another simpler example would be if Bitcoin went from 1 to 1.2.

if i understand your formula, you would say Bitcoin went up 120%. that's clearly wrong as it only went up 20%.

Indeed, 20% is correct here.

The issue with your method (6% loss + 1400% gain = 1406%) is that it assumes financial yields can be added. This is false. Consider a stock which appreciates 20%, and then appreciates 20% more. How much has the stock appreciated?

Based on your calculations, it would be 40%. However, this is not entirely correct—one can see that the stock actually appreciated 44%. This may seem counterintuitive, but we will consider some other examples to make this more clear.

**Example 1**: Imagine a stock

**A**, priced at 1

BTC per share.

**A** appreciates 20% on Monday, but then depreciates 20% on Tuesday. How much has

**A** appreciated or depreciated?

Based on a naïve addition, the answer might be:

yield = 20% − 20%

= 0%

However, if we take this one step at a time:

initial price = 1 XBT

Monday close = 1 XBT + 0.2 XBT = 1.2 XBT

Tuesday close = 1.2 XBT − 0.24 XBT = 0.96 XBT

∴ yield = −4% !!!

**Example 2**: Imagine a stock

**B**, priced at 1

BTC per share.

**B** depreciates 20% on Monday, but then appreciates 20% on Tuesday. How much has

**B** appreciated or depreciated?

Based on a naïve addition, the answer might be:

yield = −20% + 20%

= 0%

However, if we take this one step at a time:

initial price = 1 XBT

Monday close = 1 XBT − 0.2 XBT = 0.8 XBT

Tuesday close = 0.8 XBT + 0.16 XBT = 0.96 XBT

∴ yield = −4% !!!

From the above two examples, we can see that the method of calculating the yield is not additive. However, it seems to have a commutative property. We will now revisit the original goal of calculating Bitcoin's yield compared to gold.

We must understand that by comparing two commodities with no direct exchange rate, we need to introduce an intermediate commodity—in an efficient market, what we pick doesn't matter. We will choose the US dollar here.

Bitcoin, as you stated, has gone up by 1400% since the founding of this topic. Gold has gone down by 6%. These values correspond to yields of 1400% and −6%. Our goal is to calculate the

*difference* between yields. However, we have just shown that this difference is not simply additive. So we will need to break up the problem again.

To simplify this problem, we will define a Standard Bitcoin as the value of a bitcoin, in USD, at the time the thread started. Our symbol for this will be SXBT. Similarly, define SXAU as the standard value of a gram of gold. We then know the constancy, but need not calculate directly, the Standard Ratio:

r = SXBT/SXAU

By the same process as the equations in our examples.

initial price of XBT = r XAU

after +ve XBT yield = r XAU + 1400%×r XAU = 1.5r XAU

after −ve XAU yield = [1/(1−6%)]×1.5r XAU ≈ 1.596r XAU

∴ yield = 1496%

So your error was because the ratios could not simply be added. Instead, they have to be normalized and then multiplied. We will now come up with a general formula to express this.

First, we will define a normalized yield as follows:

Yn = ln(yield + 1)

It turns out that normalized yields are addable. This is because a Y

_{n} of +q is exactly cancelled out by a Y

_{n} of −q (proof left as an exercise for the reader). So the remaining difficulty is converting from a normalized yield back to a regular yield:

Yn = ln(yield + 1)

e^Yn = yield + 1

yield = e^Yn − 1

Completing the formula:

total yield = e^[ln(XBT yield + 1) − ln(XAU yield + 1)] − 1

= e^ln(XBT yield + 1)×e^[−ln(XAU yield + 1)] − 1

= (XBT yield + 1)/(XAU yield + 1) − 1

Substituting our values (1400% & −6%), we have:

total yield = (1400% + 1)/(−6% + 1) − 1

= 15/0.94 − 1

≈ 15.96 − 1

= 1496%

If you need any more clarification, I'm glad to help.