JAMES WESTON: Hi.
Welcome back to Finance for Non-Finance Professionals.
What I'd like to do in this video is start
working through some examples and practical applications
of the discounted cash flow analysis that we've been
doing in the last couple of videos.
This example is going to be about bonds, the simplest
of all financial instruments.
So let's think about what bonds are-- very simple debt instruments that
promise to pay coupons or interest payments and pay back the principal,
or what we call in bond language the face value.
OK, this is an easy application of basic DCF valuation.
What kinds of bonds are there?
Well, US treasury bonds that are offered by the US government
in order to finance the activities of government.
There is corporate bonds that corporations offer in order
to raise money to build plants and equipment,
build factories and put people to work, corporate bonds.
There are municipals-- sewage treatment plants and water
facilities and all the things that municipalities
need to make our society better.
All that money gets raised up front and then gets paid off with tax revenue
later.
Municipal bonds-- and then, of course, all kinds of different governments
all over the world raise bonds in order to finance their activities,
and we usually call that sovereign debt, bonds issued by Argentina or Greece.
OK, so what's different about each of those bonds?
Well, in a very simple sense, they're all the same.
Each of those bond contracts promises something.
They promise a series of cash flows coming
in to the bond holder in the future of coupon payments or interest payments
and face value, principal repayments.
Each of those bond contracts stipulates when cash is coming in
and what rate of interest it's going to pay.
What's different, what makes those bonds have different prices,
is the risk involved in buying those bonds.
The risk in buying a bond issued by the Cleveland school district
to build new schools might be very different from a bond issued
by the government of Greece.
Those might have different risks associated with them.
So what we're talking about when we talk about different risks is
different discount rates, right?
OK, so let's think about how we would value a bond.
Well, what cash is coming in and when?
Oh, how do we put a value on it?
We know from all the discounted cash flow
analysis that we've been doing in the last couple of videos
once we have the structure of the timeline of cash-- when it's coming in,
what discount rate to use-- all we do is hit, smack down, each of the cash
flows with that discount rate, bring them all back into the present,
add up the sum of the discounted cash flows.
That's the price of the asset.
Discounted cash flow analysis applies directly to bonds in a simple way.
So we could walk through a very simple example
of putting a value on a treasury bond in exactly the same way the treasury bond
traders are doing on trading floors across Wall Street
and across the world every single day.
What we're going to do-- let's think about a three-year bond
with a 2 and 1/2 percent coupon rate that
pays interest payments every six months, which is what treasury bonds do.
If the six-month interest rate is 1.2%, what's
the price of the bond per $100 of principal, or face value?
And let's put a price on it the way that treasury bond traders
do every single day.
We'll move to the light board and work that example together.
OK, let's start this problem by listing all the ingredients that
are going to go into solving this bond valuation problem,
and then we'll do the cooking, and we'll sort of put them all together
and do the calculation.
So the first thing we've got is a 2.5% coupon rate,
and that is the annual rate.
So every year it's paying about 2 and 1/2 percent.
So this bond is paying semiannually, so every six months.
So that's easy enough.
The semiannual coupon rate is then just going
to be 2.5% divided by 2, or 1.25% semiannual rate every six months.
OK, the other thing that we need to cook with here is the face value,
and we'll just do that for every $100 of bond value.
We'll put a price on the bond.
Cap T is 3.
It's a three-year bond, so that's going to be--
every six months, that's going to be six periods of cash flows coming in.
And then the last thing that we need is the discount rate, r.
How hard are we going to smash down in a present value sense
all those cash flows coming in over the life of the bond?
In this example, the discount rate, six months-- the six month discount rate--
is 1.2%.
Again, if we're going to have six month cash flows,
we need a six-month discount rate.
We're always matching maturity to cash flow and the discount rate.
So that's all the ingredients that we need to start cooking.
So let's see.
What are we going to do first?
Let's map out a timeline of all the cash coming in,
and then we're going to discount each of the cash flows.
Then we're just going to add them up, one, two, three.
So let's start with the timeline of where all the cash is coming
in-- period one, period two, period three, period four, period five, period
six-- six periods semiannual cash flows.
OK, now what's the first cash flow going to come in?
That's the first coupon payment after six months.
1.25% of face value of $100 gives us a cash flow of $1.25 every six months.
$1.25, $1.25, $1.25, $1.25, and then our final coupon of $1.25.
Put my little dollar signs on there.
OK, six periods, three years, six six-month periods,
$1.25 each period-- in the last period, of course,
we're also going to get back the face value of the bond,
so plus $100 at the end when the bond comes due.
OK, so how do we put a value on this thing, and how do we put a price on it?
We're going to take each of these cash flows, discount them at that 1.2%,
and then add up the present value of all the cash flows.
So we're going to discount this first period at 1
plus 1.2% raised to the first power.
That's our first cash flow.
In this one, it's 1 plus 1.2% squared.
1 plus 1.2% cubed.
I'm running out of room.
1 plus 1.2% to the fourth.
You get the idea.
1 plus 1.2% raised to the fifth.
That's that exponential discounting, right?
The further we go out, we keep raising that discounting up to the exponent,
and then this whole amount in period six by 1 plus 1.2% discount rate
raised to the sixth power.
Now again, this is a semiannual rate, so the fact
that we're raising it to the sixth power is really
just coming out that three years.
Six six-month periods equals the three years.
So a three-year bond-- six six-month periods-- we add them all up.
Now I've got all the cash flows, the series of coupon payments,
the face value, discounted each period, and that's just
going to come out to-- that's going to be $1.24.
This one's going to be $1.22, $1.21, $1.19.
Here comes that discounting.
$1.18.
That discounting smashes that value down further and further.
And then the last one is going to be-- that $1.25 comes down to $1.16,
and the $100 comes down to about $93.09.
Now, to value the bond, I've got the cash flows.
I've taken the present value of them.
The only thing I need to do now is add up the present value of the cash flows,
just put a sum between each of those cash flows.
Add them all up.
That's going to get us to a final valuation of $7.19-- that's
the present value of all the coupons-- plus the $93.09, the present value
of the face, equals $100.29, and that's the price you would pay
in the market for a bond with a 1.25% semiannual coupon rate
if discount rates were at 1.2%.
That would be the value you would pay in the market
for a bond with those characteristics.
Simple.
One, two, three.
Map out the cash flows.
Discount them appropriately.
Add up the sum of the discounted cash flows.
There is your bond price.
James WestonHouston Endowment Professor of Finance
