The exact "rule" for N% interest is N log(2) / log (1 + N/100), which has taylor series 100 log(2) + log(2)/2 N + O(N^2) ≈ 69.315 + 0.34657 N - 0.0005776 N^2 + ...
For N approaching 0, the exact "rule" becomes the "rule of 100 log 2"; but for larger N increases slightly; the "rule of 72" is exactly correct for ~7.84687% interest, and for 15% interest it only gets as far as a "rule of 74.4".
That said, the power series gives us a way to get a significantly more accurate result: Divide the annual percentage interest rate into 832 months, then add 4 months. For any interest rate between 1% and 40%, this result will be accurate to within 3 days.
(Aside: I love how useful text is a medium. In 5 minutes I can put in a quick insight to improve a decade-old article.)
72 is better (mentally) than 70 or 74 because it has more divisors.
Most of the time when I see the "Rule of 72", it's in Excel spreadsheets. I passed on investing in an infrastructure project by changing a Rule of 72 calculation to:
YearsToDoubling = ln(2) / ln (1 + Rate)
When you're in a goddamned fancy calculator, you do the full form calculation because what the hell. I mean, I don't understand why you'd be deriving NPV from "years to doubling", but you can sure as hell get it exact.
(Also, as someone who's never actually invested, do people really judge investments on that number rather than internal rate of return, payback period, etc etc?)
Internal rate of return (IRR) is the rate at which the net present value (NPV) of a set of cash flows is zero. NPV is, in essence, a scaled IRR. If you have a budget of capital to deploy, NPV is preferred. For example, if you have a $100MM budget with a cost of capital equal to zero and two investment options: a $10MM project yielding 20% and a $100MM project yielding 10%, and you cannot purchase parts, only wholes (see the appeal of securitisation?), then IRR would guide you to the former while NPV to the latter. When dealing with securities or any other sufficiently quantised investment, IRR becomes a good enough approximation.
Your assertion about NPV being useful when you have a budget confused me. If you have no budget then NPV is useful. You just do every possible NPV-positive project.
If you do have a budget, as in your example, you could not pick between those two projects without considering some missing information (the length of time). You can't just pick the highest IRR project, or the largest project. The NPV of your larger project with a slightly lower yield could be lower than that of the smaller, higher yielding project. Consider if the larger one had a length of 1 month, and couldn't be repeated, whereas the smaller one would take 5 years. Or you could flip it around. The information you gave is not enough to choose, regardless of whether or not you're budget-constrained.
Given that it made the difference between a profitable and unprofitable investment, I'm guessing that whoever was selling the investment just kept fiddling with the payoff metrics until they found a way to make it look attractive.
There's a group doing that: http://schooold.com/. They have a financial curriculum that's now part of the Kitty Andersen Youth Science Center.
Strange, so it's not taught in USA? I always assumed compound interest to be basics and taught universally. I did learn about them probably in 7th or 8th grade.
Also: It is impossible to understand the solution to a problem you don't have: http://www.jerf.org/iri/post/2943
The major obstacle is that it is much more difficult to teach thinking than it is to teach facts. So it's harder to find, train, support, and retain teachers who can do that well.
"Common Core" math (the stuff you might see mocked on Facebook) is trying to move toward teaching kids how to think mathematically, rather than memorizing equations and formats. But it's hard going (again, see the mockery on Facebook), and may not actually the right direction.
Compound interest should be taught in terms of examples people will likely use. So when they first get a credit card offer at a certain %APR, or a chance to invest, they immediately pull out law of 72 and do some quick estimath.
It is not explicitly called out in the Common Core standard, but several of the examples in the common core are interest rate problems.
Here's an example from the California Standard, search for 'interest'.
(Actually, that's exactly what we do, and when it fails, people complain about "teaching to the test" and suggest dropping the tests instead of the systematic failure to achieve core objectives of the educational infrastructure.)
Credit Card debt is really the only thing that most people will interact with that has sufficiently high interest rates to be worth worrying about and like I said elsewhere the principle on CC debt is going to be having constant significant swings which makes analysis entirely dependent on spending habits, not the interest.
It is sort of like complaining "but it is / is not in the spec!" And expecting to be absolved of any problems with the product pushed out.
People who don't rack up interest on credit cards freeload off those who do.
Edit: today's SMBC is relevant. http://www.smbc-comics.com/index.php?id=4150
Instead of large chunks of their income going to credit card companies (via interest), it could instead be spread out among more stores.
For me, the "free" service of a credit card is not worth my neighbors being saddled with debt.
edit: That cartoon bothers me more the longer I think about it; I know it's supposed to be humorous, but does the author really think that lottery tickets are driving the economy?
While that seems true it's messier than that. In your scenario people have to postpone consumption until they accumulate the cost of goods which would have a huge negative effect on the overall economy. The entire concept of interest rates is to smooth out this issue, rewarding those who can postpone consumption.
No. The joke in most SMBC comics involves taking some faulty logic and driving it to absurd conclusions.
Well, Wall Street is pretty much a giant casino, so it's not too much of a stretch to say that gambling defines a large portion of our economy.
Fine. But that means you'll pay an average of 2-3% more for items.
>Instead of large chunks of their income going to credit card companies (via interest), it could instead be spread out among more stores.
Yes. They would gain under a new system. People who are responsible now would lose under that system. It's a transfer , with credit card companies taking a cut.
I don't think better education about how badly it harms them would help much. We see present upside much more strongly than long-term future downside (see: procrastination), even if we are familiar with the nature of that downside.
Emergency funds probably would.
You also need to take into account the bank's cost of funds, the cost of administering the account, the cost of fraud, cost of disputes (do you ever do a chargeback?)
The marginal cost of servicing another person when they already have a consumer base of people with debt is very small (close to zero). I'm trying to say it's still profitable, but not where they make the bulk of the revenue.
Also I remember spending around a month learning about interest and finance-related math somewhere around 8th grade, so at least in some us schools it was being taught over a decade ago.
Maybe you should be directing your attention to those who actively exploit compound interest to take advantage of others?
I didn't write that they couldn't understand it. I wrote that it isn't being taught, so they don't understand it.
You could probably learn many things you don't already know easily, however I don't think "there are many things you shouldn't be doing" because you haven't thought to learn them. For example: empathy.
Maybe it isnt being taught universally or well enough, or maybe it just needs to be reinforced more often.
I realize self-selection bias in that it did work for me, but I went to a decent school and still had both wonderful and terrible math teachers.
Math to me is somewhat of a fractal, and if a teacher or textbook can't effectively answer "Why?" in addition to teaching blind computation then a good outcome is unlikely.
Is there someone or a group perhaps that decides who are the top few percent intellectually, maybe a fraternity who then initiate other members who can then come on HN and declare with confidence they are a part of this special group and no validation is required. Does one require to be a coder or maybe just a HN reader?
This kind of hubris of a self appointed intellectual elite is seen far too often on HN to be comfortable and apart from promoting small minded bigotry sets a dangerous kind of thinking letting these self appointed groups jump to flawed and silly conclusions about others human beings.
Billions of human beings pass exams testing basic maths like compound interest and much much more every day. Its not an achievement of any kind. There are thousands of professions in the world of which software engineering is just one.
Being a software engineer does not make you magically better than other human beings in any way. Is a civil engineer or a doctor allowed to think HN is full of idiots because they dont have a clue about their fields, or that they are an intellectual elite because of their specialised knowledge in their chosen field? How come this kind of thinking is rife here? How may engineers an doctors have you heard gloating about reading trade mags?
The ability to educate oneself is a privilege of one's enviroment, background and upbringing, the ability to do well in one specific area could be a measure of one's interests provided one has the freedom and means to follow it. None of these are a measure of one's intelligence. Individuals could go on to distinguish themselves in their respective fields and then could be part of a group of say nobel winners or some specific measure.
This 20% seems completely arbitary and self serving that puts you in a group simply because of your specific interests or chosen profession. This is not in the least scientific or valid and there appears to be no useful purpose than hubris. The bigger problem is why the great need to feel better than others and clutch at shadows? This is how bigotry works.
Surely every human being has the ability to learn and be adept at what they do given the right conditions and this is a problem the world grapples with, to provide these conditions.
An individual struggling with compound interest is an opportunity for those who know, to try to teach them in a way they can understand, not an opportunity to deride and run them down as idiots. That's juvenile and mean spirited.
There are people around here who believe that their interest in computer science is one of the things that makes them smarter than other people, but that is not a belief I am defending. In fact, I believe that is a small minority of people here. Most of the posts I saw in this thread were criticizing the education system that leaves many people without an understanding of compound interest, not calling those people idiots.
Most people are comfortable with the idea that some people are at the bottom of the intelligence heap. It is scientifically verifiable that some individuals are unable to remember new information or identify patterns. Why shouldn't this be true at the top of the heap as well? Based on my life experiences, I believe that my intelligence is significantly above average, although I realize that I have been aided by fortunate circumstances. However, I am not strongly invested in the idea: I am more interested in being happy, in being a disciplined worker, and in having strong relationships with people than I am in being a smart person. Why are you so invested in the idea that no one is smarter than anyone else?
High schools should have a mandatory personal finance class.
That said "compound interest" is and any sort of interest is kind of futile if, like a credit card, the principle is constantly changing, it's also a bit hard to scare people into frugality by telling them if they put these steaks on credit they'll have to pay $12 dollars more later. The numbers don't get scary until well into "they already knew they shouldn't do it" territory.
In case you have not yet read it, please do.
It is a good book, I recommend it.
It tells you that if market grows by 7% annually, it's going to move more volume in ten years than in all years before, combined.
For example, of country's electricity consumption grows by 7% YoY, it's going to consume more electricity in (any) ten years than it ever consumed before that, grand total.
Just in case this hasn't been said elsewhere, the rule is more precise for numbers whose factor into 72 is closer to it. So medium numbers, for lack of a way of saying it. Not 50, not 1, but 12, 8, etc.
"The Most IMPORTANT Video You'll Ever See"
(it's an old youtube obviously click-bait-style title, but it's a video of a lecture worth watching, see the description):
By the way, betterexplained.com is where I finally, after memorizing my way through just enough calculus to get a BS in comp sci, began to understand calculus. Seeing how the area of a circle was derived was wild - I'd easily memorized all that stuff, but never understood. We wait way too long to begin teaching calculus concepts.
I think of it as akin to the fluidity of a conversation between two native speakers versus a conversation with a translator in the room.
I can do calculus in my head, but I struggle with arithmetic. My colleagues who can do fast mental math but can't do any algebra to save their lives end up looking much more competent in meetings.