Simulation · 3 min
What volatility really means
Volatility sounds like a harmless number until you see how widely it spreads an outcome. Set the starting amount, the volatility, the expected return and the period: the tool rolls 200 price paths and shows you the median, the range and how many paths end below your starting amount.
What you set
Assumption: rough orders of magnitude to play with, not measured values for any particular period.
The default of 0 % is an assumption and not an opinion about the future: the model then works with volatility alone, without a direction.
200 paths in monthly steps. As long as you only move the sliders, the same random numbers stay in place, so that you see the difference your setting makes and not the difference chance makes.
Enter a starting amount above zero.
Across, the years from the start (y). Up the side, the value. Paths that rise above the top edge are cut off there.
Final value divided by the starting amount, share of the 200 paths in per cent. The four classes from the left: up to 0.5, then up to 1, up to 2 and more than 2.
Random paths from a model built on your assumptions. They are not a forecast and they describe no real price history.
Note: The paths are random numbers from a model built on your assumptions. That is not a forecast and not investment advice. The comparison values on the buttons are assumptions, not measured figures.
How the simulator calculates
Behind the curves sits a geometric Brownian motion, the standard model for price paths. The simulator works in monthly steps. Each monthly value produces the next one by the formula below, where dt is one twelfth of a year, mu the expected annual return you set, sigma the volatility and Z a standard normal random number:
| Step | What happens |
|---|---|
| Randomness | Z comes from the Box-Muller transform, which turns two uniform numbers into two normally distributed ones |
| Formula | new value equals old value times e to the power of ((mu minus sigma squared divided by 2) times dt plus sigma times the square root of dt times Z) |
| Repetition | 200 paths, each with 12 steps per year, up to 120 steps over 10 years |
| Evaluation | each month all 200 values are sorted, which gives the median and the 10th and 90th percentiles |
As long as you only pull the sliders, the same random numbers stay in place. That way you see what your setting changes rather than what chance happens to be doing. Only the "roll again" button draws 24,000 fresh random numbers. In the chart 40 of the 200 paths are drawn very faintly, because all 200 would have given nothing but a grey area. The headline figures and the distribution below always work with all 200 paths.
Why the median sits below the mean
Volatility is the standard deviation of the returns, scaled up to a year. Roughly speaking, 60 % of volatility means that in about two years out of three the annual result lands between minus 60 and plus 60 percentage points around the expected return. But because a price cannot fall below zero and is open ended upwards, the distribution of the final values is skewed. A loss of 50 % needs a gain of 100 % afterwards just to get back to level.
That is exactly what the drift correction makes visible. Set the expected return to 0 % and the volatility to 60 %, and the arithmetic mean of all paths after three years stays at the starting amount, while in the model the median sits at only about 58 % of it. The mean is pulled up by a few paths that ran very well, while the majority land below. That is why this tool shows the median in large type and not the average: the median describes the typical case.
Which figures are documented and which are assumptions
The comparison buttons for an equity index, gold, Bitcoin and small altcoins are assumptions made by this tool so that you have orders of magnitude to play with. They are not measured values for any particular period, and every volatility changes over the years. Documented, by contrast, are the price movements below, which show how large the swings have already been.
| Period or event | Documented figure | Source |
|---|---|---|
| Bitcoin, August 2015 to November 2021 | from 250 to 69,000 US dollars | Bank for International Settlements |
| Bitcoin and Ether over the course of 2022 | about 75 % fall in price | Bank for International Settlements |
| Crypto market, first half of 2022 | over 60 % fall in the market | European Securities and Markets Authority |
| The days around the insolvency of a large trading venue, November 2022 | over 20 % within a few days | Bank for International Settlements |
| Value lost across the crypto market in 2022 | over 1.8 trillion US dollars | Bank for International Settlements |
Where the model stops
The model assumes that volatility stays the same across the whole period and that the monthly returns are independent of each other and normally distributed. Neither holds for real crypto markets. In reality, calm and hectic phases cluster together, and extreme days happen far more often than a normal distribution allows. The collapse of a large stablecoin in May 2022 and the insolvency of a large trading venue in November 2022 were days of that kind. The model therefore tends to understate how bad things can get in the short run.
On top of that, the expected return is an input, not a finding. Type in 50 % and you get friendly curves to look at, without anything at all being established about the future. Fees, spreads, tax and the loss of your own keys sit in none of these figures either. The simulator answers one single question: how widely does a given volatility spread an outcome?
Frequently asked questions
What does 60 per cent of volatility mean in practice?
It is the standard deviation of the returns, scaled up to a year. Roughly speaking, in about two cases out of three the annual result lands in a band of 60 percentage points around the expected return. A third of cases land outside, and that is where the big swings happen, upwards as well as downwards.
Why does the median sit below the starting amount even though I set a return of 0 per cent?
Because price paths multiply. Minus 50 per cent and then plus 50 per cent do not give zero but minus 25 per cent. With high volatility that effect pulls the typical path down, while a few very good paths keep the mean up.
Is this a forecast for the Bitcoin price?
No. The simulator rolls paths from your assumptions. It knows no real price and no market data. It only shows how widely an outcome spreads at a given volatility. Every figure in it follows from your entries and says nothing about the future.
Why do the curves not change when I only move a slider?
The random numbers deliberately stay in place while you change parameters. That way you see the effect of the setting rather than the effect of a fresh roll. The roll again button draws 24,000 new random numbers and shows you another possible path.
What is meant by a fall of more than 50 per cent along the way?
For every path the tool remembers the highest value reached so far and measures how far the path then falls below it. The share counts all paths that lost more than half of their own peak at some point, even if they end up in profit.
Sources
- Crypto shocks and retail losses (BIS Bulletin Nr. 69, 20. Februar 2023)Bank for International Settlements, bulletin no. 69, 20 February 2023
- Crypto-assets and their risks for financial stability (TRV Risk Analysis, Oktober 2022)European Securities and Markets Authority, October 2022
- Decrypting financial stability risks in crypto-asset markets (Financial Stability Review, Mai 2022)European Central Bank, financial stability review, May 2022
Related tools
Crypto savings plan calculator
What a monthly instalment turns into under three price assumptions, with fees and total deposits side by side.
Portfolio Mixer
Five sliders for bitcoin, ethereum, large and small altcoins and stablecoins: risk score, spread and a look back at 2022.
Risk profile check
Eight questions on horizon, knowledge, liquidity and tolerance for losses, then your profile and a rule of thumb.
Further reading
BitcoinBitcoin volatility: why the price swings so widely
Bitcoin lost around 75 per cent in 2022. What volatility measures, which falls are documented, and what wide swings mean for investors.
InvestingCrypto share of a portfolio: criteria, not a percentage
No formula says how much crypto belongs in a portfolio: the criteria, measured correlations and the 5 per cent rule of thumb of a German consumer advice centre.
InvestingCrypto savings plan: what it can do and what it cannot
In a model calculation by the Bank for International Settlements buying 100 US dollars a month, over four fifths were at a loss by December 2022.