JAMES P. WESTON: Hi.

Welcome back to Finance for Nonfinance Professionals.

I'd like to work another example of discounted cash flow valuation

with you with sort of a practical application

to what are called annuities.

Annuities are a really important concept in finance.

And they're anything, really anything, with a series

of payments coming in the future that are at regular intervals.

So anything that has a cash flow that pays or gets

paid every month or every quarter or every year,

any regular series of cash flows, is what we often

call in finance some kind of annuity.

In a sense, mortgages are an annuity.

Regular deposits into a savings account.

Monthly payments on a mortgage.

Insurance premiums are kinds of annuities.

And a lot of pension plans make payments on an annuity basis.

So a lot of times when people have a pension plan and they retire on it,

the payment is a series of monthly payments.

In other words, a sort of like a substitute

paycheck that they get every month-- an annuity payment.

OK.

So let's talk about how you would put a value on different kinds of annuities.

We're going to do that by working a very simple problem.

You want to retire and maintain a monthly income

of $2,500 for the first 20 years-- for the next 20 years

let's say of your retirement.

A lot of times what you'll do is you'll purchase an annuity

with your retirement savings.

You purchase the annuity from the insurance company or the bank

and then they'll dish out regular series of payments

to you, just like a paycheck.

How much would you have to pay or what's a fair price for that annuity

if discount rates are around 4%?

And so, what we'll do is move over to the spreadsheet,

and work that problem together.

OK.

So here I've set up a little spreadsheet for us

to run that annuity example that we talked about.

So I'm going to be generating in the annuity cash flows of $2,500.

And I'm going to do that every month for 20 years.

So once a month for 20 years means-- how many payments are going to come in?

Let's work it through.

12 months times 20 years-- 240 payments.

So if I just kind of pull that down all the way down for a whole way down.

We're going to go 240 periods down.

And once we get down there-- faster, faster-- we'll see that there--

and what we're going to do is we're going to change

that annual rate to a monthly rate.

And then we're going to discount all those cash flows

and see how much that annuity would be worth.

So there's our 240 periods.

And now I'm going to copy that cash flow down,

by going into the little right-hand corner, double clicking,

copying that cash flow all the way down.

And now what I'm going to do is discount each of those cash flows.

Now I've got an annual rate of 4%.

That annual rate doesn't match the maturity of my cash

flow, which is a monthly cash flow.

So the first thing I need to do is take that annual rate

and make it a monthly rate.

So we're going to decompound using everything that we learned in week one

so far about compounding and future values and present values.

And I'm going to take that annual rate over 12 months,

and make it a monthly rate for one month.

Well if that annual rate represents 1 plus some percent times 1

plus some percent times 1 plus some percent

for 12 months, what I'm basically going to do is take the 12th root of that 4%.

So that's going to be equal to 1 plus 4% raised to the 1/12 minus 1.

OK.

So our month-- if our annual rate is 4%, our monthly rate

is-- if I put that percent and show you some decimal places-- that's

a 0.327% a month.

If I earned that-- if I earned 0.327% a month over 12 months,

I would earn 4% a year.

Now I've got a monthly cash flow and a monthly discount rate.

So now we can start thinking about taking

the present value of each of those monthly cash flows in the next column.

So, in Excel, this is going to make it easy for us.

Because instead of doing that formula 240 times,

I'm just going to do it once, and then copy it down.

So I'm going to say that's equal to my $2,500 divided by 1 plus 0.327,

my discount rate for one month, and I'm going

to put dollar signs around that cell, so that when I copy the formula,

the cell reference doesn't change.

Now it's always going to go and grab that cell.

And then I'm going to raise that to my period.

That way when I copy it down, you'll see what happens.

It'll flow through nicely.

Now-- OK.

This tells me that a cash flow of $2500 coming in one month from today

at a discount rate of 0.327 per month is worth $2,491.84.

Now if I copy that formula-- Control C-- and paste it-- Control V--

into the cell below, I'm hoping that the formula copied over correctly.

Lets check.

OK.

Now what am I doing?

I'm taking that $2,500.

I'm discounting at the-- good-- at the same 0.327 monthly percent,

but this time I'm raising it to the second power.

When I copied it down, the cell reference copied down with it.

Good.

So that's the right answer.

And if I copy and paste that again, it should now be going to the third power.

Let's see what it does.

Double check the formula.

Good.

OK.

So each period that I copy it down, it's moving down to the next row,

taking that cash flow, using the same monthly discount rate,

and then raising it to the period that I'm discounting.

Good.

OK.

So now I can just copy that formula by highlighting, again from plus sign

to little plus sign, double click, all the way down.

Now as I go kind of all the way down, what

happens that money becomes worth less, less, less, less, less, less, less,

less, right?

So as I go further and further out into the future,

I don't have to pay as much for that promise of $2,500.

82 periods from now, it's worth less and less.

Now what is that annuity-- the big question in this problem

was-- what is that annuity worth to me today?

OK.

So let's figure out what the value of that should be.

If I purchased that annuity, what am I purchasing?

I'm purchasing the right to a whole stream

of cash flows for the next 240 periods.

How much is that worth?

It's going to be worth the sum of all the discounted cash flows from there

all the way down to the end.

If I sum up all those cash flows, what's the value?

That's a big number, as it should be, $415,131.

That's a lot of money, but that's how much that annuity would be worth.

If I saved $415,000, $415,000, I could go to an insurance company and say,

hey, I'd like a retirement account that pays me $2500 a month

over the next 20 years.

Right?

A retirement annuity.

The insurance company, at a discount rate of 4%,

might come back and say to me, we'd be willing to sell you that annuity.

And they would say a fair price for that annuity is $700,000.

And I would say, no, that's way too much money.

I'm not going to pay that.

A fair price at a 4% discount rate for that annuity would be $415,000.

That's a fair price given this discount rate.

Given this cash flow, I can figure out for myself

now what insurance people might call an actuarially fair price for that annuity

would be closer to $415,000.

Now if the insurance company said, hey, we'll

sell you that annuity for $420,000 or $410,000,

I would say, hey, that's right in the right ballpark.

That's about what I would expect at a 4% discount rate

to pay for that retirement annuity.

Good.

So using the very basic principles of compounding and discounting,

we can figure out more complicated instruments like this annuity,

and price it, and figure out what the right price ballpark ought

to be for something that has a whole stream of cash flows coming in

in the future.

If I know when those cash flows are coming in,

and I know what discount rate to hit them with,

I can put a good value on it.

Great.

So that was a really easy example of a simple annuity.

And we built a spreadsheet model using our discounted cash flow technology

to put a value on it.