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According to the research discussed in this article, when the intensity of some sensory stimuli increases exponentially, our minds somehow unconsciously perceive the increase as linear. Our perception, in other words, is often logarithmic.

What I find most fascinating about this is that our conscious minds are not very good at thinking about exponential growth. My favorite example is that of the twin brothers who invest with annual gains of 12% each: one twin saves $2,000 from age 19 to 26 (a total of $16,000) and then stops saving; the other twin doesn't start saving until age 27 but then puts away $2,000 every year until age 65 (a total of $78,000). Do the math, and you will see that the savings of the second twin, growing at 12% every year, never catch up with those of the first one.

Our perception might often be logarithmic, and we might understand the math of exponentiation, but our brains are not very good at 'seeing' exponential growth!

Modern humans no more need to appreciate the economics of consistent 12% gains than they need to know how to fend off a polar bear, or how to date three supermodels on a modest budget--it just ain't gonna happen.

When I first learned about this investment problem (although, my version had 18-30 and 30-65), I worked out the math, and I think it held for as low as 2% interest. I could be remembering wrong though... Surely, someone's done a formal write up on this?

One way to simplify the problem is to figure out how much interest income the "early investor" is making at the moment the "late investor" kicks into gear. Obviously if Jane invests from age 16-20 and is now rolling $4,000/yr interest income back into the mix on her 21st birthday, John's yearly contribution of anything less than $4,000 ain't gonna cut it.

The Rule of 72 http://en.wikipedia.org/wiki/Rule_of_72 is a really handy tool the layperson can use to get a grip on how long it takes for compounding increases (which is most increases we deal with day to day--taxes, population, traffic) to result in doubling. It makes a much greater impact to say "this town's population will double in 10 years" than it does to talk about 7% growth rates.

> My favorite example is...

This exact thing caught me out a few weeks ago, it still boggles my mind. I started a pension recently and I was given examples of how much I would be worth if I started as soon as possible (my 21st birthday) or if I waited until I was 22, 23, 24... if my total monthly contributions are $2,000 and I start on my 21st birthday I will be worth hundreds of thousands of dollars more than if I started at 23, or 25, but my brain thinks that is insane and even though I know it's true, I still can't get over it. The difference between 21 and 23 is $160,000!

going even more on a tangent, it's really a great shame this sort of thing isn't taught in schools. It seems obvious now after being taught that savings can vary so much this way with so little time, but until I was explained it I had no idea. I figured waiting until 25 or 30 to start a pension made most sense!

Those estimates are often wrong though, because the idea of annual 12% returns is complete fiction. In fact the idea of any particular percentage annually is fiction. For instance, I only recently started a pension, however if you look at the stock market history (for simplicity), had I started in 2000, the net gain I could have expected was almost precisely zero, if not negative. If I could have found something that paid 12% from 2000 to 2012, sure, I would have missed out on a lot, but there in fact not only was no such beast (in the broad sense), there was also a lot of ways to experience negative gain.

This is one of my favorite visualizations ever: http://www.nytimes.com/interactive/2011/01/02/business/20110... Be sure to carefully read what it is actually visualizing; I've posted this a couple times and people often knee-jerk a reaction to it based on a complete misreading of the chart.

In the context of this discussion (and indeed as reflects how most people invest), wouldn't it be more relevant to see the average rate of return assuming regular (inflation or income growth adjusted) annual investments?

Exactly. Depending on volatility you could get a positive return investing a set amount in a stock regularly, even if the stock started and finished at the exact same price.

In addition some stocks pay dividends; reinvesting these can also improve rate of return beyond what would be expected from just looking at the stock price.

Interesting chart.

Strange, at one point it says "High inlfation led to negative returns". When money is worth less, then you get more money when you sell something. In this case your share of a company. So why would the returns of investing in the S&P be affected by inflation?

Because the nature of inflation is to devalue money. For example, let's say you put $10k under your bed to buy a car for your kid when they turn 18, and let's say inflation was at 2% for those 18 years.

Now the 18 years have speed buy and you go grab your very old stack of $10k from under the bed frame to celebrate your kid leaving the house. When you get to the car dealership, you find the equivalent car you could have purchased now costs $14,282. Your money has lost almost a third of it's value just by inflation.

Inflation affects the number that you use to value the worth of a company.

Say you buy one share at $10 of a company with a billion outstanding shares. That company is worth, on the market, $10 billion.

Let's say that next year, the company's share price is still $10. Let's also say that inflation in that period was 4%. The company is still worth $10 billion, but each dollar is worth 4% less. The company's total value has dropped.

Inflation didn't cause the total value to drop. The company's value dropped for other reasons (bad sales, company president went crazy, market jitters, whatever causes these things) and inflation simply moved the numbers a bit in the opposite direction.

In short: inflation doesn't affect your returns, but it affects the numbers you use to measure them.

You say "In short: inflation doesn't affect your returns". And thats what I said. But the linked article says the opposite: "High inlfation led to negative returns".

Inflation doesn't inflate everything at the same rate. In 1970, the DJIA was $809. 10 years later, it was $824. Inflation was more than 1%.

Starting early is only half the battle. The other half is the rate of return. If you start up a retirement account at age 17 with $2000 in it, but you just put it in a generic savings account, by age 37 you'll have . . . $2209.

People encouraging you to save often give unrealistically high rates of return. 12% is unrealistic for any investment vehicle I know of (though people sometimes say it for the stock market -- but they're crazy optimistic).

Here's a spreadsheet to show you the impact of both rate of return and time:


And of course, that doesn't account for the risk of the investment -- which, as a lot of people discovered in 2008, is a real, actual thing.

Indeed. These days, money in a savings account has a negative real rate of return.

That depends 100% on your timeframe. Start investing in March '09 and your money would have doubled in less than three and a half years.

"Savings account."

The contribution from your first year is earning interest for N years, so contributes (1 + R/100)^N (assuming fixed rate R, which is a fiction. That could be a lot of dosh...

You are not alone: www.youtube.com/watch?v=F-QA2rkpBSY

OTOH, sometimes our perception is multiplicative when it should be relative: charge me $100 extra on a house, I probably won't even notice; charge me $100 extra on a bar of chocolate, there's no way I'm going to buy it. But either way I lose $100!

You only buy a house once, you buy chocolate bars all the time. $100 on a chocolate bar every time compounds.

One-time coupon for $.50 off a chocolate bar: probably going to take the time to clip it. One-time coupon for $.50 off a laptop: probably won't bother.

You can find silly faults with individual examples all you want, but studies indicate that people have a tendency to think percentage-based about savings rather than absolute terms.

The $.50 off a chocolate bar may actually cost you money, since it will encourage you to buy the chocolate bar when you weren't planning on buying one at all. In fact worse than that, it may start a habbit of buying those chocolate bars.

>You can find silly faults with individual examples all you want, but studies indicate that people have a tendency to think percentage-based about savings rather than absolute terms.

Which sounds like the smart way to go about it.

It seems a little illogical in the chocolate vs laptop $.50 off voucher, but it too makes sense. The laptop voucher provides almost no value to you, considering you are about to spent $700 dollars on the thing anyway. $.50 could be just the gas to go to the store to buy it.

$.50 for the chocolate on the other hand, will be taken only by cash strapped people (or coupon maniacs).

I doubt people with $700 at the ready to buy a laptop would at the same time clip a $.50 coupon for a chocolate bar.

The $.50 is $.50 regardless of what the voucher is for. It provides exactly $.50 no matter what. Sure, maybe someone with $700 at the ready for a laptop won't be the kind to be clipping $.50 coupons for anything but the same tendency to consider percentages rather than absolute values remains regardless of actual numbers. People will pat themselves on the back for saving $10 on a $50 garment but won't bother going out of their way to save $100 on a $20000 car.

>People will pat themselves on the back for saving $10 on a $50 garment but won't bother going out of their way to save $100 on a $20000 car.

Which also makes sense. $100,000 to me is serious money. To Bill Gates it's small change. Money is a relative - percentage thing.

A $20,000 car is a purchase you make once in 5-10 years. Saving $100 means nothing in that context --you're already parting with 200 times the amount, so if the $100 amount meant something for you, you'd have gone for a cheaper, say $15000, car in the first place...

Savings of $10 (20%) on small items, like food or garments, on the other hand, pile up.

You spend more than $20,000 on small items per year than you do on cars...

I would clip it.

It's still a valid observation though. People don't drive to another store to save 10 dollars on a 500 dollar couch but they will drive all the way across town to save 10 cents a gallon on gas.

This isn't rational: they're clearly not buying the 100 gallons of gas for the gas-drive to make sense.

Give me 30 years and 4% interest on a loan to pay off the chocolate bar and I probably wouldn't notice either.

Dan Arielly's "Predictably Irrational" (http://www.amazon.com/Predictably-Irrational-Revised-Expande...) is a great book that covers such phenomenon.

When the intensity of some sensory stimuli increases exponentially, our minds somehow unconsciously perceive the increase as linear

By necessity, really. If you've ever spent much time with electronic sensors, you may discover as I did that seemingly half of the work is making nonlinear input linear, because nonlinear input is a mess to compute with.

SW and HW for digital imaging had/have to deal with this quite a bit.

I think your example illustrates the surprises of compound growth well, but with a realistic growth of 4.1% per year, twin A ends up with a pot of $88k and twin B ends up with $183k. So your example provides food for thought but probably shouldn't form the basis for a retirement strategy.

> What I find most fascinating about this is that our conscious minds are not very good at thinking about exponential growth.

I thought about it too, and come up with idea that some equivalent of Weber-Fechner law [1] might be working at it this level. So I have made a small write-up about this idea that you might find interesting [2].

[1] http://en.wikipedia.org/wiki/Weber%E2%80%93Fechner_law [2] http://xion.org.pl/2012/04/11/self-reinforcement-and-exponen...

What kind of investment brings 12% every year, for 40 years without the risk of losing all my money? With the investment options I have now (today), I'm better of consuming my money.

This is why the second twin should put in $5000 every year :D

on this topic - this is a video worth watching http://www.youtube.com/watch?v=F-QA2rkpBSY

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