# The most important equation or why Bitcoin has to average 30% return a year to break even with the S&P 500

In the old year, I wrote a piece about ARKK, Bitcoin and co. and showed how dangerous it is for long-term investment success to experience deep drawdowns like the ones we saw since early 2022. There, I said that the long-term result of investments doesn’t depend on returns but on the ratio between returns and volatility. Apparently, there were some readers who were surprised by this, so with apologies to all the people who know this by heart, let me spend today discussing the difference between arithmetic and geometric returns and why understanding this relationship is crucial for long-term investment success.

Most people think about their investments in arithmetic returns, i.e. the return measured in one period. For example, investment A might have a return of 10% in the first year and 10% in the second year. If you invest $100 in that asset, you will have an average return over two years of 10% and your $100 will have grown to $110 after the first year and $121 ($110 x 1.1) in the second year.

Now assume investment B has a return of -50% in the first year and 70% in the second year. The average annual return is still 10% but if you invest $100 in asset B, you will end up with $50 after the first year and $85 after the second.

The difference between investment A and investment B is that investment B has volatile returns and that costs you money over time because you have to dig yourself out of a hole (aka a bear market) from time to time.

Mathematically, the relationship between the average one period return (the arithmetic return) and the compound return (the geometric return) is given by the following equation:

You see that the higher the volatility of an investment, the more it acts as a drag on the geometric return. This is why diversification is so powerful. Take a look at the chart below, where I have calculated the average annual arithmetic return over the last five years for the S&P 500 and every single stock in the index.

If you had randomly picked one stock in the S&P 500 your average annual return would have been 15.7% compared to 11.9% for the S&P 500. That’s great. So, you would have outperformed the S&P 500 by a wide margin.

Nope.

S&P 500 vs. the average stock over the last five years

Source: Liberum, Bloomberg

Your compound return over five years would have been 8.4% compared to 9.2% for the S&P 500. How can a stock that outperforms by 3.8% per year over five years actually underperform? The reason is that the volatility of the S&P 500 was 24% vs. 37% for the average stock. Whenever your stock went down, it went down more than the S&P 500 and it took you longer to get out of that hole. Meanwhile, while the single stock investment was still climbing out of its hole, the S&P 500 investment was already making money again.

Some investors tend to complain that by investing in a well-diversified portfolio, they never have high returns, and they think they miss out on the big winners (like cryptocurrencies or the ARKK Innovation Fund in recent years). Yes, they do, but what they gain in return for lower returns is much much lower volatility.

If you do a little math, you can use the above formula to calculate the extra return you need to compensate for higher volatility. If you have an investment like a single stock, the volatility of that stock is typically 1.5x higher than the volatility of the stock market. That means that your arithmetic return has to be 1.125x the return of the index to break even over the long run.

In comparison, the ARKK Innovation Fund over the last five years had a volatility that was roughly twice as high as the volatility of the S&P 500. That means that the fund needed to achieve returns that are roughly 1.5x as high as that of the S&P 500 just to have the same compound return (some 18.4% to be precise). Unfortunately, the average annual arithmetic return of the ARKK fund was 8.3% and that means that even investors who have been with the fund through good and bad times are now worse off than with a boring S&P 500 fund.

And if we look at cryptocurrencies like Bitcoin it gets really interesting. The annual volatility of Bitcoin is about 3.8x the volatility of the S&P 500. That means that the average annual return of Bitcoin needs to be about 30% per year to make it break even with the S&P 500 on a compound return basis. Bitcoin managed to do that over the last five years, with an average annual return of 40%. But do you really think Bitcoin is going to average more than 30% per year over the next five years? Or more generally, do you think that Bitcoin is going to average returns in the long run that are 3x the return of the US stock market? If you, like me think this is too high a return hurdle then you should stay away from Bitcoin and invest in a broadly diversified stock portfolio instead.

A great explanation of a really important & often poorly understood issue

I don't think volatility is the issue. If you buy a stock at $10 and sell it at $20, you double your money. The path it takes to get there, whether that be gradual and linear, or volatile and erratic, makes absolutely no difference, if the total time is the same. It's the two end points that matter, averaging the points along the way is a pointless exercise.

The fact that you buy a stock at $10, it drops to $5, then goes back to $10, and that apparently gives you an average 25% return, just means that averages are meaningless when applied to the situation.

Another example. A King has a billion dollars, he has 1000 subjects who have $1 each. Apparently the kingdom has an average wealth per capita of $1,000,001. Nice kingdom, on average, everybody is a millionaire, right? Mathematically true, but unhelpful in describing the actual situation.