Name: Lec 12 | 麻省理工學院 6.042J 計算機科學數學，2010年秋季。 (Lec 12 | MIT 6.042J Mathematics for Computer Science, Fall 2010)
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Today we're going to start with a simple problem

that many of you may have already encountered.

or if your family took a loan out on your house,

you know, a home mortgage loan, and it's the problem

the value is the amount of money they gave you to pay MIT,

then later in life-- every year or every month-- you're

And you want to sort of equate those two things to find out,

are you getting enough money for the money

you're going to pay back monthly sometime in the future?

So let's define this, and there's a lot of variations

on annuities, but we'll start with one that's called

and it works by paying m dollars at the start of each year,

Now, usually n is finite, but not always,

and in a few minutes we'll talk about the case

They don't actually give you the million dollars.

They give you $50,000 per year for 20 years,

Life insurance benefits, you know, and so forth.

Now, if you go to Wall Street, this is a big deal.

A lot of the stuff that happens on Wall Street

involves annuities in one form or another,

packaging them all up and, in fact, we'll

It was how these things were packaged and sold

that led to the sub-prime mortgage disaster,

and we'll see how some confusion over independents, when

you look at random variables, led to the global recession,

Of course, some people understood how all that worked,

made hundreds of billions of dollars at the same time.

So it was sort of money went from one place to another.

m dollars at the beginning of each year for n years, what

How many people would prefer the $50,000 a year for 20 years?

How many would prefer the million bucks right up front?

or invest it and make some money hopefully.

So the million dollars today is a lot better,

which is why the State pays you 50 grand a year for 20 years.

It's better for them, and they call it a million bucks.

if I gave you this option-- 700 grand today or 50 grand a year

How many people want the cash upfront-- 700 grand only.

How many people want 50 grand a year for 20 years?

All right, we're almost-- that's pretty close to half way.

How about 500 grand today versus 50 grand a year for 20 years?

You know, it's this kind of a time, you have the recession,

You know, Street Wall Street didn't like the cash.

How many want, instead of a half a million

So we're going to find out what you should pay,

Now, to do that, we've got to figure out what

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Lec 12 | 麻省理工學院 6.042J 計算機科學數學，2010年秋季。 (Lec 12 | MIT 6.042J Mathematics for Computer Science, Fall 2010)

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