2768: Definition of e

2023-04-28T19:57:50Z

Explain XKCD:

{{comic
| number = 2768
| date = April 26, 2023
| title = Definition of e
| image = definition_of_e_2x.png
| imagesize = 571x186px
| noexpand = true
| titletext = Yeah, my math teacher back in high school set up the system to try to teach us something or other, but the 100% rate was unbelievably good, so I engineered a hostile takeover of his bank and now use it to make extra cash on the side.
}}

There is going to be server maintenance [[User:Explain XKCD|Explain XKCD]] ([[User talk:Explain XKCD|talk]]) 19:57, 28 April 2023 (UTC)

==Explanation==
{{incomplete|Created by 2.718 BANKERS - Please change this comment when editing this page. Do NOT delete this tag too soon.}}
In this comic the teacher [[Miss Lenhart]] is asked by the student [[Hairy]] to explain what the constant ''e'' actually means.

The mathematical constant ''{{w|e (mathematical constant)|e}}'' is known as Euler's number. It is typically demonstrated in terms of compound interest. Here, Miss Lenhart seems to be setting up such an example, but in a typical Lenhart style she is actually asking her student to deposit money.

The constant ''e'' can be described {{w|E (mathematical constant)#Compound interest|in the context of compound interest}}. For a bank account that pays interest at a rate of 100% per year, and that interest is paid ''n'' times a year and compounded, then a $1 deposit will grow to $1 * (1 + 100%/n)^n after a year. As ''n'' approaches infinity (continuous compounding), the amount approaches ''e'' dollars. In the comic, minutely compounding is used as an approximation of continuous compounding; here ''n'' = 365 * 24 * 60 = 525,600, and the resulting amount would be $2.718279, less than one part per million different from that of a straight multiplication by ''e'' (which is 2.7182818…).

As such, one would expect Miss Lenhart to say in the last panel something like "you'll have ''e'' dollars in a year". It is not clear if Miss Lenhart sees the growth of the deposited amount as answer enough to explain ''e'' or if she's just charging $1 for answering the question of what ''e'' is. The supposed interest rate the teacher can earn off this deposit, alone, is so high that the $1 principal will grow to over $22,000 in ten years, $485 million in twenty years, or $10.6 trillion in thirty years.

In the title text, a {{w|Takeover#Hostile|hostile takeover}} is an acquisition of a company against its management's wishes, by simply buying up its shares from its shareholders. A bank offering accounts with an {{w|APY}} of 172% is certain to go bankrupt almost immediately, making it a very bad investment. Banks earn money by lending at a higher rate than they pay on deposits, but it is illegal to charge such high interest rates on loans, and no one would take them anyway. Therefore the bank will lose huge amounts of money on deposits while earning essentially no revenue. The off-comic speaker is effectively buying out the bank in order to drain it of its own funds, which is both illegal and financially pointless. Alternatively, their plan may be to buy 51% of the stock, then attempt to extract a majority of the bank's reserve funds through huge high-interest deposits, which is still not profitable, since banks hold only a small fraction of deposits in reserve, and their market capitalizations (the cost of buying all the stock) are much higher than their total reserves. Even if for some reason this bank had a very high reserve ratio, and this tactic could somehow be profitable, it would still be illegal, effectively robbing the other 49% ownership of its equity through deliberately bad management.

In the title text, Randall may have also just confused a couple concepts. A bank offering a 100% rate (assuming somehow sustainably) would be an incredibly good investment, and a rational actor would shovel as much money as possible into an account at this bank. Randall may have simply misused the term "hostile takeover," which would not yield any of the benefits of the 100% rate, as mentioned above, when he really meant to colloquially describe a scenario in which one would aggressively exploit the bank's 100% rate for one's own benefit. (A perhaps unintuitive aspect about banks that might have tripped up Randall is that "assets" in other contexts become liabilities for banks and vice versa. So customer deposits become liabilities upon which a bank would have to pay such an incredibly high rate, and loans, which are traditionally considered liabilities, are assets from which banks derive income.)

The teacher who inadvertently sparked this action, male, was clearly not Miss Lenhart, who may be better at providing more memorable (if somewhat non-standard) lessons. And, as the speaker cannot even recall what the point was of the original mathematical example, it is possible that they have insufficient understanding of the numbers involved and why their attempt to profit will turn out to be ultimately illusory. A similar lack of successful education in the subjects of business and/or law could also likely come back to bite them, sooner or later.

==Transcript==
:[Hairy is seated behind a classroom desk with his hand raised asking the teacher Miss Lenhart a question. She is standing in front of him with a board behind her. Beneath the board there are ledge with writing tools on it (markers or chalk).]
:Hairy: Can you explain what the constant ''e'' actually ''means?''
:Miss Lenhart: Sure.

:[Zoom in on Miss Lenhart's upper half, as she raises one hand palm up.]
:Miss Lenhart: I have a bank account that pays 100% annual interest, compounded every minute.

:[In a frame-less, and very slim panel, Miss Lenhart is shown holding a hand up in a fist.]
:Miss Lenhart: If you deposit $1 now,

:[Miss Lenhart now has both arms down as she continues to address the off-panel Hairy.]
:Miss Lenhart: I will answer your question.

{{comic discussion}}

[[Category:Comics with lowercase text]]
[[Category:Comics featuring Hairy]]
[[Category:Comics featuring Miss Lenhart]]
[[Category:Math]]

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2768: Definition of e