Welcome back to corporate finance.
Last time we talked about compounding or
the process of moving cash flows forward in time.
Today I want to present several useful shortcuts to compute the present value and
future value of common streams of cash flows that we see often in practice.
Let's get started.
Hey everybody, welcome to our third lecture on the time value of money.
So, last time we talked about compounding or
the process of moving cash flows forward in time.
[COUGH] Excuse me.
To find their future value, whereas in our first lecture we
moved cash flows back in time by discounting to find their present value.
So, what I want to do today is I want to give you some useful shortcuts for
computing the present value or
the future value of some streams of cash flows that commonly arise in practice.
So let's get started.
The first thing that I want to talk about is an annuity.
An annuity is a finite stream of cash flows of identical magnitude in equal
spacing and time, and I've highlighted key elements of this definition.
So here's a timeline representing an annuity.
First key aspect or feature of an annuity are cash flows of identical magnitude.
All of these cash flows are the same number.
So identical magnitude.
Secondly, this is a finite stream of cash flow.
It ends at some point in time.
Okay?
And that might seem like an unnecessary or obvious assumption, but
you'll see that we'll deal with infinite cash flow streams in a little bit.
And then the last assumption is that the spacing between the cash flows
has to be equal, so it's always a year.
We always get the cash flow after one year, two months,
whatever that spacing is, it has to be the same.
And it turns out that this cash flow
stream arises in a number of situations in practice.
So insurance companies sell a product called an annuity,
representing its cash flow stream.
Home mortgages are an example of an annuity stream.
Auto leases, certain bond payments and amortizing loans are annuities.
So it's actually fairly common in practice.
Now if we wanted to find the present value of these cash flows,
we know how to do that.
We could brute force it.
We can take each cash flows and discount it back to today.
So imagine I had a second cash flow here.
I could go CF divided by 1 plus R squared.
That would bring it back to today.
I could take this cash flow, CF over 1 plus R to the T minus 1.
That would bring it.
And I do that for all the cash flows, and then I could add them up here.
That would give me the present value.
But that's a bit burdensome, especially when T is big.
So, what I'd like to show you is a shortcut or a simple
formula to compute the present value of this cash flow stream, and here it is.
We take the cash flow, CF, divide it by the discount rate and
multiply it by this term in parentheses here.
Now, if I move the R over here,
I can re-express the present value of the annuity formula as just
the annuity cash flow times this term here which is called an annuity factor.
That would give me the present value of this cash flow string.
One thing to keep in mind though is in order for this formula to make sense,
not only do all the features defining an annuity stream have to be true, but
we are assuming that the first cash flow arrives one period from today.
So, for example, if my cash flow stream looked like this.
This is an annuity stream of cash flows, but
this formula is not going to give me the present value of this cash flow stream.
Actually, what I would have to do is I could just add CF,
because this is the present value.
It's coming, today.
Let's do an example.
How much do I have to save today to withdraw $100 at the end
of each of the next 20 years if I can earn 5% per annum?
Well, step one is draw a timeline.
I'm trying to figure out how much I have to save today in order
to pull out $100 every year over the next 20 years.
Well, we know how to do that by brute force.
We can simply discount all of the cash flows back into today's
time units and add them up.
More elegant solution that we just learned, of course,
is to apply our present value of annuity formula.
Right. The cash flow is 100, CF.
The discount rate, R, is 5%.
And the time of the cash flows is 20 years.
Plugging all those numbers into the formula, and computing,
we get the present value of these cash flows is $1,246.22.
That's how much money I have to save today in order to withdraw the $100 every year.
Now let's turn to something called the growing annuity, which is as the name
suggests just like an annuity but for the fact that the cash flows are growing.
So it's a finite stream of cash flows, okay.
Evenly spaced through time, okay.
But now the cash flows aren't constant, they're growing at a constant rate, g.
And this type of cash flow stream pops up in a number of instances in practice.
Certain income streams, for example, your work.
You might imagine that your salary grows at some constant
rate or approximately some constant rate g.
Certain saving strategies, maybe you want to save a certain amount each year, but
you want that amount to grow with your growing income stream.
In corporate finance certain project revenue and
expense streams will often grow at a near constant growth rate.
So it's a really useful approximation to many cash flow
streams we'll come across in practice.
And like our annuity stream, we can represent the present value of this cash
flow stream with a simple formula as follows.
We take the cash flow [COUGH] as of the first period divided by
the discount rate less the growth rate, times this factor here.
That will give us the present value of this cash flow stream right here.
But remember,
a critical assumption is that the first cash flow arrives one period from today.
Okay.
Let's do an example.
How much do we have to save today to withdraw $100 at the end of this year
$102.50 next year, $105.06 the year after, and so
on for the next 19 years if we can earn 5% per annum?
Well, let's draw a timeline, and what we see is that our
first withdrawl of $100 occurs one year out,
then $102.5, and on and on and on.
What we can discern from this problem is that these cash flows are growing at
a constant rate, g, equal to two and a half percent per annum.
So this cash flow stream satisfies all of the requirements
needed to use the present value of a growing annuity formula.
So, our first cash flow of $100.
Our discount rate of 5%.
And, here's our growth rate of 2 1/2%.
That's going to get a present value of $1,529.69.
That's how much we would need to withdraw $100 growing at 2 1/2% every year.
Now let's talk about a perpetuity.
So a perpetuity is just like an annuity except the cash flows go on forever.
We get the same amount of money equally spaced in time forever.
So where does this thing come up in practice?
Well, oddly enough it actually does.
And something called perpetuity or consol bonds which exists over in the UK.
And interestingly enough the formula for this cash flow stream is very simple.
It's just the cash flow divided by the discount rate.
CF over R.
Let's do an example.
How much do you have to save today to withdraw $100 at the end of each year
forever if you can earn 5% per annum?
Well, the timeline looks as follows, all right?
$100 every year, forever, and clearly the brute
force method of discounting each cash flow one at a time is never going to work.
It's just impossible.
So we have to use our formula.
We take the $100, divide it by the discount rate of 5%, and
that gives us $2000.
We need $2000 to be able to withdraw $100 a year forever,
assuming that money can earn 5% annum.
And intuitively what's going on is once we get out hundred years,
two hundred years, whatever.
The present value of that money is so small, it's very close to zero,
which is why you don't need an infinite amount of money.
And the perpetuity's cousin, a growing perpetuity, is just that.
It's an infinite stream of cash flow that grows at a constant rate, g.
That are evenly spaced out through time.
So here's a visual representation.
Here's our timeline of a growing perpetuity.
What's an example of a growing perpetuity in practice,
well, dividend streams are much like a growing perpetuity.
They're useful approximation.
Don't take it literally.
Companies don't last forever, but we can treat them as such Because there is no
finite end date to most companies, absent some event such as a bankruptcy or
an acquisition or a takeover, something like that.
So, what's the formula for a growing perpetuity?
Well, it's just the cash flow that we're going to receive in the first year divided
by the discount rate minus the growth rate of that cash flow.
Again we're going to have to assume that the first cash flow
arrives one year from today to use this formula,
as well as having all the other requirements being satisfied.
The cash flow's being evenly spaced and then growing at a constant rate.
So let's do a little example now.
How much do you have to save today to withdraw $100 at the end
of this year, 102.5 next year and 105.06 the year after and so
on forever if you can earn 5% per annum?
Well, let's draw our timeline.
So we're going to get $100 here in year one.
102.5 in year two, that's growing at 2.5%,
as is the 105.06, if I had written it here.
So our g is 2.5%.
The first cash flow comes one year from today, there is our first cash flow.
This goes on forever, and the spacing is equal.
So this is a growing perpetuity to which we can apply our
growing perpetuity present value formula.
So we take the first cash flow.
$100 divided by the difference between the discount rate and
the growth rate to get $4000.
In other words, we have $4,000 dollars today, and
it's earning 5% interest per annum, every year thereafter.
We can pull out $100 next year and
have that amount grow by 2.5% every year thereafter.
Let's summarize.
We learned a couple of useful short cuts today.
We talked about an annuity and its present value formula.
We talked about perpetuity and its present value formula.
We talked about their growing cousins, the growing annuity,
the growing perpetuity, and the present value of formula for those guys.
And while that might seem somewhat esoteric and bland or
boring, we also discuss some of the applications
that you might see in practice these cash flow streams arising.
And where these short cuts are really useful,
more than just finding the present value or the future value,
is in finding the cash flow associated with the stream.
So being able to manipulate these formulas is very important.
And so in the problems you are going to spend some time in real life context,
or at least as close to real life as we can get, manipulating these formulas to
derive certain aspects of interest, whether
it's the cash flow or the amount of time, or the discount rate or the growth rate.
So tackle the problem set, it really brings the material to life.
But be very careful that applying these formulas takes care.
Don't blindly apply them to any setting, because as we discussed,
certain characteristics of cash flows have to be met in order to,
or certain requirements of the formula have to be met, in order to use it.
So good luck with the problems.