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Today we're going to start with a simple problem

that many of you may have already encountered.

For example, when you got a student loan

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

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

of pricing an annuity.

An annuity is a financial instrument

that pays a fixed amount of money

every year for some number of years,

and it has a value associated with it.

For example, with a student loan,

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

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

going to send them a check.

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

an n-year $m-dollar payment annuity,

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

and it lasts for n years.

Now, usually n is finite, but not always,

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

when it's infinite.

But this includes home mortgages,

where you pay every month for 30 years.

Megabucks, the lottery.

They don't actually give you the million dollars.

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

and call it a million bucks.

Retirement plans.

You pay in every year, and then you

get some big lump sum later.

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

look at later when we do probability.

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,

a real disaster.

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.

So it's pretty important to understand

how much this is worth.

What is this instrument-- a piece

of paper that says it will pay you

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

is that worth today?

For example, say I gave you a choice--

the Megabucks choice-- $50,000 a year

for 20 years or a million dollars today.

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

A few of you.

How many would prefer the million bucks right up front?

Much better.

OK.

Always better to have the cash in hand,

because there's things like inflation--

pretty low now-- interest.

You can put the money in the bank

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.

So that was pretty clear, but what

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

for 20 years?

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

A few.

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?

How many want half a million today?

A lot of people like the cash.

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

it's a disaster on Wall Street.

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

How many want, instead of a half a million

up front, 50 grand a year for 20 years?

All right, now we're about halfway.

All right, well that's pretty good.

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

or at least one way of estimating that.

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

$1 today is worth in a year.

And to do that, we make an assumption,

and the assumption is that there's

a fixed-- we'll call it an interest rate.

It's sort of the devaluation of the money per year.

And we're going to call it p.

Later we'll plug in values for p,

but you can think of it as like 6%, 1%.

You know, it's the money, if you put money in a bank,

they'll give you some percent back every year.

And, of course, the fact that different people

have different ideas of what this would be,

allows people to make money on Wall Street.

As we'll see, a slight difference in p

can make big differences in what the annuity is worth.

So for example, $1 today is going to equal 1 plus p

dollars in one year.

Similarly, $1 today-- how much is that

going to be worth in two years?

Say that p stays fixed, the same over all time.

One plus p squared, because every year you

multiply what you got by 1 plus p, because that's

the interest you're getting.

All right, we'll think of it in terms of interest.

In three years, $1 today is worth 1 plus p cubed in

three years and so forth.

All right.

Now, what we really care about is what's $1, or m dollars,

worth today if you're getting it next year?

So we need to sort of flip this back the other way.

So what is $1 in a year worth today in terms of p?

So if I'm going to be paid $1 in a year,

what would be the equivalent amount to be paid today?

One over 1 plus p, because what's happening here

is, as you go forward in a year, you just multiply by 1 plus p.

So 1 over 1 plus p turns into $1 in a year-- being

paid in a year.

All right.

What is $1 a year in two years worth today?

One over 1 plus p squared.

So $1 in two years is worth this much today.

Well, now we can use this to go figure

out the current value of that annuity.

We just figure out what every payment is worth today and then

add it up.

So we'll put the payments over here,

and we'll compute the current value

of every payment on this side.

So with the annuity, the way we've

set it up is it pays n dollars at the start of every year,

so the first payment is now.

So the first of the n payments is now,

and since it's being paid now, that's worth m dollars.

There's no devaluation.

The next payment is m dollars in one year,

and so that's going to be worth m over 1 plus p today.

And the next payment is m dollars in two years.

That's worth m over 1 plus p squared,

and we keep on going until the last payment.

It's the n-th payment, so it's m dollars in n minus 1 years.

And so that's going to be worth m over 1 plus p

to the n minus 1.

All right.

So we can compute the current value of all those payments,

then the annuity is computed-- the value is computed just

by adding these up, of all the current values.

So the total current value is the sum i equals 0 to n minus 1

of m over 1 plus p to the i.

And that is the total current value.

That's what you should pay today for the annuity.

Any questions?

What we did here?

All right.

Well, of course, what we'd like is a closed form expression

here.

Something that's simple so we could actually

get a feel without having to add up all those terms,

and that's not hard to get.

In fact, let's put this sum in a form

that might be more familiar.

This equals-- we'll pull the m out in front--

and let's use x to be 1 over 1 plus p to the i.

And so x equals 1 over 1 plus p, and I wrote it this way

because this might be familiar.

Does everybody remember that from-- I think

it was the second recitation?

Anybody remember the formula?

What this evaluates to?

The sum of x to the i, where i goes from 0 to n minus 1?

Remember that?

One minus x to the n.

Remember 1 minus x.

In the second recitation, I think,

we proved that this equals that.

What was the proof technique we used?

Induction.

OK?

So, in fact, there's a theorem here.

For all n bigger and equal to 1 and x not equal to 1,

we proved the sum from i equals 0 to n minus 1 x to the i

equals 1 minus x to the n over 1 minus x.

And so this is a nice, closed form.

No sum any more, just that, which is nice.

Now, induction proved it was the right answer.

Once you knew it-- we gave it to you-- using induction

to prove that theorem wasn't hard.

What we're going to look at doing this week and next week

is figuring out how to figure out this

was the answer in the first place.

Methods for doing that-- to evaluate the sum-- and there's

a lot of ways that you can do that particular sum.

Probably the easiest is known as the perturbation method.

This sometimes works.

Certainly with sums like that, it often works.

The idea is as follows.

We're trying to compute the sum S, which is 1

plus x plus x squared plus x to the n minus 1,

and what we're going to do is perturb it a little bit

and then subtract to get big cancellation.

In this case, it's pretty simple.

We multiply the sum by x to get x plus x squared plus-- I've

defined S to be that, x times S-- well, I

get x plus x squared and so forth, up to x to the n,

and now I can subtract one from the other and almost everything

cancels.

So I get 1 minus x times S equals 1.

These cancel, cancel, cancel.

Minus x to the n.

And therefore S equals 1 minus x to the n over 1 minus x.

So that's a vague method.

This gets used all the time in applied mathematics,

and they call it the perturbation method.

Take your sum, wiggle it around a little bit,

get something that looks close, subtract it,

everything cancels, life is nice,

and all of a sudden you've figured out the answer.

So getting back to our annuity problem,

we can plug that formula back in here.

So the value of the annuity is m times 1 minus x to the n

over 1 minus x.

We'll plug in x equals 1 over 1 plus p,

and we get m 1 minus 1 over 1 plus p

to the n over 1 minus 1 over 1 plus p, just plugging in.

And now to simplify this, I'll multiply the top and bottom

by 1 plus p, and I'll get 1 plus p minus 1 on the bottom.

Just gives me a p on the bottom.

I have a 1 plus p on the top minus 1 over 1

plus p to the n minus 1, all over p.

So now we have a formula-- closed form expression