Losing money on high-yield opportunities, most people are using the wrong method

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Money Machine or Death Trap?

If there's a button that, when pressed, has a 50% chance of reducing all your assets to 10% of their original value and a 50% chance of multiplying all your assets by 9, should you press it?

At first glance, this problem seems to be a classic evaluation of mathematical expectation, with a direct mathematical formula.

Suppose you have 10,000 yuan. After pressing the button, there's a 50% chance it becomes 1,000 yuan and a 50% chance it becomes 90,000 yuan. The mathematical expectation is: 0.1 * 0.5 + 9 * 0.5 = 45,500 yuan.

On average, each time you press the button, your wealth is expected to become 4.55 times the original amount. This seems to be a "money machine".

But if you pick up a calculator and do the math several times, you'll find that the "money machine" quickly turns into a "death trap".

Don't believe it? You can calculate it. Suppose you have 10,000 yuan. After pressing the button 6 times, with three times getting a 10% value and three times getting a 9 - fold increase, it becomes 7,290 yuan.

This is also a calculation problem. "Reducing to 10%" and "multiplying by 9" in each round is equivalent to a 10% reduction. The more times you press the button, the less money you'll have. After pressing the button 6 times, it's 7,290 yuan. After pressing it 14 times, half of your money is gone. After pressing it 100 times, you're left with only 51 yuan.

Why is this the case? Is the method of mathematical expectation in investment misleading?

Behind this problem lies the reason why many people keep losing money. The later you realize it, the more you'll lose. This article will discuss this topic.

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The Secret from Certain Loss to Big Profit

Using mathematical expectation to make a decision about this button is not wrong. It is indeed a "money machine", but your betting method is wrong.

The correct method should be as follows:

For example, divide the 10,000 yuan into 10 parts, with each part being 1,000 yuan. Then press the button once for each part - this is the key difference. Each part can only be pressed once, not more.

After pressing the button 10 times, we get 10 parts of money. Suppose our luck is average, with five parts getting a 9 - fold increase and five parts getting a 10% value. Now we indeed get:

0.9 * 5+0.01 * 5 = 45,500 yuan

The "mathematical expectation" doesn't deceive us. If used correctly, it's a "money machine".

This is actually a very classic financial decision - making problem. It touches on the conflict between two decision - making indicators we often use, the conflict between "expectation" and "compound growth rate". In general investment, such an extreme profit - loss ratio doesn't occur, and their differences are masked.

I emphasize that each part can only be pressed once. Why? Because the mathematical expectation calculates the average of all possible results when you press the button only once.

However, if you only press the button once, the result is uncertain. To eliminate this uncertainty, I divide the money into multiple parts. The more parts there are, the closer the result is to the mathematical expectation.

If I invest all the money at once and press the button N times, is there a way to calculate the mathematical expectation each time?

Yes, it's to take the logarithm (ln) and then calculate the expectation:

The calculation result is a negative number, - 0.0525.

It means that each time you press the button, the expected value loses 5.25%. After two times, it's 9%, which is exactly the same as the previous calculation result. If you keep pressing, you're almost doomed to go bankrupt.

Why take the logarithmic expectation? When operating on the same amount of money repeatedly, the next gain depends on the previous result. It becomes continuous compound interest after multiple repeated games, and the expectation of each time is the "per - time" growth rate. The change in assets is not addition but multiplication. To restore it, you need to take the logarithm (you need to recall some high - school math knowledge).

As long as the logarithmic expectation is positive, theoretically, you can go all - in, and there's a high probability of making money if you keep pressing the button.

To sum up, the mathematical expectation of the "money machine" only solves the problem of "whether you should press the button", while the logarithmic expectation solves the problem of "whether you'll go bankrupt" when going all - in.

So don't underestimate diversified investment. It can create returns out of thin air and change the return rate of an investment opportunity where "going all - in means certain death".

But if the "money machine" at the beginning is to be invested in batches, does it have to be 1/10 of the money each time? This kind of opportunity is usually limited, and of course, you hope to invest more money each time. So how much should you take out each time to press the button?

Many people must have thought of it. Isn't this problem the famous "Kelly Criterion"?

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Wrong Methods for the Right Opportunities

If calculated using the Kelly Criterion, for such a button, if you only bet 1/10 each time, the profit - making efficiency is too low, and you can't maximize the return.

The Kelly Criterion solves the problem of given the win rate and odds, how much principal should be bet each time to maximize the long - term return while avoiding the risk of bankruptcy?

The Kelly Criterion is as follows:

But this is a simplified formula for fixed profit - loss amounts. When dealing with the problem of "assets becoming a certain proportion", it also requires maximizing the logarithmic expectation. The process of taking the derivative is omitted. For the button at the beginning, the calculated optimal betting proportion each time is 49.3%. At this proportion, the wealth growth rate per round is 65.7%.

However, the mathematical conclusion is only theoretical. In reality, most people still can't bear the sharp asset fluctuations brought by the Kelly strategy. Think about it. Taking out nearly half of your money each time might be a bit risky. If you have three consecutive bad results, it'll be very uncomfortable. Or if you miscalculate the odds, you'll be in big trouble.

So most people use the "half - Kelly strategy", which is half of the Kelly number. In the previous example, it's 24.7%. Betting 1/4 each time, the wealth growth rate per round drops to 52.1%. The decrease is not large, but the fault - tolerance rate and asset drawdown increase significantly.

More conservative people will choose 1/4 of the Kelly number, and the return rate won't drop much.

You see, many people think that the most important thing in investment is to find high - return opportunities. The example at the beginning is actually one of the most classic conclusions of the Kelly theory. An investment opportunity can meet three seemingly contradictory conditions at the same time:

1. The arithmetic expectation is extremely high (4.55 times)

2. Going all - in means certain loss

3. Investing half of the money can lead to high - speed growth

In actual investment, high - return opportunities are not scarce. What's truly scarce is finding a betting method that has a positive expectation and won't eliminate you during the compounding process. Because the biggest risk in the market has never been making money slowly, but betting too much.

Repeat the important conclusion three times. The mathematical expectation determines whether an opportunity is worth participating in, but the Kelly Criterion determines the appropriate position size. Many investors fail at the second step - they find a good opportunity but use the wrong betting method.

This article is from the WeChat official account "lig0624" (ID: tongyipaocha), written by Thought Seal, and is published by 36Kr with authorization.