In order to test the NPB approach to bonds, we first have to somehow describe bond cash flows. In order to do so, we will put the following scheme. So, bond cash flows look like this. So, this is zero now, then there are some periods, and the first parameter is the time to maturity, T. Like I said, there are really short term government treasury bonds that may be as short as one year, but the majority of bonds, at least all coupon bonds, are longer term. Now, what happens all the time of the life of the bond? First of all, the bond has the face value that sometimes is also called the maturity value. This is the normal, it's $1,000, and this is the amount that is basically printed on the bond. Now, almost all bonds are electronic, but the principal is still $1,000. And another very important point is it's coupon. This is the interest that is promised to the investor. And these coupon payments, they occur here, and then the final is C + F. These are bonds cash flows. Like I said before, in the United States, in England, coupons are paid semi-annual. That means if the coupon rate is 10%, that means that every six months, the investor receives $50, because 10% of 1,000 is 100. You divide it by two. In between, like I said, there is no compounding, this is just an agreement, because bonds exist for a long time. Now, what else can we say here, and what is important? If all rates here, at which we discount, r1, r2, and so- rk, if they were all the same, then I would say great, we know the answer. Because what is actually a bond cash flow structure? This is the combination of one bullet payment of F at point T, plus the annuity of coupons from one to T. And we know how to calculate the PV of an annuity, and clearly, we know how to get the BV of the bullet payment if we knew all these rs. And specifically, we know that if all rs are the same, then the PV of annuity is a very nice and shortcut formula. Well, the problem, though, is that for bonds, we cannot assume that these rs are the same. And that creates a significant problem. We will discuss bonds in short in this part of the course, and then later, in our sixth week, when we talk about advanced knowledge, we will come back to them. As a result of the fact that rs are not constant, we can say that bond valuation is really a challenging problem. So, this is the first thing. And on top of that, strictly speaking, all that would be great if these Cs and F were not only contracted, but also investors will be positive that they will receive them. Clearly, bonds can default. That means that the issuer promised to pay these amounts, but for whatever reason, falls short of delivering on this promise. And therefore, that means that oftentimes the investor does not receive that. That creates risky bonds. Even if this risk is minimal, there are some other important risks associated with the bonds. I will put some of them here. Well, first of all, many bonds are callable. That means that the issuer issues a bond for, let's say, 10 years, but has the right but not the obligation. The classic option to call this bond back, and let's say, three years at a pre-specified price. So, that creates some risk for investors, because they would lend for 10 years, and they had a plan to receive these payments, but then, at a certain point in time, the issuer has the right to call the bond back. You can clearly see that the callable bond is a regular bond plus a call option. Well, there are some other things here. The bond may be convertible. Let's say the bond may be converted in the stock of the issuer again, on certain pre-specified terms. That also creates risks. And also, on top of that, we can say that sometimes coupons are not paid in cash. First of all, they don't have to be equal. They sometimes link to, let's say, commodity prices, like gold prices, oil prices. And in some cases these coupons may be payment in kind. So, instead of receiving cash, you receive proportionately more of these bonds. So, all these things, unfortunately, make the detailed bond valuation a very challenging problem. But for now, we will limit ourselves, first of all, only to riskless bonds. You can say, well, all investments have risks. Yes, they do. But the treasury bonds in United States, they serve as a very nice proxy for riskless bonds. And also we will ignore all these potential options for now. We will come back to that in the sixth week, and study some of these in greater detail. But for now, from this cash flow pattern, we have to go ahead and come up with the PV formula for bond cash flows.