So when we talked about present value, we assumed that there was a single interest rate that we could use to discount cash flows regardless of the period. So now we're going to talk about whether that assumption actually makes sense. So one way to think about whether that assumption makes sense is to think about what is the yield to maturity on bonds? The yield to maturity on bonds of different maturities tell us what the market is using to discount cash flows of different periods. So the way that the discount rate relates to the maturity of the cash flow sometimes called the yield curve. So the yield curve graphically depicts the relation between the yield to maturity and the bond maturity. So you can actually get this from bonds in the data. So here's what we normally do, we take a selection of bonds and we look at their yield to maturity. So currently, if you look at very short-term bonds at the time of this filming, the very short-term bonds have yields to maturity of about two percent rising for the longest term bonds to a yield to maturity of about three percent. Now, to construct this yield curve, only certain maturities are used. These tend to be the one month, three month, six month treasury bills. By the way, a treasury bill is a short-term zero-coupon bond and the 2, 3, 5, 10, and 30-year bonds. Now, you might ask, why these maturities? Because after all, we know that there are lots of maturities. For example, last year's 30-year bond is now a 29-year bond. But the 29-year bond is almost never shown on the yield curve, nor is the 28 year bond, nor as a 27-year bond, nor as a 26-year bond and so on and so forth. Why is that? Well, that's because the newly issued treasury bonds are the ones that trade the most. They are the most liquid and so their prices are the most reliable. The name for the bonds that are just issued is on-the-run. So note the most recently issued bonds, the ones with these maturities are on-the-run bonds and the other bonds are off the run. So the yield curve is typically constructed with on-the-run bonds. The yield curve that I just showed you was upward-sloping. Most of the time, the yield curve is upward-sloping, but sometimes it flattens. So here is a yield curve that is less upward-sloping than the yield curve up here. So people talk about this as the flattening of the yield curve. Hypothetically, you could have a completely flat yield curve. I highlight this because this was our assumption for the clip for present value. When we use the same r to discount cash flows of different maturities, we are implicitly assuming a flat yield curve. A yield curve need not be upward-sloping or flat. Sometimes the yield curve inverts and sometimes it could just be weird. So it could be hump-shaped or inverted hump-shaped. Here's what we know about the yield curve. Most of the time, the yield curve is upward-sloping. That is, we see higher yields for longer maturity bonds. Here's what else we know about the yield curve. When the yield curve looks like this, when it inverts, yikes. An inverted yield curve often predicts recessions. So you might ask, why is the yield curve upward-sloping most of the time? Note that this does not have to do with the time value of money because this already adjusts for the time value of money. That's in the definition of r. Why should it be that the time value of money is more even adjusting for the length of time for longer-term investments? So the question is, why does the yield curve upward slope most of the time? Another question is, why does an inverted yield curve predict recessions? The answer is, we don't know. We don't know the answer to these very basic questions. Hopefully, someday we will discover the answer. In the meantime, let's talk about how we might use information from the yield curve to calculate, say, net present value. Let's do an example. An investment costs one million, it will pay 0.1 million in one year, 0.35 million in two years, and 0.6 million in three years. So in other words, we have minus one million, then 0.1, then 0.35, and then 0.6. Now, we also have the following information. We know that for zero-coupon bonds, the yield to maturity on the one-year equals one percent. The yield to maturity on the two-year equals 1.5 percent. The yield to maturity on the three-year equals four percent. So the correct thing to do is to calculate our present value using these yield to maturity. So the net present value is our initial cashflow plus the 0.1 we get in one year, discounted at the one year rate. The 0.35 we get in two years discounted at the two-year rate squared, this is two years. The 0.6 we get in three years discounted at the four-year rate cubed. This, by the way, is minus 0.028 less than 0. By the NPV rule, we should reject the investment. So that's an example of how you might use the yield curve. Let's do one more example. Suppose you have the following data on zero-coupon bond prices and these are prices per $100 of face value. The one-year bond sells for a price of $93.46. The two-year bond sells for a price of $89 even, and the three-year bond sells for a price of $83.96. Given this setup, what is the price of a three-year coupon bond with C over F, a five percent per $100 of face value? So what we want to do is value something with the following cash flows. One year from now we get $5, two years we get $5, three years we get $105. So there are two ways to answer this question. One way is to back out the yield to maturities from the prices of the zero-coupon bond and discount the cash flows of the coupon bonds at those yields to maturity. So for example, given that P_1 is 93.46, the yield to maturity on this bond is 100 over 93.46 minus 1 or seven percent. Given that P_2 is 89, the yield to maturity on this bond is 100 over 89. Using the formula, the one-half minus 1 that's six percent. P_3 is 83.96, the yield to maturity is 100 over 83.96_one third, and that's also six percent. So we can now use those yields to maturity to determine the price of our coupon bond. So five, now we're going to use yield to maturity 1 plus 5, yield to maturity 2 plus 105 yield to maturity 3 equals 5 over 1.07 plus 5 over 1.06 squared plus 105 over 1.06 cubed, which, by the way, equals $97.28. So that's one way to solve the problem, but there's also a much faster way. We can use the prices themselves. So remember, the price of the one-year bond per $100 of par value is 93.46. The price of the two-year bond per $100 of par value is 89. The price of a three-year bond per $100 was 83.96. Now, suppose you were to buy not a $100 bond but a one-dollar bond. The price of that bond would be this price divided by 100. So it would be 0.9346. So if we had a one-dollar bond, the price would be 0.9346 and similarly, if we had a one-dollar bond, the price would be 0.89. Do you see what I did? I divided this by 100. If we had a one-dollar bond, the price would be 0.8396. So in other words, if you're willing to pay $93.46 to have a $100 one year from now, you would be willing to pay $0.9346 to have one dollar one year from now. So now let's take seriously what I said before about these bonds being the basic building blocks. Another way to value the coupon bond is to take each of the cashflows and multiply it by the one-dollar bond price. So what is my coupon bond? Well, it's a basket and in the basket are five of the one year one-dollar bonds and five of the two-year one-dollar bonds, and 105 of the three-year one-dollar bonds. Of course, if I calculate it this way, I get the same price as I got before, $97.28, except here I'm using the prices of the bonds. I'm not going through the yield to maturity.