Slide 1: Lecture 3 – Bond-Pricing and YTM
Welcome to Lecture 3: Bond-pricing and yield-to-maturity.
Slide 2: Two parts to this lecture
This lecture is divided into two parts: part one is a discussion of bond-pricing and part
two is a discussion on calculating YTM or “yield-to-maturity.”
Slide 3: Part 1 – Bond-Pricing
Without further ado, let’s start with bond-pricing. In bond-pricing, we say that the price
of a bond is equal to the sum of the present values of all its future cash-flows. In the
case of a ‘coupon bond,’ the present value of the bond cash-flows is equal to the present
value of its coupon payments plus the present value of its face value or principal value.
The formula we are interested in is this one right here:
PV(Bond)
= PV(Coupon payments) + PV(Face value or principal)
=PMT [(1 – (1/(1+r)^N)) / r] + [FV/(1+r)^N].
The variables required to use this formula for bond pricing are: PMT, which is the coupon
payment per coupon period, r, which is the discount rate on the bond; N, which is the
total number of remaining coupon payments, and last but not least, FV, the face value or
principal value, which is also sometimes called the ‘par value’ of the bond.
Slide 4: Cash-flows from coupon bond
Before we move on to discuss the bond-pricing formula further, let’s go back to the very
beginning. Let’s figure out what the cash-flow stream from a coupon bond looks like on
a timeline. The best way to illustrate this is with the use of a numerical example. Let’s
say that we have a coupon bond with a face value of $1,000, a coupon rate of 8 percent,
coupon payments made semiannually (i.e., twice a year), which has 9 years till maturity,
and a required return of 10%.
The first thing to do here is to figure out the numbers that go with each of the required
variables: PMT, N, r, and FV. FV is the easiest to get, as it’s simply the face value, FV =
$1,000. N is the next easiest to figure out.
We know that the time to maturity is 9 years, but we also know that the bond pays 2
coupons a year, thus, N = 9 x 2 = 18, as there should be 9 x 2 = 18 remaining
payments.
The next variable to approach is PMT, which is the coupon payment per payment period
of 6 months. The annual coupon rate is 8%, which is paid on the face value of $1000,
which means that per year, we get 0.08 x $1,000 = $80 in coupon payments. However,
since coupons are made every 6 months, each coupon payment is then $80/2 = $40.
Therefore, PMT = $40.
The more difficult variable to obtain is the r. Sometimes it can get a bit complicated
when the compounding period is different from the payment period per year. In this
example, let’s make things simple and assume that the compounding period is the same
as the payment period, i.e., the interest rate is compounded semiannually, just like the
coupon payment period. In that case, the semiannual rate is simply 10%/2 = 5%. That
was easy!
So now, we have all the variables needed to calculate the bond-price. But first, let’s draw
the timeline of cash-flows for this bond.
Slide 5: Timeline of cash-flows
We first start with a relatively straight line like this one.
Then, partition it into N = 18 equal sections.
Then, fill in the times from t =0, 1, 2, 3, …, 17, 18, where t = 0 represents now.
t= 0
1
2
3
13
14 15
16 17
4
18
5
6
7
8
9
10
11
12
We get PMT = 40 in each period. We also have one last payment of 1000 at the end of
the 18th period.
40
40
40
40 40
40
40 40
40 40+1000
t= 0
1
2
3
15 16 17 18
4
5
40
6
40
7
40
8
40
9
40
10
40
11
12
40
13
40
14
Tada! That’s our cash flow stream from the coupon bond.
Slide 6: Bond-Valuation using PVs of lump sums
A more tedious method is to calculate the present value of each cash flow one by one,
and then add them up. So, for this example, we have bond-price = (40/1.05) +
(40/1.05^2) + (40/1.05^3) + … all the way to (40/1.05^18) plus the present value of
the principal payment of 1000, which is (1000/1.05^18).
When you plug all these into your calculator, you will find the answer of 883.10, but,
there is an easier way of calculating the bond-price.
Slide 7: Bond-Valuation using separation of cash-flow streams
Valuation using separation of cash-flow stream is an easier method for bond valuation.
If we take a look at the cash-flow timeline we drew previously, we can see that there are
actually two streams of cash-flows: one is an annuity of $40 for 18 periods, and the
other is the $1000 principal payment at the end of the 18th period.
Cash-flow stream #1: Annuity
40
40
40 40
40 40 40
t= 0
1
17 18
2
3
40
4
40
5
40
6
40
7
40
8
40
9
40
10
11
40
12
40
40
40
13
14
15
16
Cash-flow stream #2: Lump Sum
1000
t= 0
1
2
16 17 18
3
4
5
6
7
8
9
10
11
12
13
14
15
This means that we can calculate the present value of the first cash-flow stream as the
present value of an annuity, and the present value of the second cash-flow stream as
the present value of a lump sum, and then add them together to get the total present
value of cash-flows from the bond.
In terms of the formula, we have:
Bond Price = PV(PMT of $40 for 18 periods) + PV(Lump sum of $1000 at t=18)
Slide 8: Numerical Example of Bond-Valuation using separation of cash-flow
streams
Now let’s do some button-pushing on the financial calculator. So… the coupon payments
are constant and this cash-flow stream is an annuity, which means that we can calculate
its present value using the PV(Annuity) formula:
PV(PMT) = PMT x (1 – (1/(1+r)^N)) / r.
We know that
PMT = $40
r = 0.05
N = 18
Plugging in the numbers for PMT, r and N for this bond to this formula, we get
PV(PMT=40) = 40 x (1 – (1/(1+0.05)^18)) / 0.05 = 467.58
For the second part of the bond-price, we calculate the PV of the face value, which is
paid in a lump sum at the end of the bond’s life. Therefore, we can use the PV(lump
sum) formula:
PV(FV) = FV/(1+r)^N.
Plugging in FV=1000, r=0.05, and N=18, we get
PV(FV) = 1000/(1+0.05)^18 = 415.52.
Therefore, as the bond-price is equal to the sum of the present value of its two cash flow
streams:
Bond price = PV(PMT) + PV(FV) = 467.58 + 415.52 = 883.10.
This is the same answer we obtained previously by doing it the long and tedious way.
Of course, after all that work, I must tell you that there is an easier way to calculate the
price of a coupon bond. Don’t get mad (I can actually hear some of the pained groans
now), the reason that we just went through ALL THAT is so that you can understand
what is going on when you start punching the buttons on your financial calculator. Also,
now that you know the basics of cash-flow identification, drawing timelines, and
calculation of PVs of these cash-flows, you won’t have to fret next time you see a bond
issue that has different cash-flows than the simple coupon bond discussed in this lecture.
Slide 9: Bond-Valuation using financial calculator
So, to use your financial calculator to calculate bond-price, you will need to enter the
following information into the calculator:
FV = -1000
PMT = -40
I/Y or r = 5
N = 18
The press the buttons:
COMP
PV
This should get you to the same answer as we had calculated before using formulas.
See how good it feels to know what’s going on behind the keystrokes?
Slide 10: Part II - YTM Calculation
Okay! Now that we know how to calculate bond-prices, let’s ask the question: “Since we
can easily find the bond-prices listed on our daily business newspaper or on the internet,
can we turn it around and ask: what is the yield on this bond?” The answer, as always,
is “Yes we can!” This question is referring to “yield-to-maturity,” which is the discount
rate that makes the present value of the bond’s future cash-flows equal to the bondprice. YTMs are nifty for comparison across different bond issues, and as a result, are an
important tool for investors.
Slide 11: Information needed to calculate YTM
First, we must ask ourselves, “what information do we need to calculate the YTM?”
Based on the definition of the YTM, we know that the proper formula to use is the bondpricing formula:
Bond price = [PMT(1 – (1/(1+r)^N))/r] + [FV/(1+r)^N].
To find the YTM, we substitute the r for the YTM/m, where m stands for the number of
compounding periods per year. So, if the interest rate is compounded semiannually,
then m = 2; if quarterly, then m = 4; if monthly, then m = 12, etc. Therefore, the
information needed to calculate YTM are: PMT, N, FV, m, and of course, the bond-price.
The bond formula to find the YTM, then, looks like this:
Bond-Price = [PMT x (1 – (1/(1+(YTM/m))^N)) / (YTM/m)] + [FV/(1+(YTM/m))^N]
Slide 12: Numerical Example – YTM Calculation
Let’s put this formula through a numerical example.
calculate the YTM of this coupon bond.
Given the information below,
Bond-price = $950
Coupon rate = 8%, paid semi-annually
Time to maturity = 9 years
Face value = $1,000
So, we have:
PV = Bond-price = $950
PMT = 0.08 x 1000 /2 due to semiannual coupon payment, which gives us 40
N = 9 x 2 due to semiannual payment, which gives us 18
FV = face value of the bond = 1000
m = 2, as we assume semiannual compounding of interest rate
Plugging in all these to the bond price formula, we get
950 = [40 x (1 – (1/(1+(YTM/2))^18)) / (YTM/2)] + [1000/(1+(YTM/2))^18]
Looking at this formula here, we soon realize that it is not very easy to solve for the YTM
mathematically.
Slide 13: Three ways to calculate YTM
Fortunately, there are three quite easy ways of solving for YTM: By trial and error,
financial calculator or spreadsheet.
With trial-and-error, we simply start with an
arbitrary number for YTM, say, 10%, plug it into the bond price formula, and check to
see if the resulting bond-price is close to the actual bond-price. If not, we try another
number. Hence, “trial-and-error.” We try to get as close to the actual bond-price as
possible by plugging in different YTMs. As you can imagine, this is a long a tedious
process and most people loathe using it.
Now, method number 3 is not too bad as long as you have some skills in using or writing
programs in a spreadsheet. If you don’t, then method number 2 is the way to go. Let’s
try that out with our numerical example, shall we?
Slide 14: YTM calculation with financial calculator
Enter this information into your financial calculator:
PV = 950
PMT = -40
N = 18
FV = -1000
Then press the buttons:
COMP
I/Y (or r)
When the answer of 4.408181445 pops up on your calculator, multiply it by 2 (the
number of compounding periods per year, m). This should give you the YTM of
8.816362889.
That is, the yield on this coupon bond with a price of $950 is
approximately 8.82%.
Slide 15: Practice question
We all know what comes next – Practice practice practice! Try your hand at this YTM
calculation and see if you can get this answer.
Given a bond with a current price of $1,020, face value of $1,000, coupon rate of 10%,
semiannual coupon payments, and 13 years to maturity. Calculate the yield-to-maturity
on this bond.
Check answer: YTM = 9.725665243%
Slide 16: Practice makes comfortable
Now try another practice question.
Given a bond with a current price of $1,020, face value of $1,000, coupon rate of 10%,
quarterly coupon payments, and 13 years to maturity. Calculate the yield-to-maturity
on this bond.
Check answer: YTM = 9.727270497%
Slide 17: Practice makes everything easy
Now try just one more. I promise this is the last one, and with that, I bid you a good
day of fun with numbers!
Given a zero-coupon bond (that does not make any coupon payment) with a current
market price of $550, face value of $1,000, and 13 years to maturity. What is the yieldto-maturity if the interest on this bond is compounded semi-annually?
Check answer: YTM = 4.652024901%