you can keep playing until the long-run law of large numbers hold (i.e. rewards converge to expected value).

Somewhere between these extremes, there should be an optimal proportion of total bankroll to bet, such that long-term wealth is maximised.

If we consider a simple example of two successive bets, one win and one loss, we see an interesting result.

At small bet sizes, the profit grows with leverage. But as leverage increases, the profit turns negative.

2x leverage earns less than 1.5x leverage.

>3.5x leverage actually loses money.

## Example 3: 52% chance of gain with equal payoffs

Consider a card game with equal payoffs and a win/loss probability of 52% vs 48%. We start with \$100.

By now, you should know that we're not going to bet \$100 on this hand.

So what proportion of our hand should we bet?

We can simulate different bet sizes over 100 trials and see the results:

If there's one thing that's clear from this graph, it's the wealth destroying effects of big bets.

The results are even clearer on a logarithmic scale:

This leads to counterintuitive realisation: given a profitable opportunity, doing more of a good thing results in a worse long-term outcome.

In other words, betting large amounts on a positive expected value bet is not necessarily more profitable.

# How should you size your bets?

The Kelly Criterion tells us the theoretically optimal proportion of one's net worth to bet.

The formula is surprisingly simple. Check out this article if you want more details.

On the left, f* is the percentage of our total wealth we should bet.

On the right,p is the probability of a win, q is the probability of a loss, and b is the odds (i.e ratio of amount we stand to win to the amount we stand to lose).

Applying this to example 2, Kelly says to bet 8.33% of total wealth.

Applying this to example 3, Kelly says to bet 4% of total wealth.

But here's the catch. Kelly represents the limit for the range of rational bets.

Betting even one penny more than Kelly would bring increased risk, increased variance and decreased profit.

Betting anywhere near the Kelly-optimal amount is irrational by most standards.

Therefore, Kelly is not the goal, but rather the boundary.

It’s often wiser to bet a fraction of the Kelly criterion.

Doing so results in large decreases in variance with only small decreases in profit. Smaller bets improve risk-adjusted returns.

For example:

• Betting 50% of Kelly returns 75% of the Kelly-optimal profit with only 1/4th of the variance
• Betting 30% of Kelly returns 51% of the Kelly-optimal profit with only 1/11th of the variance
• For money managers, slow and steady wins the race. 0.10x-0.15x of the Kelly-optimal bet size is a good rule

# Conclusion

Whether investing in startups, trading the markets, or making business or career decisions, understanding bet sizing is foundational.

Whilst it's a useful first step to be an expected-value bible-basher, EV becomes 10x more useful when you understand bet sizing.

A trader with mediocre strategy and a great risk model will become fairly successful. A trader with a great strategy and a mediocre risk model will become bankrupt.