0:19

In the following way, this is just an algebraic manipulation.

Â one over 1 plus F 3, 6 equals so that's

Â bringing that to the other side, and then bringing this 1 plus

Â R 0, 3 all over 1 plus R 0, 6. Okay, do I get that right?

Â 1:14

Now it's two months ahead, because we're now the

Â time 1, but it's starting at time 3, okay.

Â There's going to be a forward rate, and that is going to be defined as 1 plus R,

Â now 1, 3, and 1 plus R, 1, 6.

Â 1:47

[SOUND]

Â In period two there's going to be another if we

Â rode our forward contract in period two.

Â For one period ahead now, okay, it's just going to be the ratio of

Â this one period interest rate and this

Â three period, four period interest rate here.

Â Okay?

Â And it's going to, in general be a different number than that number.

Â 2:37

sorry.

Â 1 plus R, 3, 6.

Â So just by manipulating the definition, what we see here is that the forward rate.

Â Okay, at maturity, you know at period three

Â is exactly equal to the spot rate, okay?

Â So this thing here, where the, the forward rate is greater than

Â the expected spot rate over the lifetime of a forward, the, a forward

Â contract that's written at maturity is going to be exactly equal to the, equal

Â to the spot rate, so this is convergence sort of, the happens here.

Â Okay. This changing forward rate, okay,

Â means.

Â The, the I'm just, just sort of I guess motivating it here.

Â That this contract between the firm A and the bank, okay, changes value.

Â Over its lifetime.

Â It doesn't si, it doesn't stay.

Â At, at the time that we, at the time that

Â we brought it, okay, it was a zero value transaction, right?

Â We just swapped IOUs. I owe you

Â for three months, you owe me for six months

Â and that was it, and they were of equal value.

Â And tha, that's why no cash flow had to happen, okay?

Â So, was a zero value transaction, okay.

Â But from then on, it's not a zero value transaction at all, okay?

Â One side wins and the other side loses. Okay, it's a zero sum transaction, okay?

Â In the sense that.

Â Maybe interest rates will rise, okay, over this period, okay?

Â In that case one side wins.

Â Maybe interest rates will fall over this period.

Â . In that sense one side looses.

Â Now we know, because of this sort of stylized

Â fact, that in general, okay, the bank wins, right?

Â 4:24

Because the bank has locked in a lending

Â rate that's greater than the expected spot rate.

Â That's why the bank is doing this.

Â The bank says, this looks good to me, I really doubt that I'm going to

Â get anyone to borrow for that higher rate, you know, come the future.

Â So I'm going to lock this in now, as a source of profits now.

Â 4:45

The firm on the other side says, well I'm willing

Â to pay a little more, I understand this thing here,

Â because I'm worried that the spot rate, that there's fluctuation

Â around this expectation and that fluctuation might be me, okay?

Â And this thing that I'm facing is,

Â is infact a very serious.

Â Survival constraint, and I want to push, I want to make sure that I take

Â care of it now, and I'm willing to pay a little premium for that.

Â So, so everyone's happy, okay?

Â Firm A and firm, and, and, and bank and

Â the bank are, are happy with, with this arrangement.

Â So, on average, we expect the bank to win if, in that sense, okay?

Â That if the, that,

Â that, if, if.

Â Interest rates turn out, as they typically are.

Â Okay?

Â It will turn out that the firm would have been better

Â off waiting, and not locking in, but waiting for three months.

Â Okay?

Â But he had peace of mind, and so he didn't lose in that sense, okay?

Â There's psychic, there's psychic gains, and possibly

Â real economic gains, from knowing having that taken

Â care of, allows you to pay attention to, to other, to other things, and not worry

Â about it.

Â Let's, let's have a continuous time version of this

Â [NOISE]

Â so that we can talk about it, okay? Take this item here,

Â okay, and call this the what do I call it?

Â 6:22

okay?

Â 1 over 1 plus an interest rate is a price, okay?

Â We, we recognize that, okay?

Â And this is in kind of price terms. Okay.

Â And then we have, let's call this term.

Â Let's separate these out.

Â Okay. Let's call this term.

Â Okay, the spot price,

Â 6:54

okay, because it's the price of a six period bond, okay, it's

Â a, it's a spot price. And then this is an interest rate over the

Â next three, three periods and we can write this, this eh, equation in the following

Â terms. K equals S, 0, E, to the R, T.

Â [SOUND].

Â Okay.

Â This is just a continuous time version of that equation there.

Â 7:31

Where I'm doing it in terms of prices instead

Â of yields also, so there's two, there's two changes.

Â That's why I did it 1 over there, in anticipation of that.

Â So doing the prices, you, so this is to link up, if you've

Â had a finance course or if you go on to have a finance course.

Â you're going to see equations like this, okay?

Â And I just want you to be aware that this is the same thing as that.

Â It's a, it's a, it's a transformation of it.

Â I'm using discrete time. I'm, I'm being

Â more specific about the timing of cash

Â flows because we're doing money, money and banking.

Â 8:02

you'll see, you'll see this.

Â This is in the notes will be called equation 1.

Â Note that there's no time subscript on K

Â here, as opposed to my time subscript there.

Â The forward rate is established at the moment that you do

Â the contract and then it stays the same for three months.

Â Even though everything else is changing, everything else is changing here,

Â that means the value of the forward contract changes over time.

Â And, and how does it change?

Â 8:57

just make sure that I don't mis-write this.

Â No, this is right.

Â K e to the minus r, T minus t. Okay.

Â So what we've done is take this factor to

Â the other side and then subtract, and then subtract them.

Â So this is the, so the things that cause the value of the

Â forward contract to, to change, are changes in the spot, in the spot rate.

Â Which we've seen here, here, here,

Â here. and changes in the term of that is, that

Â is left to go, that's these, these, these here as we're

Â discounting by less, discounting by less as we get closer to maturity.

Â Okay? So now.

Â 9:40

I think the, the intuition of that is, is enough, you know, we're.

Â This isn't a finance course.

Â I just want you to see the factors

Â that effect this. And now, think about time.

Â 10:10

I hear whispers, and their right. Zero, yes zero.

Â Okay, and you can

Â see that just by plugging in.

Â If this is zero here, okay, and that zero there.

Â Then these two terms are the same and so ft is, is, is zero.

Â Okay.

Â The value that for, so we'll start here at 0.

Â 10:32

At time zero the value that forw, forward contract is zero.

Â I said, it's swap of IOUs, right? There's no, it's a zero value contract.

Â Okay?

Â That seems, I know weird. That's, that's why banking is

Â hard often because it seems like how can anything add value that has zero value,

Â And it's almost always the case that we're dealing with things like that in banking.

Â And so it, it's a little mind blowing.

Â Here's another one, okay, but it's not going to be 0 typically from then on.

Â Okay?

Â 11:00

From then on, there's going to be fluctuations

Â in the spot rate and so forth.

Â And so let's, let's just say that this causes like this, or

Â something like this.

Â And then we come to time T, the maturity, okay.

Â 11:38

Okay?

Â That's the typical case with these kinds of interest rate contracts.

Â By the way, all of this apparatus about forwards and futures is also

Â true for, for commodities and things like that, that have costs, costs of storage.

Â There, there's elaborate versions of this, okay?

Â but we're really just thinking about.

Â Discount bonds, because we're in the money market.

Â Okay.

Â So we're not talking about coupons or anything.

Â This formula gets all messed

Â up when you add coupons.

Â We're not talking about storage if you dealing with wheat.

Â Okay.

Â You've got to worry about the cost of storing wheat, and.

Â this formula would be different, if you're talking about different commodities.

Â But the basic principles are the same.

Â The basic principles they were, they were, they were talking about.

Â 12:20

I'm showing here fluctuations over the life of

Â the forward, where sometimes the long side is,

Â is ahead, sometimes the short side is ahead,

Â but in the end the long side is ahead.

Â But there's no cash flows until here, okay.

Â 12:36

Here at the end what happens? Here at the end, what happens,

Â okay, is that there's this, this is S, T minus

Â K, right, because when T's a capital T, this term goes away.

Â And so it's the, spot rate at termination

Â minus the forward, the forward rate at inception, okay?

Â And that's how much the longside wins, okay?

Â It's the, it's the divs between the realized spot rate and the spot rate that,

Â that the spot price and the spot price that you, you locked in the forward price

Â that you locked in at the beginning.

Â 13:14

But there's no cash flows, there's no cash flows until right here.

Â [NOISE]

Â In, in many forward contracts, what, what, what your obliged to pay.

Â Is one side gives whatever, you know the bourgeois.

Â Okay, and the other side gives the price that they locked in.

Â You know.

Â So, one side gives K, the price. The other side gives the bourgeois, okay.

Â In financial contracts, it's typically not that, okay.

Â It's more that one side gives, gives the value of, of the,

Â of the, of the, of the bond, S, T.

Â And the other side gives K, and so they're netting.

Â There's a cash netting, and this is what's paid from the shorts to the longs.

Â in the event that we wind up here.

Â If we round up here, okay, it would go, it would go the opposite way.

Â