Hello everybody, today we are starting our lecture number 7 and as you can see it combines chapters 8 and 9, the valuation of stocks and bonds. In your textbook you have them as two separate topics but think it's better to combine them in just one long lecture. I try to keep most of the basic information, most important information and some of the details which are not very important are left out.
so don't think of lecture 7 as really chapters 8 and 9 in its entirety that would be way too much so think of lecture 7 as just the most important stuff from chapters 8 and 9 okay also by the way notice that we skip chapter 7 that's by design so we don't want to do that we want to jump to our valuation of securities right away so why do we want to know the valuation principles of securities such as bonds and stocks so let me try to explain where we are heading if you remember we looked at for example net present value right and we said that it's minus cash flow zero plus present value of future cash flows which would be let's say cf1 over 1 plus r plus cf2 over 1 plus r squared and so on on now the question is what is r so so far up until now r was always given to you but i mentioned to you before that r is related to risk uh the question is risk to who and the answer is risk to the investors to people who invest people agency banks shareholders bondholders people who give money to the firm to use for a particular project so R in sense is a rate of return required by investors so again R is a required rate of return or rate of return required by investors the question is who are those investors and it turns out that investors are stockholders or shareholders and bondholders okay so what's going to happen stockholders will require their own return depending on the riskiness so that would be our s and bondholders will require their own return we're gonna call it rb or rd for debt but i prefer to use just the names of securities now what's going to happen is most of the time company uses both debt and equity so the company is dealing with both stockholders and bondholders. Therefore, it doesn't look at RS or RB only. The company combines those two and calculates something that is called RWAC, or weighted average cost of capital. So RWACC is so-called weighted average. cost of capital so in other words you could think of it as some weighted average rate of return required by stockholders and bondholders where the weights are the percentage of finance and that comes from stocks versus percentage of financing that comes from bonds so if i want to have 1 billion dollars to finance my new project if i use only bonds if i only issue debt then my overall required rate of return will be what the bondholders want.
If, on the other hand, I issue only stocks, then I need to cater to the needs of the stockholders, so I will have to discount the future cash flows from my project using RS, which is required rate of return needed by stockholders. But most of the time, companies use some kind of mixture of stocks and bonds. And by the way, you have to understand... we will talk about it later when i say stock i mean equity in general which means that you don't have to issue shares even if you use your retained earnings that still required rate of return on equity because you need to make sure that you satisfy your existing shareholders you got to be very careful there but anyway if you use both equity equity and debt then it will be some kind of weighted average cost of capital so that's where we are heading and you will see later on uh that we will address the weighted average cost of capital but before we go there we need to understand the nature of rs and rb and in order to do that we need to understand the nature of securities itself we need to understand the valuation of stocks and bonds in other words we need to understand what the investors really want from those securities so if i'm a stockholder what am i concerned about if i'm a bond holder what am i concerned about how do i determine the appropriate price how do i value a stock or a bond and that's exactly what we are doing today okay so we are talking about general principles of valuation valuation of securities so we start with bonds and then we're gonna switch to stocks so okay what is a bond so let me just describe the main parameters first so a bond is just a legal binding agreement between a borrower and the lender that specify the par value coupon payment coupon rate maturity date maturity and yield to maturity that is required by the market so let me just go over each of those items So okay, the first one is par value or face value.
So this is fixed. So you could think of par or face value as the principle. So if you buy a bond, and it has $1,000 face value, or par value, it means that at maturity, you are entitled to receive that face value back.
Okay, no matter what you pay for a bond, sometimes you pay above face value, sometimes you pay below, sometimes you buy it at face value. Regardless what you pay for it, at maturity, you're going to receive that face value back. And that face value does not change, no matter what. It's fixed amount.
The second part is coupon payment. So together with face value, you're also entitled to receive some regular coupon payments so you could think of coupon payments as regular interest payments and you receive those interest slash coupon payments up until maturity on a regular basis usually every six months or maybe once a year most bonds especially government bonds have six month coupon payment semi-annual so again coupon payment is fixed so i'm gonna write fixed by the weight That's why bond is called fixed income security because all of the income that you're getting from it, aside from price fluctuation, is fixed. So if you buy a bond and hold it to maturity, you're guaranteed to receive fixed coupon payments plus face value in the end.
Okay, now, since coupon payment is fixed and face value is fixed, the coupon rate, which is the coupon payment divided by face value, is also fixed. Okay, so for example, if coupon payment is $100 a year, okay, and the face value is $1,000 at maturity, that means that your coupon rate is 10% a year. Okay, so the bond has coupon rate, which is the coupon payment divided by face value. Okay, also the bond has maturity date.
Okay. which is also fixed unless it's restructured somehow and it's close to default and maybe the the issuer of a bond renegotiates the deal but that's you know that's outside the original contract the the normal bond has maturity date that is fixed which means that if it says that it's mature January 1st 2028 that's exactly when it matures and that's when you're gonna get your final coupon payment plus the face value okay and then you have maturity don't confuse those two maturity date is just point in time maturity on the other hand is number of years left until maturity date so if you buy a bond today right and it has maturity you know um 30 years from now all right that would be maturity date let's say today's uh you know let's let's imagine that today is january 1st 2020 so that it's easy for me to calculate let's say maturity date is january 1st 2030 so you're talking about 10 years to maturity five years from now it will have only five years to maturity and so on okay but maturity date is fixed and finally the last important parameter the last important characteristic of a bond is the yield to maturity we're gonna call it ytm so yield to maturity is the required market interest rate on the bond so think of yield to maturity as required a rate of return on a bond so the required rate of return means that it fluctuates because the people's mood towards that bond might change they might require higher rate of return they might require lower rate of return depending on market conditions so yield to maturity changes with market conditions and risk. Excuse me. Riskiness of the bond.
So that's the key here. Coupon rate does not change, but yield to maturity does change. If the company issued bonds 30 years ago, the risk was at a certain level and overall market conditions were very good.
different. Now when the bond is only five years to maturity, things have changed. Economy is completely different. The risk perceptions might be totally different.
So the yield to maturity, the required rate of return might be much higher or much higher. lower so what happens is again even though a bond in general is a fixed income security it has fixed face value fixed coupon payment fixed coupon rate the yield goes up and down up and down every day the question is how is it possible that everything is fixed but the yield is different okay so if I buy a bond today but somebody wants to buy that same bond a year from now okay the parameters of the bond did not change same face value same coupon payments everything is the same but somehow that person will get higher or lower yield to maturity it will depend on change in market conditions and the answer is it's the price okay it's the price of the bond that will always adjust and make sure that whoever buys a particular bond at particular point in time gets the yield to maturity that is consistent with market conditions regardless of the coupon payment the face value the maturity the price will always adjust the price will make sure that ultimately the holder of the bond the buyer of the bond derives the yield that is consistent with particular market conditions okay so you will understand it a little bit later uh it might be not obvious to you but once we consider the pricing model of a bond you will see why the yield to maturity is always guaranteed by the price so now let's talk about the pricing so in general the price of any financial asset is simply the present value of all the future cash flows that that asset will generate and that is true for any financial asset not just a bond okay so if i'm here in period zero and i'm buying a bond What I'm really buying is the coupon payments and the face value. Okay, so I'm buying a cash flow stream. So if I'm buying a cash flow stream, I need to discount those cash flows using appropriate required rate of return. So think of it as cash flow stream that needs to be discounted to present value terms.
And then that present value would actually be the appropriate market price. So... Let's consider an example here.
So let's say you have a typical bond that has, here's your time 0, here's your time 1, 2, all the way to time n. n is your date of maturity. So you get coupon payment C, coupon payment C, and then you're going to get C plus face value. Okay, so again C is your coupon payment FV is your face value. So C is Coupon payment And FV is face value.
or par value whatever you want to call it so i'm going to claim that the price today p0 would be equal to the present value of this cash flow stream which would be equal to c divided by 1 plus r but now appropriate r okay for a bond is called yield to maturity so r is yield to maturity okay so it's it's just a different name conceptually it's it's r it's required rate of return by a buyer okay so if you buy a bond you want certain yield it just happens that in bond markets that particular yield is called yield to maturity that's just a terminology so i'm going to divide by ytm yield to maturity plus c divided by one plus yield to maturity squared plus all the way to C divided by 1 plus yield to maturity to the power of N. And then finally, plus face value divided by 1 plus YTM to the power of N. Okay? So, if you look carefully at this cash flow stream, you will see that this is just present value of annuity, right?
Because you have cash. Cash flow C, coupon payment C, C, C, all the way to period N, fixed coupon payment, you know, up until fixed maturity time, finite. So that's typical present value of annuity. And then separately, you have present value of face value. So, not surprisingly, the formula for bond pricing is just this.
That's your present value of annuity. previous discussion and the second part is just present value of face value okay now it turns out that you don't need that formula luckily excel has very nice way to do it it's called price function you need to know how to use it i'll show you later but again i still want to show you formula simply because it teaches you the logic okay if you simply use excel without understanding the fundamental principle of value you will not really deeply understand how the bond prices and pricing is determined but looking at this formula now you see that it's pretty much discounted value of cash flows associated with coupon payments plus discounted value of face value and it turns out that it's equal to present value of annuity plus your regular present value of single amount now if you look at this formula you will see actually let me uh let me me go here if you look at this formula here you will see that when yield to maturity goes up the price of a bond pb goes down by the way i'm going to call this pb okay and vice versa when ytm goes down the price of a bond goes up so it's very obvious mathematically but let's talk about it intuitively so to do that that let's consider a very simple example so let's say you have a very simple bond that basically has only one period okay so you have c and then you have face value right here that's it in period one so in other words the bond has only one period until maturity you could think of it one year to maturity okay so So, all right. So let's suppose you look at the price of a bond and it's equal to C or 1 plus yield to maturity plus face value divided by 1 plus yield to maturity. Okay.
So suppose that C is equal to 100. Let's say face value is equal to 1000. That's very typical. So now what it means is that the coupon rate. If you want to calculate it your coupon rate is equal to 10%, which is 100 over 1000, and it's equal to 10%. Now, imagine that market yield to maturity is equal to 12%.
Okay, so what does it mean? It means that, you know, the bond has only one year to maturity. But remember, that bond may have been issued 20 years ago, when interest rates were lower.
And as a result, its coupon rate is only 10% but now the markets require high rate of return they want 12% so what I'm going to claim now is that if you buy this bond today even though your coupon rate is 10% your actual yield is going to be 12%. How is it possible? Well, the price will adjust.
So what's going to happen if I plug in that 12% yield to maturity in the price of a bond equation, you will see that the price will be lower than face value. As a result, you will be buying a bond that gives you $100 coupon payment and $1,000 face value, but you're buying it at a price that is bigger. below face value you're buying it at a discount and because you're buying it at a discount you derive the yield that is actually consistent with market condition so you're getting more than 10 coupon rate so let me prove it to you so if i do now pb it would be equal to 100 divided by 1 plus 0.12 plus 1000 divided by 1 plus 0.12 so overall it's 1100 divided by 0.12 i'm sorry divided by 1 12 so let me quickly do the the calculation here so it's 1000 excuse me 1100 divided by 112 that gives you 982 $982. So what happened? Think about it this way.
You're getting 10% relative to face value. Your coupon rate is 10% relative to face value, but you're not buying this bond at $1,000. okay you're buying it below face value so what you're doing is whatever you're missing on your coupon rate whatever you are lagging in terms of your coupon rate you're making it up by buying bone bond below face value so your overall rate of return now is going to be 12% how to check that let's reverse the logic let's think about it so you buy A bond at $982 in one year you get $1100. What is your rate of return?
Well, very easy. your R is going to be equal to 1,100 minus 982. That's actually what you make divided by what you invest, which is 982. And without any calculations, I can guarantee you that is going to be equal to 0.12. Okay? So what's going to happen again, even though you are buying a bond that has coupon rate that is lower than your expectation, you are buying the bond. at a lower price compared to face value so you're getting the market yield the opposite will be true for example I encourage you to do example that shows that if yield to maturity is equal to let's say 8% the price will be way higher so what's going to happen the price will go up above face value so in general and you will see it later but I'm going to give it to you right now when yield to maturity is greater than coupon rate then the price of a bond is above face value I'm sorry below face value Okay, we call it discount bond.
Okay, I'll show that to you later on the slide So in general when yield to maturity is higher than coupon rate the price is below face value again. Think about it this way Coupon rate is not competitive the market yield is higher so the investors want higher rate of return compared to coupon rate the only way for that to happen is if the price goes down below face value so the mechanism of that happening is just simple demand and supply so when yield is high compared to coupon rate nobody wants that bond at its face value so supply of that bond will increase meaning that everybody wants to sell the buy will uh the uh the demand will go down excuse me yes the mail will go the demand will go down because nobody wants to buy so what's going to happen the typical demand and supply the market forces will push the price down until the market reaches equilibrium such that the price is just right to make sure that whoever buys that bond actually gets yield to maturity consistent with market expectations and market conditions okay so when yield to maturity is Less than coupon rate, the price of a bond is actually going to be higher than face value. We call it a premium bond because now the opposite is happening. Your coupon rate is too good. The markets are willing to accept lower rate of return, but your coupon rate is pretty high.
So everybody wants that bond. The demand shoots up. Nobody wants to sell.
The supply goes down. that pushes the price up until again it reaches the equilibrium. And finally, when yield to maturity is exactly equal to coupon rate, we say that the bond is priced at par value. And again, coming back to our example, if you imagine then yield to maturity is 10%, you could see that if I substitute and put 0.1 here and here you would see that i would have 1100 divided by 1.1 and that would give me exactly 1000 so you would see that Okay, when yield to maturity is 10% which is equal to coupon rate the price of a bond is going to be exactly equal to face value Okay, so those relationships are not coincidence. Okay, but the point is The bonds may be issued years ago and the coupon rate is fixed The face value is fixed.
The coupon payment is fixed But what makes a bond always competitive is the fact that its price goes up and down to reflect the true market conditions as a result, given the buyer of a bond, the market rate of return, which is called yield to maturity. Okay. All right.
So now let's talk about different types of bonds. So now that we understand the general principle of bond pricing, let's talk about different variations. What kind of bonds exist in the market?
well the simplest one is so-called pure discount bond so what that means is that there is no coupon payment okay so if you look uh let's just say you know your typical bond is supposed to have coupon payments ccc and so on and then face value okay c plus face value period one two three all way to period n okay now so the present value or price would be equal to c over 1 plus yield to maturity plus c or 1 plus yield to maturity squared and so on all the way c plus yield to maturity to the power of n plus face value one plus yield to maturity to the power of n now imagine that c is equal to zero you might think wait a minute what's the point if there are no coupon payments who wants that bond after all you're not getting any interest and you're correct you're not getting interest but that does not mean that the bond is bad because Because remember, it's all about price. So even though there is no interest, even though I'm not getting any coupon payments, so all of them are zeros. Okay.
So, zero, zero, zero. So, what's going to happen? This part is going to be gone.
So, the price of a bond would simply be equal to face value divided by 1 plus yield to maturity to the power of n. So, what it means? It means that the bond is still viable. It's just it's clearly obvious that as long as yield to maturity is greater than zero, the price of a bond will always be below the face value okay so that's why it's called zero coupon or pure discount or you know a zero but again the pure discount bond that's the key that's the clue the reason why it's always a discount bond is because it cannot possibly be sold above face value. Potentially, it could be sold at face value if yield to maturity is exactly equal to zero.
Now, that being said, there are crazy times sometimes, and sometimes yield to maturity might be negative. We call it negative interest. But again, that's a separate story.
We could discuss it sometime, but not today. That actually means that sometimes you are being paid, okay? to actually you know you pay in order to to borrow money from somebody okay so that that's just really weird and usually it happens with government type of borrowing and lending but again don't worry about it normally we assume that yield to maturity is greater or equal to zero so let's not worry about negative interest rates although it does happen and the bad economic conditions sometimes it happens Okay, so that would be your pure discount bond. A very typical example of pure discount bond is treasury bill.
And usually pure discount bonds are shorter term. Okay, longer term bonds normally do have some coupon payments because the investor still wants some kind of liquidity. They still want some sort of cash flows before the maturity. So here's the pricing. I already showed you that.
So here is just an example. example let's suppose you have a 30-year zero coupon bond that have one thousand dollar power value and yield to maturity equal to six percent now this is just an example in reality you're not going to see a 30-year zero coupon bond as i mentioned most zero coupon bonds are very short term up to one year so so the good example is treasury bill but anyway hypothetically if you did have a bond like that the price would simply be the face value discounted at six percent using 30-year maturity and that's it is 174 dollars okay very simple now the level coupon bond would be uh your typical one the one that we started with so you do have coupon payments you have those cccc and then you have face value so the price is just present value of annuity plus present value face value the only thing that you you need to be aware of is that very often the level coupon bonds have semi-annual coupon payment okay so the coupon payments are typically semi-annual if you remember from our discussion of effective annual rate when you have a frequency of payments that is higher than annual your effective rate of return on that investment is a little bit higher than the stated yield but remember that yield to maturity will still be quoted on annual basis so yield to maturity would always be quoted quoted on annual basis okay that's just a convention it's annualized yield that's why if you look at bond quotes and you see treasury bill that matures i don't know a week from now but the yield is still let's say two percent and annually so what they do they always show you the annualized version of it now it's your job to figure out what is the actual rate of return okay but in this case it's not even about maturity let's say you have have multiple years until maturity it's more about frequency of your interest payments frequency of your coupon payments so let's say yield to maturity is equal to i don't know six percent but you have semi-annual coupon payments So in this case, what's going to happen is your effective annual rate is going to be a little bit higher than 6% because you're getting your coupon payments more frequently and you're able to reinvest them more frequently. So how much higher your effective yield is going to be? Well, your EAR, effective annual rate, would be equal to 1 plus, your R is 6%, your M is 2 to the power of 2, and then minus 1. So let's see what is... is going to be so I'm going to do my calculations here so I have 1.03 I'm sorry 1.03 to the power of 2 okay and then minus one so it's gonna be a little bit higher 6.09 so that would be equal to 6.09 again it's not trivial in finance nothing is small okay if you're talking about billions of dollars of investment the difference between six percent and 6.09 percent is is quite significant okay so that's just something to keep in mind so now let's consider an example so let's say you have a us government bond uh as of jane January 1st 2016 just for simplicity let's assume that's the date the bond has six and three eight coupon rate now that's not uncommon for a bond to specify its interest rates and yields to maturity in fractional in in simple fractions as opposed to in decimals okay so that's just the convention so six and three eight you just need to understand that that's actually six point four seven five i think i might be wrong but i think that's about it anyway we'll figure it out okay so the bond matures uh in five years in this case okay so as you can see it looks like it's four years but it's january 1st to december 31st so you're talking about five years and the yield to maturity is five percent now i want you to recognize right away without any calculation that because yield to maturity is lower than coupon rate I can tell you right away and you again you should be able to recognize that that the price of a bond is going to be greater than face value so that's going to be so-called premium bond and again it makes sense because it's coupon rate is higher than market wants market tolerates five percent but the bond gives you six and three eighth so everybody wants that bond at face value they wanna you know at face value everybody wants to buy it in fact people want to buy it even above face value how much higher well that would be your price so the price will show you how much higher than face value you're gonna go okay so coupon payments are made semi-annually so what's going to happen you're going to multiply six and three eight times one thousand and then you're going to divide it by two because if you take 0.06 and i think it's 0.06475 something like like that um and then you multiply by 1000 uh okay you're gonna get uh 600 475 hold on a second hold on a second let me just do that because i keep confusing you so let me just make sure what 3 8 is i always get lost here 375 okay i'm sorry okay so Okay, so that actually is equal to 6.375, okay, percent.
But you're getting half of that, okay? So what's going to happen, you're going to take 6.375%, you're going to divide by 2, and that would give you 3.1875. So that's going to... to be your actual coupon payment semi-annually so you got to be very careful with that now that's 3.1875 so multiplied by one thousand dollars and that's what gives you these coupon payments right here okay so you're going to have 10 semi-annual coupon payments so even though we have five years to maturity you're gonna have 10 coupon payments because they are made semi-annually and also at maturity you're gonna get $1,000 face value the question is what is the price of that bond and here we go here is your present value of annuity adjusted for the fact that you yield is not five percent is five percent over two because it's semi-annual and also your time adjusted for the fact that it's not five years it's 10 semi-annual periods same happens with here that's your face value so overall the price turns out to be equal 1060 dollars and 17 cents as you can see is greater than face value which is exactly what we predicted so the the markets are willing to pay a premium they want want to pay more for that bond because its coupon rate is very attractive okay now whoever that bond at $1,060 holds it to maturity even though that person will receive you know what is it six and three eight coupon rate annually reality the actual yield will only be 5% annually and the reason for it is because they overpaid they paid more for that bond compared to face value okay and that's how I want you to think about it I want you to visualize it and understand it intuitively okay so now let's talk about Excel so how do we do that with with spreadsheet so turns out that it's very simple function it's called price so there are several parameters so you have settlement so i encourage you to put today's date or let's say the date of purchase the date of purchase okay whenever you're ready whenever you buy a bond or if you want to just hypothetically imagine some date just write that date that's your date of purchase maturity obviously is maturity date okay then the rate is your coupon rate and the yield is your yield to maturity okay so so far very self-explanatory now redemption that's a little bit tricky so redemption is your percentage of face value that you receive at that you receive at maturity so it's percentage of face value that you get at maturity so as you can see Excel does not ask you the actual dollar amount of face value it asks you the percentage of face value so if your face value is $1,000 and you fully expect to receive that $1,000 you would put a hundred over there so redemption would be your hundred percent so I would say normally it's equal to 100 okay very rarely you have situation when it's you know more than 100 or less than 100. usually it happens if the terms of the of the bond are renegotiated or if there are special conditions in the contract that allow the issuer to either reduce the face value or allow the borrower excuse me the the lender to increase the face value so that the the the borrower and ends up paying more again it happens very very rarely so for our cases always make redemption equal to zero excuse me to 100 meaning that you do get full face value which is 100 then frequency is your frequency of coupon payments Okay, and finally you have basis but again don't worry about basis.
It's just Use the default. Okay again basis is more about when your cash flows actually happen So we always assume that it's default which means that your bond price is done such that you buy it today but you receive your first coupon payment exactly one period from now okay so all right so now let's just um um Let me just do an example. So let's just do our example here. Okay.
So I hope I remember all the parameters. So, okay, let's see what's going on here. me just open the new Excel spreadsheet so here's what I normally do I try to put all the input in Excel first and then I make references so let's let's do it so we're gonna have um so we had uh january 1st 1 1 2016. again it could be any date it doesn't matter as long as the difference between those dates is going to be five years but i'm going to use what you guys have so 1 1 16 and another one was uh 12 31 2020 okay so i'm gonna call this one Settlement.
And I'm going to call this one maturity. All right. Okay.
Then for rate, you're going to have 5%. So 0.05. And for yield to maturity, YTM, we're going to have 6.375.
Now notice, oh, I'm sorry. I'm sorry. It has to be 0.06375. Now notice, I don't make any adjustment. I don't divide it by two.
I didn't divide. by two either the reason for it is because excel does it for you okay that's the beautiful part about excel so once you tell excel what's the frequency it will automatically make all the necessary adjustment so that's why the next item would be frequency and let's make it uh two okay all right and uh well if you want to let's just put redemption here as well And make it 100. Just to be complete. So that we have everything here in cells.
Now, anytime we make any adjustment, we will see the result. So, equal to. And I'm going to say price.
And as you can see, Excel tells me already that is a function, that is a known function. So I have this view here so that you can see what the parameters are. So for settlement, here's our settlement.
For maturity, we're gonna put this one for rate we're gonna put this for yield this is my yield to maturity now redemption 100 and finally the frequency is 2. that's it again as i said we don't need basis so we click enter and uh that is interesting for some reason we got it uh below face value hold on a second what happened here uh 0.06 oh i'm sorry i mixed up the uh the uh it's the other way around so well now now but now you see uh you learned your lesson see that's how you need to think about it so i got the price below face value so i knew something is wrong right away so what happened is i switched rate and yield maturity so let's just do it the right way then so the rate is actually 0.06375 by the way when i do it that way you could you could see that it's uh it's pretty much uh face value but let me explain in a moment so here is your 0.05 okay all right now we got it right one thing that you notice is that the price is not in dollars the price again is in percentage of face value so it says that the price is 106 percent of face value now if I know my face value which is $1,000 and if I know that it's hundred and six percent of that automatically i know that the price is going to be equal to 1060.14 cents so all i have to do is to pretty much uh you know uh equal to i can actually multiply this by 10 and that would give me the right that's a shortcut but that will give me the right answer okay all right because you know because you multiply by a thousand and you divide by 100 so that's basically results in in multiplying by 10. so as you can see i got exactly the same answer $1,060.14 now the very nice you know feature of doing it in Excel is that now I can play with parameters and you can see how it affects the results so again in our case the coupon rate is high higher than yield to maturity so the bond is selling at a premium it sells above face value now imagine the opposite happens let's say the market conditions change let's say there is some huge inflationary pressure in the economy so that the yield to maturity comes dramatically let's say from five percent to i don't know eight what's going to happen so if i click enter you should see that the face value i mean the price of a bond has to drop below face value and as you can see that's exactly what's going on so now the price is 934 dollars the logic is very simple the yield to maturity is very high it's higher than coupon rate the bond is not competitive anymore at any price that is at or above face value so the price has to drop how far well apparently has to drop to $934. At this price, the bond actually yields 8%. Even though coupon rate is 6.3, the bond actually yields 8%.
And finally, another very important threshold, let's imagine that the yield is actually equal to coupon rate. Now, as you can see, the price is equal to $1,000 by all means. Okay, so again, that's not a coincidence. It's always going to be the case that when the coupon rate is exactly equal to yield to maturity, the bond is... sold at par value or at face value and this is usually what happens when the bond is first issued because remember if you are the issuer of a bond if you're a corporation when you issue a bond you designing the appropriate coupon rate.
It's your job to figure out the coupon rate. How do you figure out the coupon rate? Well, you observe what is the current yield to maturity on bonds of similar risk and you match that yield to maturity by designing the coupon rate that is exactly equal to expected yield to maturity. So that is why normally when bond initially is issued it's sold at par.
It's normally sold at face value. because the issuer, the corporation, makes sure that the coupon rate is exactly equal to expected yield to maturity. But when the time goes by, the coupon rate does not change anymore, but the market conditions change, and sometimes yields go up, sometimes yields go down. And that is why when bond starts trading, especially after a few months or a few years even, the price of a bond may be dramatically different from its face value.