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by Anthony XieJune 26th, 2018

Half-way into 2018, it’s become clear that we’ve strayed far away from the jubilant exuberance that characterized the peak months of December and January.

Total Market Capitalization from Jan. 1, 2018 to June 30, 2018 aka Mr. Bones’ Wild Ride

Since then, every attempt to reascend the peak has been thwarted, sending both traders and HODLers alike tumbling downwards.

The question on everybody’s mind right now is:

Just how far down are we going to go?

Many people are predicting that this is merely a small bump in the road, and we will be at an all-time-high in December. But for the sake of conjecture, let’s consider this:

Monte Carlo simulations are used to model the probability of different outcomes. Rather than simply making a prediction based on a single average number, Monte Carlo simulations employ randomness to generate millions of different outcomes.

We then inspect the distribution of all the total outcomes and use that information to aid our decision-making.

For every day between January 1, 2018 and June 28, 2018, we’re going to calculate the daily % change for the entire cryptocurrency market cap. Then we create a density plot that spans daily returns for the entire time period.

The area under the curve of a density function represents the probability of getting a value between a range of x values. If the width for a particular section is tiny, the height can be much higher than 1 without violating any the rules of probability. I.e. 4 * 0.01 is just 4%.

The plot forms a probability distribution from which we can draw random samples. The tallest area of the graph is going to be the most likely result when we do a random draw.

We make sure to replace the data after taking each sample. This way, the first sample is independent of the second one. In financial terms, this means the next price movement is “conditionally independent” from past price movements.

**Running 1 Simulation**

There are 184 days between June 30, 2018 and the end of the calendar year. So for one simulation, we take 184 random draws from our probability distribution.

1 Total Market Cap Simulation over 184 Days

**Running 100,000 Simulations**

To get a better sense of the entire realm of possibility, we’re going to run 100,000 simulations. Each line is its own prediction of the market from today until the end of the year.

100,000 Total Market Cap Simulations over 184 Days

*Simulated daily returns are normally distributed*

In total, 18.4 million days are simulated. As we would expect, the probability distribution for all randomly drawn daily returns looks nearly identical to the one we started with. And at a glance, it looks like our returns are normally distributed.

*Simulated Final Market Cap Follows a Lognormal Distribution*

However, our end of the year market cap distribution does not look normally distributed at all.

*Why is this the case if our returns are normally distributed?*

Think about it this way. An asset price cannot be negative. If you lose 10%, you are left with less to lose next time. On the flip side, continuous positive returns have a compounding effect so a few simulations will have extremely large final values.

These conditions creates a lognormal distribution with a sharp drop off to the left of the mean, and a highly skewed long tail to the right of the mean.

**Total Market Cap Predictions at the End of the Year**

If we take all 100,000 predictions for total market cap at the end of the year, we can compute some interesting summary statistics.

The median total market cap from our simulations is **$99 billion**. This is 58% down from today. Talk about a bear market!

Because our standard deviation is so high, there is actually a massive $2 trillion difference between our minimum observed value, and the maximum observed value. Obviously the likelihood of such extreme outcomes are very low. But our Monte Carlo simulations tell us that these outcomes are still possible.

**Value at Risk**

While the simulations force us to consider a wide range of possibilities, it can definitely be overwhelming.

We can frame the way we consider risk by looking at the “Value at Risk” (VaR) metric. VaR estimates how much capital we may lose, at a given probability, over a set period of time.

For example, 10% of the simulated outcomes finish with a total market cap value under $34.1 billion.

If the market continues to decline in the same manner, we can say there is a 90% chance our value at risk will be between $0 and $212 billion (today’s market cap — $34.1 billion). Conversely, there’s a 10% chance that a loss will occur outside of our VaR threshold.

Another way we can model our Monte Carlo simulations is by using Geometric Brownian Motion (GBM). This is a very popular way of modelling asset prices in finance. In fact, the well-known Black–Scholes model assumes that asset prices follow GBM.

Whoah! Scary math stuff.

It sounds fancy, but GBM is actually quite simple. There are two major components to understand.

One is the “drift”, which is a long-term trend in a positive or negative direction. Second are ‘random shocks’ that are added or subtracted from the drift.

The asset price follows a series of steps, where each step is a drift plus or minus a random shock (the random shock is a function of the asset’s standard deviation)

The drift and standard deviation are both derived from an asset’s historical performance.

*Between Jan. 1, 2018 and June 30, 2018:*

- Daily mean return was -0.27%
- Daily standard deviation was 6.1%

We’re going to determine the magnitude of the random shock by multiplying the standard deviation with a random draw from a standard normal distribution. Most random shocks are small, but some can be very large.

Normal distribution with mean of 0, and a standard deviation of 1

**Running 1 Simulation**

There are 184 days between June 30, 2018 and the end of the calendar year. We’re going to apply our GBM formula to every previous day, in order to get the value for the next day.

1 Total Market Cap Simulation over 184 Days

**Running 100,000 Simulations**

100,000 Total Market Cap Simulations over 184 Days

*Returns are normally distributed*

Again, all 18.4 million daily return simulations are normally distributed.

*Predicted Market Cap Follows a Lognormal Distribution*

But our total market cap predictions follows a lognormal distribution.

**Total Market Cap Predictions at the End of the Year**

The summary statistics for all 100,000 simulations are as follows:

The median total market cap from our simulations that use GBM is $69 billion. This is 71% down from today. Ouch!

**Value at Risk**

10 % of our total simulated outcomes finish with a total market cap equal or under $24.5 billion.

Therefore according to our model, we can say there is a 90% chance our value at risk will be between $0 and $222 billion (today’s market cap — $24.5 billion).

I hope this article was informative and sheds some light on what we could expect from a prolonged bear market.

While Monte Carlo methods are great for showing us a large range of disparate outcomes, the simulations are heavily influenced by their initial assumptions.

I simplified this analysis to make it an easier foray for readers. A 6 month period is too short to be certain we’ve seen the variance because in the real world, returns tend to follow fat-tail distributions.

This article should be taken as a light-hearted exploration of the Monte Carlo method rather than a serious prediction on EoY conditions. I’ll happily admit that at the end of the day, I’m just as bad at predicting the overall market as anyone else.

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