Latest / How Canadian Markets Work / Episode 38: Bond Pricing From Scratch

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Episode 38: Bond Pricing From Scratch · How Canadian Markets Work

0:00
So I want to start today with a really simple
0:02
question. If I offer you an envelope with $1
0:06
,000 in it today, or I offer you that exact same
0:10
envelope in three years, which one are you taking?
0:12
Oh, I'm taking it today, yeah. Right. Every time.
0:14
Exactly. It's a complete no -brainer, right?
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You take it today. But that split second, like
0:19
intuitive choice you just made, that's actually
0:22
the entire theoretical bedrock of a multi -trillion
0:26
dollar global bond market. It really is. Yeah.
0:30
Yeah. The fundamental reality that money today
0:32
is intrinsically worth more than money tomorrow.
0:35
That drives every single fixed income transaction
0:40
on earth. Right. That simple preference. That's
0:42
the foundation. Well, welcome to today's Deep
0:44
Dive. We're kind of stripping away all the jargon
0:47
today and really looking closely at the absolute
0:50
bedrock of fixed income investing. So our mission
0:53
today, relying on our comprehensive source notes,
0:55
is to deconstruct the exact math behind, well,
0:58
present value, the time value of money, and basically
1:01
how to price a bond from scratch. And we're going
1:03
to take this nice and slow. Yes, please. Because
1:06
the math we're exploring today, it requires a
1:09
very deliberate pace. Yeah. We really want to
1:11
understand the structural gravity of finance
1:14
here. Right, the gravity. I love that. So let's
1:18
go back to that envelope for a second. The reason
1:21
you want the $1 ,000 today rather than three
1:24
years from now, it basically comes down to two
1:27
very specific forces. And the first one is inflation.
1:31
$1 ,000 today buys a certain amount of groceries
1:34
or pays a certain amount of rent. In three years,
1:38
well, that same thousand dollars is almost certainly
1:40
going to buy you less. Exactly. The purchasing
1:43
power just naturally erodes while you wait. Yeah.
1:46
And then the second force is opportunity cost.
1:49
Like if I hand you that money today, you can
1:51
put it to work. Yes. You can invest it. You can
1:53
earn a return. And over those three years, your
1:55
money actually grows. But if you wait for the
1:58
envelope, you surrender all of that potential
2:01
growth. which means mathematically we need a
2:04
way to quantify that lost time. Since we inherently
2:09
know that a dollar in the future is just worth
2:11
less than a dollar in our hands right now, we
2:14
require a really reliable mathematical way to
2:17
travel backward in time. Travel backward, right?
2:19
Yeah. We need a mechanism to look at a future
2:23
cash flow and kind of shrink it down to its precise
2:26
value today. And that mathematical time travel,
2:29
That's what the industry calls discounting, right?
2:32
Yes, exactly. Discounting is simply the inverse
2:35
of growth. OK. If you have cash today, and you
2:38
can safely earn, say, a 6 % return in the market,
2:42
well, calculating present value is just running
2:44
that growth engine in reverse. If we were projecting
2:48
forward in time, we would multiply. We would
2:50
take our money and multiply it by 1 .06 for every
2:53
single year that it grows. Right. But because
2:55
we want to move backward from the future to the
2:58
present, we just do the mathematical opposite.
3:00
We divide. Exactly. We divide by 1 .06. OK. Let's
3:04
unpack this for a second. Think of discounting
3:07
like paying a toll to travel backward in time.
3:11
Oh, I like that. Yeah. So every single year you
3:13
want to pull that future money closer to the
3:15
present day, the market charges you a toll. Right.
3:18
And in this case, it's a 6 % toll. That is an
3:21
incredibly helpful way to visualize it. You're
3:25
paying a penalty for early access to the value.
3:28
Yeah. And that 6 % toll, that is what we call
3:31
the discount rate. But the market doesn't just
3:34
pick 6 % out of thin air to charge us, does it?
3:36
No. No, not at all. The discount rate is thoroughly
3:40
grounded in reality. Okay. It is the prevailing
3:43
return available in the broader market on an
3:46
investment of comparable risk. Conquerable risk.
3:49
Yeah. It represents your next best alternative.
3:52
So if you can get a 6 % yield somewhere else
3:54
with the exact same level of safety, well then
3:57
6 % naturally becomes the baseline hurdle for
4:01
any new injectment you might consider. Okay.
4:03
So because a bond isn't just, you know, one single
4:06
lump sum payment, pricing one actually requires
4:09
us to calculate a a whole series of these tolls.
4:13
Right, because a bond is essentially a schedule
4:15
of future promises. And those promises are scattered
4:18
across different points in time. Which means
4:20
we have to pull multiple cash flows backward.
4:23
And they all have to cross completely different
4:25
distances in time. Right. So we are going to
4:27
walk through that exact math together. Yes. We
4:29
are going to price a single bond entirely by
4:33
hand, just relying solely on the structural logic
4:36
of discounting. So let's set the variables really
4:40
clearly here. We are looking at a standard three
4:43
-year bond. It has a $1 ,000 face value, and
4:47
it pays a 5 % annual coupon. Let's just translate
4:51
those percentages into actual cash flows, just
4:53
so we can see exactly what we were buying. Right.
4:55
So we know how a standard bond works. 5 % on
4:58
$1 ,000 means we are expecting $50 a year. Right.
5:01
So the schedule is pretty clear. We receive $50
5:04
at the end of year one. We receive another $50.
5:07
at the end of year two, and then at the end of
5:09
year three, we receive the final $50 coupon plus
5:13
the return of our $1 ,000 principle. Making that
5:17
final year three payment a rather substantial
5:20
$1 ,050. Exactly. $1 ,050. Now we need to determine
5:25
the market's demand. Our bond pays a 5 % coupon,
5:28
but let's assume the broader market currently
5:30
demands a 6 % return for this exact level of
5:33
risk. So our discount rate, our toll in your
5:35
analogy, is 6%. Right. We will divide by 1 .06
5:40
to pull those cash flows into the present. OK,
5:42
let's take this nice and slow and start the arithmetic
5:45
with year one. We have a $50 payment sitting
5:48
exactly one year in the future. Yes. So it only
5:51
has to travel backward through one single year
5:53
of time. So we take that $50 and we divide it
5:56
by 1 .06 just a single time. And the result of
6:00
that division, let's see, is $47 .17. Right.
6:04
So that is the present value of our first coupon.
6:08
We would pay $47 .17 today to receive $50 a year
6:13
from now. Exactly. Now, we have to address year
6:17
two. OK. We have another $50 payment, but this
6:20
one is sitting two full years away. which means
6:23
it has to pay the toll twice. Right. So we take
6:25
that $50, we divide it by 1 .06 for the first
6:28
year, and then we have to divide that result
6:29
by 1 .06 again for the second year. And because
6:33
of the compounding effect of time, that means
6:35
it shrinks even further. Right. That second coupon
6:38
is only worth $44 .50 today. Wow. Just simply
6:42
because we have to wait a whole extra year to
6:44
get our hands on it. That's the time value of
6:46
money at work. Yeah. Okay, now we reach year
6:50
three. This is the big one, the final massive
6:53
payment of $1 ,050. And it's sitting three years
6:58
out on the horizon. Yes. This one has to travel
7:01
the furthest distance backward. Right. So we
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take the $1 ,050 and we divide it by 1 .06 three
7:08
separate times. And after passing through three
7:11
years of a 6 % discount, that large payment is
7:15
whittled all the way down to a present value
7:17
of $881 .60. OK, so I am looking at these three
7:22
present values now. We have $47 .17 for year
7:26
one. We have $44 .50 for year two. And we have
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$881 .60 for year three. So to find the fair
7:38
price of the bond today, we simply add those
7:41
three components together. Right. And when you
7:43
gently sum those present values, the total comes
7:46
out to exactly $973 .27. We just priced a bond.
7:50
We did. Without any complex algorithms or trading
7:54
software, just the pure logic of the time value
7:57
of money, the market would pay exactly $973 .27
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for this bond. So when you look at how those
8:03
numbers broke down, what stands out to you the
8:07
most? I mean, honestly, it is the sheer weight
8:09
of that year three payment. Yeah. Because by
8:12
the time it hits the present day, it is worth
8:14
$881. So if you look at the total bond price
8:18
of $973, that final principal repayment, it makes
8:22
up over 90 % of the bond's total present value.
8:25
Right. And that is such a crucial realization
8:27
for investors to internalize. Yeah. Because the
8:30
$50 coupons along the way, they feel very tangible,
8:33
right? Right. It's cash in hand. Exactly. They
8:35
provide a nice steady stream of income to your
8:38
account. But mathematically speaking, they are
8:42
really just a sideshow. Just a sideshow. The
8:45
vast overwhelming majority of your actual wealth
8:48
is locked inside that final envelope. Just waiting
8:51
patiently at the end of year three. Yeah. You
8:54
are essentially just waiting for a massive balloon
8:56
payment. You really are. So we arrived at a present
8:59
day price of nine hundred seventy three dollars
9:02
and twenty seven cents for a bond that, let's
9:04
remember, promises to pay back one thousand dollars.
9:07
Right. So why is the price lower than the face
9:09
value? Well, because the mathematical reality
9:12
of the market just forces it to be lower. I mean,
9:14
the bond only pays a 5 % coupon. But we established
9:18
that the market has better alternatives. The
9:21
market demands a 6 % return. Right. If a rational
9:24
buyer can safely earn 6 % somewhere else. Yeah.
9:27
they are certainly not going to hand you a full
9:30
$1 ,000 for a bond that only yields 5%. Exactly.
9:34
The only logical way a buyer will accept the
9:37
inferior 5 % coupon is if they receive a discount
9:41
upfront. Right. So they only pay you $973 today.
9:45
But at the end of year three, the contract dictates
9:47
that you still have to hand them a full $1 ,000.
9:50
Exactly. So they collect their 5 % coupons along
9:52
the way, and they also get a built -in capital
9:54
gain at maturity. I see. And when you gently
9:57
blend those two sources of profit together, their
10:00
total return magically equals exactly 6%. That's
10:04
amazing. So the price just drops until the total
10:07
return matches the market's demand. It's like
10:10
a self -correcting equilibrium. It is entirely
10:13
self -correcting. Yeah. So let's actually run
10:16
the alternatives just to see how the equilibrium
10:18
shifts. OK. Imagine the economic environment
10:21
changes. Instead of demanding a 6 % return, What
10:25
if the broader market only wanted a four percent
10:28
return? Oh Okay, so if the prevailing market
10:31
rate drops to four percent suddenly our bonds
10:34
five percent coupon looks Incredibly attractive
10:37
exactly it is generating more yield than the
10:39
alternatives and because it is mathematically
10:41
superior to the alternatives buyers will naturally
10:45
scramble to own it. Right. They will actively
10:47
bid the price up. So if we run our exact same
10:50
time travel arithmetic again, but this time we
10:52
use a four percent toll. So we're dividing by
10:54
one point zero four instead of one point zero
10:56
six. The price of the bond actually rises. Right.
10:59
It rises to one thousand and twenty seven dollars
11:02
and seventy five cents. Yes. Because the coupon
11:04
is so generous relative to current conditions,
11:08
a buyer has to pay a premium to acquire it. A
11:11
premium. Yes, that is the literal definition
11:14
of a premium bond. That makes perfect sense.
11:16
And what if the market wanted exactly 5 %? Well,
11:21
if the market demands 5 % and your bond pays
11:24
exactly 5%, you have perfect harmony. Right.
11:27
The coupon matches the demand perfectly. Absolutely,
11:31
no capital adjustment is required. The bond just
11:34
prices perfectly at par, exactly $1 ,000. Yes.
11:38
And I love going through this because it completely
11:40
demystifies the terminology we hear in financial
11:43
media all the time. Oh, absolutely. People use
11:45
words like premium and par and discount as if
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they are these rigid classifications handed down
11:52
by some regulatory committee. Right, like they're
11:54
fixed categories. But they're nothing more than
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arithmetic outputs. They literally only answer
12:00
one single question. How does this specific coupon
12:03
compare to what the market currently demands?
12:06
If you understand that, you just understand bond
12:08
pricing. Exactly. A skeptic might ask, you know,
12:11
if a trading terminal can execute this discounting
12:13
math in a fraction of a millisecond, why did
12:15
we just spend all this time mapping it out so
12:17
slowly by hand? It's a fair question. But there
12:21
is a profound difference between memorizing a
12:24
financial rule and genuinely understanding a
12:27
structural truth. Yeah, that's so true. You hear
12:29
the phrase bond prices fall when interest rates
12:31
rise constantly. People memorize it for a finance
12:35
exam, they nod when they hear it on the news,
12:37
and they quickly forget why it actually happens.
12:40
But by slowing down and actually performing the
12:44
division together, it becomes an undeniable logic.
12:47
When the market's demanded rate goes up, the
12:50
discount rate gets larger. And when you divide
12:52
a fixed future cash flow by a larger number,
12:55
the present value inherently shrinks. The price
12:58
must fall. It isn't just a theory. No. It is
13:01
mathematical gravity. Gravity. I love that. Now,
13:04
our baseline math today, it used a very clean,
13:07
simple scenario. But the real world obviously
13:10
introduces layers of complexity that tweak this
13:13
foundational arithmetic. It does, yes. Let's
13:15
walk through those real world layers, starting
13:18
with the timing of the payments. Because we used
13:20
annual payments for our bond today, assuming
13:22
we received cash just once a year. Right. But
13:24
in practice, the vast majority of real world
13:27
bonds actually pay interest semi -annually. So
13:30
you receive cash twice a year. Right. So how
13:33
does that... alter our baseline calculation.
13:36
The structural concept is completely identical.
13:38
It just requires kind of softly adjusting the
13:41
variables. Okay. Because you are getting paid
13:43
twice as often, you basically just double the
13:46
number of periods in your calculation. So a three
13:49
-year bond becomes six distinct periods. Six
13:52
periods, okay. And because the periods are shorter,
13:56
do we just have the discount rate, like if the
13:58
annual demand is six percent, we discount each
14:00
six -month period by three percent? Exactly right.
14:02
You're simply paying the toll every six months
14:04
instead of every year. Right. The present value
14:07
changes slightly due to the nuances of compounding,
14:10
but the logic remains entirely untouched. That
14:13
makes total sense. Now, the second complication
14:15
is a bit more nuanced. It involves the yield
14:18
curve, because in our manual we used one flat
14:21
discount rate. We assumed the market demanded
14:24
6 % for year one, 6 % for year two, and 6 % for
14:27
year three. Right, but time is rarely priced
14:30
that uniformly. Yeah. In reality, you do not
14:33
use one single blanket discount rate for every
14:36
cash flow. The market rationally recognizes that
14:40
tying your money up for three years carries a
14:43
completely different profile of risk than tying
14:46
it up for just one year. Well, yeah, because
14:49
more things can go wrong over three years. Exactly.
14:52
Inflation could spike, economic policies could
14:54
shift drastically. Right. Therefore, the market
14:57
usually demands a slightly higher rate of return
15:00
for longer time horizons. So you discount each
15:04
year's cash flow by its own specific rate, which
15:07
is dictated by the actual shape of the yield
15:09
curve. Ah, I see. So we might apply a 5 % discount
15:13
rate to pull back the year one payment, but we
15:15
might have to use a 6 % rate to pull back the
15:17
year. Yes. It requires a few more steps, but
15:21
it provides a much more precise valuation of
15:23
the risk across time. Right. It's just much more
15:25
accurate to the real world. Which naturally brings
15:27
us to the third major real world complication.
15:30
Right. Credit risk. Oh, credit risk. Okay, wait.
15:33
So let me get this straight. If a stable sovereign
15:36
government and a highly leveraged kind of risky
15:40
corporation both issue bonds that promise the
15:43
exact same cash flows. You know, $50, then $50,
15:47
then $1 ,050. Are their bonds really worth completely
15:51
different amounts today? They are worth substantially
15:53
different amounts, yes. Wow. The promise of the
15:56
money matters just as much as the amount of the
15:58
money. Because the corporate bond carries the
16:00
looming threat of default. Yes. Like the very
16:03
real anxiety that they might simply run out of
16:06
cash and fail to pay me back. Right. The cash
16:08
flow schedules look completely identical on paper.
16:11
But to consciously take on the anxiety of lending
16:15
to a struggling corporation, a rational market
16:18
demands a much higher expected return. They demand
16:20
a risk premium. Exactly, they demand a risk premium.
16:23
And based on the math we just slowly walked through,
16:26
demanding a higher return means applying a higher
16:28
discount rate. Which mathematically forces a
16:30
significantly lower present -day price. You might
16:33
happily pay $970 for the government's reliable
16:36
promise, but you might only offer $800 for the
16:39
risky corporation's identical promise. Because
16:42
the exact same future dollars are valued entirely
16:45
differently depending on whose signature is actually
16:48
on the contract. Precisely. That paints to such
16:50
a clear picture. Bond pricing is this this quiet
16:54
kind of relentless tug of war between a fixed
16:58
unyielding promise standing out in the future
17:00
and the constantly shifting emotional demands
17:03
of the present day. Yes, that is beautifully
17:06
put. Every time the market's assessment of risk
17:08
or inflation changes, the present value of that
17:11
future promise flexes in response. It is the
17:14
specific mechanism that translates future uncertainty
17:16
into a clean present day price tag. Well, we
17:19
have spent our time today pulling future dollars
17:22
backward, paying the toll of the discount rate
17:24
and finding true present value. But before we
17:27
conclude, is there a final, maybe provocative
17:30
thought from our sources that we can leave the
17:31
listener to chew on? Yes, actually there is.
17:34
There is a concept. that goes a bit beyond the
17:37
strict arithmetic, and it centers back on that
17:40
crucial 90 % realization we discussed earlier.
17:43
Oh, the fact that almost all the present value
17:45
is locked inside that massive principal repayment
17:48
waiting patiently at the end of the maturity
17:50
schedule. Exactly. We essentially rely entirely
17:52
on that final envelope. Now, the math we perform
17:55
today is utterly flawless. Right. If you buy
17:58
a treasury bond, and you hold it to maturity.
18:01
The arithmetic absolutely guarantees that you
18:04
will receive your nominal dollars back. You will
18:07
get a deposit for exactly one thousand dollars.
18:09
Right. And it's often called a risk free return
18:11
for that exact reason. It is. But consider what
18:14
happens if the currency that principal is denominated
18:17
in undergoes a sudden systemic loss of purchasing
18:20
power right before that maturity date arrives.
18:23
Oh, wow. So like a massive unexpected spike in
18:26
inflation fundamentally devaluing the currency
18:29
itself. Exactly. The math ensures you get your
18:33
$1 ,000. But the actual real -world value of
18:37
that massive final payment, the groceries, the
18:40
rent, the tangible goods it can actually secure
18:43
for you, that can just vanish. The arithmetic
18:47
protects your nominal wealth perfectly while
18:50
leaving your real wealth completely exposed.
18:52
Oh, man. Proving that even the most pristine
18:55
mathematically risk -free investment has this
18:59
profound blind spot when it collides with the
19:01
messy reality of the global economy. Yes. Like,
19:04
you can calculate the present value down to the
19:06
penny, but you simply cannot secure its future
19:09
purchasing power. No, you can't. It is a very
19:11
sobering reminder that financial mathematics
19:13
measures numbers, but it does not inherently
19:16
guarantee value. What a fascinating place to
19:18
leave it. Thank you for taking this slow, deliberate
19:20
journey through the bedrock of fixed income with
19:22
us today. We really hope this deep dive provided
19:25
a new lens on how time and value intersect. Until
19:28
next time, keep gently questioning the math and
19:30
look closely at the true value of your time.