Date: April 22, 2026
Topic: Swaps, Forward Rates, and Final Course Integration
This session is not really about introducing something radically new.
It is a compression of the entire course into a single logic:
Finance = present value of future cash flows.
The professor starts exactly there, deliberately:
Everything reduces to:
\[\text{Price} = \text{Present Value of future payments}\]This is the backbone. Swaps are just another expression of it.
Before entering swaps, he reinforces a key idea:
But the structure is identical:
He even revisits two-stage valuation:
The key message is not the formula itself, but this:
There is no magic formula. There is only structure and understanding.
If you understand cash flows and discounting, you can solve any variation.
Before touching swaps, the professor goes to what actually matters:
Forward rates
This is crucial.
Because swaps, in practice, are just:
You are given:
Question: 👉 What is the forward rate between year 2 and 3?
Core idea:
\[(1 + S_3)^3 = (1 + S_2)^2 \cdot (1 + f_{2,3})\]This is pure arbitrage logic:
He is very explicit:
This WILL be asked.
You must be able to:
And understand:
Once you have spot rates, you can price anything.
Example: Bond with coupons
Cash flows:
Price:
\[P = \frac{50}{(1+S_1)} + \frac{50}{(1+S_2)^2} + \frac{1050}{(1+S_3)^3}\]Key insight:
Each cash flow is discounted with its own rate.
This is term structure pricing.
After all that preparation, swaps become trivial.
A swap is a contract where two parties exchange cash flow streams.
That’s it.
A swap has:
At initiation:
Later:
Example:
Used for:
The most important type
Structure:
Professor emphasizes:
This is not about actual money moving.
Important clarification:
Notional ≠ cash exchanged
Only differences are exchanged.
You have:
→ no payments
→ floating pays fixed
→ difference = 0.5% × 100M
→ fixed pays floating
→ difference = 0.5% × 100M
Swaps transfer interest rate risk, not principal.
This is one of the most important conceptual parts.
Swaps are NOT for:
Swaps are for:
He uses a powerful analogy:
The best insurance is the one you never use.
Same with swaps:
He explicitly says:
This will NOT be asked in the exam.
But conceptually:
\[PV(\text{fixed leg}) = PV(\text{floating leg})\]At initiation:
So you solve for the fixed rate that equalizes both sides.
The swap rate = geometric average of forward rates
This is the key connection.
You don’t need pricing formulas.
You need to understand:
So the hierarchy is:
22 de abril de 2026, 5:09p.m.
1 h 3 min 17 s
Welcome, bye.
Thanks.
Yeah.
Yeah.
Before starting, have any questions regarding the whole course regarding the final? I mean, I mean regarding equity, regarding loans.
I don’t know. Okay, no, let me go. Overwhelmed. Yeah, let me go quick.
Little ideas.
Because, yes, being alone, I think we can, we need make, we got quick, quick, yes, all course.
Imagine that I have a financial instrument.
One financial instrument.
Okay.
You promise? Hey, two years.
A 100, yes.
I’m not selling from in that.
Did you hear that?
Will pay me 100.
How can I get the price?
Please.
And you present that?
I need to take is that this 100 to present value.
100 / 1 R rise to the second.
Yeah.
The whole course.
Do we need to do?
This probability by some value of things that will be made in the future.
If we are talking about its go.
We know for sure what future payments are going are going to be.
Same income, yeah.
On the other hand, we are talking about equity valuation.
We don’t know for sure what these payments are going to be, but we can estimate it.
Take me future payments.
Are we going to be called the events?
But it’s an evening that I will proceed because of holding.
So, so I will receive.
Yes, it’s here.
Homans.
No, no.
I see the amount of money that I will receive because of only a stock.
No.
How are we going to?
How are we gonna calculate?
Because I so must.
I can read in person, Mario future minutes, yep.
The most simple thing.
I’m gonna have a constant dig there.
That is going to be paid.
For it. A constantly didn’t know, for example, K.
And.
This is a perfectly.
Or are we going to calculate the present value?
Then, open up.
For one payment.
One plus R, right? Do you want to say the true value for all?
Then, you went up.
In order to understand this in an implicit way.
I have the price, and each payment will represent the percentage of the price.
Okay.
This is the most simple.
Yeah.
This one.
This is the most simple method.
In Alba.
Okay.
I can, I can also, OK, you need them to know.
That seems, you can go at that all some break. Yeah.
You.
It was actually.
**.
Okay.
That would be like different, like 10, 12.
And then repeat our conference, but in an indirect code.
Well…
Eleven.
I have these payments.
We are going to calculate the price.
With true strength, yes?
I think this, I think this, we have this.
From year three.
On year three, the begin constant payment of 10, yes, third year two at 12, and second year 11.
In case we have this.
You can use, you can go.
Good.
Take two ways, every way.
You were gonna.
It’s under perfect in class.
True.
One plus R + 1.
One plus R rise to the second. Make sense?
What?
Because here you got 10 + 2, 10 + 1, so I continue with the perpetuity and also add this two and this one. Make sense?
On the final.
This one like this? No. Like this one? No.
But I’m gonna tell you just right now, yes, but I want you to think and to play with numbers.
What can be in the China?
This exercise can be the fat. This, this what? I mean, this same exercise.
This exercise like this can be defined out, but the way in order to solve it.
Is.
Here, this is on Tuesdays.
Here, we can play prize in here to us then over our.
Yes.
Once we got.
Nice in here, too.
Okay, I wanna complete.
Nice today, as we.
When?
One plus.
Yes.
One plus R raised to the second.
Last.
You too, and you stand up over one plus R, right? Yes.
Please number.
This number.
Look at me.
Same result.
What I’m doing, I’m calculating present value of future values.
I have to stop the movie, Alibi.
How are you going to bring the place of the store?
I can play the present by you off in 20 minutes.
This time, everyone.
Not only you can find something similar to this.
Probably, you will find something similar to this, because it’s what you can find it, it’s easy to calculate. Is there a question like this on the practices, so I can practice with it?
Right, right. I mean, this is working with that other one.
You are looking for a formula, Troy. It is not a formula, it is just understanding. What I’m doing is, what I’m doing here, calculating present value of this, calculating present value of this, calculating present value of this.
Okay.
Calculating present value of all feature test flows all together.
Why did you stop at the current one? Should you also go to the open?
Why do I stop? Why do you stop at 11? I mean, this, I have 9. I mean, you want to receive 12 euros, you want to receive 11 and then 10 forever, so once we do that.
Like, we don’t need to, it doesn’t stop at your file, yeah, no, no, so, like, something that goes on.
Okay.
And something that would be paid forever is called perfect. Perfect. Yes.
I thought it stopped at year 5, but if it so, if the question like stop at year 5, then you would have to continue finding for year 4 and year 5 by doing like 1 by R, three, one, yeah, yeah.
What we are doing is just calculating present value, and…
The exercise that is just like this is the one that you have a one of the second class. There was one exercise that was the Starbucks exercise.
Mhm.
Even one of the winter of girls, it’s a big no.
Pizza.
Good morning.
You have gone through these kind of exercises.
Oh, thanks. I mean, here is the mouse.
I love you.
Ohh.
I think.
So good. Personally, I have just one or two. Four now. Yeah, absolutely. Sam, the exercise, there are several exercises like this.
The one that we got, the one that we got on for the class is just.
If we evaluation 2, if we evaluation 2, I don’t remember.
And.
Where are you?
Trying to see anything because it was 15, what, what exam 15 years after 15, 16 is it would be one and 17 is it would be two. I don’t know. Yeah, I mean this is.
This is called two states.
First state, first state is calculating this one and this one. The second state, the second state is calculating the rest, yes.
Okay.
I am told to tell you this in class. I think you have received the evaluations. You can fill them up.
Oh, I see the map.
Yes, pudding, good place, me, and the peace and…
OK.
And Sam, if you feel it, I will also give you a grade because you are just thinking about maths. OK, plus.
I was looking at the beginning, I had just some, so I have start reviewing them with evaluation.
Regarding equity evaluation, what I was telling Sami is that you are going to have.
Future payments.
Please coach the whole course.
Hello?
I believe present value of future payments.
What we have been doing when talking about the multi-valuation calculating present value of future payments. What we have been doing when talking about fixed income value ratio, calculating present value of future payments. In the case of fixed income, future payments are fixed, are certain. In the case of
Equity evaluation, future payments are being called dividends, and we don’t know what the dividends are going to be.
Yep. Also, I have shared with you in the WhatsApp group, problem set, sorry, for example, final one and sample final two solution.
And has has me and people.
I strongly recommend you to start reading from the final examples without looking at the source.
Because, not looking at the solutions for distance and extract, yeah, and where?
Any questions regarding?
Every evaluation or…
Can you, or?
Is it both?
Now, next week, we are going to dedicate both days of class to review everything.
Time now will be this Wednesday, no, the week after.
So we have the whole week in order to review. We will talk about technique, we will talk about physical, and we will talk about data. Are we going to go over the practice exams next week or just like general review? We are going to go over some more final one.
And, depending on time, sample final two, but I will start asking you.
Do you have any questions? Do you want me to go through anything in particular?
You don’t have any questions, and if you don’t want, I will pay for a few seconds, and then I will do the exercises. Yep.
Okay.
Fixing long, equity valuation.
This week, we can review it, yes?
Then we have been talking about derivatives.
We’ve been talking about the river.
Options.
Now, to understand, first time you see them.
After working with them for two or three days, options are so simple.
Yeah, you should know how to calculate the day of my call, my boot.
Then knowing the binomial model is absolutely important, binomial model, and if we did one exercise in class, you have another exercise in problem set 6.
And you guys might need one or two exercises in sample files.
I know the exercise. Listen, calculating a portfolio. A portfolio that combines a stock.
Come on, in order to regulate your surprise. Make sense?
Then, also, you have what are the fixing code in the station exercises, but again, it’s calculating one portfolio.
Do you remember exercises?
Wish you.
The one we did the day after.
That one fixed payment that will happen in year 50, you have one with higher maturity with lower maturity, and you have a with lower.
Absolutely important to know how to calculate the ratio. This is OK.
Now, today, we are going to talk about Schwabs.
What’s?
Hey.
And why I’m talking about Schwartz?
We will review.
For what?
We will review this, the new proof.
It’s all great, and we will review also.
Campus.
Let me start, instead of talking about swaps, let me start with the most important thing we are going to review from today. Yes?
Do you remember reading Wolf?
Yep, great.
I’m talking about the deep proof.
Ohh.
Now, I’m going to do an exercise, because we are a few. I’m going to do one exercise that will combine your move forwards and spot rates.
Also.
Phone Always.
An exercise like this can be asked in the pipe.
These same exercises can be asked.
You have.
Yeah, one, two, three.
One person.
1.5%
And 2%, yes.
This is the new book.
Pizza Campus.
What is?
For one rate, we be here to agree.
Going the spot in year one, the spot in year two, and the spot in year three. Make sense, Vega?
Why is we so worried? No way.
How much is this knowing that this is 1.5% and that this is 2%, yes?
How will you calculate this for what?
I do it.
What plus the support?
Year 2, one class, year 2 raised with the second.
Thanks, one plus the forward I’m looking for.
Yes.
What was this? Thanks. This one?
It’s going to be one, two.
One plus 2% rise to the third. Make sense?
1 + 1.5.
And one plus one.
Power between year two and three, if between two and three.
What I’m doing at the end is like, Ana, this is a geometric average.
I want to break this one.
In order to calculate this one, how I want to calculate this one?
And this plus this is equal to this.
One plus 1.5% times this one.
Yep.
What kind of exercise can you find in the final?
Once one where, for example, I give you the power rate, I give you the spot.
You are asked to calculate the other conference.
Or, you are asked to calculate for work.
Makes sense.
And now, with this data.
Once you know this, I told you.
Please, go play.
Fifty, 50, 1000. Yes, once you know this is for rates.
Please help me.
The price of people.
What? Sorry, I don’t know. This is the spot rate in your one.
The spot rape in year two, and the spot rape in year three.
Did you?
And that was a bit easier for you.
One 1015 is here. No, no, no, no, please, please.
There are two things to them.
First thing.
You.
Deal.
Cool.
What is the new proof?
What is the deal proof?
Yes, I have.
Leaving the time.
Will give me their name?
I am great. Yes.
You are one, you are two, you are three.
Wait, how much return I will get if I wait for wedding?
One person. How much return I will get if I wait for two years, 1.5%. How much return I will get if I wait for three years, 2%.
This is great.
I have been called Scott Prince.
Thanks.
But we know this for grades.
You got a great, what rates?
rate is between spot rates, spot rates are between zero and year one, zero and year two, zero and year three.
Race.
What rates are rates between year one and year three, year two and year three?
First thing you need to know is…
Giving forward rates, how to calculate the spot rates, or giving the spot rates, how to calculate forward rates.
Is this?
Last terms of exercises in sample final one and sample final two regarding power and spot rates.
Yeah, I mean, in order to calculate over rate, you need to know to spot rates, and if you…
Have one for water rate, you need to know another spot rate in order to public for water.
Next week, we are gonna do two, three, 4 exercises like this.
Are careful because there is one thing more that will complicate things.
These are rates, even the game.
Do you remember the exercise regarding RV drugs?
The Army cuts, we did several exercises with balls and rings.
Once we know the race.
We can calculate price of things that will happen in the future. I have future payments.
I have your payments, I can calculate the price.
For example, on a boat, yes.
This is the school in year one, year two, and year two.
Once we know this, and we take another color.
The.
No.
Let me take another color and let me give you one of these.
Okay.
Once, I know this is for grace.
What can I ask you?
Please calculate.
The Price of Avon.
With 5% coupons and 1000 face value.
Google.
Who come payments is gonna be 50?
50 and 1000.
Hello!
I have this group of payments.
And I am asking you to calculate this price.
How will you calculate this price?
Price is going to be equal to.
Is it not your first payment, 50 over?
One plus 1%, rise to one.
Yes.
Class.
Fifty over one plus.
1.5% rise to the second.
That’s Australian.
One 1050 / 1 2%.
Right, do the same.
Yep.
Yeah.
You should fully understand this, why? Because…
I can give you the price, and if I give you the price, you should translate the rates. You can, we can go from one in one way or into another way. You have in the exercises, you have several examples of this. Tell me, but like, how can you calculate like?
Call Nepal.
No, no, I mean, again.
But, several things, you are not going to find this with more than two years.
And no money.
Do we know the rate in year 2?
You will be asked to regulate the way to India one.
Try to do the exercises, and then we will try to do the exercises, sample final one and sample final two, because in these exercises you have examples of all of these, and those exercises are there in order to make you think.
Excess.
Okay.
Most important thing we are going to see today is to review this idea regarding polls.
Where is that swap?
I never have trips.
In which context have you had all the clubs?
Which type of jobs?
There are three principal uses for swaps.
I want, let me start with the…
Less, we call you.
This.
Please come on juice.
Regarding commodities, we’re talking about swaps.
You will always have, I mean.
Skype.
You will have shoe legs, yes?
You have two commodities, oil and gold, for example, this.
How will you get the price of this one?
How will you making the present value of both plays the same? Yes.
You have one leg and another leg.
Today, when you close the contract, today nobody pays anyone.
Like, bye in the future.
If you buy a picture, you don’t pay. Just assume one position in the contract. Makes sense?
When you buy a picture, don’t pay anything; you just close a company.
OK, goodbye, future. Sorry, you.
You get the price. What does to get the price mean? You calculate present value on one side and present value on another side. We’re talking about commodities.
We’re talking about commodities.
You assume that present value of one part of the contract would be the same as present value of the other side of the table. Make sense?
Okay.
Second type of swaps.
Wait.
You will have donuts.
Be paid in the future, you will have EUR to be paid in the future.
And you assume that the amount of dollars that you will receive in the future will be equal to the amount of dollars that I will receive in the future. And wish what you will give me.
A fixed number of EUR in the future, and I will give you a fixed number of dollars in the future. Yes? Do you remember from last class how was regarding currency?
Forward regarding courage, one exercise that we did last class. We will repeat. I will show you and we will see.
is the same. Calculating a total for currency is the same as calculating a swap for currency.
And…
The most important use for swaps, most important use for swaps.
Interest rate risk.
It is great, Ruiz.
I cannot believe, Viva, that you are enjoying my class so much.
You are smiling a lot.
But please, not for I mean, I don’t care much, but I think that you are not enjoying it because of my explanation regarding shops.
Probably because that’s something.
Be careful with this. When you look into something different than what you are seeing in the class, don’t smile much. Because the teacher can think that you are somewhere else.
Personally.
Vega.
IRS interest rate swap. I want all of you some.
It is great. I want all of you to hear at least one in your life, please.