0:13

Hi everyone.

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

Â And just how markets influence bond yields.

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

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

Â This is why most bonds are called fixed income securities

Â because they provide the investor with a reliable return.

Â In fixed payment periods.

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

Â 0:50

Face value, this is the denomination of the loan

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

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

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

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

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

Â which is 70 percent of 1000.

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

Â 2:02

Now the maturity.

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

Â to the lender or the bondholder.

Â The maturity can range from very short period of time,

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

Â There are exceptional cases of bonds that are outstanding for

Â 100 years or in perpetuity.

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

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

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

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

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

Â So let's suppose that coupon rate of three and

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

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

Â 3:13

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

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

Â If they hold that bond until it matures.

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

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

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

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

Â And all of these again are exerted by the changes in

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

Â And those are the real component, the inflation component and

Â the risk premium component.

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

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

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

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

Â And we know from course one,

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

Â Semi annual compounding would mean compounding occurring

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

Â daily compounding 365 times a year and so on.

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

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

Â This is why we multiply the time period by m and

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

Â 5:27

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

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

Â by investing in this bond.

Â And the receive typically two things.

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

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

Â 5:57

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

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

Â bonds that were issued on January 2010.

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

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

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

Â And junk status is usually B or triple C.

Â This signifies a very speculative status that the company may or

Â may not pay the interest.

Â Or even the loan back.

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

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

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

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

Â Because the yield is lower than the coupon rate.

Â So what is the price on January 1, 2016?

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

Â just referred to in the definitions that we had.

Â So let's begin first with face value.

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

Â Okay, the coupon.

Â The coupon rate [COUGH] in this particular

Â problem is given to be 12%.

Â 12% of $1,000 is going to give

Â us of course a dollar amount Which is 120.

Â 7:57

The next item.

Â The maturity of the bond.

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

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

Â years ago which means that the remaining maturity,

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

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

Â and so we will focus on the remaining time period.

Â That takes us to the all important yield to maturity.

Â Remember the yield to maturity was given in this problem.

Â It is right now assumed to be 10%.

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

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

Â since this bond pays coupon every six months, so

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

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

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

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

Â One of them is going to be the time period.

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

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

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

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

Â expressed annually.

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

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

Â 9:41

Okay, let's do that right away.

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

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

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

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

Â 10:10

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

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

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

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

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

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

Â What are we getting in each of these periods?

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

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

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

Â 11:06

by receiving these future cash flows.

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

Â the bond, okay?

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

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

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

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

Â 11:40

And that essentially is what the pricing does.

Â Now how do we actually compute the present values

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

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

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

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

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

Â multiplying it by the present value of the annuity,

Â using the yield to maturity as our interest rate and

Â the time periods to give us the factor.

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

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

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

Â This particular factor was computed as 1 minus 1

Â over 1 plus r to the power t over r.

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

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

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

Â we know that the coupon is 60.

Â 13:13

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

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

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

Â Right? This is our

Â first part of the equation as you can see here.

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

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

Â 13:46

If you work out these factors with these equations,

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

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

Â this by 60 and this one by 1,000.

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

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

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

Â 14:25

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

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

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

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

Â And we saw the coupon rate to be 12%.

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

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

Â Of course,

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

Â Now, if you have that financial calculator,

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

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

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

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

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

Â If this number goes up to 20%, okay,

Â if it increases by 20%, well the price

Â is then going to be equal to $1417.

Â Now if the bond is selling at $1417,

Â if I include that as the new value, right?

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

Â I have 60.

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

Â Hang in there.

Â So let me work with the pink marker here.

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

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

Â That's my question mark.

Â But for the same time periods,

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

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

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

Â And I solved for the yield.

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

Â of course, first, a semi-annual rate.

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

Â just about, just a little over 4%.

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

Â Notice what happened to the yield.

Â 17:32

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

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

Â It's ratings drop.

Â They go down to double C or something like that.

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

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

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

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

Â 18:04

Well, again I can plug the numbers in, and

Â here you can see something very interesting going on.

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

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

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

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

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

Â The price which had climbed to 1,417

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

Â 18:47

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

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

Â is the inverse relationship between bond yield and market prices.

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

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

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

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

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

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

Â they want something safe to invest their money.

Â And historically that safety comes from the government and

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

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

Â to print more money to meet their debt obligations.

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

Â So let's go back to our example.

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

Â And the yield to majority is

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

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

Â So let me recall the information.

Â We had face value of 1,000.

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

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

Â This is a 0 coupon bond.

Â And the maturity we said is 24 years.

Â And the yield to maturity, right?

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

Â 20:39

Let's not forget the frequency.

Â And the frequency denoted by M.

Â We said for that bond, was twice a year.

Â So remember what we did last time.

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

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

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

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

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

Â 21:12

In our timeline, what do we have?

Â Well, again, we have from zero right up

Â to 48 periods and we can denote these periods And

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

Â you get absolutely nothing in each of these periods, But

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

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

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

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

Â in terms of its present value or its market value.

Â 21:54

So the bond value equation,

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

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

Â So, we can just plug the numbers in here.

Â This is actually quite easy.

Â 22:12

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

Â Present value annuity for a 48 periods at 1%.

Â Kind of irrelevant because the coupons are zero.

Â Zero times any number will be zero.

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

Â lump sum factor at 1% for 48 periods.

Â 23:08

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

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

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

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

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

Â Okay.

Â What do we see in today's marketplace?

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

Â 23:45

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

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

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

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

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

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

Â I feel the market is too risky.

Â That's why I want to keep my money.

Â And they keep driving the price up.

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

Â right down to negative territory.

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

Â just by using this equation and solving for

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

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

Â what does that really mean?

Â It means that you are willing to pay for

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

Â So you're not even getting your principle back.

Â And that's what negative yields mean.

Â