Chapter 3: Bond Valuation & Interest Rate Risk

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Welcome to the Deep Dive, where we take the crucial concepts you need to know in finance, strip away the academic jargon, and give you the foundational insight that sticks.

Today we are getting into the real engine room of corporate growth.

We're talking about the debt markets.

Specifically, we're focusing on bond valuation.

That's right.

I mean, think about it.

If a big company wants to grow, that costs serious money.

Building a new factory, upgrading a huge fleet of trucks, a multi -year R &D project.

That's significant capital.

Huge capital.

And if they can't just use their own profits, their retained earnings, and they don't want to issue new stock the owners.

They borrow long term.

They borrow long term.

And that means they issue bonds.

So our mission for this deep dive is really central to being a good financial manager.

We need to get a solid grasp on bond valuation.

Yeah.

And that means starting with the absolute foundation of finance, present value.

How are bond prices actually set?

But it's more than just the math, right?

We also need to get the dynamics.

How do those prices react when interest rates change or when the market suddenly sees more risk?

Exactly.

For a financial manager, that's everything.

It's the difference between managing your risk and getting completely blindsided by volatility.

And this isn't just for managers.

For an investor, it tells you the fair price to pay.

For the company, it tells you your cost of capital.

What it's going to cost to raise cash.

It's foundational, period.

Bonds are really just packages of promised payments.

And if you value those correctly, you can't value anything else.

Okay.

So before we jump into the, let's say the messy world of corporate bonds,

our sources say we need a starting point, a benchmark.

And that benchmark is always government bonds.

Things like US treasuries or in Europe, maybe the French OATs.

Why start there?

Because they're considered the safest.

Their interest rates are, you know, the baseline for all other interest rates in the economy.

So just to be clear on that point, a corporation will always have to pay a higher interest rate than the government.

Always.

There's more risk.

But the key is that the corporate rates will generally move in proportion to the government rates.

If the benchmark rate goes up, the company's cost of borrowing goes up too.

Exactly.

So for our roadmap today, we've broken this down into a few key areas.

We're going to start with that present value formula, get really comfortable with it.

Okay.

Then we'll get into interest rate sensitivity and a really important concept called duration.

After that, we'll look at the term structure, why a 30 year bond has a different rate than a two year bond.

And finally, we'll tackle the two big risks that every bond investor worries about.

Inflation and of course, the risk of default.

That sounds like a solid plan.

Let's jump right in and start with the basic anatomy of a bond.

Okay, let's start at square one.

In its simplest form, what is a bond?

What does it actually promise the person who buys it?

It's really just an IOU.

It's a contract.

And that contract promises you two main things.

First, you get regular interest payments.

Those are called the coupon.

You get those every year, or maybe every six months, right until the bond matures.

And then second, on that final maturity date, you get your last coupon payment, but you also get the full loan amount back.

And that's called the face value or the principal, or sometimes the par value.

I mean, the same thing in the US, it's almost always a thousand dollars.

Okay, let's use the European example from our material, the French OAT, because it keeps the math nice and clean with annual payments.

Perfect.

So let's imagine we're looking at an OAT with a face value of a hundred euros.

And let's say it has a 3 .50 % coupon and it matures in 2026.

Okay, so if we're looking at this from 2020, that's six years of payments.

Exactly.

Every year, you, the bondholder, get 3 .5 % of a hundred euros, which is three euros and 50 cents.

And then in that final year, 2026, you get your last 3 .50 coupon plus your hundred euro principal back.

For a final payment of 103 .50.

Correct.

So here's the core question.

We have this stream of future cash flows.

How on earth do we figure out what that bond is worth today?

We have to apply the most fundamental rule in all of finance.

Time value of money.

Time value of money.

The value of that bond is simply the present value, PV, of all those future payments.

And to calculate that present value, we need a discount rate.

Where does that come from?

That rate is the investor's opportunity cost of capital.

It's the rate of return you could get on another similar investment out there in the market with the same level of risk.

Okay.

Let's put a number on it to make it real.

Let's say the market rate for similar French government debt is now 5%.

Our bond only pays a 3 .5 % coupon.

How does the math work?

We have to discount every single one of those six cash flows back to today using that 5 % market rate.

So the first five payments are 3 .50 each.

And then that last payment is the big 103 .50.

Right.

So the formula looks like this.

You take the first 3 .50 and divide by one point zero down.

Then you take the second 3 .50 and divide by 1 .05 squared.

And you keep doing that for all six payments.

You're treating each payment as his own little mini valuation problem.

That's all it is.

And when you do all that math and add up all those separate present values, you get 92 .39.

So even though the bond has a face value of 100 euros, it's only worth about 92 euros today.

And that makes sense, right?

The market is demanding a 5 % return, but the bond's coupon is only 3 .5%.

The only way for an investor to actually earn that 5 % is to buy the bond for a lower price.

At a discount.

At a discount.

There's also a slightly easier way to think about this using the annuity shortcut.

Instead of six separate calculations, you can view the bond as two different pieces.

First, you have an annuity of the coupon payments.

So the six payments of 3 .50 are?

Right.

And second, you have a single lump sum payment of the principal, the 100 euro at the very end.

I see.

So you find the present value of the 3 .50 annuity for six years at 5%.

And then you just add the present value of that final 100 payment.

Exactly.

And when you do that, the PV of the annuity is 17 .76 and the PV of the final principal is 74 .62.

Add them together and you get 92 .39.

It's just a helpful way to visualize the bond as a package.

Okay, so now let's flip the question around, which is what happens in the real world?

We usually know the price.

If we know the price of this OAT is 92 .39%, what return will an investor actually earn if they hold it all the way to maturity?

Now you're asking if the yield to

YTM.

The YTM.

The YTM is the single interest rate, let's call it buy, that makes the present value of all the bond's cash flows exactly equal to its current market price.

So since we just calculated the price using 5%, if the market price is 92 .39, the YTM has to be 5%.

It has to be.

There are two sides of the same coin.

And in this case, the coupon rate, 3 .50%, is lower than the YTM of 5%.

Why?

Because you're paying less than the face value today.

You pay 92 .39 to get 100 euro back at the end.

That difference, that 7 .61, is a guaranteed capital gain.

So that capital gain gets added to your coupon payments and together they bring your total return up to that market required 5%.

Precisely.

And this leads us to the three main ways we classify bonds.

First, you have a discount bond.

That's our OAT.

The price is less than the face value, so the YTM is greater than the coupon rate.

Then you have a premium bond.

Right.

The price is above face value, say 105%.

In that case, the YTM has to be less than the coupon rate because you're guaranteed to have a capital loss over the life of the bond.

You paid 105 to get 100 back.

Exactly.

And finally, the simplest case,

a par bond.

Price equals face value and the YTM equals the coupon rate.

Now, we have to talk about something really strange that's happened in the real world, especially in Europe and Japan over the last decade or so.

Negative interest rates.

Yeah, this just breaks your brain a little bit.

The sources mentioned that in late 2020, something like 17 trillion dollars of debt was trading at a negative yield.

Which means you are guaranteed to get less money back than you put in.

So why would anyone do that?

Why would a huge corporation accept a negative return when they could just hold physical cash and get a zero return?

The answer is all about I mean, for you or me, storing $10 ,000 in cash is pretty easy.

For a giant corporation with billions in cash, it's a massive security and logistics problem.

Right.

There's that amazing anecdote about Pablo Escobar.

He was reportedly losing 10 % of his physical cash every year to rats and water damage.

10 % just because it was so hard to physically store and protect it.

Exactly.

So when you think about that, a 10 % loss on physical cash,

suddenly a big institution accepting a, say, minus 0 .4 % yield on a super safe government bond doesn't seem so crazy.

It's the lesser of two evils.

The bond becomes a secure, if slightly loss making place to park huge amounts of cash.

Right.

You're paying for safety and liquidity, not for a return.

And the math still works.

If the yield OKI is negative, the discount factor actually becomes less than one, which pushes the present value above the face value.

OK, before we move on, we have to touch on the specific way the US bond market works, because it's a little different.

The big difference is semi annual coupons.

They're twice a year.

Twice a year.

A US Treasury bond with $1 ,000 face value will pay its coupon in two installments.

And this leads to a really important rule that everyone has to remember, which is the YTM quoted on US bonds is twice the semi annual yield.

It's a true compounded rate.

So if a bond is quoted with a 4 % YTM,

you can't just plug 4 % into your annual formula.

You have to use 2 % as your discount rate for each six month period.

You have to.

You use Y divided by two.

This is just an old market convention, but it means the quoted YTM is not the same as the effective annual yield or EAY.

The EAY would be the true compounded return, which in that 4 % case would actually be 4 .04%.

Correct.

So you have to be really careful when comparing a US bond to, say, a European bond that pays annually.

You need to convert to EAY to do an apples to apples comparison.

This all gets summarized in the general bond valuation formula, equation 3 .1.

Right, which just says the present value is the sum of each cash payment, C divided by one plus YN raised to the power of the period.

The N is just the number of payments per year.

So for US Treasury, N would be two.

For example, the 2 .5s of 2024.

That's a 2 .5 % annual coupon on a thousand dollars.

So $25 a year, which means you get $12 .50 every six months.

Right.

And if the YTM is quoted at a tiny 0 .252%, the semi -annual discount rate you use is half of that.

So 0 .126%.

And if you have, say, eight of those half year periods left, you discount all eight of those $12 .50 payments plus the final thousand dollar principle.

At that 0 .126 % rate per period.

And you get a price of $1 ,089 .41, a big premium.

Which again reinforces the rule.

When interest rates are super low, bond prices are high.

And you have to pay very close attention to those compounding periods.

That's a perfect transition actually.

Because thinking about how small changes in yield can affect the price bring this to our next big topic,

volatility.

Right.

How much does a bond's price actually move when interest rates change?

We've already established the fundamental rule.

The inverse relationship, it's non -negotiable.

Bond prices and yields move in opposite directions.

Rates go up, prices go down, vice versa.

It's just baked into the present value formula.

Exactly.

And that brings us to interest rate sensitivity.

The key insight here, the one every manager has to know, is that long -term bonds are much more sensitive to interest grade changes than short -term bonds.

And they're more volatile.

Much more volatile.

If you think rates are about to rise, the last thing you want to own is a 30 -year bond.

I remember seeing the chart for this, the price yield curve.

For a 30 -year bond, the curve is really steep.

Whereas for a, say, 4 -year bond, it's much flatter.

That's the visual.

A small change in the yield, say from 5 % to 6%, causes a huge price drop for that 30 -year bond, but only a small dip for the 4 -year one.

Why is the effect so much bigger for the long -term bond?

It's all about the power of compounding over time.

A change in the discount rate doesn't have a huge effect on a cash flow you're getting next year.

But when you're discounting a cash flow that's 30 years away, that change gets magnified year after year after year.

Massively.

The distant cash flows, especially that big final principal payment, get hammered by a higher discount rate.

We saw a really extreme example of this in the real world back in 2008 during the financial crisis.

Oh, it was an incredible event.

Example 3 .2 in the text walks through it.

As the whole financial system seemed to be melting down, investors just panicked.

And they fled to safety.

The ultimate safe haven.

U .S.

treasury bonds.

This huge wave of demand sent long -term bond prices soaring.

There was this one 30 -year bond, the 4 .375s of 2038.

Its price went from 96 .38 % of par.

All the way up to 138 .05 % in just seven months.

That's almost a 46 % return on a government bond.

In seven months.

That's like a hot tech stock, not a boring bond.

It was unbelievable.

And if you include the coupon payment you received, the total return for an investor who timed that perfectly was 45 .5%.

It's a powerful lesson that even safe government bonds can have enormous interest rate risk.

Which in that case paid off spectacularly.

For the lucky few, yes.

Okay, so long maturity equals high volatility.

But you said earlier that maturity is actually a bit of a clumsy measure of risk.

Why is that?

Because of the coupons.

A 30 -year bond that pays a high coupon is returning your cash to you much faster than a 30 -year zero coupon bond.

So its effective life is shorter.

Exactly.

We need a more precise measure of the bond's average time to receiving its cash.

And that measure is called duration.

Sometimes you'll hear it called McCauley duration.

Okay, so let's use the example from the text to make this clear.

We have two 7 -year bonds.

Both have a 4 % yield to maturity.

But one has a low 3 % coupon and the other have a high 9 % coupon.

Same maturity, 7 years.

But which one is the longer -term investment?

It has to be the 3 % coupon bond.

Why?

Because the 9 % bond is paying out so much more cash in the early years.

It's front -loading the returns.

The 3 % bond, most of its value is tied up in that final principal payment way out in year seven.

Perfectly said, its effective life is longer, which means its price will be more volatile when interest rates change.

Now, there is a formula to actually quantify this, equation 3 .2.

There is.

And it can look a little intimidating, but the concept is really simple.

Duration is a weighted average time.

Okay, so what are we waiting?

You're waiting each point in time to, by the proportion of the bond's total value that comes from that time's cash flow.

Let me see if I get this.

You calculate the present value of, say, the year one coupon.

Then you see what percentage of the total bond's price that PV represents.

Right.

Let's say it's 6 % of the total price.

Then you multiply that weight 6 % by the time, which is one year, and you do that for every single cash flow and add them all up.

That's all it is.

Let's walk through the 9 % seven -year bond.

Its total price is $1 ,300 .10.

The first year's coupon has a PV of about $86.

That's about 6 .7 % of the total price.

So you take one year times $0 .067.

And you keep going.

But the big one is the final payment in year seven.

Huge.

The PV that final payment is over $828,

which is almost 64 % of the total bond price.

So you take seven years and multiply it by that big weight 0 .64.

And when you sum up all seven of those weighted time periods, you get the duration.

For the 9 % bond, it's 5 .69 years.

Which is, as we expected, shorter than its seven -year maturity.

And for the 3 % bond, the duration is longer, at 6 .40 years.

Confirming it's the more volatile longer -term investment, even though they both mature on the same day.

So duration is the true measure of a bond's term.

But for a manager, how do you translate that number of years into actual price risk?

That is the perfect question.

And the answer is a small but critical adjustment called modified duration.

Okay, what does that do?

Modified duration gives you a direct estimate of the percentage price change in a bond for a one percentage point change in its yield.

So it's the number managers actually use for risk management.

It's the go -to tool.

The formula is simple.

You just take the duration and divide it by one plus yield.

So for our 9 % bond, the duration was 5 .69 years and the yield was 4%.

So modified duration is 5 .69 divided by 1 .04, which gives you 5 .47%.

And what does that 5 .47 % mean practically?

It means that if the yield on that bond suddenly jumps up by 1%, you should expect the price to fall by about 5 .47%.

If the yield falls by 1%, the price should rise by 5 .47%.

Wow, so it's an immediate back of the envelope tool for quantifying your risk exposure.

It's incredibly useful.

Of course, in the real world, nobody does these long calculations by hand anymore.

You use a spreadsheet.

You use spreadsheet functions like price, YLD, duration, and M .D.

variation.

But you have to know the theory behind it to use them correctly and to understand what the output is telling you.

Okay, so far, we've been using a single interest rate, the YPM, to discount all the bond's cash flows.

But we all know from watching the news that the interest rate for a one -year loan is

almost never the same as for a 30 -year loan.

Exactly.

This is where we move from a simplified model to how the market really works.

We have to start thinking about the term structure of interest rates.

Which means we're moving away from a single discount rate and toward a whole series of different rates.

A series of rates for different maturities.

And the proper term for one of these rates is the spot rate.

The spot rate, let's call it a TERT.

Right.

It's the interest rate today for a risk -free payment that you're going to receive at a single specific point in the future, time and T.

So there's a one -year spot rate, a two -year spot rate, a three -year spot rate, and so on.

Correct.

And that entire series of spot rates is what we call the term structure.

So if I'm valuing a two -year bond, I can't just use one rate anymore.

I have to use the one -year spot rate to discount the first coupon.

And the two -year spot rate to discount the second coupon and the principal.

Each cash flow gets discounted by the spot rate that corresponds to its specific maturity.

So once I use all the correct spot rates to find the bond's correct price, then I can work backwards to figure out what single rate the YTM would give me that same price.

Now you've got it.

The YTM is an output, not an input.

It's just a complicated average of all the underlying spot rates.

This is a huge insight.

So many people think the YTM is the fundamental rate, but it's not.

The spot rates come first.

They do.

And this is why professional valuation experts always use the spot rate curve for accurate pricing, especially if the curve is really steep or inverted.

The YTM can be a really misleading average in those cases.

But then why does everyone from financial news to fund managers still quote the YTM all the time?

It's convenient.

It's a single number that gives you a quick summary.

But for doing serious valuation, especially for complex instruments, relying on YTM alone is, as we'll talk about later, a very dangerous oversimplification.

Okay.

So if spot rates are the true foundation, how do we actually find them?

Most bonds have coupons, which makes it risky.

We need to find bonds that only make one single payment.

And those exist.

They're called stripped bonds or just strips.

Zero coupon bonds.

Right.

Since they only pay out at maturity, their price gives you a direct reading of the spot rate for that specific maturity.

So if a five -year strip promises to pay you $1 ,000 in five years and it costs $950 today, that ratio, 950 divided by a thousand, gives you the five -year discount factor.

And from that, you can solve for the five -year spot rate or five.

In that case, it would be about 1 .03%.

So if you have a market for strips at every maturity, you can map out the entire term structure.

You can perfectly map it out.

And the reason this works so well is because of a core principle of efficient markets.

The law of one price.

The law of one price.

It just says that two identical assets have to sell for the same price.

In this world, it means that any risk -free that's promised on say April 15th, 2030 has to be discounted at the exact same spot rate, no matter which bond it comes from.

And if that law gets violated, even for a second, you open the door for.

Arbitrage.

A risk -free profit.

The proverbial money machine.

The source material has a great conceptual example of this.

It's a little extreme, but it makes a point.

Imagine the one -year spot rate is 20%, but the two -year spot rate is only 7%.

Yeah, that should immediately set off alarm bells.

It's saying that money you get in one year is worth less today than money you get in two years.

That makes no sense.

So what does the arbitrage do?

She makes two trades simultaneously.

First, she buys a one -year strip.

She spends say $833 today to get a guaranteed $1 ,000 in one year.

At the same time, she borrows money for two years at that cheap 7 % rate.

She can borrow $873 today in exchange for promising to pay back $1 ,000 in two years.

Wait, let's track the cash today.

She spent $833, but she took in $873 from the loan.

Right.

So she has an immediate risk -free profit of $40 in her pocket right now.

And what happens in the future?

Well, in one year, she gets the $1 ,000 from the strip she bought.

She just sits on that cash for a year.

And then in year two, she uses that $1 ,000 to pay back the $1 ,000 loan.

Her future cash flows are a perfect wash.

She has zero net obligation, but she made $40 today for free.

It's a money machine.

And in real functioning markets, computers would spot that and trade on it so fast that the opportunity would disappear in milliseconds.

The prices would adjust until it was gone.

So arbitrage is what forces the term structure to be consistent.

Now let's talk about the shape of that structure.

It's usually upward theories that try to explain this.

The first one is the expectations theory.

It says that the return from investing in one long -term bond should be the same as the expected return from investing in a series of short -term bonds and just rolling them over.

So an upward sloping curve can only exist if the market expects short -term interest rates to rise in the future.

That's the core idea.

Let's use the example.

If the one -year rate is 5 % and the two -year rate is 7%, the theory says that has to be because the market expects the one -year rate next year to be much higher.

How much higher?

Well, the math shows that for the returns to be equal, the expected one -year rate for that second year has to be 9 .0%.

That expectation is what's priced into today's 7 % two -year rate.

But that theory alone doesn't seem to cover it because historically, long -term bonds have almost always paid more than short -term bonds.

Exactly.

Which suggests there's persistent reward for holding them.

And that brings us to the second theory,

interest rate risk.

We just talked about this with duration.

Long -term bonds are way more volatile.

And investors hate volatility.

So to convince them to hold a risky 30 -year bond instead of a safe one -year T -bill, you have to offer them a higher expected return, a risk premium.

A liquidity risk premium.

Right.

That premium helps create that upward slope.

Even if nobody expects future rates to change at all, you're just getting paid to take on more risk.

And what's the third factor?

Inflation risk.

The uncertainty about future purchasing power.

Precisely.

If you buy a 20 -year bond, you have no idea what inflation will look like in 15 or 20 years.

Your real return is very uncertain.

Whereas if you stick to short -term bonds, you can just roll them over at whatever the new higher interest rates are if inflation suddenly spikes.

You're protected.

So long -term investors demand an extra bit of yield and inflation risk premium to compensate them for that uncertainty.

Those two premiums together are why the term structure is almost always sloping upwards.

That discussion of inflation risk is a perfect segue into our next topic.

We have to be really precise about the difference between the dollars a bond promises you and what those dollars can actually buy.

This is the critical distinction between nominal and real values.

Okay, so nominal dollars or nominal rates.

That's just the actual amount of money on the contract.

It's the stated amount what the piece of paper says you're going to get.

And real dollars or real rates are about purchasing power.

They're adjusted for inflation.

Right.

What can you actually go out and buy with that money?

That's what really matters.

So let's talk about converting these.

Let's say I get a nominal payoff of $1 ,100 one year from now.

But over that year, inflation was 6%.

What's the real value of that money?

The formula is pretty simple.

You just take the nominal cash flow and divide it by one plus the inflation rate.

So $1 ,100 divided by 1 .06.

Which gives you a real value of $1 ,037 .74 in today's purchasing power.

So my actual wealth only went up by about $38, not the $100 I thought it did.

Exactly.

And we can do the same conversion for the interest rates themselves.

The formula is that one plus the real rate equals one plus the nominal rate divided by one plus the inflation rate.

Okay.

So with a 10 % nominal rate and 6 % inflation, we get 1 .7 Wagarow divided by 1 .06.

Which is 1 .03774.

So the true real rate of return is 3 .774%.

Now, I always hear the shortcut, the rule of thumb, that you just subtract inflation from the nominal rate.

So 10 % minus 6 % gives you 4%.

That's pretty close.

It's a decent approximation when rates are low.

But the higher the rates, the less accurate that shortcut becomes.

For professional work, you should always use the full formula.

Okay.

So since most regular bonds have this problem, they pay fixed nominal amounts.

So your real return is uncertain.

Governments came up with a special type of bond.

They did.

In the U .S., they're called TIPTS, which stands for Treasury Inflation Protected Securities.

And what's special about them?

They're designed to give you a fixed real return.

The way they work is that both the principal value and the nominal coupon payments actually adjust up or down with the CPI, the official measure of inflation.

So if I own a TIPTS bond and inflation is 10 % one year, my principal amount gets adjusted up by 10 % and my coupon payment for that year also gets a 10 % boost.

Correct.

Your nominal payments rise with inflation so your purchasing power, your real return stays constant.

So the yield on a TIPTS bond,

that is the real yield to maturity.

It is.

And this is incredibly useful for analysts because you can compare the yield on a TIPTS bond to the yield on a regular nominal Treasury bond of the same maturity.

And the difference between those two yields tells you what the market's expectation for future inflation is.

Exactly.

It's the market's best guess.

The text mentions that in May 2020, a 10 -year TIPTS was yielding negative 0 .45%.

A negative real yield.

Right.

While a regular 10 -year Treasury was yielding positive 0 .65%.

The difference between them was about 1 .1%.

So the market was pricing at about 1 .1 % inflation per year for the next decade.

That's what it implied.

But that negative real yield is fascinating.

It means investors were willing to accept a guaranteed loss of purchasing power just to own a super safe inflation protected asset.

Why would they do that?

Are they just paying a massive premium for safety in a crisis?

That's a huge part of it.

In a world of uncertainty, a guaranteed liquid inflation hedged asset is incredibly valuable, even if its return is negative.

It's about capital preservation, not high returns.

Okay, so finally, what actually determines what the real rate of interest should be?

The classic theory here comes from the economist Irving Fisher.

And what was his argument?

He said, the real rate depends on the fundamental supply and demand for capital in the economy.

The supply comes from people's willingness to save, and the demand comes from companies'

opportunities to invest in profitable projects.

So if there's a tech boom, and suddenly there are tons of great investment opportunities,

the demand for capital goes up.

And the real rate has to rise to encourage more people to save and provide that capital.

And what was Fisher's big insight about how this real rate connects to the nominal rate we see in the market?

His core idea is that a change in expected inflation causes an identical change in the nominal interest rate, leaving the required real rate unchanged.

Let's use the Apple analogy from the text.

If I want a 3 % real return, that means I'm willing to give up 100 apples today to get 103 apples next year.

If there's no inflation, the nominal interest rate is 3%.

Simple enough.

But now, let's say everyone expects the price of an apple to go up by 5 % over the next year.

So to get my 103 apples next year, I'm going to need more dollars.

A lot more.

You'll need enough dollars to buy 103 apples at their new 5 % higher price.

The math shows you'll need about 8 .15 % more dollars.

So the nominal interest rate has to jump from 3 % to 8 .15 % just to keep my real return at 3%.

Exactly.

The nominal rate rises almost one for one with the change in expected inflation.

And historically, the data shows this relationship holds up pretty well.

The returns on treasury bills have tended to track the inflation rate very closely over the long run.

Everything we've talked about until now has been based on risk -free government bonds.

But now we have to get into the real world for most financial managers, which means dealing with corporate bonds.

And that introduces a huge new complication, default risk.

The risk that the company just won't be able to pay you back.

Right.

A company can go bankrupt.

Their promises are conditional.

And because of that risk, investors are going to demand a higher yield to compensate them for the chance they might lose their money.

The range of this risk is just incredible.

The text notes that in late 2020, a bond from a super stable company like Johnson & Johnson was yielding about 1 .12%.

Very low risk.

But on the other end, a bond from AMC Entertainment, the movie theater chain, was yielding a jaw -dropping 36 .01%.

And that 36 % isn't a gift.

That's the market screaming that there is an extremely high probability that AMC is going to default and that bondholders are going to lose a massive chunk of their investment.

So how do investors and managers get a handle on this huge spectrum of risk?

How do they categorize it?

They rely heavily on bond ratings from the big rating agencies.

Moody's, S &P, Fitch.

Exactly.

These agencies do a ton of research on a company's financial health to judge its credit worthiness.

And they assign a rating like AAA, AA, and so on.

Right.

The really important dividing line is between investment grade bonds, those rated BBB or higher, and what are called speculative grade or junk bonds.

Or sometimes you hear the term high yield bonds.

Investment grade means the default risk is thought to be low.

Junk means there's a significant chance of default.

But even an investment grade rating is no guarantee, is it?

Absolutely not.

The classic cautionary tale is WorldCom.

In 2001, WorldCom was investment grade, and they issued billions in new debt.

And less than a year later.

They filed for bankruptcy after a massive accounting scandal was uncovered.

Bondholders lost over 80 % of their money.

It's a reminder that a rating is just an opinion about risk, not a promise of safety.

The market has its own way of pricing this risk, which we can see in the yield spreads.

The yield spread is just the difference between a corporate bond's yield and the yield on a safe government bond of the same maturity.

And since the corporate bond is always riskier, that spread is always positive.

Always.

And the size of the spread tells you everything about the perceived risk.

And AAA -rated bond will have a tiny spread.

But a BBB -rated bond, the lowest investment grade, historically has had a spread of around 2%.

And in a crisis, like in 2008, those spreads can blow out.

They exploded.

The spread on BBB bonds shot up to over 6 % because investors were panicked and just dumping anything that wasn't a treasury.

Is that spread purely about default risk, though?

Not entirely.

A big piece of it, especially for bonds from smaller companies, is a liquidity premium.

Meaning the bonds are just harder to sell.

Much harder.

They don't trade very often.

So investors demand a higher yield just to compensate them for the hassle and cost of trying to sell that bond quickly if they need to.

Okay.

Last thing.

We have to touch on sovereign bond default.

We've been calling U .S.

government debt risk -free.

But that's not true for all governments.

Not at all.

And the risk really depends on what currency the government is borrowing in.

Okay.

So what's the first scenario?

Foreign currency debt.

This is when a country like Venezuela borrows in U .S.

dollars.

They can't print U .S.

dollars.

So their ability to pay depends entirely on their ability to earn dollars through taxes and exports.

And if they run out, they default.

There have been over a hundred of these kinds of sovereign defaults since the 1970s.

The second scenario is when they borrow in their own currency.

Own currency debt.

Here, default is much less likely because they can always just print more money to pay the debt.

But that creates a different problem.

Hyperinflation.

Right.

So a country might face a terrible choice.

Default on your debt or bring so much money that you destroy your entire economy.

Russia chose to default on its ruble debt in 1998 to avoid that fate.

And the third case is the unique situation in the eurozone.

Eurozone debt.

This is a tricky one.

A country like Greece or Italy uses the euro, but they can't print euros.

Only the European Central Bank can do that.

So they're in a similar position to a country that borrows in a foreign currency.

Very similar.

It makes them financially constrained.

And we saw what happened in 2012 when Greece was on the brink of bankruptcy.

They were forced to write down a hundred billion dollars of their debt, imposing huge losses on their bondholders as part of a bailout deal.

So whether it's a company or a country, at the end of the day, the yield investors demand is a reflection of a really complex assessment of the odds and the ways that it could fail to pay you back.

Hashtag tag outro.

So we've covered a huge amount of ground today from the basic math of present value all the way to the complexities of sovereign default.

And it seems like there are three really essential principles that a manager or an investor has to take away from all this.

I think that's right.

The first is just that bond valuation is all about present value discounting.

You have to use the general formula, but you also have to be really aware of the market conventions like semi -annual coupons in the U .S.

Okay.

That's number one.

What's the second essential?

Interest rate risk is not about maturity.

It's about duration.

Duration is the true measure of a bond's effective life.

And modified duration is the tool that tells you exactly how much your bond's price will change when rates move.

It's critical for risk management.

And the third principle.

That interest rates themselves are complex.

They're built up in layers.

You have the fundamental real rate.

Then you add a layer for expected inflation.

And then you add more layers to compensate for all the different kinds of risk default risk, liquidity risk, inflation risk.

Which brings us back around to our final thought for today.

It's something we touched on earlier.

We now know that the yield to maturity, the YTM, that single number everyone quotes, is really just a messy average of all the underlying spot rates.

It smooths over all that important information about the term structure.

And it completely hides the detailed risk dynamics that are captured by duration and all those different risk premiums.

So given everything we've learned about the volatility of long -duration bonds, the uncertainty baked into the term structure, and the need to separate default risk from liquidity risk,

when is relying on that single, simple YTM quote a dangerous oversimplification for a financial manager?

The answer is probably, almost always.

The more you understand what that single YTM number is hiding, the better you're going to be at actually managing financial risk in the real world.

A perfect place to leave it.

Thank you for joining us for the Deep Dive.

We trust this exploration provides a solid and actionable foundation for your understanding of debt markets and sound financial decision -making.