efficient frontier

CFA Level III Portfolio Management
#1

Totally don’t have a clue: Vol 2, P307, Q10A. Basically, the question says efficient frontier B is the true underlying EF. EF A lies north-west to EF B. It asks if EF A can be 1) the conventional mean-variance EF 2) the resampled EF.

The answer says 1) the true EF must be superior to the conventional efficient frontier, and a superior EF should lie above, so A can’t be the conventional EF.

  1. A true EF must lie above the resampled EF. So A can’t be the REF.

What do they mean???

#2

Part A)

Conventional MFE Frontier - uses estimates of returns of various parameters. Since Frontier B reflects the “true return parameters” and hence does not have any “estimation errors” built in … Frontier B will be ABOVE the conventional MFE frontier.

So Frontier A lies above Frontier B which lies ABOVE the Conventional MFE Frontier. So Frontier A is NOT a Conventional MFE Frontier.

Part B)

They ask if Frontier A is the Resampled Efficient Frontier. Resampled Efficient Frontier averages weights across different portfolio simulations and thus addresses estimation error. Now so Frontier B will lie ABOVE the Resampled Efficient Frontier as well (since Frontier B has the true return parameters - again no error).

So Frontier A > Frontier B > Resampled Efficient Frontier.

By the above - Frontier A is NOT the Resampled Efficient Frontier either.

#3

As far as I understand:

  1. The question relates to the shortcomings of the conventional mean variance optimisation procedure that lies behind the optimal potfolios which taken together become the efficient frontier.
  2. The inputs to the MVO procedure are EXPECTED returns and the variance- covariance matrix. Notice these inputs are estimates (our best guess as to what the returns/variances WILL be in one future period).
  3. Like all estimation processes, there is a certain amount of error involved.The conventional MVO process latches on to this error and results in portfolios that MAXIMISE ERRORS rather than returns.
  4. The MVO procedure systematically overweights the assets with the highest expected returns, and these are also the assets most likely to contain positive estimation error. Therefore, the MV framework weight heavily the assets with the highest estimation error.
  5. So far we have only looked at the conventional efficient frontier (CEF) that is constructed from estimates,estimates which by their very nature are flawed.
  6. Suppose we knew with PERFECT CERTAINTY what the inputs to the MVO process are and that there is no estimation error (i.e.we know the future price/returns/variance/covariance of every asset with CERTAINTY and do not have to estimate anything). Then we can construct the True Efficient Frontier (TEF).
  7. This TEF will be a collection of portfolios with asset weights that reflect the fact that we know WITH CERTAINTY what future returns/variances/covariances are and hence there is no estimation error.
  8. Clearly the TEF is superior to the CEF because there is no estimation error for the MVO to latch onto.
  9. What does it mean to be superior? It means we are creating portfolios that have the same variance but higher expected returns…this places the TEF northwest of the CEF.
  10. The resampled efficient frontier (REF) is an attempt to address the drawback of the MVO procedure of error maximisation. Resampling creates portfolios that have lower expected returns for a given level of risk.
  11. Hence REF lies below CEV lies below TEF.

To summarise:

TEF : certainty in paramaters -> no estimation errors ->superior portfolios ->efficient frontier lies towards northwestern corner.

CEF : uncertainty in paramters (estimation error in returns/cov) -> MVO latches on to error in estimation ->inferior portfolios ->efficient frontier lies below TEF.

REF : mitigates error maximisation problem through sampling and averaging across many efficient frontiers -> efficnet frontier lies below CEF.

…Hence : TEF > CEF > REF

hope this is right!

#4

Thank you both for the notes.

I now get why TEF >= CEF and TEF >= REF, but Alladin, why does it have to be that CEF > REF? I guess REF is the avg of a series of CEF.

#5

In the REF estimation error is removed by the process of averaging across a bunch of simulated portfolios.

The CEF - has a higher return - due to the problem of “return maximization” that Alladin described above. So the positive error estimation causes the CEF to be “in general” higher than the REF.

#6

Still don’t get it cpk. Am I right to say REF is in some sense an average of a bunch of CEFs? If so, REF is superior to some CEFs, and inferior to others.

#7

simulated thro’ MC (Monte Carlo) - so first of all the effect of the Higher return which is present on the CEF will not be present. So REF will be lower than CEF.

#8

Original and Simulated Efficient Frontiers

Here is a picture of the CEF (red) compared to a series of Simulated Efficient frontiers(blue) that can be averaged to yield the REF. As cpk said, monte carlo simulation is key to creating each of the blue lines which are then averaged to yield the REF (not shown in the above image). Basically it works like this.

To get the CEF : you feed the inputs (expected return;variance covariance matrix) to the MVO procedure and it gives you a set of asset weights or minimum varaince portfolios. Each point on the red line corresponds to a combination of such weights.

To get one simulated efficient frontier (SEF): you conduct a montecarlosimulation as follows. you take a random sample from the return distribution under the assumption that

  1. the mean of the distribution is equal to the expected return used for the CEF
  2. the variance of the distriubtion is equal to the variance/cov used for the CEF

Now we have sample mean and risk estimates which we can feed to the MVO procedure as before to create one SEF (one blue line).

To get another SEFs: you resample from the distribution above to get another sample mean and risk estimate;feed them to the optimisation procedure and get another SEF (another blue line).

Repeat this process many many times ( that is the simulation part) to get many many blue lines. Then average across all the SEFs(blue lines) to get the REF.

:th_squish

#9

Alladin, thanks for the very clear description how the REF is obtained. But what makes you thing the REF is below the CEF? If you create an infinite number of SEFs to get the REF, will the REF fall exactly on the CEF?

#10

“infinite” … does not mean “Sample”.

This statement in the book:

The portfolios resulting from the resampled efficient frontier approach tend to be more diversified and more stable through time than those on a conventional mean– variance efficient frontier developed from a single optimization.

(Institute 250)

Institute, CFA. Level III 2013 Volume 3 Capital Market Expectations, Market Valuation, and Asset Allocation. John Wiley & Sons (P&T), 6/18/2012.

tells me it is lower. More Diversified = Lower Std. Deviation for same level of return. So it will lie below.

#11

Please check if I get these right:

  1. CEF: with the expected return and variance for each asset class, construct the CEF by optimizing class weights for highest portfolio expected return at each portfolio std level.

  2. SEF: given each class’s expected return and var, draw a return randomly from each asset class, construct the SEF by optimizing class weights for highest portfolio return at each portfolio std level, where portfolio return is based on class weights and the returns drawn this one time from the return distributions.

  3. REF: given a bunch of SEFs, at each portfolio std level, avg the portfolio returns of the SEFs.

So the only difference between a CEF and an SEF is that the CEF uses expected return while the SEF uses a return drawn from the normal distribution described by expected return and variance. Then why should REF, an average of SEF, lie below CEF?

#12

did you read my post above?

it is an average of Sampled Efficient Frontier - and is MORE DIVERSIFIED. So that means for each level of return it has a lower Std Deviation - and hence would LIE BELOW THE CEF.

The portfolios across various Sampled Efficient Frontiers created are ranked - the same ranked ones across multiple samples are averaged - and that creates the Resampled Efficient frontier. This process builds in the higher diversification.

#13

If the REF has higher diversification (lower std) for a given return, then it should plot left to (above) the CEF.

#14

Cpk, as ever your expertise on the technical aslects of the curriculum is astounding.

I recall similar posts when you were on the L2 forum.

I am surprised you are on here after the 2012 exam.

Would you have time for an insight as to which part of the exam was the issue?

What would you have done differently or any change in approach this time?

My only concern is that I am unlikely to ever attain a knowledge such as your’s, so I may as well burn my books now!

cheers.

#15

yup cpk is the man!

#16

really? What is on your x-axis? On mine it is Std Dev and return on Y axis.

so for a 10% return - CEF would have e.g. 10% Std dev, while REF would have a 8% Std Dev.

so if the CEF where the B portfolio — this one (REF) would be BELOW it.

Original and Simulated Efficient Frontiers

#17

Yeah, I have return on the y-axis, and std on the x-axis.

“For a 10% return - CEF would have e.g. 10% Std dev, while REF would have a 8% Std Dev” – doens’t it mean the REF lies left to the CEF? That is, point (0.08, 0.1) lies left to (0.1, 0.1). If not, I must be missing something very fundamental…

#18

coffee! where were you when I put foot in mouth!

#19

I take that back that higher diversification means lower std dev at the same level of return.

Doesn’t higher diversification mean reduced risk - and since risk and return are correlated - lower risk = lower return. so at the same level of risk - it will provide a lower return with the REF. and as a result that would lie below the CEF now.

#20

Sorry cpk, still unconvinced. There could be some rather complex math behind this which I think I’ll skip for now.