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Hi everyone.

So, for starters, in this segment we look at the mechanics of how bonds are priced.

And just how markets influence bond yields.

A bond is just an I.O.U, a loan to get someone else's money for

typically a very long period of time, that is repaid in fixed time intervals.

This is why most bonds are called fixed income securities

because they provide the investor with a reliable return.

In fixed payment periods.

This makes this quite easy to value so let's start by defining some common terms.

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Face value, this is the denomination of the loan

the amount of money a bond holder will receive back once it matures.

It's also called the par value and it's usually expressed in units of 100 so

you can easily compute a percentage of the actually face value.

Note, however, that most bonds have a $1000 face value.

So if a bond is trading at 70, it's actually worth 700,

which is 70 percent of 1000.

This bond is trading at a discount, which is below 100.

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Now the maturity.

This refers to the time period when the face value of the bond is paid back

to the lender or the bondholder.

The maturity can range from very short period of time,

let's say several months to a very long period of time, typically 30 years.

There are exceptional cases of bonds that are outstanding for

100 years or in perpetuity.

With no maturity date, and this was the case of the British consuls

that were issued by the bank of England, post world War II.

Now we can define yields, and there are principally two types of yields.

The first one, known as the current yield, is the bond's annual rate of return.

Based on the coupon divided by the discount or premium price.

So let's suppose that coupon rate of three and

a half percent is being paid when the bond price is $700.

This would give us a current yield of 5%, which would be $35 divided by 700.

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It's really important because incorporates the time value of money.

And provides the investor with the actual, total return for a bond.

If they hold that bond until it matures.

Now YTM is a market determined rate, which means that it's going to be fluctuating

all the time depending on market forces that exert its pressures on the rate.

It's taking into account the coupon rate, the maturity of the bond, the frequency

of compounding and of course the current price influenced by supply and demand.

And all of these again are exerted by the changes in

the general level of interest rates which you will recall include three components.

And those are the real component, the inflation component and

the risk premium component.

So as you will see in examples below, when the yield to maturity is greater than

the coupon rate the bonds are going to sell at a discount and if the yield

to maturity is less than the coupon rate, the bonds will sell at a premium.

Okay, so let's talk about frequency, or the frequency of compounding.

And we know from course one,

this refers to the rate of compounding that occurs within one year.

Semi annual compounding would mean compounding occurring

twice a year where as monthly compounding would mean 12 times a year,

daily compounding 365 times a year and so on.

The frequency is denoted by the letter M So a 10% coupon paid semi annually

means that you're actually getting 5% every six months.

This is why we multiply the time period by m and

we divide the coupon and the yield to majority by m.

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Finally, lets talk about the market value of the bond.

This is simply the present value of what the bondholder will receive,

by investing in this bond.

And the receive typically two things.

One is the coupon down the road until the bond matures and,

of course, the repayment of the loan itself, which is the principle amount.

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Lets assume we're looking at the balance sheet of an actual drug company.

And from its financial statements we find the particularcies of 30 year junk

bonds that were issued on January 2010.

These bonds have been deemed junk because agencies like [INAUDIBLE] and Fitch

rate them below investment grade, which means they are deemed to be very risky.

The best rating of course is a AAA rating, followed by AA, A, then BBB, BB, etc.

And junk status is usually B or triple C.

This signifies a very speculative status that the company may or

may not pay the interest.

Or even the loan back.

So to make it attractive, this bond has a 12% coupon, it's relatively high.

That is paid semi annually on June 30th and December 31st.

And we see that it's yields to maturity has decreased to 10% on January

1st 2016 driving the price up.why is the price driven up while you can already see?

Because the yield is lower than the coupon rate.

So what is the price on January 1, 2016?

Well let's first isolate all of the information that I've

just referred to in the definitions that we had.

So let's begin first with face value.

We said face value Is in denominations of $1,000.

Okay, the coupon.

The coupon rate [COUGH] in this particular

problem is given to be 12%.

12% of $1,000 is going to give

us of course a dollar amount Which is 120.

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The next item.

The maturity of the bond.

The maturity of the bond is when you get your face value back.

In this case, the bond which was originally 30 years was issued six

years ago which means that the remaining maturity,

the coupon that are going to be paid down the road, pertain to 24 years.

That's the difference between, of course, 30 and the 6 years that have gone by,

and so we will focus on the remaining time period.

That takes us to the all important yield to maturity.

Remember the yield to maturity was given in this problem.

It is right now assumed to be 10%.

Now, let's not forget the frequency of compounding.

The frequency of compounding we defined by looking at the variable m and

since this bond pays coupon every six months, so

semi-annual compounding, the value of m is equal to 2.

So before we move any further, it's very important to take into account

the impact of frequency right away with the information that's been provided.

So this is always going to affect at least two things.

One of them is going to be the time period.

The time period, remember, is expressed in years, but

since now we are going to be compounding every six months, or

twice a year, then we must take the time period and multiply that by the frequency.

Whereas, the other impact is going to be on the interest rates which are, again,

expressed annually.

But now we're going to be compounding semi-annually so we will divide both

the coupon rate by m and we will divide the yield to maturity by m.

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Okay, let's do that right away.

If we divide the coupon rate by m which is 2, we get $60 dollars every 6 months,

and the years now become, of course, 48 periods.

And the yield to maturity, semi-annually, is now going to be 5%.

Now we have all the data and we can work out the bond value today.

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Again, before we do that, we've learned that visualizing this information is

very important to get a sense of what exactly we're up to.

So why don't we put all this data on a timeline.

And if we do that, let's say the timeline looks something like this where we have

time 0 today, 1, 2, 3, and we can go all the way to 48 periods.

That's the maturity of the bond in this example.

What are we getting in each of these periods?

What we're getting in each of the periods is the coupon rate which as we saw is 60.

So we're getting 60 every single 6 month period, all the way till the end.

Plus we're going to get our money back, which is 1,000, right at the end.

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by receiving these future cash flows.

We can calculate that price quite easily by setting up the equation for

the bond, okay?

So the equation in common sense terms would mean the bond price today

at time 0 would be the present value of the coupon, so we can write that,

present value of the coupon, plus we're going to get the present value

of the future principal amount, present value of the principal amount.

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And that essentially is what the pricing does.

Now how do we actually compute the present values

in terms of the equations we've learned in time value of money.

Well, notice that we got two series of numbers, an annuity and a lump sum amount.

You'll recall that to calculate the present value of an annuity,

I'll just put up that formula to remind us.

The present value of an annuity, okay, so we're going to be taking the coupon and

multiplying it by the present value of the annuity,

using the yield to maturity as our interest rate and

the time periods to give us the factor.

Plus, we're going to take this principal amount and simply multiply that by

the present value factor, at the yield and the time period.

And this is just a refresher, so I don't want to go into too much detail here.

This particular factor was computed as 1 minus 1

over 1 plus r to the power t over r.

Whereas this particular factor is simply 1 over 1 plus r raised to the power t.

So all we have to do is plug some numbers in here.

And if we depict that on our equation with the information we have,

we know that the coupon is 60.

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We also know that the factor we're looking for, so this is going to be

present value annuity factor at a yield that we've already computed,

5% for the time period, we already know, 48 periods.

Right? This is our

first part of the equation as you can see here.

And we need to add to this the face value of 1,000 and

multiply that by the lump sum present value factor, 5%, for 48 periods.

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If you work out these factors with these equations,

this value is going to work out to 18.0772 where as this value

is going to work out to 0.0961 so we just have to multiply these now,

this by 60 and this one by 1,000.

Okay and that gives us the bond price of 1,181.

That's the value of a bond that promises to pay you these payments.

Notice the bond is above 1,000, so it is selling at a premium.

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We can say it's selling 18.1% above face value.

The intuition is that if you're paying a premium that means the real

return you're earning on the bond, that is the yield, which in this case we know on

an annual basis is 10%, is lower than the coupon rate.

And we saw the coupon rate to be 12%.

So it's important to note that bonds that sell at a premium always

will have a yield that is less than their coupon rate, okay?

Of course,

we don't have to do all of this math if we work with a financial calculator.

Now, if you have that financial calculator,

then you can simply plug the values in for the coupon, for the face amount,

for the interest rate and then you can just hit the key for the market value.

Now what happens in this example if the price goes up.

Suppose this particular price, 1,181, actually goes up.

Let's just make an assumption it goes up by 20%.

If this number goes up to 20%, okay,

if it increases by 20%, well the price

is then going to be equal to $1417.

Now if the bond is selling at $1417,

if I include that as the new value, right?

So if I say that the same equation here, okay?

I have 60.

I'm running out of ink with this one, so I'm going to choose another one.

Hang in there.

So let me work with the pink marker here.

Let's say this same coupon payments, but this time I'm going to be looking for

a yield an annuity value based on a yield, I don't know yet.

That's my question mark.

But for the same time periods,

the 48 periods plus I will get my 1,000 and then the factor.

Again, looking at a yield, I don't know yet for 48 period.

This should be equal to this new price, it's trading at $1,417.

And I solved for the yield.

If you solve for the yield, what you end up getting is,

of course, first, a semi-annual rate.

The semi-annual rate for this yield works out to be 4.02% or

just about, just a little over 4%.

And then annually, of course, this works up to 8.05%.

Notice what happened to the yield.

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Now, I want to continue this example but do exactly the opposite.

Let's say this company is going through a lot of difficulty.

It's ratings drop.

They go down to double C or something like that.

And the yield now jumps up, because it's a risk yield bond, it's in difficulty

this company, people start to sell this bond, what's going to be the price?

So this time, if I'm going to work with the yield, if the yield to maturity has

now increased to 12%, okay, what's going to be the price?

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Well, again I can plug the numbers in, and

here you can see something very interesting going on.

The yield is, in fact, exactly equal to the coupon rate.

And when that happens, when the yield is exactly equal to the coupon rate,

you can imagine what happens, the bond is going to sell at face value or par value.

Which in this case is going to be 1,000.

So as the yield goes up, the price goes down.

The price which had climbed to 1,417

with a lower yield is now going to dive down to $1,000, okay?

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So, what's the take away with this example?

The take away with the example is the very important principle that

is the inverse relationship between bond yield and market prices.

So they are inversely related and the result is rooted in the time value of

money and generally applies to all well-functioning bond markets.

Now we just looked at an example of a very risky junk bond,

let's look now at the other end of the bond spectrum that is bonds with little or

no risk, which of course will offer very little yield.

Most people, especially older people don't want risky junk bonds,

they want something safe to invest their money.

And historically that safety comes from the government and

those Government Bonds are almost risk less, because unlike private companies if

Governments are in a jam, they tend to just raise taxes to collect revenues or

to print more money to meet their debt obligations.

Especially, if they have an entire economy to back their claims.

So let's go back to our example.

Let's assume the bond does not pay any coupon.

And the yield to majority is

pretty close to what you see in the marketplace these days, 2%.

And we'll keep the rest of the variables exactly the same.

So let me recall the information.

We had face value of 1,000.

That's the repayment we get at the end of bond maturity.

We had coupon, in this case we're going to go simply to 0%.

This is a 0 coupon bond.

And the maturity we said is 24 years.

And the yield to maturity, right?

I just mentioned to you is very low 2%, okay?

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Let's not forget the frequency.

And the frequency denoted by M.

We said for that bond, was twice a year.

So remember what we did last time.

As soon as we have frequency, we make adjustments to our time and our rate.

For time, we multiply it by two, so this is going to be again 48 periods.

For the coupon rate, we don't have to make any adjustment, it's already 0%.

And the yield of course, will be divided by 2 which now becomes 1%.

So like before, let's put this on a timeline.

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In our timeline, what do we have?

Well, again, we have from zero right up

to 48 periods and we can denote these periods And

for each of these periods, remember now this is a zero coupon bond so

you get absolutely nothing in each of these periods, But

you do get your principal back of 1,000 in the 48th period.

This zero coupon bond also is known as a deep discount bond.

It would be a deep discount bond if the interest rates were higher but

because they are low, we will see now what impact this will have

in terms of its present value or its market value.

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So the bond value equation,

you will recall was simply the price, today is the present value

of the coupon payments plus the present value of the principle amount, okay.

So, we can just plug the numbers in here.

This is actually quite easy.

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We have coupon payments of zero, so we're simply looking at zero dollars.

Present value annuity for a 48 periods at 1%.

Kind of irrelevant because the coupons are zero.

Zero times any number will be zero.

Plus, we have 1000 multiplied by the present value factor,

lump sum factor at 1% for 48 periods.

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So if you pay the government $620 right now, today,

they will pay you back $1,000 24 years from now or 48 periods from now,

and this investment will earn you a compounded yield of as we said, about 2%.

Again, during this time period if the yield goes up,

the price will go down, if the yield goes down, the price will go up.

Okay.

What do we see in today's marketplace?

We see that government bonds are yielding less and less and less.

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We already know if the yield was equal to the coupon rate, if this goes down to 0%,

we know the price is going to be equal to the face value.

So yield goes down, the price will indeed go up to $1000.

But any price higher than $1000 would imply a negative yield.

And that's what people are trying to get their heads around.

So if the demand for this bond, if people said, I want this government's bonds.

I feel the market is too risky.

That's why I want to keep my money.

And they keep driving the price up.

If the price keeps going up, the yield will go down,

right down to negative territory.

In fact, if this price goes all the way up to 1,272,

just by using this equation and solving for

the yield, that implies a yield of -1%.

So you would be earning -1% if you're willing to pay $1,272,

what does that really mean?

It means that you are willing to pay for

a price that is greater than what you're going to get back.

So you're not even getting your principle back.

And that's what negative yields mean.