Continuation from last class: move from single assets to two-asset portfolios.
Next class will add a risk-free asset; today is risky A & B only.
Asset A and B each have:
Expected return: E[RA],E[RB]E[R_A], E[R_B]
Variance / Std. dev.: σA2,σB2\sigma_A^2, \sigma_B^2 and σA,σB\sigma_A, \sigma_B
Portfolio weights: wAw_A and wB=1−wAw_B=1-w_A.
Expected return:
E[Rp]=wA E[RA]+wB E[RB]E[R_p] = w_A\,E[R_A] + w_B\,E[R_B]
Variance (risk):
σp2=wA2σA2+wB2σB2+2 wAwB Cov(A,B)\sigma_p^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2\,w_A w_B\,\mathrm{Cov}(A,B)
Linking covariance & correlation:
Cov(A,B)=ρAB σA σB\mathrm{Cov}(A,B)=\rho_{AB}\,\sigma_A\,\sigma_B
so
σp2=wA2σA2+wB2σB2+2 wAwB ρABσAσB\sigma_p^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2\,w_A w_B\,\rho_{AB}\sigma_A\sigma_B
ρAB=+1\rho_{AB}=+1: no diversification. σp=wAσA+wBσB\sigma_p = w_A\sigma_A + w_B\sigma_B (a straight line).
ρAB=−1\rho_{AB}=-1: perfect hedge possible; can reach zero risk for a specific weight mix.
−1<ρAB<+1-1<\rho_{AB}<+1: partial diversification; portfolio risk curve is bowed inward.
Vary wAw_A from 0→1 to trace all feasible portfolios (risk on x-axis, expected return on y-axis).
The curve forms the investment opportunity set.
The Minimum-Variance Portfolio (MVP) = lowest σp\sigma_p point.
Efficient Frontier = set of portfolios above the MVP (highest expected return for each level of risk).
Fix cells for parameters (expected returns, std. devs, ρ\rho); compute Cov via ρσAσB\rho\sigma_A\sigma_B.
Generate a column of weights wAw_A (e.g., 0%, 1%, …, 100%), set wB=1−wAw_B=1-w_A.
Compute E[Rp]E[R_p], σp\sigma_p row-by-row; chart σp\sigma_p vs E[Rp]E[R_p] to visualize the curve.
Use shortcuts and anchoring ($) to avoid mistakes; sanity-check endpoints:
wA=1⇒(σp, E[Rp])=(σA, E[RA])w_A=1\Rightarrow (\sigma_p,\;E[R_p])=(\sigma_A,\;E[R_A])
wA=0⇒(σp, E[Rp])=(σB, E[RB])w_A=0\Rightarrow (\sigma_p,\;E[R_p])=(\sigma_B,\;E[R_B])
Volatility, standard deviation, and (in this context) risk are used interchangeably.
“Return” variants from earlier units still apply; today’s focus is expected (mean) return and variance.
Weights must sum to 1. Shorting allowed ⇒ negative weight possible.
Diversification benefit is entirely about covariance/correlation.
For same risk, pick the portfolio with higher expected return; for same return, pick lower risk.
You can’t “choose” correlation; it’s an empirical property of the assets.
You have.
Your production, it’s higher.
On the other hand, you have more time and also when you automatize things, you feel things that, oh, I’m happy of doing that, I didn’t.
Start with the transcript. I don’t like to do this after.
OK.
And today I need the I need the link getting group because we are going to work with Excel.
I I was looking for the other.
This is I want to work with.
And you are.
Yeah.
Right.
Bye.
Yes.
Oh.
Yes.
OK.
I’m.
It’s Excel. I don’t like it. It says.
So the other day I like.
You didn’t came, Isan, but the other class, the other day, the class was not one of the classes I felt more comfortable with. You. You know what? Oh, you don’t know it. They know it.
Because there was one person from the stand. She was lovely, Carol. Oh, she was watching you. Yes. And at the end it makes you be more confident what she could be thinking about what I’m and it makes things a little bit flowing.
But whatever, normally, usually I should have written one formula. I didn’t do it. Why I should have written this one formula in there? There is one formula that I have written on this that is or is invaluable. See what future value over one plus.
We are not going to work with this formula. This formula will have an impact during the whole course when talking about portfolio theory.
Not that much. Not that much much. I have already worked working, yes. What are we going to do today?
Today, the other day we were talking about portfolio. What is a portfolio? At least two stocks. Today we will work with a portfolio for my shoe stocks.
Next next day we will include our grease free asset.
What we’ll see next day what is a rich bee asset. We make a spoiler. A rich bee asset is and I said that no matter.
What will happen? Go ahead. It will give you a fixed return. So the variance of the risk recipes will always have same return of the risk recipes 2%.
What does it 100% make sense?
And the covariance between the and all of the arsets in the war is here because makes sense to me. At least for what are we going to do today? What with two stars?
The other day, we were talking about one stock, another stock. We can read the other days.
Today, instead of talking about the average, we will talk about the expected return.
Then variance variation. Well, I mean you have to not only have variance variation and inspect the return for each asset. Not only you have variance of each asset, but also you got.
All should be put.
And.
I have.
Let me this is.
Specter return a message. Hey, yes.
Take the return, nothing 8.
Stigma, stigma, variance when I said hey, yes.
And I said hey.
This is deviation, the square root of variance deviation. Make sense.
On the other hand, I’m going to have expect the return when I said P.
Valiance, deviation, all of you are with me.
Yeah.
And also I’m gonna have.
Covariance between A and this.
OK.
Now what we are going to do today?
Today we are going to combine. Today we are going to combine asset A and asset B in a portfolio.
I have 1000 euros.
How much do you want me to buy or I said pay?
Big weather.
Of 10,000, so one person.
If I buy 1% on asset A, how much I will have on asset B, be raised 99. Make sense.
Let me call.
Wait a the person dates I went by and I said hey.
And we go way B.
Oh, the backpack was touching, but then Nicole will be the person. Make sense.
OK, considering this.
They combine weight date and will be in a portfolio is.
The question is what is first?
Expected return on this portfolio and 2nd what is the variance on this portfolio?
The specter returned.
On the preference.
Inspect the return on the portfolio.
If the waves were 50% at 50%.
The person.
Expected return would be 50% of this and 50% of this.
Yes.
Yes.
Are you following me?
93% on one and 4% on the other one. I am taking one and one. I will be here. I just add expect to return on one and expect to return on the other one and I will get the result.
And if instead of having 100%, I have have, yeah, I just move it back like the ice.
Wait a thanks. Expect the return on a plus. Wait B times expect the return on B. Make sense.
This.
Another it. Away there, yes.
OK.
Because I have more walls, more black walls, I’m going to move here. All of you.
Sorry.
OK, regarding this vector return on the portfolio, no problems.
This it’s a little bit more complicated when talking about the variance of the profile. Why? Because.
When considering variance of the portfolio.
Not only have the variance, but also covariance. The covariance has an impact in the variance on the portfolio. OK.
How do I calculate the variance on the portfolio?
The variance of the portfolio is variance. The first one that’s variance on the second that sometimes covariance. I’m careful because careful because I have ways. Yes, let me write it here.
Variance on the portfolio, yes.
It’s going to be equal to weight a raised to the square. Variance a raised to the variance. Sorry.
You gonna follow me?
Plus.
Weight B rise to the square covariance, yes.
Plus.
Two times weight A, weight B.
Go by and between A and B. Make sense.
I’m here with me.
You’re going to see this formula in the next classes. I’m going to write it more than 1020 times and in Excel today more than more than 20 times that formula. Yes, don’t worry because you will get used to that formula.
I’m going to make a spoiler. Yes, in order to do that’s up of and then we’ll come back. We’ll we’ll go later. Covariance, covariance.
What can covariance tell me about the relationship between this true stuff? Do you remember from the other class the Monday’s class? What covariance tell me?
But that’s when it’s good.
Yes, I’m.
Yes, and the sign sign. If it is positive or negative, it will be something. Yeah, they go together. Yes, if it is positive, they move together. If it is negative, yours.
Hello.
And if covariance is 100, does it tell me something? Now what should I do? Correlation coefficient? Do you remember correlation coefficient? Correlation coefficient? What is the relationship between covariance and correlation coefficient?
Correlation coefficient between A&B is equal to correlation coefficient is equal to covariance between A&B over deviation A deviation B. Yes, do you remember?
Let me.
From here let me covariance between A&B. It’s equal to correlation coefficient times.
Where do we meet?
So knowing this.
Let me rewrite this equation.
Let me.
We write it’s a question and instead of writing here.
Mobile audience.
I’m going to write deviation A times deviation B times.
Correlation between me and me makes sense.
Correlation coefficient is a number that could range from -1 to 1.
OK, I’m gonna clean this up.
All of you will be noticed. It is the variance on.
Me.
Write this into another way.
Yes.
Saying no.
Same.
Grade A.
Variation A times weight B. Variation B, yes.
Are you following me? Imagine that correlation.
Coefficient is correlation coefficient would be one.
The correlation coefficient will be one.
Don’t you see that piece?
Is the same than this and this is the same than this, yes.
If correlation coefficient, let me one.
Do you let me just for one second call this A and call this B?
Job.
This is a.
And this is me, yes.
Correlation coefficient is 1.
I have.
No.
Variance on the portfolio is a square.
Isn’t it?
Plus b ^2 + 2 * a * b * 1.
Does that formula look familiar to you?
We go here, just one second. A plus B squared. A plus B squared is equal to.
That’s B and this is equal to.
A square B square plus AAA times BB times a. So 2 * a yeah, yeah.
Yeah.
Sorry for having done this.
So correlation coefficient is what variance on the portfolio is weight a times variation a.
A+ weight B times deviation B makes sense.
This is equal to a + b square, so the square root of this is equal to this. Make sense?
And then?
If correlation is negative, yes, negative one. Perfect and negative I have.
The same but instead of plus minus no. So I have the other one that is exactly the same but minus minus -2 times AD that is exactly the same. So I will have here.
That the variant deviation deviation on the portfolio is weight A times deviation A minus weight B times deviation B. Make sense.
Yep, OK.
Have you understood these these numbers?
You have understood this. You are done with today’s class. I’m going to review everything. Yes, the weight. Yeah, weight is a percentage. OK, and and thanks for that question.
Because I have forgot writing one more equation, weight A plus weight B is equal to.
1% and 99%, 100%, yes, it’s weight.
OK. And glasses somewhere here.
I don’t know where I have thoughts enable anything but.
I don’t need to add anything portfolio theory. What are we going to do today?
We are going to go work with.
We are going to introduce the concept of portfolio, then portfolio expected return and the variance on the portfolio.
And once we have, we’ve got this, we are going to work with one sample profile, OK.
What is a portfolio?
A portfolio is a combination of a number of assets.
And what is the weight? What is the weight? The percentage of the asset that you have?
Then.
The sum of all the weights is equal to.
But i’m saying
Where it is, you have 100 euros, 1000, yes.
Would you have a negative weight of buenosit?
Jake is saying no.
Common sense tell us that no, but.
Imagine that I have asked Perla and Perla has told me 1% of 1000. What if Perla would have told me?
-20%.
What does to have a 20% on one message? It means that I am buying with my 1000 a percentage. I’m I’m dedicating 1000 to buy.
If a positive weight means buying.
A negative weight will mean.
If you buy, you can wait for how long you buy, you can wait for a long time.
Can you sell something without having it?
Can you sell something without having it?
Yes, you can. You can sell something without having it.
But if you sell something.
And when I save you.
Amazon stocks.
I will gave you the stuff this night.
What should I do?
When I give you the stuff, that’s what I do. I’ve already sold it.
What should I do?
Proceed I don’t buy them. I will go to jail.
You see what I mean? A positive way means buying. A negative way means selling shortly.
Short because of short time. Have you ever have a long and short, long and short? Long is buying the buyer’s position and short is the seller’s position.
We will talk a lot about this. We will talk a lot. And not only we will talk when talking about the theory, but also when talking about them.
OK.
Hey, hey, hey.
What is the return on our portfolio? We have already gone through this, yeah, wait times return, wait times return and so.
If I have what? What about the variance on a properly? If I have two? If I have two, this is a form graph that I have already showed you. What is this? This is the covariance, isn’t it?
This is the covariance and if I have more than two two classes, we will go with more than two what we will have here.
Looking this way, looking you look in this way, probably you cannot compare that what we have is.
One we have two big covariance between 1:00.
And two covenant between one and three covenant between two and three.
One that is the covariance between one and two and the covariance between one and two is the same. That’s not what I mean.
Is 1 is a symmetric matrix 2-3 this we have the regarding this you don’t need to. We are not going to work a lot. Now we will move just a little bit in the interim. We are not going to ask. I’m not going to ask you about the covariance matrix.
This has to do more with the statistics, but if you are working with big amount of data, this will be. But this has more with the statistics. Make sense? What is this? The covariance matrix at the end. If you look at this formula, it’s the multiplication of amatrics. Get it.
But today we are working with just two stops. You don’t need to worry about more than two stops.
OK, and here I’ve got one example.
Let me.
I’ve been teaching this class before. OK, I have.
Two stops.
USA.
And to us, one from US and another one from Japan, yeah.
Expected.
The expected return on US is 13.6% and the expected return on Japan is 15.0% for it.
OK.
Now volatility. Careful again. Careful with language. Do you remember when talking about the value of money? HCRERAER. First time you look these things.
It looks completely. It can make you feel misunderstood.
But after you get used to these things, we know that volatility, variation and risk, volatility, variation and risk stands for the same and then valiances.
Volatility, variation or risk rise to this work. Careful with risk because later we will see that we can call risk other things.
But language is important, yes.
Thanks.
Sorry, no.
The sooner I have told you that volat volatility.
Volatility is deviation, yes.
50.4%.
And 23% and then what else do I have here?
I have the corporation coefficient that is 27%.
And that’s it.
OK, now in the correlation coefficient I can calculate the covariance.
What is covariance correlation coefficient times?
Deviation one times deviation 2 makes sense.
OK, now I’m told that I have a portfolio with weight A and weight B.
Well, A is 60%.
Makes sense.
With bees.
1 minus weight day that will be 40%.
OK, now I’m going to calculate the return on the portfolio.
The return on the portfolio.
Simple. The return on the portfolio is going to be way day. Thanks.
Return it.
I’m going to fix.
This. Yes. Why? Because later I will drop. You understand what I’m doing by thinking. Have you used Excel?
And do you know how to fix yourself?
Yes, Perla. Perla. No, don’t worry. No, don’t worry because you will not need to use Excel. But understanding this is important and in order to understand this Excel would be really interesting.
Weight A plus weight B times.
Yeah.
Fix this.
Function a four that is dollar.
14 point and the USA all in those asset 8.
And Japan, let me call it asset B. I’m also I don’t want to a nervous say fast. I like to do this at the beginning or at least. Oh, this is a new class session.
Sorry, anyone that I have reconsession with this.
One of the bin and two in the middle. OK, thanks. And session 6.
Makes sense.
What is the return?
Wait times return plus wait times return. The formula that you’ve got there now.
Let me calculate.
They.
Yes, yes.
Variation on the portfolio is that formula. Yes, I don’t like how I have is that formula.
Two times weight a aviation. Two times weight a will be provided, yeah.
This formula, first thing I should write, I want to calculate the deviation that is the square root of the variance. No. So let me start by saying SQRT square root.
And now inside.
Do you know what what is chorizo in Spanish?
But with a different taste. Chorizo is sauces, but with a point of pepperoni.
Let me look for cholizos.
Have you ever heard the word chorizo? Yeah. Chorizo also means bad. Who do you call? Lather. Lather. Lather. Yeah, Robert. Yeah. Chorizo. Also, if you call someone chorizo, he would see you are calling him Robert.
And this is actually, yes, yeah.
Also I call we are we are going to see formulas bigger than this. I used to call them choritos also.
It’s a victory, yes. Yeah.
OK, sorry for this culinary, if anyone of you is better.
I am. Please forgive me for putting sorry. If you are not beggar, anyone is not. No, not you is beggar. So you those are wonderful. Don’t begun this because they could be.
I I know that I cannot and I will never ask you about religion, about gender, about sexuality, or about ideology. Yes, ideology.
I don’t know if big Megan has to do with ideology. Is there any shifters in the class? Shifter. Yes. Swifty. Swifty. Swifty. Sorry. Yeah.
Are you 50? I I respect her a lot though, but I just don’t listen to her music. OK, now it’s because I cannot ask about politics. I mean, I would never ask, but asking about 50, I don’t know if it goes inside politics. What do you think? Sorry for this parenthesis.
I’m trying to talk with. I mean, if anyone is a Swifty, I don’t want to go over there.
OK, let’s start with the Toledo. Yes, SQRT and now.
Look, wait A.
Thanks.
Polativity A and I’m gonna fix it, yes.
Rise to this word plus.
Wave B.
Thanks for the TV. Let me fix it, yes.
Price to the square plus two times times.
We they times, we we times.
Covariance that covariance is variation times variation times correlation. And let me fix this one also. I close the parenthesis of the area of the square root.
And let’s see what happens.
Perfect 14.60.
All of you are with me.
What is this? A portfolio that has a 60% of US and a 40% of Japan. And I have this variation and this is Pedro Rico first thing.
I’m going to do it and check my numbers are correct or not. Oh, I’m going to check this.
When I write here, see the person.
If I have a 0% on US, how much I will have on Japan? Yes. And what should be the deviation on the return on my portfolio?
Luis Garvía Vega ha detenido la transcripción