[CROSSTALK]. Let me start from this equation here. This is forward interest parity. We know it and we love it. We think it's basically true, okay. It's basically true. Okay. I'm going to rewrite that equation. In the following way, this is just an algebraic manipulation. one over 1 plus F 3, 6 equals so that's bringing that to the other side, and then bringing this 1 plus R 0, 3 all over 1 plus R 0, 6. Okay, do I get that right? It's just algebra. Haven't changed that equation at all. What I'm going to now point out, is that this is the forward rate. At time 0, adding a little 0 subscript there. And I'm going to say that at time 1 [SOUND] there is also going to be a forward rate. Now it's two months ahead, because we're now the time 1, but it's starting at time 3, okay. There's going to be a forward rate, and that is going to be defined as 1 plus R, now 1, 3, and 1 plus R, 1, 6. There's no reason that this is the same as that. There's no reason that this is the same as that. 1 over 1, you see what I'm doing now, right? [SOUND] In period two there's going to be another if we rode our forward contract in period two. For one period ahead now, okay, it's just going to be the ratio of this one period interest rate and this three period, four period interest rate here. Okay? And it's going to, in general be a different number than that number. And just for completeness, let's right 1 plus F, 3, 3, 6. And the interest rate from period three to period three is, is basically zero, okay? So really this is just 1 over 1 plus F. sorry. 1 plus R, 3, 6. So just by manipulating the definition, what we see here is that the forward rate. Okay, at maturity, you know at period three is exactly equal to the spot rate, okay? So this thing here, where the, the forward rate is greater than the expected spot rate over the lifetime of a forward, the, a forward contract that's written at maturity is going to be exactly equal to the, equal to the spot rate, so this is convergence sort of, the happens here. Okay. This changing forward rate, okay, means. The, the I'm just, just sort of I guess motivating it here. That this contract between the firm A and the bank, okay, changes value. Over its lifetime. It doesn't si, it doesn't stay. At, at the time that we, at the time that we brought it, okay, it was a zero value transaction, right? We just swapped IOUs. I owe you for three months, you owe me for six months and that was it, and they were of equal value. And tha, that's why no cash flow had to happen, okay? So, was a zero value transaction, okay. But from then on, it's not a zero value transaction at all, okay? One side wins and the other side loses. Okay, it's a zero sum transaction, okay? In the sense that. Maybe interest rates will rise, okay, over this period, okay? In that case one side wins. Maybe interest rates will fall over this period. . In that sense one side looses. Now we know, because of this sort of stylized fact, that in general, okay, the bank wins, right? Because the bank has locked in a lending rate that's greater than the expected spot rate. That's why the bank is doing this. The bank says, this looks good to me, I really doubt that I'm going to get anyone to borrow for that higher rate, you know, come the future. So I'm going to lock this in now, as a source of profits now. The firm on the other side says, well I'm willing to pay a little more, I understand this thing here, because I'm worried that the spot rate, that there's fluctuation around this expectation and that fluctuation might be me, okay? And this thing that I'm facing is, is infact a very serious. Survival constraint, and I want to push, I want to make sure that I take care of it now, and I'm willing to pay a little premium for that. So, so everyone's happy, okay? Firm A and firm, and, and, and bank and the bank are, are happy with, with this arrangement. So, on average, we expect the bank to win if, in that sense, okay? That if the, that, that, if, if. Interest rates turn out, as they typically are. Okay? It will turn out that the firm would have been better off waiting, and not locking in, but waiting for three months. Okay? But he had peace of mind, and so he didn't lose in that sense, okay? There's psychic, there's psychic gains, and possibly real economic gains, from knowing having that taken care of, allows you to pay attention to, to other, to other things, and not worry about it. Let's, let's have a continuous time version of this [NOISE] so that we can talk about it, okay? Take this item here, okay, and call this the what do I call it? forward contracts are promises to deliver goods at a future time T at a given price K, okay? So this is the forward price, okay? 1 over 1 plus an interest rate is a price, okay? We, we recognize that, okay? And this is in kind of price terms. Okay. And then we have, let's call this term. Let's separate these out. Okay. Let's call this term. Okay, the spot price, okay, because it's the price of a six period bond, okay, it's a, it's a spot price. And then this is an interest rate over the next three, three periods and we can write this, this eh, equation in the following terms. K equals S, 0, E, to the R, T. [SOUND]. Okay. This is just a continuous time version of that equation there. Where I'm doing it in terms of prices instead of yields also, so there's two, there's two changes. That's why I did it 1 over there, in anticipation of that. So doing the prices, you, so this is to link up, if you've had a finance course or if you go on to have a finance course. you're going to see equations like this, okay? And I just want you to be aware that this is the same thing as that. It's a, it's a, it's a transformation of it. I'm using discrete time. I'm, I'm being more specific about the timing of cash flows because we're doing money, money and banking. you'll see, you'll see this. This is in the notes will be called equation 1. Note that there's no time subscript on K here, as opposed to my time subscript there. The forward rate is established at the moment that you do the contract and then it stays the same for three months. Even though everything else is changing, everything else is changing here, that means the value of the forward contract changes over time. And, and how does it change? the equation, the value of the forward contract, will use little ft, okay, is the difference between the spot rate, which is changing over time. This, this, this, see this changing over time? And minus the K, just make sure that I don't mis-write this. No, this is right. K e to the minus r, T minus t. Okay. So what we've done is take this factor to the other side and then subtract, and then subtract them. So this is the, so the things that cause the value of the forward contract to, to change, are changes in the spot, in the spot rate. Which we've seen here, here, here, here. and changes in the term of that is, that is left to go, that's these, these, these here as we're discounting by less, discounting by less as we get closer to maturity. Okay? So now. I think the, the intuition of that is, is enough, you know, we're. This isn't a finance course. I just want you to see the factors that effect this. And now, think about time. And think about the movement of this value of the, of the, of the forward contract over time. Check your understanding. At time zero, what's the value of the forward contract? Times zero what's the value of the forward contract? I hear whispers, and their right. Zero, yes zero. Okay, and you can see that just by plugging in. If this is zero here, okay, and that zero there. Then these two terms are the same and so ft is, is, is zero. Okay. The value that for, so we'll start here at 0. At time zero the value that forw, forward contract is zero. I said, it's swap of IOUs, right? There's no, it's a zero value contract. Okay? That seems, I know weird. That's, that's why banking is hard often because it seems like how can anything add value that has zero value, And it's almost always the case that we're dealing with things like that in banking. And so it, it's a little mind blowing. Here's another one, okay, but it's not going to be 0 typically from then on. Okay? From then on, there's going to be fluctuations in the spot rate and so forth. And so let's, let's just say that this causes like this, or something like this. And then we come to time T, the maturity, okay. Which is 3, okay. In our, in our simple example here. and long side wins. Okay? Okay? That's the typical case with these kinds of interest rate contracts. By the way, all of this apparatus about forwards and futures is also true for, for commodities and things like that, that have costs, costs of storage. There, there's elaborate versions of this, okay? but we're really just thinking about. Discount bonds, because we're in the money market. Okay. So we're not talking about coupons or anything. This formula gets all messed up when you add coupons. We're not talking about storage if you dealing with wheat. Okay. You've got to worry about the cost of storing wheat, and. this formula would be different, if you're talking about different commodities. But the basic principles are the same. The basic principles they were, they were, they were talking about. I'm showing here fluctuations over the life of the forward, where sometimes the long side is, is ahead, sometimes the short side is ahead, but in the end the long side is ahead. But there's no cash flows until here, okay. Here at the end what happens? Here at the end, what happens, okay, is that there's this, this is S, T minus K, right, because when T's a capital T, this term goes away. And so it's the, spot rate at termination minus the forward, the forward rate at inception, okay? And that's how much the longside wins, okay? It's the, it's the divs between the realized spot rate and the spot rate that, that the spot price and the spot price that you, you locked in the forward price that you locked in at the beginning. But there's no cash flows, there's no cash flows until right here. [NOISE] In, in many forward contracts, what, what, what your obliged to pay. Is one side gives whatever, you know the bourgeois. Okay, and the other side gives the price that they locked in. You know. So, one side gives K, the price. The other side gives the bourgeois, okay. In financial contracts, it's typically not that, okay. It's more that one side gives, gives the value of, of the, of the, of the, of the bond, S, T. And the other side gives K, and so they're netting. There's a cash netting, and this is what's paid from the shorts to the longs. in the event that we wind up here. If we round up here, okay, it would go, it would go the opposite way. The cash flow would go the opposite way.