Linear Approximations

The shortest distance
Lines have two parts [y = mx+b]: they start somewhere [b] and with unwavering speed [m] they get somewhere. Pretty simple.

Since I am simple, lines describe me pretty nicely too. I have 50 dollars and each redbull vodka costs 10 dollars. 5 redbull vodkas later, I am numb to my net worth.

Relationships can suffer over distance
To estimate how much money you would make investing at 8% a year with 100 dollars, see that you make 8 dollars in your first year. The amount of money that you have changed by only 8% so far, so figuring that you will make about 8 dollars the next year on the same investment is only 8% off. After two years, I have made about $16.

After a greater number of years, the compounding of error would become a problem. After all, what is the difference after 10 years. ($100+$80) - [100*1.08^10]. The error gets worse and worse and we lose love for the line.

Over that line
A threshold is a point where everything changes. I was a jello crazed child. One thing was very clear to me: if there is anything better than one cup of jello, it is two cups of jello (rational by any economic sense). One way or another, I got enough money to satisfy my dreams. I ate an orgy of jello. Unlike the linear promise of joy, I threw up -- violently. The formula for maximizing pleasure can be tricky.

Call me
Some things can be described well linearly. Take for example how much you gain or lose on a stock when it goes up by a dollar, but what happens when you add a threshold to the payoff? A call is similar to holding a stock contract except that it only entitles you to money if the stock is beyond a certain threshold like 100 bucks (when the contract ends). Of course, you could sell your call contract before the end of the contracts life. Instead of getting dollar for dollar returns, you have to value your call with an equation that is much more difficult (at the top of the post). All due to a little threshold. After all, if I can hook up with an 18 yr old, why can't I hook up with a 17 yr old? Thresholding can be difficult to understand.

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Igor 'I only post when you make me' Schmertzler