efficient frontier

[Alladin]

@kazec

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?

1. CEF: correct

Using the inputs (expected returns for each asset,expected covariance matrix across assets), the objective is to find a set of asset weights that maximise expected returns for a given level of portfolio risk,or equivalently minimise portfolio variance for a given level of expected returns. If certain assumptions are met, this is equivalnt to maximising investor utility. So the portfolio optimisation process (conventional) can be stated in a number of ways

1. Max portfolio return given risk and other constraints
2. Min portfolio risk given return and other constrains
3. Max investor utility under certain assumptions

2)SEF : correct ( i think)

When we construct a SEF, we are treating the expected return/expected covariance inputs as true population parameters and sampling from this return distribution. Each sample is described by a sample mean and variance which become inputs to the MVO process to yield an SEF. We resample from this return distribution many times under monte carlo, each time we resample we get a new sample estimate for return and risk,and hence for every sample we get a new SEF.

3)REF

To find the SEF we need to average SEF portfolio weights for each level of risk. For example imagine drawing a vertical line across all blue lines in the figure above. At the intersection between this vertical line and all blue lines we have different SEF asset weights (e.g. for the lowest blue line we may have wlow1=0.2,wlow2=0.8,for the higher blue line we may have whigh1=0.5,whigh2=0.5). To find the REF portfolio for this level of risk we average all the SEF weights. We repeat this process for all levels of risk to get the REF.

Regarding why REF lies below CEF we have to adopt the perspective of the REF constructer :

The reason why REF lies below CEF is due to construction. Remember, when constructing the SEF we are assuming that the inputs to CEF are true population parameters. When we are sampling from the return distribution described by these supposed true parameters, we are incurring an estimation error (e.g. true mean = 5,sampled mean=2,error=3). Estimation error make estimated efficient frontiers suboptimal. Hence REF lies southeast of the CEF. When we are constructing this REF every point on it is some weighted average of the ‘true population parameters’.

The CEF is higher because we are feeding ‘true population parameters’ into the MVO process. It is as if we know with certainty what the parameters are.

The REF is lower because we are feeding ‘sampled parameters’ into the MVO process. Uncertainty with regards to return/risk estimates.

From the perspective of the REF creator,the CEF created here is the true efficient frontier and estimation error obviously makes the REF inferior.

This is probably the source of confusion :

Why would you rely on a suboptimal REF portfolio when the CEF is superior? Well again remember the point of reference and the issue; we assumed that the CEF encompassed true parameters for the purposes of the monte carlo sim. In reality only god knows the true parameters(future returns,variances)and only He can construct the GEF (Godly efficient frontier). GEF > CEF becasue man has to estimate what god knows without error. CEF>REF because we are sampling from what is considered true parameters.

So why use REF portfolios if the CEF > REF?

Eventhough the REF lies southeast of the CEF,the composition of the portfolios on the REF are much more diverse over the range of risk levels. Where CEF portfolios have extreme weights (80% asset1,20%asset2), REF portfolios have more sensible weights (10%asset1;20%asset2,30%asset3,etc,etc)

When you then combine the REF weights with actual future returns these REF portfolios tend to outperform CEF portfolios.

I think this is the twist…to get better performing portfolios, you need to apply the weights of the suboptimal REF portfolios to future actual returns.

[kazec]

Alladin, thanks for checking my understanding.

I agree that assuming the CEF uses the true parameters (return expectations, var, cov), the REF will be inferior to CEF. But when the TEF enters the comparison, like in the textbook question, the only true and optimal EF is the TEF. Then there is no guarantee the REF will be inferior to the CEF. Is that so?

One more thing I’m not sure of. People all say that the REF more likely gives meaningful asset allocation than the CEF. Any intuition behind this? I’m thinking of running a Matlab simulation, but that may not explain as much.

[Alladin]

i am not taking L3 so i cant check up on the text book

I don’t know if there is a guarantee that the REF is always below the CEF but…i haven’t come across anytime it wasn’t (not that i have read alot of journals though!)…but if the CEF is treated as the TEF from the perspective of constructing the REF, then by definition the REF portfolios must be on or below the CEF/TEF when expressed using the original inputs

As far as i understand, when you run a conventional MVO on estimated return/risk parameters,the resulting asset weights are very extreme. MVO significantly overweighs those assets that have large estimated returns, small variances, negative correlations with other assets, and underweighs those with small estimated returns, large variances and positive correlations. These assets are most likely to suffer from large estimation errors. That is how the MVO latches onto estimation errors and introduces a bias in the optimised portfolio with extreme asset concentrations that are not sensible.

Perhaps the averaging process with respect to SEF weights reduces this bias somehow ( don’t know the math…probably never will with my brain!), resulting in less extreme allocations than the original optimal portfolio.

Meh…i don’t even know what i am talking about anymore

[kazec]

Now I’m happy enough with my understanding of these concepts. Alladin, cpk, thank you both for the great help.

[BBUI]

I read the reference article given in the text ( Scherer 2002). The conclusion at the end seems to be that.

1. REF is an interesting heuristic to deal with error maximization, meaning more work to be done to confirm its validity.

2. What remains to be investigated:

• Why averaging the ranks give optimal solution?

-.Whether empirical results that REF beats CEF (Markowitz) out-of-sample can be generalised.

(the empirical evidence is that for long-short portfolio, REF coincides with CEF. Only for long-only portfolio that REF beats CEF ( again empirical))

• Relevant benchmark for REF is not CEF but Bayesian EF.

• Some limitations of REF:

1. Lower Sharpe ratio

2. EF may have turning points

3. Need at least 3 assets.

[QNA2012]

I dont understand why there is a veryyyyyyyyyy lengthyyyyyyy discussion on this topic.

It is very clear from the reading that REF improves on CEF as it addresses the latter’s mean return sensitivity (which is the main source of error).

Rapidquizzer succinctly made a point: On a chart of expected return vs. standard deviation, a curve is superior to the one below it. REF is superior to CEF curve as it improves the latter.

I also notice cpk as a smart guy but I’m afraid to say in this one he got them transposed.

[QNA2012]

@cpk123

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.

im afraid the second paragraph is totally wrong just like how you took back an earlier mistake in the first paragraph.

this topic is too beaten up that i dont care anymore what you guys say since it seems to me you overcooked your analysis in your mind in your desire to be really ___.

[BBUI]

We wanted to understand the subject.

You can take from the reading that REF is superior to CEF, end of story; or ask why.

There is no theoretical foundation to the approach. What we have now is a handful of empirical results.

This is important as well because the LOS is to discuss strengths and weaknesses of the methods, not just to accept the result.

[Teresasxn]

I found the following answer from this website(http://forum.cfaspace.com/viewtopic.php?f=5&t=23329), hope it helps.

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One of the reasons is that sampled frontier will always overestimate standard deviation of the TRUE frontier, thus also lie to the right of the true frontier (thus under it). Remember from level I, when you don’t know the real population standard deviation and have to estimate it from sampled data. The distribution of sampled data will take on t-distribution shape which is fatter (wider than true distribution which is normal distribution). As n approaching infinity, the sampled distribution resembles more and more normal distribution and the stddev of sampled data tightens (i.e., stddev lowers). For the same reason, the sampled stddev is higher than real stddev for each of the portfolio and thus for the whole frontier, so the sample frontier will also be to the right of the true frontier (or below it)