>>Chapter Seven: Bonds and Interest Rates. In this chapter we are again continuing what

we did in chapters five and six. The last part of chapter six was on different

types of loans. In chapter seven, we will continue looking

at loans. Let me start with the big picture of what

chapter seven and then chapter eight will be on. Chapter seven is on a real-life example of

an annuity. So we will see how the annuity calculations

feed into a specific type of loans called bonds. Bonds are loans that large corporations have

in order to raise money for their projects. And then in chapter eight that will follow,

we will look at a real-life example of a perpetuity where our perpetuity calculations can be used. And that’s the topic of stocks or stock shares. Bonds are financial securities sold by companies

or governments. To borrow money from the public on a long-term

basis. So think about how you can borrow money to

buy a car, right? You can, you know go to your bank and ask

for a loan for let’s say $20,000. Or maybe at the car dealership, they have

a financial services department which can also offer you a loan. Right? Well imagine now a large corporation. A large, a large corporation may want to borrow

one billion dollars, right? Maybe it would like to acquire its competitor. Do you think a large corporation that needs

one billion dollars can walk into a local bank and ask for that kind of money? It’s probably not going to happen, right? Because it’s a lot of money. And so imagine if there was a way for that

large corporation that needs one billion dollars for some major project. Imagine if it was possible for it to borrow

a little bit from you, a little bit from me. A little bit from whole bunch of people in

this country. Or maybe in other countries. And then, all those small loans would add

to a total of one billion dollars, right? So this is exactly how large corporations

can borrow money. In this world. They sell whole bunch of bonds. Each bond is sold for some amount of money,

relatively small. Let’s say around $1000. Plus/minus. And by selling a lot of such bonds, the company

is able to borrow the large amount of money that it needs. It could also be a government like local California

government or federal government that needs to borrow money. And it, this can also be done by issuing bonds. So bonds is like a long-term loan. So let’s say company that sells bond is essentially

borrowing money from whoever bought the bond from the company. And in return, the company will be paying

it off with interest on top like with any loan. Over let’s say ten years, maybe 20 years. It could be shorter than that, but generally

it’s a long-term, you know commitment for the company. Bonds are also known as debt securities because

they represent a company’s debt. A company owes money to those investors who

bought its bonds. And it will be making payments back that will

also reflect interest for up to 30 years since the issue. So the seller of bonds which could be a company. It could be government. Is also known as a debtor or a borrower. And those investors or maybe other companies

or other governments that buy these bonds. They’re lending money, right? To the issuer. So they’re called lenders or creditors. This is an example of America’s debt. So it’s a little bit old. It goes back to May 2016. A figure I found on CNN.com. Who owns America’s debt? So this is federal debt that the federal government

issued as bonds. And these bonds were sold to whoever would

like to purchase them. And the buyers came from all over the world. So who owns America’s debt for that time? Which is $12.9 trillion dollars? China is the number one lender, $1.3 trillion. Then goes Japan, $1.1 trillion, and then whole

bunch of other countries. And they’re other, you know investors inside

the United States which make up the rest of this large $12.9 trillion dollar amount. So this is kind of to give you an idea of

who may be you know the people. Or you know the agencies, the institutions

who buy bonds. Issued by let’s say government. Although in this chapter, we will be primarily. You know all our examples will be about corporations

rather than governments. So let’s say company or a government borrows

money from the public by selling bonds, right? And so the price at which each bond is sold

at is exactly how much was borrowed by that company or government. With each bond sale. And in return, kind of like with interest-only

loans that were covered in the previous chapter. The corporations or the government will be

paying back you know regular loan payments that reflect interest. It’s actually more like interest-only loans. So the company of the government will be paying

interest periodically. And at the very end, it will also pay the

borrowed amount in one large chunk. Bonds have their own terminology. Face value or par value or simply par. Coupon rate, coupon payment or simply coupon. Maturity date or time to maturity. And yield to maturity. Or simply yield. So let’s discover on the next slide what these

terms refer to. The payments that pretty much reflect the

interest payments, only borrowed amount of money are known as coupons or coupon payments. When the last coupon is made, at the same

time the borrower will also pay, you know. The borrowed amount of money which is known

as the face value. It’s actually may or may not be exactly equal

to how much was borrowed. It’s time zero. We will see that in the next examples. The time between today and when the last payment

will be made is known as the time to maturity. Abbreviated as TTM. Face value can also be called the principal

value or the par value or simply par. In most of the par examples, we will be using

$1000. So even if it’s not given, you should always

assume that the face value is equal to $1000. And this is true for most corporations and

the government bonds. Some government bonds have a much larger face

value. But unless specified, you should assume $1000. Next, how are coupon amounts calculated? There’s also one explanation that is given

known as the coupon rate, some percentage, let’s say five percent. You take that percentage, the coupon rate

and multiply it by the face value. Which as was just explained is usually $1000,

right? So let’s say the coupon rate was five percent. You multiply it by $1000 you get $50. So that’s the coupon that will be paid at

the end of the first, at the end of the second year, third year, and fourth year. Every year the same dollar amount. And one more thing that is typically given

for bonds. That’s the discount rate or IY in the financial

calculator. When we talk about bonds, we call that yield

to maturity. Abbreviated as YTM. Or simply yield. So this is also something I found on Morningstar.com. It’s also a little bit old, about year or

so old. It shows the capital structure of Starbucks. For that time. The orange area reflects what is known as

equity. And the dark blue area reflects the amount

of debt. So this figure shows that in March 2017, the

proportion of debt. How much money Starbucks owes to others is

a little over 41% of total money that it has. On its hands. And the historical proportions of debt to

equity are similar to that. Now let’s look at a little bit more detail

on, you know debt. What exactly it looks like for Starbucks. So at that time, March 2017, this is the kind

of bonds that Starbucks had. And notice that I didn’t say bond. It’s bonds. And by plural I don’t mean that there are

many bonds. There are actually many different types of

bonds. And each type has many bonds, right? Like the count. For example, we see one, two, three, four,

five, six, seven. Seven rows in this little table. These are different types of bonds. For example the percentage here, 3.85%, is

the coupon rate, right? It’s also repeated in the coupon percentage

column: 3.85%. The second row shows 2.1% coupon rate, right? So you would take this percentage and multiply

by $1000 face value. That would give you the coupon amount paid

every year. Then maturity date, another column. For example, the first type of bonds matures

on October 1, 2023. This means that this is when the very last

payment will be made by Starbucks to the bond holders. In the second row, the maturity date is February

4, 2021. Again, that’s when the last payment would

be made. Then we also see different amounts in millions

of dollars, right? So there were different amounts of money borrowed

at different points of time. And in the maturity date column, we see when

the last payment would be made for each group of bonds. And so on. So there are you know different characteristics

of this. One, two, three, four, five, six, seven different

groups of bonds that Starbucks owns. And in the last column you can also see how

the yield to maturity which is the discount rate or IY differs for all these different

categories. For example, you may notice that the yield

to maturity is by far the highest, 3.99% for the bonds which have a very long-time left

until they mature. Which is the year 2045. And this is probably because there’s so much

uncertainty about what will happen in the economy. Between now and the year 2045. That whoever buys such bonds [inaudible] need

to basically lot of uncertainty. Between now and the year 2045. And because of so much uncertainty, or in

other words, risk. Those bond buyers require a compensation which

comes in the form. Of a higher return on their money. Which is essentially the yield to maturity

on such bonds. So there are like a lot of stories that could

be told when we compare different groups of bonds. So now let’s look at like a different side

of bonds. So let’s say today there is a company that

sells bonds. So again, let’s talk about, let’s continue

talking about Starbucks. So Starbucks decided to sell bonds. Why? Because it needs to borrow money for something

that’s important for its business. Let’s say it sells bonds to Mr. Connery. Then Mr. Connery he bought these bonds. Maybe they have 20 years to maturity. But he’s actually not obligated to hold these

bonds for such a long time. He’s free to sell these bonds at any time

he likes. And let’s say at some point he sells his bonds

and their new buyer is Mr. Brosnan. And then Mr. Brosnan again can sell his bonds

at any time he likes. And let’s say they end up in Mr. Craig’s hands. And so on. So bonds switch hands. The ownership changes, you know all the time. Have you noticed the strange last names? Well, remember, James Bond? Hint, hint? Bonds. That’s what this chapter is about, right? Okay, so back to financial securities, right? So what am I trying to say with this, you

know sort of timeline? What I’m trying to say is that bonds can be

resold. Somebody sells them and other investor buys

them, right? One thing remains. When these bonds mature. When the last payment would be made by Starbucks,

to the bond holder. Another thing that remains constant throughout

the years is the coupon rate. Let’s say five percent that determines the

coupon payment every year. As well as the face value. Which is the additional dollar amount that

will be paid at maturity on the very last date. But one thing that will be changing throughout

these years is the price on the bond. Right? So maybe Mr. Connery sold the bond for $900. But Mr. Brosnan then sell these bonds for

$1200 to Mr. Craig. So the price fluctuates. Okay, so now let’s look at the price. What determines that price on the bond? Well the price should reflect how much a bond

is worth or the bond value. .And each bond should be worth no more than

how much money it brings you in the future, right? So you would never want to pay more than what

you’re expecting to get from it. From the company that issued the bonds. So what is the bond value? How much should each bond sell for? How much money does a firm raise or borrow

with each bond? These three questions ask you to calculate

the same exact thing, right? So we usually call it bond value, calculating

bond value. Or bond valuation. And like was shown earlier, what you get in

the future when you buy a bond is annual coupon payments as well as the face value at the

very end. And you continue receiving those coupon payments

until maturity. So you look at the amount of time that’s left. And then that becomes part of our calculations. So what are the calculations that we need

to do? Well, the coupons are constant. They will be repeated every year, and they

will eventually end. This is the definition of an ordinary annuity. So we have that as part of the cash [inaudible]

from the bond. And we also have the face value at the very

end. Like a single cash [inaudible], right? You know sort of by itself. So to find how much you would be willing to

pay. Or the maximum you would be willing to pay

for a bond today. If you are on the investor side, on the buyer’s

side. And if you’re on the firm’s side that’s selling

the bond. The question could be rephrased as how much

the firm should sell each bond for. So that becomes nothing but the present value

of all the coupons. The annuity of coupons, plus the present value

of the face value at the bond’s maturity. So we just add up the two sort of components

of the bond value. And so the bond value is equal to the sum

of the present value of the annuity of coupons. And the present value of the face value. At maturity. And the formulas are from you know the annuity

formula from chapter six for the coupon part. And for the face value part, it’s kind of

going back to chapter five. Discount and a single cash load, back to time

zero. Okay, let’s do our first numerical example

on bond valuation. Golden Eye Company, that’s one of the James

Bond movies, right? Issued bonds with a coupon rate of ten percent

in annual coupons. The par value is $1000 and the bonds have

five years to maturity. The yield to maturity on similar debt issued

by similar companies is 11%. What is the value of each bond? This question could be put in different words. How much money can the company raise with

each bond? Or how much should the company’s bonds sell

for in the market? It’s all one and the same thing. So ten percent coupon rate determines that

coupon amount every year. The face value is given as $1000 which is

the same thing as the par value. These could be completely [inaudible] in which

case you would assume that the face value or the par value is $1000. And another important thing that I haven’t

mentioned yet. Is what’s important for bond valuation is

how many years are left until maturity? For example, maybe these bonds were sold originally

seven years ago, right? Which means seven years until now plus another

five years from now until maturity, makes it originally 12-year bonds. But we don’t use 12 years for our bond value

calculations. But we only use how many years remain until

maturity. That’s important. So always make sure you’re using the number

of years between today and maturity date. The remaining time until maturity. Okay, so the coupon amount is always calculated

the same way. You take the coupon rate, and you multiply

them by the face value. Both are given, .1 coupon rate multiplied

by $1000 face value. That gives the annual coupon amount of $100. And we have now everything we need to compute

the bond value. We can either plug in the numbers into the

formula or use the financial calculator. So let’s see how that’s done. If you use the formula, then you will $963.04. And now the formula. The formula is, sorry. Now, the financial calculator. The financial calculator uses the same keys

as an annuity. You enter the coupon amount, $100 is the PMT. And so on. But one extra thing that needs to be entered

now is the $1000 future value. Okay, so let me bring up the financial calculator. And explain how we solve this kind of problem. This is the financial calculator. Turn it on. It’s always a good idea to clear everything. Even if you used the calculator a few days

ago. Second, plus/minus/enter does a full reset. Now we are ready to go. Maybe we also want to increase the decimal

place. So let’s do that. Second decimal place. Let’s make it not two but maybe six. I press six, enter, it’s done. And if I turn the calculator off and turn

it back on, it’s keeping six decimal places. Okay, now let’s begin. Again, in no particular order as always. We enter the information that is given. And now there are four numbers that need to

be entered. With annuities, in chapter six it used to

be three numbers: N, IY, and PMT. Now there is an extra number that needs to

go into the calculator. The face value or the future value, $1000. So our coupon payments will be $100 each every

year. You put 100 as usual, we make it negative

and save as PMT. Then not only will we be receiving hundred-dollar

coupon payments if you buy such bond. But we are also going to receive $1000 at

the very end. So let’s enter that as our next piece of information. One thousand. And because it’s, just like coupons, received

by the bond holder, you use exactly the same sign. So we made PMT negative. You’re also going to make your future value

negative. So you press the plus/minus key. And then you press FV. So FV stands for future value. But here it’s also the face value when you

talk about bonds. And then in no particular order, you enter

the discount rate, 11% which is also known as the yield to maturity. When you talk about bonds. So I press 11 IY, and five, four, N. Five

is how many coupons there will be between now and the maturity date. So five N. If we want to compute the present value of

this bond: $963 and a little bit more than four cents, right? So the new thing here is entering the face

value which is FV, future value. And it’s always $1000. And the important thing is, like I explained

on this slide, both the PMT. The coupon amount and the FV the face value. Must be entered with the same sign. So I showed how you enter both as negative. What if you didn’t? What if you left them as positive? The only thing that would be different is

that when you get your bond value, the answer, right? On the display, it would be displayed with

a negative sign. Otherwise it would still be the same dollar

amount, $963.04. So let’s continue. In chapters five and six, we saw that whenever

the interest rate, or IY, is higher, the present value is lower, right? So there was this negative relationship between

the discount rate or the interest rate and the present value. And it remains true when we talk about bonds,

right? Why would it be any different? It’s the same. So IY, the interest rate is called the yield

to maturity. And the present value in the context of bonds

is the bond value, what we just did. So there is again a negative relationship

between the yield to maturity and the bond value. So let’s actually you know kind of prove it. What we just did was an example with the yield

to maturity, the interest rate, right? Of 11%. We have five years to maturity, $1000 face

value on the bonds. Ten percent coupon rate. And the present value that we found earlier

was $963.04. What if we lower the yield to maturity, the

interest rate, from 11% to 10%? The negative relationship tells us that, that

should make the bond value higher. And if we repeated the financial calculator

keys and only changed the IY from 11 to 10. We would get the higher bond value, and that

would $1000. If you make the yield to maturity the IY in

the financial calculator even lower, then the bond present value will become even higher. Because of the negative relationship between

the two. And if we redid the calculations in the financial

calculator. We would, we would get the following answer:

$1079.85. That’s the bond value at eight percent yield

to maturity. Right? So as we lower the yield to maturity, we increase

the bond value, so that’s a negative relationship. Just like what we saw in chapters five and

six. Where the single cash flow or be it multiple

cash flows. Now notice how in the second example on this

slide with the 10% yield to maturity. The bond value became exactly $1000. Is it a coincidence? No, turns out that whenever the coupon and

the yield to maturity are identical, right? In our example, they are both ten percent. You will always get the bond value of $1000. You will always have $1000. Even if the coupon rate is let’s say 15% and

the yield to maturity’s 15%. As long as they are the same, you will always

be getting $1000. So if you know this trick. If you’re given the same percentages for the

coupon rate and the yield to maturity. Don’t even spend your time calculating the

bond value. One thousand dollars is your answer. We also see that the bonds that sell at $1000

which is their face value or par value. These bonds sell at par. If they sell for less than their par value,

then they’re known as discount bonds. Right? So $963.04 is a discount bond. So it’s like if you’re an investor and you’re

buying such bond. Which was issued by some company, you’re kind

of buying it at a discount. Because when in the future you’ll be receiving

your interest payments known as coupons. And then the last lump-sum payment at the

very end which is $1000 it’s kind of like, you know you’re borrowing. You’re lending less than what you’re getting

back. So you’re lending $963 but you’re getting

back $1000 in future value, in face value. Together with all the interest until then. If instead the bond present value is above

$1000 like in the last case. With the eight percent yield to maturity,

right? One thousand, seventy-nine dollars and eighty-five

cents. That kind of bond is called a premium bond. And just to repeat again, whenever the yield

to maturity or IY, right? Like the discount rate is above the coupon

rate, you’re always going to get the bond value below $1000. That’s a discount bond. If the yield to maturity and the coupon rate

are the same percentage, you will always be given $1000. These bonds are said to sell at par. And premium bonds are the ones that are worth

more than $1000. And that’s always going to happen when the

yield to maturity is less than the coupon rate. So this is like you know a little rule that

you could always keep in mind when you do the calculations. So why does it even? Kind of let’s, let’s make a step back. What we just did was calculating the bond

price or bond value for different discount rates. Why did we actually have to do all that? And that’s because in the real world, interest

rates are always fluctuating. We are going to cover this towards the end

of this chapter. Why, you know there are different things that

are happening. That are, that get reflected in interest rates

and loans. So at this point, let’s just, you know take

it as given. Interest rates fluctuate over time. And because they do, bond value will also

constantly fluctuate over time. And what this in turn implies is that if you

bought the bond yesterday and paid some amount of money for it. Then maybe if the interest rate went up overnight,

then your bond value went down overnight. So what you paid a lot of money for yesterday

is not worth as much money today. So as an investor, because of these constant

fluctuations in interest rates on bonds. You’re facing this risk of your bond value

going down. Unexpectedly. Technically in finance, we would, we would

use the word risk when the price unexpectedly goes up. So any sort of fluctuations are referred to

as risk in finance. And we call it interest rate risk. So people who buy bonds, issued by corporations

or governments. They face this so-called interest rate risk. So it’s the fact that the value of the bonds

on their hands will be fluctuating. Whenever interest rates change. Turns out that when you buy bonds, right? So now you’re a bondholder, you own bonds. So turns out that your bond value will be

going up and down, right? So fluctuate significantly more if the bonds

that you hold have a longer time to maturity. Or if they have a lower coupon rate. So let’s see [inaudible] behind this, you

know two factors. That affect the interest rate sensitivity

of bond value. Let’s consider this figure. This is figure 7.2 in our textbook. On the horizontal axis we have the interest

rate. That’s the yield to maturity. The IY in the financial calculator. On the vertical axis, we have bond to value

in dollars. Then we know that the relationship that when

the interest rate and the bond present value is negative, right? We just had it on one of the earlier slides. And that’s why we see the line sloping down

and to the right. Now why are there two lines? Why not one? Because we are comparing two bonds. The green line is a down sloping line for

a bond that has 30 years left to maturity. It’s a 30-year bond. The blue line which is a lot less steep. Is for a different bond which has only one

year left to maturity. So it’s kind of you know a very long-term

bond. And a very short-term bond. What do we see going on here? Imagine you changed your interest rate, the

yield to maturity. The IY in the financial calculator. So you change it let’s say from five percent,

increase it to 10%, then increase it to 15%. Then increase it to 20%. And recalculate the bond value for each of

the two bonds. When you do that, so you will be getting new

numbers for the bond values, right? If you plot the results on this figure, you

would get these two lines. The blue line and the green line. The blue line for the one-year bond which

is almost flat. And the green line for the 30-year bond which

is a lot steeper. Right? So it kind of comes from the calculations

that you can actually you know try doing yourself. It could be a good practice. And so bottom line. The steeper line for the 30-year bond, the

bond with more time left to maturity. Has a much higher interest rate sensitivity. So the bond value is very sensitive to changes

in the interest rates. In the yield to maturity, right? Whereas the one-year bond with the flatter

line has little fluctuations in bond value when the interest rate is changed, right? There is almost no effect on the bond value. It’s always around $1000 on this figure. Even if we double or triple the interest rate. So make it from not five percent but ten percent. Or 15%, right? The bond value still, is still, you know very

close to $1000. So very small drop in bond value. So that’s the interest rate sensitivity depending

on how many, how much time is left until maturity. That’s what this figure summarizes. And [inaudible] so I mean I just said that

you can verify that this is what will be happening. If you try changing the interest rate and

recalculate the bond value and see how big the changes are. But there’s also a different kind of an intuitive

explanation for why this is happening. So consider a longer time to maturity bond. If it’s let’s say a 20-year bond or a 30-year

bond. Which will see coupons being paid over 20

or 30 years, right? Then the $1000 face value is at the very,

very end of this long time period, right? And the $1000 face value is by far the largest

cash flow on our timeline. Because the coupon payments are significantly

smaller. So those would be maybe $50 a year, maybe

$100 a year. And because the largest dollar amount, the

largest cash flow, the $1000 face value is at the very end. When we discount it as you know part of finding

the bond value. We end up discounting it very heavily. And so any slight change in the interest rate

will dramatically affect the present value of the face value of the bond. So if the interest rate goes up, the present

value will go down by a lot. And so because it’s a huge portion of the

you know the sum of all cash flows on the bonds timeline. We end up observing a higher sensitivity of

bond value to changes in the interest rates. Turns out there is another factor that plays

a role. We get exactly the same higher sensitivity

of bond value to interest rate changes. If we have a bond with w lower coupon rate. And actually, the interpretation is, you know

the, the interpretation is exactly the same as earlier. With longer-term bonds. So cons–think about what a lower coupon rate

bond really means. If the coupon rate is low, right? Some small percentage like three percent,

two percent. Then relative to the coupon payments, the

$1000 face value will seem like a huge number, right? So the face value, it’s a much higher fraction

of all bond payments on the bond payment timeline. And then the rest of the [inaudible] is just

like on the previous slides. Where we looked at the fact of the time to

maturity. So because the face value’s at the very end,

it’s discounted very heavily. And since it’s a huge fraction of all bond

payments between now and maturity, any slight change in the interest rate. Dramatically affects the face value, the present

value of the face value. Which in turn plays a huge role in determining

the bond value, right? And so a bond with a lower coupon rate will

fluctuate in its value a lot more. When interest rate changes. So we observe a higher sensitivity of bond

value to changes in the interest rate. Next, this is, this topic is called current

yield. What is the current yield on the bond? Think of it a one-year return on your money. When you buy a bond that some company or government

issued. So a one-year return is what you get back

in one year, divided by what you paid for that, you know financial security today. So you take, to calculate the current yield,

you take the coupon payment which is the money you will get between now and end of the first

year. And you divide it by the price of the bond

that you paid today. So this becomes that first year return on

your money. So let’s say the coupon payment is $100, right? And the price is $900. You take 100, you divide by 900 and you get

something around eleven and something percent. So that’s your first year return on your money. It’s not a current yield sounds a lot like

just yield which is another term for yield to maturity. But it’s not the same thing as yield to maturity. In our previous examples, with a coupon rate

of 10% that determines the coupon amount, $100. And then we varied the interest rate, the

yield to maturity from 11% to ten percent, and then to eight percent, right? And we got three corresponding bond values. So to calculate the current yield in those

three similar scenarios. Where the only difference was, you know the

interest rate, the yield to maturity. We take the $100 annual coupon payment and

divide it by, separately by each of these three prices. And we get the current yield of 10.38%. Ten percent, and 9.26%. In this country, bonds usually pay coupons

semi-annually. So two coupons every year. One after the first six months, and the second

one after the second six months. Let’s do an example. Let’s say we have [inaudible] Corporation,

another one of James Bond’s movies. Is raising money for new investments by issuing

corporation bonds. The bonds have the following characteristics:

ten years to maturity, eight percent yield to maturity, 12% coupon rate, and coupons

are paid semiannually. Same question as in the previous problems. How much can this corporation raise with each

bond? In other words, we are asked to calculate

the bond value. We have a 12% coupon rate. So as always, we can multiply it by a $1000

face value which will give us $120 in annual coupon. Except because our coupons are now semiannual,

we get half of that every half a year. So every half a year, whoever buys such bond

will receive a $60 coupon. We have ten years left to maturity. Which means there will be total of ten times

two, or 20 coupons between now and the maturity date. And this is what we will use for N in the

financial calculator. Because remember in the financial calculator,

N refers to how many recurring payments we will have. Our recurring payment is $60 repeated over

and over, every half a year. And we need to use N that reflects how many

of those we are going to have. And to be consistent, everything throughout

our calculations of the bond value. Should reflect this semiannual frequency. So for the IY, the interest rate, the yield

to maturity. We need to use the semiannual interest rate. If per year it’s eight percent which is given. Then we need to divide it by two. And four percent is the correct discount rate

that we will use. So plugging these numbers into the bond present

value formula. Will give us the bond value of $1271.80. This is a premium bond because it costs more

than $1000. And of course you can do it in the financial

calculator. So let’s do it together. Let’s clear everything we had from our earlier

calculations. Let’s also increase the decimal places. Let’s try second decimal place. Let’s make it six, enter. Let’s begin. So in no particular order, we need to enter

the future value or the face, the $1000 face value. PMT the dollar amount of each coupon payment. The number of coupon payments, N. And the

discount rate or the yield to maturity, IY. So let’s start with the, the face value. One thousand, we are making it negative, right? Press the plus/minus button and save it as

FV. The same sign should be used for the coupon

payment. What is our coupon payment? Every half a year, it’s $60. So you put 60, you change the sign to negative. By pressing the plus/minus key. And then you save it as your PMT. Then how many of those recurring coupons will

there be? A total of 20 coupons. You use that for N. Twenty N. And the fourth

number that is entered is the discount rate. It should also reflect the same semiannual

frequency. Four percent yield to maturity per half a

year. So I put four IY. And then I’m computing present value: $1271

and about 80 and a half cents. So technically, if it’s rounded to two decimal

places, then that would be 81. If it’s rounded to one decimal place like

on my slide, it’s just .8 at the end of the answer. So keep consistency. If you have semiannual coupons, you need to

adjust kind of three things. The coupon is the annual coupon divided by

two. The number of coupons, number of years left

to maturity multiplied by two. And the discount rate is the annual yield

to maturity divided by two. These three things, all three must be made

in order to do the calculations correctly. So for yield to maturity, or YTM—sometimes

like in real life, we actually know how much bonds sell for. Which was our bond value calculation from

the earlier slides. And sometimes what investors are interested

in is calculating the implied yield to maturity which is like. The rate of return on your money every year. Or the IY in the financial calculator. One way to calculate it is trial and error,

using the formula. And trying different interest rates until

you get the present value that is known. But of course there’s a way to do it very

easily in the financial calculator. Let’s look at the following example. Skyfall Company issues corporate bonds to

raise money for a new investment project. So it would like to borrow money, essentially. The bonds that this company issues have the

following characteristics. Ten years to maturity, nine percent coupon

rate, coupons paid semiannually, and the company can raise $880.50 with each bond. What is the yield to maturity? Okay, in the financial calculator, we are

given ten years. But we are going to have to input 20 for the

number of coupons, right? Just like before. For the PMT, the coupon amount every year,

we need to do nine percent coupon rate multiplied by 1000. Which gives $90 per year. And then to make it semiannual, we divide

$90 by two and get 45. So that’s our PMT. Then as always, the future value is $1000

and this time, we are given the present value. Which is $880.50. And we are going to have to compute the IY,

the yield to maturity in this problem. Notice one important thing. We are given three dollar amounts: the PMT,

the FV, and the PV. What do we do with the signs? Everything that’s today is one sign. Everything that’s in the future is with the

opposite sign. For example, the way I you know put my signs

here. Is I left both PMT and the future value, which

are both in the future. As positive, and I changed the sign to a negative

for the present value. But you can also do it the other way around. When you do that, you will get 5.5%. It looks like this is our answer, right? Actually, it’s not. We are missing one more step. Why? What’s going on here? Because everything in our problem is semiannual,

the interest rate we get is also semiannual. So 5.5% is for half a year. And what’s really meant by yield to maturity,

is the annual interest rate. The annual yield to maturity. And so the last step that we need to do is

multiplying our answer by two. To convert the answer to its annual equivalent. And so 5.5% for half a year multiplied by

two gives 11%. This is the right answer. Let’s, let’s do this math in the financial

calculator. Let’s bring it up, turn it on. First let’s make sure it’s hasn’t stored anything

from any earlier calculations. Let’s reset it. Second plus/minus, answer. We can also increase the decimal places. And we do that by pressing second, decimal

place. Let’s make it six. Six answer. And I just turned the calculator off and then

back on. And the calculator is now going to round everything

to six decimal places. So we need to solve for the yield to maturity

or IY in the financial calculator. And we need to enter the number of coupons

which is 20. So 20 M. Then the coupon amount itself or

the payment, PMT, is 45. Forty-five PMT. Because I left it as positive, my $1000 face

value or the future value should also be positive. So I put $1000 and without changing the sign,

I press FV. And only for the bond price or the present

value. I’m going to change the sign to negative. So I put 800 [inaudible] .50 negative [inaudible]. And I’m computing IY, compute IY. So we get 5.499965. If you rounded to two decimal places, you

will get 5.5 like on my slide. And then you multiply that by two and get

the annual yield to maturity, 11%. Bond futures. So a bond is like a loan. A company that sells a bond, borrows money

from whoever bought the bond from the company. Indenture means a written agreement between

the company that borrowed money and bondholders who leant money, the creditors. This written agreement includes a lot of you

know conditions and terms. Let’s look at the most important ones. The most basic ones are terms of the bond. For example, what is the face value amount? Usually it’s $1000, but it could be something

else. For example, $100,000. There are bonds like that. Then how many years are left to maturity? When, on which dates the coupons will be paid

to the bondholders? What is the coupon rate? And so on. Security: bonds are called secured if they

have some sort of collateral in case the company cannot meet its obligations and pay the coupon

and the face value to the bondholder. So what could be collateral? Maybe the company cannot pay cash as coupon

payments. It instead pays in stock shares or maybe it

uses something else as a collateral. Maybe some sort of accounts receivable. Unsecured bonds are also known as debentures. And this is basically what this whole chapter

is on. We never talk about any sort of collateral. So it’s more like unsecured bonds are like,

you know just a promise from a company, from the company that sells the bonds. That it will make the required coupon payments

and the face value at the very end. Technically in this chapter we call all the

different types of bonds as bonds. But anything that has less than two years

to maturity is also known as bills. Two to ten years to maturity is also called

notes. And anything with longer than ten years to

maturity is just bonds. But all of this combined is also called bonds. Another thing that could be included in the

indenture is you know the type of seniority on these bonds. Bonds are, bonds are called senior bonds if

they are older bonds. They have been issued way back when. So maybe a company issued bonds a few years

ago. And today decided to issue more. Those bonds that were issued a few years ago

are senior bonds. Those that are about to be issued are junior

bonds. So those investors who bought the senior bonds

that were issued a few years ago. They have the preference of receiving payments

over those investors who are going to be buying the junior bonds. So the junior bonds are basically more risky. They present more risk to those who buy them. Because there is a chance they will not receive

the promised coupon payments and the face value. Since they get the second priority. Repayment. So we have a company that sells bonds. We have investors who buy those bonds. There could be third party, a sinking fund. Like a third party between the company and

the investors. The role of a sinking fund is the company

will be making annual payments into this sinking fund. So it could be some sort of bank. And then this bank, this sinking fund will

use the money that will accumulate in it to sort of speak retire part of the bonds. In other words, it will buy them back from

the market, from the investors’ hands. They’re no longer circulating, you know among

investors. They are removed from the market. And this is called as calling bonds. And the point of that. So why would a firm want to use this option

to call the bonds? Because if there are fewer bonds in the market. Then it’s like the firm has less debt. So this may improve its credit score. Which is good for all sorts of different things. The sinking funds may be used after several

years. And, and some bonds have this provision of

you know the possibility that they would be called sometime in the future. Types of bonds. So let’s look at different types of bonds. But of course we are not going to be talking

about James Bonds. But instead, corporate bonds or government

bonds. So bonds can be of different type. On this slide, I’m summarizing the types of

bonds by issuer. Who may issue a bond? A corporation or a government. So there are corporate bonds, there are government

bonds. The government bonds can in turn be issued

by the federal government and those bonds would be known as Treasury bonds. Or Treasuries. Or they can be issued by more local governments

such as State of California, City of Pomona, and those are known as municipal bonds or

munies. And again, there are different variations

in how bonds may be called. Bills if they have less than two years to

maturity. Notes if they have two to ten years to maturity. And simply bonds if they have a longer time

to maturity. How do corporate bonds and different types

of government bonds differ in terms of risk of default? So what is risk of default mean? Defaulting means you’re not able to make your

payment. So who are we talking about? The issuer, a corporation or a government. So in which case do you as an investor who

bought a bond. Face a, you know this risk of default? If you bought a corporate bond, you face a

low to high risk of default. Because any company can go out of business. It may go bankrupt. It’s out of money. So you may not get your coupon payments. In case of the government bonds, the risk

is smaller. And there is no risk of default if you buy

federal government bonds. In other words, in other words, you will get

your coupon payments and the face value. In the future. But if you buy local government bonds or state

government bonds. There may be, you know a chance that the state

is out of money. The state budget is, you know low. And you don’t get your coupons. Maybe not for a while. The second thing that makes corporate bonds

and government bonds different from each other. Is whether you as an investor who buy such

bonds need to pay taxes on your coupons. Because coupons that you receive is your,

you know additional income, right? Is it taxed or is it not taxed? Yes, it is taxed if you bought corporate bonds

or federal government bonds. But your coupon income is not going to be

taxed if you’re buying state and local government bonds. So see how all the different bonds, you know

are different in many different ways. Let’s look at this example in which the whole

taxability of coupons on bonds is analyzed. Specter Corporation has bonds that give a

seven percent return per year so that’s their yield. Municipal bonds that an investor could buy

instead yield a five percent return per year. So let’s say we have investor James. Who is in the 30% federal income tax bracket,

right? So any income is taxed at 30% rate. Would he prefer buying Specter Corporation’s

bonds or the municipal bonds? So at the first glance, it looks like his

return would be higher if he buys the Specter Corporation’s bonds, right? So we are talking about seven percent being

above five percent. But wait a second. Remember, corporate bonds are taxed. So if could be that after taxes, the return

on these corporate bonds is below five percent. So let’s check that. So we need to compare the after tax of the

effective returns to this investor. Corporate bonds are taxed. And so after taxes, the return will only be

4.9%. Which is calculated by taking the before-tax

return, seven percent which is given. And multiplying by one minus .3, the 30% tax

rate. And the municipal bonds are not taxed. The coupons received from municipal bonds

are not taxable. And so the effective rate is as given, five

percent. And now you can see which bonds would be more

desirable for this investor, James. It’s the municipal bonds with the higher after-tax

return. There is another type of calculation that

could be done. When we are comparing bonds with taxable versus

bonds with non-taxable coupon payments. For example, we can calculate at which federal

income tax rate an investor would break even. In other words, be indifferent between buying

corporate bonds and municipal bonds. And what we need to do is basically set equal

the after-tax yields. So after-tax yield on the corporate bonds

with the seven percent yield that was given. Would be equal to seven percent multiplied

by open parenthesis, one minus some tax rate, closed parenthesis. And that needs to equal five percent. Which is given for the municipal bonds. So we are asked essentially to calculate the

tax rate that would make both sides equal to five percent, right? The after-tax return on the municipal bonds

is five percent as given. And so is the after-tax return on the corporate

bonds. So to solve this little equation, we can first

move you know .07 to the right. And mathematically we would be dividing .05

on the right-hand side by .07. And now on the left-hand side, we are just

left with one minus T. And then we want to have T equals, and then everything else. So we want to get rid of one. And the easiest way to do it is to leave T

on the left-hand side. Actually I first simplify the fraction. So .05 over .07 equals .7143. And then T will equal one minus .7143. And when you simplify it, you get .2857. In other words, the tax rate that you would

need to face in order for you to be indifferent between buying corporate bonds and municipal

bonds. Would be 28.57%. And is kind of the general formula that you

would apply in problems like this. So to find the break-even tax rate, you do

one minus a fraction where in the numerator you have the percentage yield on municipal

bonds. And in the denominator you have percentage

yield on corporate bonds. Next we can group different bonds into how,

into different categories determined by how coupon payments are made. Now coupon bonds is what we have had in this

chapter so far. And we will look at coupon bonds until the

end of this chapter. That’s when there are coupons, and that’s

the most common type. So there are coupons periodically. Typically every half a year, right? Semiannual coupons. With a face value paid at the very end. There are bonds which have no coupon payments. They’re known as zero coupon bonds or zeros. So there’s just the face value at the end,

and one large interest payment together with the face value. Then bonds that do pay coupons can further

be differentiated by the coupon rate. Level coupon bonds have a level coupon rate. So what does level mean? On the same level, flat. So unchanged or fixed. And that’s also what we’ve been using in all

our numerical examples in this chapter. So it’s the same coupon rate which we then

use to calculate the same amount of coupon payment paid regularly. Other bonds in real life have a floating coupon

rate. So those are known as floating-rate bonds. Or floaters. So the coupon rate sort of floats, it goes

up and down. In years when it’s high, we have a high coupon

payment. In years when the coupon rate is low, we have

a low coupon payment. So the coupon payment is adjustable over years. What could be linked to it? It could be linked to sort of the cost of

living or inflation. So if the prices are higher, then investors

would want to be compensated with a higher coupon payment to them, right? And in those years, the coupon rate will be

adjusted up. Now that we mentioned the word inflation,

let’s talk more about inflation. In general, when we talk about interest rates. We can talk about so-called nominal interest

rates and so-called real interest rates. Nominal interest rates is what we’ve had so

far. Same thing as a [inaudible] rate or a stated

rate. The rate is given. What is a real interest rate? It’s the rate that’s adjusted for inflation. Let me start with a simple example. Let’s say you put your money in a bank account

that offers a five percent interest per year, right? And so at the end of the year, you’ll be five

percent richer, right? If you, if you leave your money there for

the full year. But let’s say something else is going on during

this year. Everything you typically buy gets more expensive. Let’s say everything is three percent more

expensive at the end of the year. So how much richer do you really get? With a five percent increase of your money

at the bank and three percent higher prices? You probably already figured out that you

feel only two percent richer. Because three percent is kind of eaten by

inflation. And this is correct. So we say the nominal interest rate is five

percent that you earn in your bank. The real interest rate is only two percent,

and the different is the three percent inflation. That sort of erodes that value of your money. Now let’s look at it a bit more formally. Because actually the math I just explained

is only approximate. So let’s see how we would do it more accurately. Let’s start with something simple. You deposit $100 in your bank account. The interest rate that the bank offers to

its depositors is ten percent per year. We also know that prices are expected to rise

over the next year by three percent. That’s inflation. The inflation rate is three percent. The question is what rate of return do you

really earn at your bank? In other words, what is the increase in your

purchasing power? So how much wealth there do you really get

by the end of the year? So let’s imagine this timeline with two points

of time marked on it. This year or right now, and next year, one

year later. And let’s talk about this whole purchasing

power concept. What it really means is how much stuff you

can buy, so in units. For example, let’s say you, you’re buying

jeans. And since you’re talking about hundred dollars

put in your bank account. Let’s say to buy one pair of jeans today,

you would need to spend $100. So what is your today’s purchasing power in

terms of pairs of jeans. You have hundred dollars. You divide it by the price to buy a pair of

jeans. So $100 divided by $100 gives one. Your purchasing power. The purchasing power of your $100 today is

one pair of jeans. You can use $100 that you have now to buy

one pair of jeans, right? And now let’s calculate how that would change

in one year. We’re going to use the same steps. Your purchasing money will equal the money

you have in your bank account in one year. Divided by the price for the pair of jeans

next year. We know that the bank pays you ten percent

interest. So your money in the bank will increase by

ten percent compared to today’s $100. Right? So we have $100 times one plus ten percent,

just like in chapter five. A single cash flow, future value. And we divide by the price for a pair of jeans

next year. We were given a three percent inflation rate. So the jeans will cost $100 times one plus

three percent which is $103. So now we have $110 divided by $103, and that

equals 1.068 pairs of jeans. So today your $100 can buy one pair of jeans. Next year, you will be able to buy only a

little, you know, 1.068 pairs of jeans. A little bit more, which is a good thing. These two circled numbers, so ten percent

increase in the money in your bank, right? And 6.8% increase in your purchasing power. Are the two types of interest rate that we

are now differentiating on these slides. Ten percent like a [inaudible] increase in

the money you have, right? In the bank is the nominal interest rate. And you might recall that the word nominal

came up when we were covering the topic of effective annual interest rate. So that’s how the financial calculator calls

the state rate. It was nominal. And then the increase in the purchasing power

which is 6.8%, right? Over one here. That’s how, that reflects how much wealthier

you really get. Considering the increase in the cost of living. And that’s the real interest rate. And if you look at you know the numbers that

we used to find the real interest rate, you can kind of cancel out the $100, right? They become unimportant. And so what really determines the real rate

is the fraction with one plus the nominal rate. Divided by one plus inflation rate. Another thing you can notice is that the different

between the ten percent nominal rate and the 6.8% real rate is 3.2%, right? And that’s approximately, you know it’s very

close to the three percent inflation that was given. So one plus the real rate is calculated by

taking one plus the nominal rate and dividing that by one plus expected inflation rate. Using notations, the real rate is lowercase

r. The nominal rate is uppercase R. And you can

think of it as the nominal rate typically being higher, so it’s in uppercase. And you divide by one plus H. So H stands

for inflation rate. Because there are three notations in this

formula, there are three things you might be asked to solve for. Either the real rate or the nominal rate if

the real rate is given. Or the inflation if both the real and the

nominal rates are given. So we can rearrange this formula to solve

for lowercase r, uppercase R or H. This formula here that calculates the nominal interest

rate is known as the future effect. So the nominal interest rate, uppercase R

equals one plus lowercase r. And then that whole thing is multiplied by

one plus H. And then we subtract one. If instead we rearrange the original formula

to solve for the real interest, or the lowercase r, you would get a fraction. With one plus R in the denomin–in the numerator. One plus H in the denominator. And then you subtract one. If instead you want to solve for the inflation

rate which is lowercase H, then you also have a fraction. With a one plus capital R, nominal rate, in

the numerator. And in the denominator you have one plus lowercase

r, or the real interest rate. And then you subtract one from the whole thing. These three are like the most accurate ways

to compute the real rate. The nominal rate, or the inflation rate. The proximate versions are these. For the real rate, you do nominal rate minus

inflation. To find the nominal rate, you do real rate

plus inflation. And to find inflation rate, you do nominal

rate minus real rate. This will give you approximate answers which

are a little bit off. So in our example, we have a ten percent nominal

rate, a three-percent inflation rate, right? If you’re asked to find the real rate, we

could do it using the exact inflation effect or using the approximate inflation effect. By plugging in ten percent into these formulas,

we get .068. Like we had earlier. And .07, right? So this is the approximate real interest rate. Let’s look at this example. These days the interest you earn in your checking

account is pretty small. It’s very close to zero. So let’s say last year, in the year 2018,

you kept your money in a checking account which was offering a half-percent interest. I looked up the data in the official source. The Bureau of Labor Statistics to see what

the inflation was like. In the year 2018. And this figure here shows that for all the

things you are buying for your everyday life. Which includes food and you know paying for

gas for your car and everything else. The prices went up by approximately 1.9%. What does it mean? So let’s answer the following question. How much richer did you really get in 2018? If you earned a half a percent interest on

the money in your account? But prices went up by 1.9%? So let’s do the math both exactly and approximately. Let’s start with the approximate solution. So find the approximate real rate of return

you need to take the nominal rate and subtract inflation rate. That’s .005 minus .019 which gives a negative

.014. That’s the same thing as a negative 1.4%. So you lost 1.4% of your money. What does it mean? You might as well just have spent it last

year. Because if you didn’t, you would have lost

it, and you did. This is how much you really lost. That’s how much less the money could buy you. The exact formula gives a very similar number. And this is actually the most accurate result. Negative .0137 or negative 1.37%. So this is the loss in your purchasing power. This is how much less stuff your money was

able to buy you at the end of the year 2018. Relative to how much you could buy at the

beginning of the year 2018. Let’s look at another example. You want a ten percent real return. You expect inflation to be eight percent. And the question is what should be the nominal

rate of return on your investment, right? So what should be the nominal rate of return? So that your real return is ten percent considering

the eight percent expected increase in the cost of living? So here ten percent is the real return. And we need to find the nominal rate. So it’s the capital R that we need to solve

for. The approximate solution is real rate plus

inflation. So we add together ten percent and eight percent. And get 18%. The exact solution will be just a little bit

more complicated. So you would do one plus the real rate. Then you multiply that number by one plus

inflation rate. And then subtract one. So 1.1, that’s one plus .1 in decimals. The real rate. Times 1.08 which is the same thing as one

plus .08, the inflation rate. And then you subtract one. And when you simplify it you get .1880 or

18.8%. So approximately 18%, exactly 18.8%. The two answers are pretty close. I would say always use the exact formulas

because that’s how you get the right answer. Anything else is only approximate. So don’t use the approximate formula unless

you’re asked to find an approximate answer. Just always stick to the exact versions for

all the formulas we have. And the last topic in chapter seven on bonds

is related to term, is called. Term structure of interest rates. So let’s recall what we had so far in chapter

seven on bonds. The look at what bonds are. It’s basically a way to borrow money. A company can borrow money by selling bonds. Or a government can borrow money by selling

bonds. We looked at how to calculate bond value. Then the topic of inflation that we just finished

has to do with yield to maturity. The IY in the financial calculator. Right? The return on your money when you buy a bond. Now this topic, term structure of interest

rates goes in a little bit more depth. Into this you know, the how the nominal interest

rate on bonds or IY is determined. Term structure of interest rates. So the word term means the number of years. So time to maturity, how that determines what

the nominal interest rate on a bond would be. There are three important factors. That affect the nominal interest rate for

different bonds. First, the real interest rate. So when investors are buying bonds, why do

they do that? Because they want to end up with even more

money in the future. They want to get some return. So there, there’s this bare minimum return

that investors would like to get. That’s the real return, right? The real interest rate. We already know what it means. So they want to be able to buy more stuff

with their money. So the real interest rate captures that, you

know adjustment to any possible inflation in the future. So we, regardless of how long the time to

maturity in the bond date. Whether it’s only one year or ten years or

30 years. We want some bare minimum. So you can imagine there’s you know flat line

with the years to maturity on the bond on the horizontal axis. And the nominal interest rate on the bond

on the vertical axis. The second factor that affects the nominal

interest rate for bonds is called inflation premium. Inflation premium means let’s say investors

think that. Or there’s this general belief that prices

will keep going up. Which is probably true in real life, right? Then if you’re buying bonds with more years

to maturity, you will want to get higher return so you are compensated for that increase in

cost of living in the future, right? If you instead hypothetically expect the prices

to go down over time, or deflation. Then bonds with a longer time to maturity

can offer a lower return to their investors. Right? We don’t need that compensation for the higher

cost of living. Because that’s the opposite of the expectation

of inflation in the future. And now you can combine the first two factors. And sort of imagine stacking these lines on

top of each other. So you have the flat line from the real interest

rate that we desire. And you take the [inaudible] sloping inflation

premium line and add it on top. So now we have this upward-sloping line, you

know shooting up by the amount of the real rate. The third thing that’s also very important

is something which we already had earlier in chapter seven. Interest rate risk premium. So what we had earlier in this chapter was

the idea that. Turns out that bonds have a longer time to

maturity fluctuate in their price a lot more when there’s any slight change in, interest

rates. And so what this implies is that if you are

an investor. If you are buying bonds, and the bonds you’re

buying have more time to maturity, you know that your bonds will be fluctuating in their

value up and down, up and down a lot. Compared to bonds with fewer years to maturity. And so because you’ll be facing this interest

rate risk, you will want to be sort of compensated or reward with what? Rewarded with a better return on your money. That’s why we have this upward-sloping line. Which basically summarizes the idea that. When you’re buying a bond with more years

to maturity, you want to be compensated with a higher return, right? So offset this interest rate risk. And now we have one, two, three factors. And you can sort of stack the three lines

we get on top of each other. And get something like this. Upward-sloping term structure or relationship

between the time to maturity on the bond and what the nominal interest rate for it should

be. The line slopes up where in which the three

factors are sort of put together. It could be a downward slope in term structure,

a downward-sloping line. But that’s only because we expect deflation. So the inflation premium goes down. I would say this is an unrealistic case because

we all know how things are only getting more expensive over time, right? Most things we buy, they get more and more

expensive. Apart from those three factors that determine

the nominal interest rate for bonds with different times to maturity. We have three more additional factors. But they’re not true for all bonds. They’re true for some bonds, but not all. For example, on one of the earlier slides

in which we looked at different types of bonds. We saw that corporate bonds and municipal

bonds have a risk of default. So corporations and local governments that

issue bonds may go bankrupt. And so investors who buy those bonds may not

get their promised coupon payments. Right? And so if you are an investor who are buying

corporate bonds, then you’re aware of the fact that you’re facing this default risk,

right? And so to, to, you’re only going to be interested

in buying the corporate bonds if you’re compensated appropriately. And the compensation will again come in the

form of a high return on your money. So corporate bonds and municipal bonds. Those that you know include this default risk,

right? They have this default risk embedded in them. They will be offering a better return. So a higher nominal interest rate. Then the other types of bonds. So what’s left here? The federal government bonds. Then number five. Again, this is a factor that is true for some

bonds but not all. So we have corporation bonds, federal government

bonds. And municipal bonds. Corporate bonds and federal bonds pay coupon

payments that investors will need to pay tax on. Investors that buy local government bonds

they don’t need to pay taxes on you know that additional income, right? The coupon payments that they receive. And so in order to you know make corporate

and federal government bonds attractive to investors, the return must be higher. That’s the taxability premium. And the last thing that actually never came

up anywhere in this chapter. Is known as liquidity premium. So what else could investors require compensation

for? Some bonds are difficult to find. They’re not traded very frequently. That’s corporate bonds and municipal bonds. Federal government bonds are actually traded

very actively. So they are very easy to buy or sell. And so corporate and municipal bonds that

don’t sell very frequently are said to be not very liquid. And because of that, investors who are willingly

buying corporate or municipal bonds would only be, are only doing that because they

know they are compensated. So liquidity is premium is basically that

extra, like little bit of return. So maybe another half a percent. Or, or whatever that might be. That investors receive when they buy bonds. So to summarize all this words premium. So what does premium really mean? It’s an additional return. A little bit more, higher percentage. So maybe another quarter of percent here and

there. So that’s the extra percentage that some but

not all bonds have. Depending on whether the bond issuer may default

or not. Whether you as an investor are required to

pay taxes on your coupon income or not. And whether the bonds you’re buying are liquid,

so easy to find. Easy to buy or to sell or not. And here we see an example of term structure

of interest rates for federal government bonds. So on the horizontal axis we see one month,

three months, five years, ten years, 30 years. That’s time to maturity, right? And on the vertical axis we have the yield

in percent. That’s the yield to maturity. That’s the nominal interest rate for bonds

with different maturity. The line slopes up which means it’s an upward-slope

in term structure. So for all those various reasons. And we have a total of six of those summarized

on the last couple of slides. Federal government bonds that mature after

more years. Offer their buyers, their investors who buy

them a better return. A higher nominal return. The lower line on the bottom here is the real

rate. So the different between the higher line,

the nominal rate. And the lower line, the real rate is the expected

inflation. So we can even calculate it. It’s roughly at two percent inflation 30 years

from today. And this graph is about a year old, so it’s

from end of December 2017. Let’s summarize chapter seven. We looked at bonds. That’s how companies and governments can borrow

money. For bond calculations, we did two important

things. How much a bond should sell for or finding

the bond value. For that we need to know the coupon rate that

allows us to find the coupon payment, the PMT in the financial calculator. The time left to maturity. So between today and when the bond will mature. That allows us to find the number of coupon

payments, or N. Yield to maturity that’s another term for discount rate or interest rate. We also refer to it, to it as the nominal

interest rate in this chapter. That’s IY in the financial calculator. And we also need to be given the face value

which becomes our future value, FV in the financial calculator. The face value is typically $1000. So you should always assume it’s $1000 if

it’s not even given. Then we looked at how we would instead calculate

the yield to maturity, IY. When the bond present value is known. We also looked at so-called interest rate

risk. So bonds which have either a longer time to

maturity. Or a lower coupon rate fluctuate in their

value a lot more when interest rates change. We looked at different types of bonds. Some bonds don’t pay any coupons. They’re known as zeros. Other bonds that pay coupons pay varied months

of coupons. They’re known as floating rate bonds. Then coupons are taxed, so you as an investor,

when you receive your coupons, do you need to pay taxes or not? You do need to pay taxes. Do you need to pay taxes on the coupon payments

you receive or not? You do have to pay taxes on your coupons if

you bought a corporate bond or the federal government bond. If you bought a municipal bond, you do not

need to pay taxes. So for municipal bonds, the coupons are not

taxed at the federal level. Then we looked at the topic of inflation. And the format we have is known as the Fisher

Effect. It says that when you earn some return on

your money. Whether it’s in a bank or some other investment. But when at the same time, the cost of living

is going up, your ability to buy stuff with your money is sort of. You know eroded by this inflation rate. And so the real or the true interest rate

you earn in the bank or in some other investment is approximately equal to the nominal interest

rate that the bank quotes. Minus the inflation rate. This is the approximate Fisher Effect. And we also saw a different version of it

in which the calculation of the real interest rate is done more accurately. And then we looked at the so-called term structure

of interest rates. So that’s six factors that determine the nominal

interest rates for bonds with different time left until maturity. And to finish up the [inaudible] on bonds. This is the James Bond Island. It’s in Thailand. I traveled there in December 2011. This is me right here. And I saw this very famous island. I think it appeared in actually a lot of movies. But in particular in one of the movies with

James Bond. And that’s where this informal name came from.