NYU | February 9, 2026
Instructor: Luis Garvía Vega
Duration: 1h 01min
Session 7 is a bridge session.
If Session 6 explains what is possible,
Session 7 explains what is chosen.
Despite initially stating otherwise, the session begins with a deliberate and fast review of Session 6.
Reason:
[ E(R_p) = w_A E(R_A) + w_B E(R_B) ]
Linear, intuitive, no diversification effect here.
[ \sigma_p^2 = w_A^2 \sigma_A^2
This formula (the “big chorizo”) remains the structural core.
By varying weights:
We obtain:
All feasible portfolios formed by combining A and B
Plotted in:
Key observation:
This is pure diversification.
Among all portfolios:
This is the Minimum Variance Portfolio.
Important insight:
Only an extremely risk-averse investor would choose the MVP.
Definition:
Rational investors only choose portfolios on the efficient frontier.
Anything below it is strictly inferior.
Now the key question appears:
Given many efficient portfolios, which one do I choose?
Answer:
This moves the analysis from:
Same logic applies to:
Properties:
Interpretation:
Each curve represents combinations of risk and return that give the same satisfaction.
Key parameter:
Higher A:
Lower A:
Utility: [ U = E(R) - \frac{1}{2} A \sigma^2 ]
Used only to show:
Crucial point:
Assets are not “good” or “bad” in absolute terms.
The investor chooses:
The portfolio where their highest indifference curve
is tangent to the efficient frontier.
This point is the optimal portfolio.
Optimal portfolio:
Intersection of what markets offer
and what investors prefer
This is the first time the course:
After Session 8, everything becomes cleaner.
Before next class:
9 de febrero de 2026, 5:04p.m. 1 h 0 min 59 s And. Yes, you got it now. Another work. Great. Happy to have you. And were you finance? These are you. This is days and also. Best thing we can, best thing we can do today is to review. Yes, yes. You see what I mean? We are not going to review. We are not going to go over last class again, but the best thing we will do place that taking last class going through that and. I’m gonna go quick, yes. The other day we had one stroke. We expect a return. Hey, let me. 5% for example, yes. Then we have another stock with respect to return on B is 10%. Risk variation, volatility, standard variation. Yes, risk of risk on stock A is we made 5%. And standard deviation. Stop, please. 12% let me write here 6% this in order to have different numbers. What else is missing? I have stop A, stop B. What else is missing? Covariance between AMD correlation coefficient is for example 10%. 10% yes, one say no. Correlation coefficient. I can say that covariance between A&B is correlation coefficient times. And the division is under division. Make sense. OK. What did we do last class? We combined these two stocks. We do first. We start with one portfolio and then we combine it with two B portfolios. Yes, we draw as class the investment opportunity set. But I do it again quickly, yes. I’m going to start with a portfolio that is weight 60%. Weight B 40%. I’m going to start with that portfolio and then I will draw the entire investment opportunities, OK. Yeah, I was thinking for myself. I said that best thing we can do is to review last class. I I said that I was not going to do it, but before saying that I’m going to, I’m going to do it everything again quickly. But I important thing, I’m not the one. Who who needs to do this review? You are the one that should do this and the more times you see formulas a number. Oh what is let me just right here present formula future value over 1 plus. Right. So this formula, we are not going to use it today. This we’re not going to use it, but I have right and the formula expected return on the portfolio is weight A, expected return A plus. Why expected return? Yes, this is the formula for expected return. I’m there. Variance on the portfolio, yes. I will look for the square root of this one. This weight A square, variance A plus weight B square. Variance B, yes, plus. Times weight A, weight B times covariance between A&B and I can write covariance as this. Same formula again and again and again. Now we know these numbers. Return. Hey, hi, everyone. I was missing you. Return me. Is it better? I have full house. Great. Ivan, last class, last class was absolutely important and what I’m doing now is to review last class. I’m going through what did we do in last class. We have two stops with each covariance and we create first one portfolio and then withdraw the investment opportunity. So that is what I’m going to do. Yes, I have. Office hours, but now is one day. Today is one day that what we are going to see on last or or next class will be construct over today’s class, not today, sorry over last day class. So it’s important return a is 5%. Have you made made-up? 5%. We don’t be 10%. 10%. For me and. 6%. 12% and. Or a Latium coefficient is 10%. And global audience. These covariance. Always happens this and now. Variance is correlation times is deviation times this deviation, yeah. Now I’m going to create a portfolio. Weight B. I said 60%. Weight B is 1 minus weighted. Yes, and here I’m going to write return. We don’t on the portfolio. And return on the portfolio is wait eight times. Return A. Let me fix it plus weight B times return me and let me fix it. Make sense. This is simple. Return is the weighted average, the weighted average and let me calculate. Deviation on the portfolio. Deviation on the portfolio is going to be square root of the formulas also I introduce. On last last Wednesday, I introduced the concept of chorizo. Justice. This formula is a big stream. It’s. I call it chorizo. Chorizo in Spanish stands for pepperoni. It’s not pepperoni, but chorizo. Yep. Have you heard of chorizo? OK, SQRT. And let me start with a big chorizo, yes. Thanks. Deviation May. Let me fix this one. Rise to the square. Yes, what I have written. It’s first part. Let me go class. Wait B time deviation again. Let me fix this. Rise to the square, Yep. Plus two times. Someone finds a typo. I am missing something. Let me see what has happened last. Two times times… … … Variance function. Yes, and this should be the liaison on the portfolio and the relation on the portfolio. What is the calling? OK, let me take it. This is correct 0. And 100% how I’m checking that is correct because if weight is 100% on a. Deviation is 6%, return is 5% and everything matters. On the other hand, if this is 0, this is asset B. Make sense. OK, now I’m going to say this plus 1%. And I’m going to copy paste. This and with this I’m going to run here. Would I have a weight higher? Would I have a weight higher than 100%? Do you remember from last class? Can I have a highest weight than 100? Can I buy more than 100% of one asset? If I short the other, I can be long if I short the other. This negative weight means that instead of buying asset B, I’m selling asset B, yes. And I’m going to take these two things. I’m I’m will not, Sir. When I insert this. OK, what is this? These are the combination of all possible portfolios, yes. Let me look for. Let me look for. Asset A as a return on five. Asset A Specter return on five. Asset A should be somewhere and standard deviation of 6%, yes, but you see that this is asset A. Which one is asset? B. Standard deviation 10%. Sorry, we don’t. This is SSB. Make sense. Yet I said this point. This point. This point is as a B, this point that cross standardization 6%, yes. What are all these points? Portfolios that consist on shorting, yes. OK, let me start with A in negative. 50%. I’m going to start this negative 50% and what do I have here? This is as a B. Let me guess, 12% should be here. Yeah, 10% here is SMB, yes and all these stocks. All these portfolios are the resultant of sorting as it be. Make sense. I’m. Be right here, -20. Yes, here I got 6%. This is I would like to paint this one, this point. I don’t know how to do it. Sorry for not knowing how to point one point. How to pay one point? Yes, I would like. There should be a way. And I hope to do it. Sorry for that. I’m lucky. I feel like a millennial and I I need a prompt there and ask Excel, please paint this point. You you see my feeling, no? Whatever going back. Not a millennial. What is your generation? How do you call yourselves? OK, I feel like again, whatever. I don’t know. I have. I have seen Harry Potter growing now, now. So whatever. OK, all of you are with me. I’m going to take this out and I want you to see several things. Expected. This is expected return, yes. This is effective return and this is deviation. What does this shows? This shows that I said a. 5%. And six person, yes. This is asset A, yeah. 10% under the ACMB’s. Per person. Make sense? What do I have to draw? Well, why? What did I draw? The investment opportunity said yes. And I can show both what you have there is not just reaching this point, but also is going farther. Yeah. What is that? The investment opportunity set all possible portfolios that I can get by combining two stores, yeah. OK, among all these portfolios you have, let me call this one. Oh, let me call. The name is A and B. Nine that you can choose between mine as an A itself or. Find this portfolio what you will choose. Same risk, but higher return. So you will always choose a portfolio that lays above. Do you remember holding we call this portfolio? Oh, did we call the portfolio with minimum variance? We were so creative. We call it NBP. MVV stands for what we call the minimum variance portfolio. This is the minimum variance portfolio. Yes, imagine that I am a complete. Risk Affairs Investor. Which portfolio would you choose if you were at home a complete freeze coverage investor? Let me what is it? Let me make up when you say. OK, because I don’t know if it’s zoom or whatever. In Spanish it’s zoom. Let me make a zoom, yes. This is the minimum balance portfolio. You see what I mean? If you are absolutely risk averse. I want the less risk possible. You should take one, but I want the one with the less risk possible. You will choose this. I think you choose this one. Who are in Russian? What does this mean to Russia? It will never make sense to take the lowest unless you are an absolutely risk averse. Why? Because. Look, this is return. Yes, this is risk. In an infinitesimal way, all maps. O man’s return when you get. A lot, a lot, a lot. I am in an infinitesimal way. You will get Infinity return, assuming yes, almost nothing risk. That’s all I’m saying. Don’t need it. You are paying, you told me. I will offer you, yes. You are absolutely risk averse. You told me, Luis, I don’t want, I want the less risk possible. So you will choose this one, yes. But why I am calling you irrational? Because if you just assume a little, almost nothing, you assume nothing more risk you will get. A lot of return. No. They’re just a little risk lower or you don’t care just to bear a little bit of risk will be there. Hope you understand what I’m saying. What are we going to talk about today of this day between Chris and we go, Chris and we go. This is just a warm up and personally. I prefer to talk about last class. Yes, let me go deeper into last class and then I will start with today’s. How are we going to call this effect? The effect is that here has got a portfolio that is better. Or are we going to call this effect diversification? Diversification is getting by combining 2 portfolios is getting a better portfolio. And what important thing the more the better, the better. There are no discussions on this. The more differs. The better. And if this is not philosophy, this is fact. More diverse, the better. If I have just one stop, I can only get this if I have two stops. We will see. That I can get portfolios that are better than this one by combining this with another one and we will see our next class that also by combining this with a risk free asset. We can get portfolios that are better than these two stocks it serves, so it is better always be with someone that be alone. And more different, yes. Mhm. Teams is telling me that. It does not recognize my English. Your English is not English, it’s his. I’m talking Maltese. Maltese is the language he’s been talking in Malta. Sorry for for things when talking in Spanish, he also said. He that’s in Brooke amazed me also as whatever understood. Pivant, Garth and the rest. So what is going to happen now? I cannot repeat last day’s class, but imagine that there were no diversification when there won’t be diversification. Sorry, covariance. No. There won’t be diversification when correlation coefficient would be 100% if both stocks will behave the same. The investment opportunity set would be alive. What I’m going to do, I’m going to start changing. Oh, here was 90. AD. 78. 60 You see that as I make to stop different, the less correlated there are, the more diversification I can get by combining them. Let me go to 0%, yes. And let me continue increasing. 10%. 90%, yes, -95%. We finish last day’s class by getting this portfolio, yes. If both stocks are perfect, negative, correlated, what can I get? I can get a specific portfolio that has zero risk. I can get a risk free portfolio. Yes, this is theory in practice. This not you will never get these numbers barrels, but you can find for example gold. Correlates negative with all in normal circumstances. Nowadays all this maximum SP500 is maximums. Nowadays life is not absolutely not. Bitcoin is doing just the opposite that what it should do. Strange times, but whatever you understand what I’m saying in theory, no, at least any questions. What I have done, I have done a review, a complete review of last class. Why? Because last class, two reasons. Last class was important and today’s class I don’t care too much about today’s class. OK. What are we going to talk about today? Risk return trade off. Then we will talk about indifference curves and then I I will finish with OP optimal portfolio choice. Yep. OK. What is the point? Hello. What are we looking for? I have I know I need to invest. Question? One and the other is. Depends on on your risk profile, yes. Depending on your risk profile, I have already told you if you are actually a risk averse investor, you will choose a portfolio here, yes, and then you will always choose a portfolio that we lay on the. Efficient frontier. What is the efficient frontier? All points, all those that are above, that are above the MBP, the minimum variance for point. Optimal portfolio choice. Any invalance investor should choose an efficient portfolio. Which efficient portfolio is optimal depends on the investor’s preference, in particular fair risk aversion. Make sense. OK now. Which invest? Which? Acid. Do you prefer? We just said to prepare. Yeah, the question that I’m asking is. Which one do you prefer? This one? Or this one? Same standard deviation. This case is 10% instead of 6% that same standard deviation. The more return the better, yeah. Take one question. Which one do you prefer? In this case here. I don’t. I don’t have a proper example, but the question is which one do you prefer? Let me call this C and this B. Which one do you prefer, C or B? The next. The less risky, the better. Make sense. Do you prefer? OK, now in this case we have a problem, but not a problem because it will depend on your risk profile. The question is which one would you prefer, B or B? You flex on what? Because B. Has higher return, but more risk. It will depends. Yep. Today we will talk about preferences. OK, at the end. On one hand, on one hand, we have the supply. On the other hand, we have the demand, yes. Supply and demand love. Regarding the demand, what investors prefer more, the more return possible, on the other hand, the more return. You should pay for. You pay for more risk, the more return, the more risk. Make sense. The other hand, you have supply, whatever. Let me show you. Let me show you the investors. Wait, do you remember? Do you remember a different scores? You remember any transports? You don’t remember the little work because in this case so simple. In this example I have And beer. What is the idea you give me? The more chocolate and the more beer, the better. The more things I’ve got, the more utility I will get, yes. So the more the better. In this case there are no problems. In the case you give me more chocolate and more beer, it would be more. We believe it for me. Once we are. In a point, once we are on a on a in a point, there is a trade between chocolate and beer. You have for example 2 units of beer and two units of chocolate. Yes, reaching this point. What is the question? OK, you want one unit of chocolate. How much beer are you going to give me in order to have the same utility? You can like more beer than chocolate, or you can like more chocolate than beer. Depending on your preferences, depending on your preferences, the slope will be like this or like this, yes. If you. Let me think about this, because my dyslexia is talking about chocolate and milk. This person. Lights chocolate a lot or at least dislikes beer. On the other hand. This is the one that’s like chocolate. Hello. We’d like to. These are two people, but normally it’s person. Each person has its own. Relationship. Make sense? And now the more chocolate and the more beer and we have. We have these different schools, yeah. I have never seen in life in different school and I have never seen anyone in Mercadona, in Walmart or in Venark when making. I’m talking throw in different schools. Yes. Oh, I have. I’m gonna buy. Because of that, probably I don’t like too much today’s class. I have never seen a different school. No boy. Next day I will introduce Sharp Razor. A Razor is something that deserves a. Can I read the rigor more or even the different schools you can need to help understand the community schools or work on the community schools, Vicent. Do you see this? Yep. What is the the problem we have? We’re talking about risk. ** beer and chocolate are deserved. The more beer, the more chocolate, the better. What is the problem we have with risk that we don’t like risk? Yes, so. So let me write here. Let me write here. Return and let me write. Dear Risk, yes. Thinking about return. The more return, the better. Yes, thinking about risk. So. So. Between nothing and and almost nothing. Would you care to bear a little bit of risk? Yes. Oh, so if we are talking so close. close nothing and almost nothing will mean same yes. And we are approaching to pain points. Thanks you. Do you see what I mean? We approach two paying points at the beginning. I don’t care a little bit, but us. You start in Greece and Greece, there is one point and he said no. I could also because for almost nothing. I like these ones better, yes. Oh, a little return. I compared to play lottery. You are telling me to bet my salary and the odds will check. What are these in different schools? What is the name of? Again of today’s class. I’m gonna make a big expoiner. 1234. Which one would you prefer? Cool. Make sense. Let me take this one here. Let me call this A. Let me call this B. We call this B, yes. Here is 1 portfolio and here is another one, yes. Between this portfolio or this portfolio, which one would you prefer? Between this portfolio and this one, which one would you prefer? What do you what you will feel difference? That. Yes. Between this portfolio and this one, or between this and this, you will always prefer portfolios that lay on indifference proof too, yes. What happened with the difference group 3? Difference group free standing. You see that as I move there, two points get so close and there is one business group that is standing with investment opportunity. I cannot get a better portfolio than this one considering my profile. Probably I would like a portfolio that lay on different school for. I would like $1,000,000. That is not possible, yes. So, what I have done? I’ve taken on one hand, in blue, my desires, and on the other hand, I have taken reality. I miss reality with my desires and I get the. Tangent portfolio. The tangent portfolio is so, so, so important, not for today’s next day. We will look the new portfolio and let me just I have already said but let me say that what is going to be the name we will give to the. Slow of the tangent portfolio and phrase and this would be necessary. Excuse me, yes, in order for you to become familiar with that, yeah. OK, I think I have to spoil the whole class with this. I’m ready. OK. This is an indifferent school for beer and chocolate. And these are indifferent schools considering risk, considering risk as something we don’t decide. The less risk, the better. Yep. No. We will never use a utility function. Yes, I’m going to do this example just because. Example is here. This is a utility function. What is A? A parameter, the higher A, the more risk averse. This low at the end of this low. Make a deep this low. So as the flow is negative. Now the slope is positive. A is positive. Variance. Whatever. Don’t really care. Let me compare 2 assets. Yes, with this example you will understand one idea and that’s it low. There is 2 assets. Let me call asset. I said. Low risk and high risk L. Low age. Hi. 7 and 30 percent, 7%. 30%, yes, and standard deviation is 5% and 20%. 5% and 30% return. And. Make sense? What do we have? Two assets. Also I have two investors. Investor A has a risk aversion of two. Investor 2 has a risk aversion of five higher a the higher the risk aversion, yes. Let me investor 1-2 and five. Best of Juan. A. If you. Investor 2. A is fast. Make sense? What are we going to do? We are going to calculate how well Is how you you. What is the utility? I want to take out this one. What is the utility for? It’s investor of each one. Let me write here investor. One, yes. For investor one, asset L and asset H, just apply the formula. Yes, for investor one. 7 -, 0.5. Thanks. Hey, let me fix. This one, yes. Time variance variance is this deviation right to the square. Make sense? What is this number? The utility for investor one utility 6.75. And for the investor one asset A asset 8 is 9%. Which one has a highest number? The highest risk. Make sense? Investor. 1. Which asset will prefer? Eight. Make sense? Let me make same numbers for. Investor to yes. Investor 2 return 7% -, 0.5 times. Hey, let me fix it. Rise to the square. Yep. I’ve made something. Oh, yes. Hey! Let me see this time. Yeah, I forgot the issue. It’s fine. 38 Control C. And based that investor to which asset will investor to prefer? Which one is has a higher risk aversion? Investor 2 has a higher risk aversion. And investor 2 will prefer asset with low risk compared with the other one. Make sense. Have you understood this? One question, and it’s the only question I’m going to ask regarding this. Wheat’s acid is better. I said L or I said H. Which semester is better? Bye. What we can say is that for investor one. Asset H is better for investor two. Asset L is better, but we cannot assume anything regarding asset L or asset H. Or investor one. We cannot use utility curve in order to compare investor one with investor 2. Expense. Let me say this because I want to say this more than because of this exercise, because previous one today’s session. Today’s session 7. Make sense? Now putting it all together on one hand we have. We are looking for an optimal portfolio. Yes, on one hand we have someone’s indifference curve and on the other hand we have efficient frontier. We have the efficient frontier, the marginal rate of substitution, how much will you? Hey, in terms of return, forgetting a little bit more of risk, yes. And on the other hand, how much? The investment opportunity set offers you you want a little bit more priest, so you will support. You will get if you want little more of return, you should support a little bit more of risk. Make sense. When this match in the tangent portfolio, yes. More questions? Have you understood today’s class? Best part for you of today’s class is the review from last class. Less. Next day we will continue talking about the tangent portfolio and I will repeat and I will give you sub ratio. I will tell you what is sub ratio, you will see sub ratio and that’s it. Make sense? Conclusion. Call Lapu. Conclusion. Everyone is looking for a portfolio that lays on the physician frontier, yes. Location will depends on your risk profile. And next day we will introduce the risk free asset and once we consider the risk free asset, we will see how things are becoming a little bit more simple. But in order to understand next day, you should fully, fully, fully understand the big, not the mean, the small cholitho formula, the one that the values of the portfolio formula. Nations, please for next day. Walk over. I will. I wish this were not true, but the 20 minutes that I’m going to give you today, I will take back. Little by little on next classes. Yes, thanks a lot. We are done. Any questions? Oh yeah, which ones? Are not in bright space. In less than 10 minutes you will find them and and also I will upload next Wednesday and next week. The sooner you’ve got it, the better, no? I’m going to do it. Sorry. If you find this on Sunday, send me a WhatsApp and I will upload it. Yes, because I have all materials with one week of advance. Also, you can find in Brightspace solutions to Yep. These problems are in the are not in BKM police are not in the. Probably this is from a last an old edition, but you have you have. Probably the numbers have changed, but say I can look for them. Yep, difference between the marginal rate of substitution and marginal rate of substitution. I mean supply and demand, supply and demand. This where is, where is, where is. This one, the marginal rate of substitution, is based on investor preference is demand. And this one is based on the offer supply, so sorry on the supply. Welcome. Here, let me start this with you. And I’m gonna run.