When investing (long term) the most important part of the analysis is seeing the potential and outlook of an asset. This is what everyone always talks about. But there is an important second part: Risk management. And very few people really talk about it beyond the very simple 'diversify' mentality.

The mathematical basis: Probability Distribution Functions (PDF's)

A PDF is a function that categorises events by their respective probabilities of occurring. For a simple dice, the pdf is

or simply a 1/6 chance for any side to come up when rolling.

Form a pdf we can compute useful statistics. For example the expected value is the sum over all outcomes weighed by their propability: 1/6 * 1 + 1/6 * 2 + 1/6 * 3 + 1/6 * 4 + 1/6 * 5 + 1/6 * 6 = 3.5, or whenever you roll a die, the mean is 3.5.

Another example is the variance, telling us about the randomness of the event. It is computed as a sum of all the squared differences from the mean, weighted by their probability:
1/6 * (1 -3.5)^2 + 1/6 * (2 - 3.5)^2 + 1/6 * (3-3.5)^2 + 1/6 * (4-3.5)^2 + 1/6 * (5-3.5)^2 + 1/6 * (6-3.5)^2 = 2.917
Which tells us that most (68%) of the rolls will lie within a range of 2.917^(1/2) = 1.5 from the mean or within [2,3,4,5] with the far outliers 1 and 6.

This is only the simplest case of independant random variables and it can get far more complex that this. Especially when investing in multiple assets that are correlated. But we will leave this for later.

When we talk about the price of an asset and not dices, the outcome is not a set of categories but a continuous spectrum. The only difference is that we replace sums by integrations and the pdf is no longer a table, but a function g(x), where g is the probability and x the outcome. We then have:

What most people naively do is only estimating the mean and if that is higher than the current price of btc, they will conclude that buying now is a good choice. But this neglects all randomness and the associated costs .

What is the investment goal

The simple goal is of course to make money. And we expect to make money as soon as the mean is higher than the price. But is that really enough? Lets make an experiment. Suppose I run a lottery and promise to pay five dollar when you roll a 6 and the price to play is one dollar.

The pdf is 5/6 : -1 (loose the entry fee) and 1/6 : 4 (win 5 dollar minus fee). The mean is -1/6 dollar So we expect to loose one sixth of a dollar every time we play. Its a bad deal.

Now assume I pay 7 dollar as the price. The mean is 1/6 and positive!

Finally assume I promise to pay you 1 billion $ but your chance to win is only 1/ (1 billion - 1). The mean is 1 /(1 billion - 1) so it positive and you expect to make a (tiny) gain. But the chance of actually winning is so small that by playing you likely loose all your money and have to stop playing long before you win because you are bankrupt.

This game thus ends with a very high chance of you loosing everything versus a tiny chance of getting more money than you will ever need. Even though the mean is positive, this is not a game that should be played (unless you have significantly more than 1 billion dollar, but then why care wasting your time to win a few dollars).

The question is how do we analyse risk in games so that we find reasonable answers that are better than just trusting the mean.

Utility functions

The solution to this problem is called utility function. This function states your quality of life as a function of how much money you have.

There are a few steps in creating your personal utility function. First, the more money you have the less of an impact it has. If you are poor, money is very important but when you are rich it only buys you useless stuff. This is the diminishing return of money. Secondly the utility function takes into account your personal goals.
As a very simple choice one might use u(x) = ln(x/s), where s is an amount of money that is neither negligible, nor life changing. But this does only serve as an example and should not be used for any real analysis.

Instead of looking at the mean, we know compute the expected utility, computes as the integration over u(x)g(x). This measure takes into account the risk of losses and weights it with the chance of gains in a non-linear way.

Lets go back at the two examples. Assume that we have 1000 dollar and our utility function is ln(x/100). We play with a bet of 100$. The first game is 5/6: 900, 1/6 : 1600. The mean is 1016.67$ and higher than our starting funds. The utility before is ln(10) = 2.30, after paying it is 5/6 * ln(9) + 1/6 * ln(16) = 2.29 slightly lower than initially. Even though we expect to make money, the likely pain of loosing 100$ dollar outweighs the chance to win 600$, but we do not need much more to accept the deal.

This only becomes worse for the game that is extremely risky. You loose real money for the fantasy of being rich. The logarithm kills the win of a few billions and the utility is close to ln(9) = 2.20 which is equivalent to a guaranteed loss of 100$.

The opposite can also be true. Suppose you have a debt with the mafia and need 10.000$ by tomorrow but you only have 1000$. Your utility function will now be close to a step function around 10.000$, meaning that you will suddenly favour risky games and if you have no better idea you should really go to the casino to play.

Don't follow traders

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Utility functions make a huge difference for investing. They specify how much risk is right for you and that will differ greatly from person to person. The only way to make good decisions is knowing your personal utility function. Don't use the log above from the example, it is just a toy model that will not hold in real scenarios. Instead think what you would/would not do given the money. How big is the impact on your life?

People that already are successful traders have a very different financial situation than most of us. They will be looking for very different trades. The chances are high that what they do is not optimal for you.