1. Pricing a Bond
If the owner of a bond was to sell the bond before the
maturity date, what would the value or the price be?
To answer this critical question, the investor can use the discounted cash
flows framework. The price of the bond today is the sum of the
present values of all the future cash flows of the bond.
To see why, let's look at the bond issued by CFI that we
introduced in the last lesson. It's a five year bond with a par
value of $100, and a coupon rate of 5%.
So the annual coupons are $5. In the final year or in year five, the
investor also gets their $100 par back. So the final cash flow is
actually $105. To find the value today, we need to discount these future
cash flows to find their present values today.
The discount factor formula using DCF is one over[one plus I%]
all raise to the power of N. Now,
when we price a bond, the I% is the annual return investors are
getting if they hold the bond until maturity.
This return has a really special name, and it's called the yield to
maturity. This bond has a yield to maturity of 6%,
meaning each cash flow is giving the bond investor a 6%
return. The discount factor for year one, is one over one plus 6%,
all raise to the power of one, which is 0.9434. Multiplying this by
the future value of $5 gives a present value of 4.72. Now,
we can do this for the next four cash flows, changing the N in
the discount factor each year, so that we can calculate all the present
values. Summing these up gives the price of the bond today at 95.79. We
have another workbook for this chapter, and the workbook is called Bond
Pricing Template, and you can find it in the downloads section for this
course. If you haven't downloaded it already, pause the video now,
download it and when you ready, press play. I'm going to walk through
this previous bond pricing example in Excel, and it would be great if
you follow along with me. So here we are in the bond pricing
template workbook, and I moved to the second sheet called demo.
Let's price the bond that we were just discussing. It's a five year
bond, and the par value was 100, but coupon rate was 5%
paid annually and this bond was yielding 6%.
So the first thing that we need to do is we need to
calculate the future values, these are going to be either coupon payments.
The coupon payments are 5%, and I'm going to absolutely reference that,
times the par value of a 100. And again, I'm going to absolutely reference
that. And when I fill down, I've just got to remember to go
to the fifth year, activate the cell by pressing F2 and I'm going
to add the par value of the bond back. All right. So, that's in cell...
Let me just pick it up in D5. So let me just type D5. All right. The discount
factor. The discount factor equals one divided by open bracket one plus
the yield to maturity, which I'll absolutely reference, I'll close the bracket
and raise it to the power of N. And so, by selecting the
column E Ctrl+D to fill down, you get the five discount factors really
easily. The present value is easy to calculate. We've done all the hard
work, we take the future value and we multiply that by the discount
factor. And again, we can highlight the column Ctrl+D to fill down and
the shortcut to put the sum at the bottom of a highlighted column
is Alt+=, and we can see that the price of this bond is
95.79.
2. Let's practice!