Corner portfolio

CFA Level III Portfolio Management
#1

What is the concept behind corner portfolio and adjacent corner portfolio

#2

That any efficient portfolio between adjacent corner portfolios will be a linear combination (technically, a convex combination) of those corner portfolios; if the corner portfolios are A and B, any portfolio X that lies on the efficient frontier between A and B can be written as:

X = wa(A) + wb(B)

where:

wa + wb = 1

0 ≤ wa, wb ≤ 1

#3

You may want to start with these.

http://www.analystforum.com/forums/cfa-forums/cfa-level-iii-forum/91318033

http://www.analystforum.com/forums/cfa-forums/cfa-level-iii-forum/91337908

http://www.analystforum.com/forums/cfa-forums/cfa-level-iii-forum/91319264

TP = Tangency Portfolio ; EF = Efficient Frontier ; MVO = Mean Variance Optimization ; GMVP = Global Minimum Variance portfolio

Takes awhile to fully wrap your head around the concept.

The key to CP = Corner Portfolios is that it is a way to find efficient portfolios without using the relatively complex MVO. I.e. you only need a limited number of corner portfolios to identify all efficient portfolios.

#4

To be frank, this is one of the easiest concepts in LIII. Tougher roads are up ahead.

#BetterLuckIn2017

#5

Thanks “Band 6”, this is exactly what I was trying to understand and I could not find any confirmation of my understanding.

I understand that you could define any efficient portfolio as a corner portfolio (CP). There is no “absolute” corner portfolio as such. The idea is just to define a certain number of efficient portfolios instead of drawing the whole efficient frontier which is composed of an infinite number of portfolios.

If you have set a certain return or certain risk tolerance, and you are searching for the optimal asset allocation, you will search within the CPs. Maybe one of them will happen to be the efficient portfolio you are searching for. If not, you need to take the two CPs that represent the adjacent portfolios to the one you are searching for (i.e. the ones that have the closest expected returns to the your optimal portfolio, just above and just below. Or same reasoning with expected standard deviation) and you deduct the portfolio you are searching for as being a linear combination of the asset weights of these two CPs.

It is an approximation as it does not take the convexity of the efficient frontier into account. The more CPs you define at the beginning, the more precisely you will be able to define your efficient portfolio as the error due to convexity will be smaller.

Do you agree?

#6

This is really an important topic for LIII. The thing here is that the efficient frontier is non-linear (it is a curve). However, due to constraints, the plausible efficient frontier is a set of interconnected lines with corners. Hence the corner portfolios. Mostly theoretical BS (which nobody uses in real life).

#7

I think daharmattan1 is the only close answer.

the ‘concept’ is that you would like your computer to generate a portfolio which will be on the efficient frontier. you do this by constructing a least squares model with corner portfolios as the input, and the output is a so called ‘adjacent corner portfolio’.

probably it’s called adjacent, since it will likely be very close to one of your corner portfolios (the efficient frontier is an almost straight line in reality - not curved like a wing).

it is used in real life - I’ve seen plent of presentations with the results of this kind of optimisation. my take on this topic, is the steps are important, and the cludges it involves. there are plenty of other ‘minimum finding’ techniques and least squares is really something from the late 80s.

#8

That’s not how I understand it if I may. Adjacent is a mathematical concept that means, among other possible definitions, “the nearest to… on one side or the other”. It is the other way around: you find your two adjacent portfolios to be used as inputs (the way I explain it in my post above), and t he output is the minimum-variance portfolio that you are looking for.

The curriculum definition quite clearly confirms that: "In a sign-constrained optimization, the asset weights of any minimum-variance portfolio are a positive linear combination of the corresponding weights in the two adjacent corner portfolios that bracket it in terms of expected return (or standard deviation of returns)".

#9

This is also how daharmattan1 means it. Adjacent is just an adjective you can use for anything. Here it just means that, among all the corner portfolios, you take the two that are the closest to the one portfolio that suits your strategy. Those two are called adjacent corner portfolios (adjacent to your own optimal portfolio).

#10

I’m glad others have questions about this topic! What I’m trying to wrap my head around is how do you figure out which investments might be included in a corner portfolio, if you didn’t have fancy software.

For example, in Reading 17, Exhibit 17, we start off with a simple portfolio made up of 100% U.K. equities. It’s defined as Corner Portfolio 1. Corner Portfolio 2 is made up of U.K. equities and real estate. Why real estate?

We know that there is benefit to adding an investment if it’s sharpe ratio > the sharpe ratio of the portfolio excluding it x its correlation with the portfolio. This criteria is met for all of the alternative asset classes presented. How would the software “know” to optimize CP1 by adding real estate rather than something else like ex-U.K. equities?

Thanks!

#11

Hi Lammy,

I have the same problem. I totally see your point. Corner portfolios have to be on the efficient frontier. I kind of see in practice how you can figure out the efficient frontier (although it guess it takes a lot of computer power, and I have never done anything like it). And then you spot the corner portfolios on the frontier (basically any portfolio on it works, you just have to start with the first on the left and then move towards the right and select additional portfolios here and there, just making sure the gap between two of them is pretty much constant, to have a good coverage of the curve).

But corner portfolios are actually meant the other way around. They are meant to help you save time: you don’t have to define the whole minimum-variance frontier; you just need to know a few corner portfolios, and from there you can approximate any other portfolio on the frontier thanks to a weighted average of two adjacent corner portfolios.

Well, cool, but I have no clue how you can figure out any corner portfolio if you don’t have the whole frontier first…

And why exhibit 17 starts with 100% UK equity, and how do we know that this lies on the frontier, (maybe it is subefficient) I don’t know. It’s probably just an illustrative example.

#12

Thanks, Myriam2222-

Corner Portfolio 1 is 100% UK equities because it has the highest expected return and highest standard deviation.

I’m guessing Corner Portfolio 2 is made up of UK equities and real estate because although it has the same Sharpe Ratio as ex-UK equities but a lower correlation w/ UK equities… Not sure if that’s the right reason why though.

#13

lammy - you do need fancy software. but all the software does is solve (linear) equations.

i.e. if you are contemplating the weights of 10 different assets, then you need 11 equations (10 unknowns, 11 equations). obviously the more equations, the better your minimum finding algorithm will work.

. so your second question.

“As the minimum-variance frontier passes through a corner portfolio, an asset weight either changes from zero to positive or from positive to zero.”

this is simply choosing equations with a parameter set to zero. the assumption being that an adjactent portolio adds no value to solving the equation.

e.g.

1 + X = 11

1.1 + X = 11.01 doesn’t help to solve the first equation.

#14

Daharmattan1 , Thanks. This is totally clear. This is what I was trying to say in several posts here as well (but I cut it short because it was not my point. And maybe my English is too approximative. Sorry about that).

I am not asking how to solve for your optimal portfolio given corner portfolios. This I understood. I am asking how to find corner portfolios if you don’t have already the efficient frontier. But you are right, this is not part of the exam.

So let’s not bother because we know already enough for the exam. Corner portfolios, if needed for the exam, will be given to us. Thanks Daharmattan1 for this info.

**However, for those who are interested (**and continuing my discussion with Lammy because I think it was your point as well) here’s my view on it: I think in practice you need to know the whole efficient frontier to define a few corner portfolios. However, an asset manager won’t have all the data related to the whole frontier, but the department that may be in charge of defining it and making this type of research and running the model will just give him a few corner portfolios for him to be able to deduct the portfolio he needs to meet his investment objectives any time without having to deal with a complicated model, by simply doing a weighted average. I don’t know if that’s clear. It’s just my guess. Maybe you have another view. Or maybe someone works in this field FOR REAL and can tell us how it is in practice :wink:

#15

@onlysimon:

Again, I think Lammy and I are talking about something different from what Daharmattan1 and you are talking about. What you are saying I think is clear to us.