Topic: Extreme exact values for the number of Bitcoins Tesla bought and sold in Q1 (Read 116 times)

RESULTS:
Since I hate it when people wait with the interesting imformation until the very end, I will give the results first, explaing how I have calculated them afterwards:

Tesla bought 46,726.50 Bitcoin for an average price of $32,101.70 per Bitcoin in January and
Tesla sold 4,544.75 BTC for an average price of $59,849.27 per Bitcoin late in March.



EXPLANATION:
Well, we got to know the exact amounts of money Tesla has spent for their Bitcoin purchases in January as well as the exact amount of gains they made with their sells in March.
However, Tesla has never revealed the exact amount of Bitcoins they bought and sold, neither has Tesla revealed their average entry and exit prices.
There were some guesses of these numbers floding around, which were all based on trying to estimate the average entry and/or exit price by looking at the Bitcoin chart and considering the the likely time spans of these buys and sells, but the different results of these approximations were ranging in an intervall of a lenght of a few thound Bitcoin up to even more than 10k Bitcoin.


With the numbers given in their latest report for the SEC (see https://ir.tesla.com/_flysystem/s3/sec/000095017021000046/tsla-20210331-gen.pdf), Tesla has essentially revealed the exact numbers (up to rounding errors), though not directly. But the numbers Telsa has provided here make it possible to exactly calculate the missing numbers of their Bitcoin trades.
So here we go:


First, let's define the unkown variables that we want to know:

x₁: Number of Bitcoins Telsa bought in January
x₂: Number of Bitcoins Tesla sold in March
p₁: Average buying price
p₂: Average selling price
pᵢₚ: Impairment price



Next, let's introduce abbreviations for the numbers given in the report and
an abbreviation for the Bitcoin price at the end of March (Daily Close of March, 31th of "Bitcoin / U.S. Dollar Index, taken from tradingview.com)
c₁ = $1.50 billion, c₂ = $272 million,
c₃ = $1.33 billion, c₄ = $27 million,
c₅ = $128 million, c₆ = $2.48 billion
p_Q₁ = 58,793.16 USD/BTC



Then let's formulate the corresponding equations for the unkown variables by looking at what the report tells us:

(I) "During the three months ended March 31, 2021, we purchased an aggregate of $1.50 billion in digital assets, comprised solely of bitcoin." (see page 14):
x₁ * p₁ = c₁

(II) "In the first quarter of 2021, we also sold an aggregate $272 million in bitcoin." (see page 38):
x₂ * p₂ = c₂

(III) "As of March 31, 2021, the carrying value of our bitcoin held was $1.33 billion, which reflects cumulative impairments of $27 million." (see page 16):
(x₁ - x₂) * pᵢₚ = c₃

(IV) "During the three months ended March 31, 2021, we recorded $27 million of impairment losses on bitcoin." (see page 16)
x₁ * (p₁ - pᵢₚ) = c₄

(V) "We also realized gains of $128 million through sales during the three months ended March 31, 2021. Such gains are presented net of impairment losses in Restructuring and other in the consolidated statement of operations." (see page 16)
x₂ * (p₂ - pᵢₚ) = c₅

(VI) "Net of such sales, the fair market value of our bitcoin holdings as of March 31, 2021 was $2.48 billion." (see page 38)
(x₁ - x₂) * p_Q₁ = c₆



Finally, we need to solve the euqtions (I)-(VI). I have done it, but I don't want to go through all the mathematical details here.
However for those of you interested: Noting that we have five unkown variables and six equations and noting that the transformation x₃ := 1/p₁, x₄ := 1/p₂ and x₅ := 1/pᵢₚ will give us an overdetermined system of linear equations, the way to go is solving the normal equations to get the best least squares approximation.
The results are as follows:

x₁ ≈ 46,726.50 BTC
x₂ ≈ 4,544.75 BTC
p₁ ≈ 32,101.70 USD/BTC
p₂ ≈ 59,849.27 USD/BTC
pᵢₚ ≈ 31,538.14 USD/BTC