 Hello and welcome to this session. This is Professor Farhad in which we learn how to find the price of a bond. This topic is covered on the CPA, BEC section as well as well as the FAR section. It's also covered on the CFA exam. This topic on the CPA, it's a little bit tricky and the reason is simple. Because when you learn about bond pricing, you need to know about the time value of money. So this lecture is basically for students who are taken principles of investments or essential of investments, which is relatively speaking an advanced course. But if you don't know how to deal with the time value of money, you can go to farhatlectures.com or if you're looking to supplement your CPA preparation or your CFA preparation, please go to farhatlectures.com where I teach you all about this. Now, if you're taking a CPA exam, I don't replace your CPA exam. I cannot claim to do so. What I do is I supplement. My lectures will help you keep up with your CPA prep course. So I do have lectures about many topics. Please, if you like my lectures, please like them and share them. Connect with me on LinkedIn, subscribe to my YouTube. I have 1,800 plus lectures. Also, you could connect with me on Instagram and Facebook. And again, on my website farhatlectures.com, you'll find additional resources for this course as well as other courses. So let's talk about a bond and how does a bond work? Well, let's do a quick review about a bond. What is a bond? Basically, a bond is when the company borrows money and they give you a bond. So simply put, we have a company here. The company needs money. They will go to the market and we have lenders or investors and the company will give them the bond. The lenders will give the company back the money that they need. Then the company will pay them interests with pay interest to the lenders. So what does a bond looks like? So this is what the bond would look like. The bond will have a face value or par value, something called face or par value. And I'm going to keep it very simple. I'm going to assume the company needs to borrow $1,000, which is one bond. The bond face value or par value is the amount that the company will have to pay back. So the lender will give them the money. Now the company will have to pay back the face value. The face value is also called the maturity value because when it matured, when the bond mature, you have to pay back this $1,000. Now within the bond, you have what's called the stated rate, stated rate or another word for it or coupon rate goes by other terms, but we're going to go stated rate or coupon rate. What does that mean? It's a rate. It's a percentage. For example, here it's 8%. It means this bond would pay interest 8%. Now the interest on the bond is paid semi-annually. What does that mean? It means each six months, they will pay you 4%. So to find out how much you get paid off this bond, you will take $1,000 times 4%. You will get paid $40 every six months. For two annual, you'll get the $80. Now let's assume this bond has a life of three years. If this bond has a life of three years, it means it's going to have one, two, three, it's going to have six payments. And this is what we mean by the payments. So it's going to be $40, $40 for year one, $40, and $40 for year two, $40, then $40. So it's going to make six payments of $40. Each six months, they'll pay you $60. So what happens is every six months, the holder of this bond, they will clip the coupon and they will take it to the company, think about it that way, and the company will give them $40. Once all these coupons are gone, which is three years later after they redeem all the coupons, they will go back and get the face value of the bond. Now that's very important to understand this picture because this is what's happening when you buy a bond. Why is this important? Because when you want to price a bond, you have to understand what you are buying in the first place. So when you invest in a bond, you are buying two things. Notice when you invest in a bond, you are buying two things. You are buying the face value of the bond and you are buying the payment. What does that mean? It means you're going to be giving money today and you're going to get back $1,000 and throughout the life of the bond, you're going to be getting the payments. So to find the price of the bond, what you are really doing is you're finding the present value of the coupons. Here are the coupons, $40. So you have the coupons, $40 times the present value. Well, let's think about it. The coupons are the same, $40, $40, $40, $40, $40 and $40. We call this an annuity. So basically you are looking at the present value of an annuity. Now every, well, we know this present value, the period equal to six, you're getting six payments. The question is which interest rate do you use for this annuity? In other words, when you discount, you have to use interest rate. We have to be very careful here. When you discount, you have to use what's called the market rate or the yield. Well, this is going to be giving. So I'm going to put this on the side. Just kind of to remind you, this is something different. The market rate or the required rate of return. So when you, when the investor is trying to buy a bond, they will need to find the present value of the coupon payments first. When they find the present value, which interest rate do they use? They don't use this stated rate, unless the market rate is eight. And for the sake of illustration, I'm going to assume the market rate is eight. So here I'm going to assume the market rate is eight and the coupon rate is eight, just for the sake of illustration. But the market rate doesn't have to be as the coupon rate. And oftentimes it's not. And I will explain the implication of that shortly. Therefore, the market rate is eight. You have to multiply it by half a percent. I'm not half a percent by one half because the bond pays interest semi-annually. Therefore, the interest rate is four percent. So what you do is you find the present value of the inwardly. That's one part of the bond. This is part two of the bond right here. So we did this. Then you have to find the present value of the par value. What's the par value plus the par value equal a thousand. But remember, the par value, you're going to get it only once. So also you're going to find, you're going to multiply the thousand dollar by the present value of one. The present value of one, it means you're going to be getting this money only once and equal to six. Whatever you use for the annuity, you would use for the present value for the phase value and I equal to four percent. So this is basically the big picture. Don't worry, we're going to work an actual example to win this. But this is basically what you do when you buy a bond. The bond is composed of two components. The phase value is component number one. You will discount the phase value based of all the discount thing happened based on the market rate. And for now, I'm using eight percent for both the bond and the market rate, which it doesn't have to be. And often it's not then the present value of the coupon payment. So let's take a look at another formula saying the same thing, basically the price of the bond, the bond value equal to the sum of the coupon payments divided by one plus R, the interest rate raised to the end, which is the present value of the coupon payment of the annuity plus discounting the phase value. Same thing as this formula or the price of the bond equal to the coupon times the annuity factor. This is what I put down plus the power value times the present value annuity factor of the power value. Basically saying the same thing. Now the best way is to show you an example. Now we're going to work on Excel. Let's assume we are dealing with a bond that pays eight percent. It's a 30 year bond with a phase value of a thousand. This bond is making 60 semi-annual payments because it's 30 years. It's being twice a year. Therefore it's 60. The $40 is a thousand times eight percent. Well, 80 divided by two equal to $40. So we're going to be making the payment. So here's what we do and don't worry. We're going to work the computation in a moment. We're going to find the present value of the sum of all the payments discounted plus the thousand dollar discounted. And what we find out when we do so, because we're using eight percent as a discount rate and the coupon rate is eight percent, we're going to get exactly the phase value. Now we're going to see why later on you will not get the phase value. So it's easy to confirm that the present value of the bond, 60 semi-annual payment of $40. So the payments present value of the annuity is 904.94 and the present value of the thousand dollar payment 30 years from now is $95.06, which in total will give you, this is the present value of the par value. Together they will give you a value of a thousand dollar. Now you could use a financial calculator to find this, to find the prices and a financial calculator when the annual market interest rate, again we're assuming annual market interest rate at age is to make this simple. So you'll put the information in the following order. First you put in the number of periods and you'll put the number 60. And why it's 60? Because the bond had 30 years, 30 years times two payments a year. Then you have I, you put I the interest rate equal to four percent. Remember the market rate is eight percent since the payments are semi-annual. You have to adjust your interest rate. Then you put the future value is a thousand dollar. The bond will provide one time cash flow of a thousand dollar when it mature. Then the payment, you click on PMT, the payment is $40. And I come, how do you come up with the payment? It's one thousand dollar times eight percent, the coupon rate. And I emphasize here the coupon rate. Don't confuse this eight percent with this eight percent. This eight percent is for the market rate, not for the coupon rate. Now on most calculator, you would, you would punch the word compute or COMP or CPT and then enter PV to obtain the bond price. Now what's going to happen? The PV, it's going to, going to show as a negative one thousand because you have to pay a thousand dollar to make the investment. So the negative sign signifies that while the investor receive cash flow from the bond, the price to pay the bond as a cash outflow or a negative cash flow. Think about it. When you start your education, when you start to make investment in your career, you have to pay. It's a negative outflow. That's why the calculator will show you, will show you the number as a negative because it's the price of the bond that you have to pay today. It's a negative outflow. Corporate bonds typically issued at par. What does that mean? Now, I would not agree with the statement, but let's assume the issue at par. At par mean, let's assume the company wants to borrow a thousand dollar, the face value, they will get a thousand dollar for that. But often that's not the case. This means that the underwriter of the bond, of the bond issue, must choose a coupon rate very closely approximate market yield. It's the market rate. Remember, if the bond, remember, we have the bond coupon rate, coupon rate, let's do this now. It's a good, it's a good idea. We have the bond coupon rate and we have the market rate or the market yield. And here's what's going to happen. Let's assume the company is offering 8% and the market rate is 8%. What does that mean? It means this company is telling the investors, I'm willing to pay you 8% if you invest $1,000 with me. These investors, they are not dummy. They're going to go to the market and find out what is the market rate for similar bonds. If the market rate for similar bond is 8%, we would say that the bond will sell at par value. Par value means that the company will get exactly the par value for their bond. It means if the par value is $1,000, they will get $1,000. Now, let's assume the company is offering 8%, the market is offering 10%. Now, why would we offer 8% and not 10% like the market? Because we cannot afford the paying 10% cash outflow, just we can't. We can only offer 8%. Now, what's going to happen is this, investors are going to look at your bond and you're going to say, you're paying 8%. I'm not going to pay you the full $1,000. I can take my money somewhere else to the market and earn 10%. So what happened is under those circumstances, your bond will sell at a discount. What does that mean? It means rather than getting $1,000, the company will sell the bond for $930. I just made up this number, something less than 1,000. Let's assume the company is offering 8% and the market is offering 6%. What happened under those circumstances? Well, everyone's going to look at the market, then look at your bond. They're going to say, wow, this company is offering more than the market. What happened under those circumstances? Your bond will sell at a premium. What does that mean? It means rather than getting the $1,000, the company will get $1,050. I'm just making this number up more than 1,000. Okay. So of the rate, if the coupon rate is an adequate, investors will not pay the power value for the bond. What does it mean? They don't pay the power value. It means they pay something less than the power value. Well, if the coupon rate is more than adequate, then they will pay more than the power value, which is they pay a premium. Okay. After the bond are issued, bondholders may buy or sell the bonds in the secondary market that has nothing to do with the company. Once the bond is selling in the secondary market, the company is out of this. Okay. Now we have to understand in the bond market, in the bond market, the main determinant of bond prices is market interest rate. Again, back to the same concept that I just told you. If you have a bond, let's assume you have a bond that's paying 8% and suddenly market rate, the market rate over all the market rate, the Federal Reserve reduced the interest rate and the market rate went down to 5%. Now anyone that's selling bonds, they only have to offer 5% and your bond already pays 8%. So if you carry this bond, your bond will sell at a premium or the bond price will go up. So notice as interest rate went down, fluctuate inversely as interest rate as interest rate over all interest rate goes down, your bond value will go up. And let's switch the scenario. You're offering a bond paying 8% and now the market rate, the Federal Reserve increased the market, the interest rate and now the rate is 12%. Well, you're carrying a bond, you already invested your money and you're carrying a bond that's only paying you 8%. Now investors can go to the market and buy a similar bond paying 12%. What's going to happen is now interest rate went up, your bond value will go down and don't worry, we're going to see this in numbers in a second, but this is the idea. So this is a central feature, not only for bond for any fixed income security. So when we're dealing with fixed income, what fluctuate the fixed income is the ongoing interest rate and let's take a look at these figures and an Excel sheet to see how this all fits together. Let's assume we are looking at this bond. This bond is paying 8%, annually, semi annually is 4%. The face value for this bond is $1,000, just one bond, number of years is 30, number of payment is 60, which as we already know this and coupon payment is $40, which is the face value times 4% or the face value times 8% divided by 2. So this is the 8% and the market rate semi-annually is 8%, so the market rate is 8 and the market rate semi-annually is 4%. So notice we're looking at two different things. This 8% here that I highlight in red, this 8% is what the company is offering. This 8% is printed on the bond, stated on the bond, it's what the company is offering. This market rate semi-annual that I have it in yellow, this is what the ongoing market, what similar companies are selling for, 8%. They happen to be the same. So let's find, using the Excel sheet, the present value or the bond price. Here's what's going to happen. Well, before I do so, I should know that my price should be 1000 because if the company is offering 8 and the market is 8, it should sell exactly at 1000, at par value. So here's what's going to happen. First, I'm going to find the present value of the annuity. What's the present value of the annuity? First, you have to put the rate. The rate is MB10, always you would use the market rate. So notice the rate is cell B10. Cell B10 is the semi-annual interest rate of 4%, then comma B6. What's B6? The number of payments. Okay, number of payments, number of payments right here. Then comma B8. B8 is the coupon payment of $40. When I find the present value and notice I put negative here because if I don't put the negative, the answer will be minus. So I put the negative, so the answer is plus. So when I find the present value of the $40, so the present value of the $40 alone is $904.94. Let me do it, let me show it to you one more time, like what we did it earlier. So what I said earlier when we started the session, I said the price of the bond is the present value of the annuity. So the annuity here is $40 and the present value of this annuity equal to $904.94. So simply put, you're going to be receiving, let me show it to you on a timeline. You're going to be receiving, let's assume those dashes, we have total of 60 dashes and each dash represent a payment of $40. When I discount, when I find the present value of all these dashes, they equal to $904.94. Now, I'm going to have to find the present value of the face value, the face value is 1000. How do I find the present value? I'll take 1000 and I'll take the face value divided by one plus the interest rate B10 raised to the 60th power raised to N. So notice it's the formula is the face value divided by one plus I, the market rate raised to the Nth power. Okay. And if I do so, I will find out that $1,000 is worth $95.06. So I'm going to be receiving $1,000 30 years from now. If I discount that $1,000 till today, it's worth by itself $95.06. So it's very important to see the bond as two different pieces. One is the present value of the annuity. The other one is the present value of the face payment. Now, what I'm going to do is this, I'm going to show you what's going to happen to this bond, if we, if the market rate changes. So let's assume I'm going to change the market rate from, let's make this a premium bond. Premium. So let's assume the market is paying, the market is paying 6%. If the market is paying 6% and I'm paying eight, notice what happened. The present value of the payments are worth $1,107.02 and the present value of the face value is worth $169.00. Together they're equal to $1,276.00. Notice the bond sells at a premium. Why does it sell at a premium? Because the ongoing rate is lower than what the company is offering. So I'm going to keep the coupon rate fixed. Let me show you how you come up with a discount bond. If the company is paying 6%, but the market rate is 10%. 10% means 5% semi-annually. If I discount my payment using 5%, the payments alone are worth $7.57.17 and the face value is worth $53.54. Notice my bond is selling at a discount because my rate is not competitive now. Now once again, you can go ahead. Let's change this to 2%. Well, if I change this to 2%, if the market is paying 2% and your company is offering 8%, everybody's going to run to your bond and wants to buy your bond, your band will sell at $2,348.00 at a really, really high premium. Also, if the market is offering crazy 20% per year and your company is offering 8%, I'm just showing you extreme examples so you see the point, well, your bond will sell at a discount and you would only be able to get $401.00 for your bond. And this is the same thing. So when you buy a bond, so the key point here is to understand when the investor buys a bond, the investor look at two things. They would look at the future payment, which called the annuity, and they will discount the annuity using the market rate. I'm going to put here the market rate. Then they would look at how much they're going to get in the face value. They will take the face value and they will discount it using the market rate. Now I'm showing you this as a simple example, but the concept is the same. Even if you buy a bond three years later or four years later, how do you find what's the bond? What's how much is the bond is worth? Well, you would look at how much payments am I going to be getting into the future? You discount them. You'll find what's the face value that I'm going to be getting in the future. I will discount that face value. Those two numbers together will be the bond price. Once again, if the coupon rate is higher than the ongoing rate, your bond should, should sell at a pretty, you should sell at a premium. If it's lower, it should sell at a discount. If it's exactly the same rate, it should sell at par. So this is all what I'm going to do in this session. We're going to be working in the next session. I'm going to look at the bond yield because that's very important to understand this concept. What is the bond yield? Always, I'm going to remind you to like my lecture, share it. If it benefits you, it means it might benefit other people and then don't forget to visit farhatlectures.com for additional resources for this course, as well as other courses. If you're studying for your CPA exam, this is your chance to have some additional resources to help you pass the exam. Good luck and study hard.