Futures contract works exactly like this, except that I'll put equation three in between here. Every period the value is set to zero again, by changing the futures price. So instead of fixing this K forever, for the lifetime, we change it every day. We change it every day to get back to a contract that has zero value. And the relevant equation there is Ft = spot rate times e. So if you had that set equal to zero, you can see what we're doing is just replacing K with this futures price here, replacing K with this futures price here. Which fluctuates over time and is quoted in the newspaper every day. And on your problems set, there's a lot of futures prices. This is the futures price that we're talking about. Now, we change the price of this contract and at the time we change this price of this contract, we move cash from the losers to the winners. Somebody wins every day, somebody wins. Any time the futures price changes, there's going to be cash flow from one side to the other. So here all along, just let's think about the same history here, all along here, the long side has positive cash flow. Oops, I got that wrong, didn't correct me. It's to here. The futures price is rising. The futures price will be rising whenever the value of the forward is rising. They're the same thing. So as long as the value of the forward is rising, there is cash going from the short side to the long side. But then as the value falls, it's the reverse, the cash flows back. Short side has positive cash flow. And then if it goes, then we have the long side again. So the big difference between forwards and futures from the point of view of money and banking course, is cash flow. That the forwards, the only cash that changes hands is at the very end where one side ultimately wins. And who is winning in the middle? It doesn't matter. All that matters is what happens at the end. In the futures, it's not true. OK. They're showing you some fluctuation here but it can be much wilder than this. There could be very large cash flows. There's no particular reason to think that these things are, that the biggest fluctuation is at the end. The biggest fluctuation could be in the middle. So if you're entering into a futures contract, you have to be prepared to absorb those kinds of cash flow fluctuations. There's liquidity risk involved in a futures contract that is not involved in a forward contract. And that's maybe why you expect the price of the futures not to be exactly the same as the price of a forward. Even though in these formulas, I've sort of said it that way just for intuition sake, and empirically they're often not. There's an example in Stigum pages 718 to 722 something she calls the cash and carry arbitrage. When I first started teaching this course, I read those pages and I said, "There's something deep in these pages and I don't understand it." Because you shouldn't be able to make money on this cash and carry arbitrage. I knew enough about finance to say you shouldn't be able to make money doing this. And yet she's showing real world examples of actual numbers and people are making money, and she's letting you, showing you how to calculate the money that you're making there. And so, I said, "OK, I've got to figure this thing out." This lecture is a consequence of 10 years of me figuring this stuff out. So here's what I think about these pages, these four pages here, I always tell my students, you know, if you, this is one way to produce knowledge, is to find something that doesn't seem right and just keep worrying at it until you can figure it out. Just keep worrying at it and eventually, you'll figure it out, or not. But certainly you won't figure it out if you don't keep worrying at it. So, that's a strategy for intellectual production. That's my strategy for intellectual production.