Hopefully, you've had the time to stare at what that bond is trying to say and so on and so forth. We did the timeline and formula, just remember, match the interest rate with the periodicity. Always make your interval of thinking the compounding interval and it'll be pretty obvious when you're looking at the bond or whatever the idea and so on, what it is. Match the compounding interval with interest rate. Let's do some calculations. Here I'm going to go over to my friend, the Excel. There are some numbers from the past which we'll ignore for the time being. I'm going to stick in cell C. If you can see me, hopefully, you're fine. What I'm going to do is I'm going to do PV. Why? Because I'm trying to do a PV of a bond. What was the interest rate of a similar bond? Remember, it was 6 percent, but we are doing everything semi-annual, 0.03. How many number of periods? Twenty. What is the PMT? Thirty. What is the future value? You can do everything. What do you notice? I purposely picked this example first, construction. What do you notice? The price of the bond is equal to the face value? Yes. Do you notice that? Let's go here and I'll show you the intuition for it. The price of the first bond is the face value. That is today, the price exactly equal to the face value, even though that face value is coming 20 periods from now. The reason for it is, what do you see the relationship between coupon, which is like a percentage, and the interest rate? They are both what? The same, 30 over 1,000 is 0.03, and what is my interest rate? 0.03. What's happening? Very intuitively, the coupon is the numerator, it's coming to you, but time is hurting you at the same rate, so they cancel each other and what are you left with? A thousand dollars. This is a neat thing that they'll say in the real world, that the bond is trading at face value, right? At par. Par, P-A-R is face value. Let's assume now that the interest rate in the real world is actually 4 percent per year, which is what per month? Two percent. What has happen to the price? It's gone up, it's greater than face value. It's called trading at a premium. Premium doesn't mean you're paying a higher price than you should, premium simply is relative to thousand. Discount is relative to thousand too. Okay, 1,163. Quick question, why did this happen? Forget about the exact price and this is what I want you to have in your gut. A feeling for finance, not just numbers. You know, the answer has to be greater than thousand, why is that? Notice the coupon is still 30 and it cannot change once you've issued the bond. Because what's the coupon? Who determines the coupon? The person issuing it. We are assuming the government is not going to default on the 30 at least, right? But what can change is the market rate that we are using. The market rate is now 2 percent and the coupon is 3 percent. What is going to do? It's going to make the value of the bond go over a thousand simply because the value you are getting it at a higher rate coupon than the discount rate. Now suppose the price is such that the rate of return that you're getting in a similar instrument out there in the market is actually eight percent and half of eight is what? Four percent. What will happen to the price? It'll go below a thousand worth. This is called selling at a discount. Zero-coupon bonds always sell at a discount because there's no coupon compensating for it. Whereas coupon-bearing bonds can and do sell at a discount. But what has to be true? Let's look at the parameters. The face value is $1,000. I'm not changing that. The coupon is 30. I'm not changing that. The n is 20. I'm not changing that I'm standing today. What have I changed? The interest rate. Now I'm hurting the bond's price at a higher rate than the coupon flowing in, so what happens? You get less than face value. This tells you something about the pricing of bonds, which is this, interest rates go up, price goes down. But for a coupon paying bonds, government bond, there's this neat relationship where the interest rate and the coupon rate are the same, the price of the bond has to be the face value, and then if the interest rate is higher, it's lower, and the interest is lower then the coupon rate price is higher. Let's try to take this learning and graph it. We've done the calculations, and I would really encourage you to go back and do those calculations again, but I'll write out what you should find. If coupon rate is greater than r, price will be greater than face value. If coupon rate is less than r at a specific time, the price will be less than fail face value, and if coupon rate is equal to r, price will be equal to face value. This is what you should find and just mess around with this because it'll help you. What I would like to show you now is a graph, and this graph is important. Price, r, zero, it looks like this. The price of a bond falls when interest rates goes up, that's what it's showing and that makes sense. If the interest rate is six percent or three percent per six months and the coupon rate of this bond was three percent, what will the price be? Thousand, we just did this. One thing I really want to emphasize about this, coupon rate is not a market thing, it's fixed by the government. Don't ask me why because that's getting into detail. It's something determined by their ability to pay periodically versus face value. This curve is telling you the relationship between bond prices and interest rates. Let me just give you one little bottom-line thing, price goes down if r goes up. Second, short-term versus long-term, which bonds are more price-sensitive? In other words, if interest rate changes, which bonds will be more sensitive bonds around? Which bonds? Long-term. Finally, whose price will jump around most, zero-coupon or with-coupon bond? Let me for simplicity, keep the maturities the same. Whose price will bounce around more and it's related to point number 2? If you have a zero-coupon bond, where is all your money coming from and supposing it's 10 years maturity? All your money is coming 10 years from now. You have a 10-year same face value, coupon paying bond. What is happening with this bond? Some of its money is coming earlier and not all is coming at the end. The zero-coupon bond will be more sensitive to changes in interest rates compared to an otherwise identical coupon paying bond. It's not perfectly identical, but the only difference is the coupons. Let's take this break because this is an interesting point to think about, and I want you to know this stuff inside out because this is the simplest kind of bonds, and this is what the world talks about all the time. Every government issues these and so on. We'll come back to issues related to it. But in the next chunk, I will go dig a little deeper as we go further.