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Before answering your questions directly i would like to briefly restate the idea of the resampled efficent frontier:

One of the problem with classical mean variance optimization is (even if the multivariate normal assumption holds) that you cant estimate $\mu$ and $\Omega$ (which is usually denoted as $\Sigma$) exactly. Thats why you incur estimation errors $\hat{\mu}-\mu$ and $\hat{\Omega}-\Omega$ when you deviate from the "population means". This error can change your portfolio dramatically (since MVO is not very robust in general). So the idea is to estimate $\mu$ and $\Omega$ to end up with $\hat{\mu}$ and $\hat{\Omega}$ and sample from the $\text{MVN}(\hat{\mu},\hat{\Omega})$ distribution. From all these samples, calculate the efficient frontier and the optimal portfolio. The RE frontier and RE optimal portfolio will be the mean of those.

This method is a bootstrapping method (where you strangely gain information for adding uncertainty).

Another nice way to think of it is (and here i quote the paper you mentioned): "Every simulated MV efficient frontier [...] is the right way to invest given a set of inputs. But the inputs are highly uncertain. How should an investor deal with portfolio optimality uncertainty?"

And the answer is: One should use the REF to obtain a sub-optimal (for one estimated set of parameters) portfolio which is more prone to estimation error and therefore should result in a better portfolio than from estimation alone. (look at Figure 4 in the paper you mentioned!)

As for your questions, you got everything correct up to step 3.):

3.) I don't know what you mean by futures. I would call them assets. What you should do is you take your values $\hat{\mu}$ and $\hat{\Omega}$ and calculate the RE frontier. To do that you draw lots of samples from $\text{MVN}(\hat{\mu},\hat{\Omega})$, do MV optimization on each of them and compute the mean (both of the optimized pf and the frontiers).

4.) As you said, repeat the steps several times.

5.) You are right here. Just "score" all the portfolios (i think you meant $\Omega$ instead of $\Omega_n$).

6.) The plot shows the average in-sample and out-of-sample efficient frontiers. in-sample: You take all MV efficient frontiers for $\hat{\mu}$ and $\hat{\Omega}$ and build the average. That gives you the "Classical in-sample" curve. Computing the average over the REFs for the same parameters gives you the "REF in-sample" efficient frontiers. In the chart this means that if you consider the average of your sampled parameters as the true population parameter, the MV efficient frontier is always abvoe the resampled efficient frontier. out-of-sample: You take all calculated MV and REF weights but now you calculate the efficient frontier with respect to the true population parameters $\mu$ and $\sigma$. Of course (since they are optimal for another parameter set) this efficient frontiers are lower then their in-sample counterparts. Now comes the important point: The MV efficient frontier is now inferior to the REF. Thats what the whole point of the bootstrapping technique was.

Hope that enlightened you.

Before answering your questions directly i would like to briefly restate the idea of the resampled efficent frontier:

One of the problem with classical mean variance optimization is (even if the multivariate normal assumption holds) that you cant estimate $\mu$ and $\Omega$ (which is usually denoted as $\Sigma$) exactly. Thats why you incur estimation errors $\hat{\mu}-\mu$ and $\hat{\Omega}-\Omega$ when you deviate from the "population means". This error can change your portfolio dramatically (since MVO is not very robust in general). So the idea is to estimate $\mu$ and $\Omega$ to end up with $\hat{\mu}$ and $\hat{\Omega}$ and sample from the $\text{MVN}(\hat{\mu},\hat{\Omega})$ distribution. From all these samples, calculate the efficient frontier and the optimal portfolio. The RE frontier and RE optimal portfolio will be the mean of those.

This method is a bootstrapping method (where you strangely gain information for adding uncertainty).

Another nice way to think of it is (and here i quote the paper you mentioned): "Every simulated MV efficient frontier [...] is the right way to invest given a set of inputs. But the inputs are highly uncertain. How should an investor deal with portfolio optimality uncertainty?"

And the answer is: One should use the REF to obtain a sub-optimal (for one estimated set of parameters) portfolio which is more prone to estimation error and therefore should result in a better portfolio than from estimation alone. (look at Figure 4 in the paper you mentioned!)

As for your questions, you got everything correct up to step 3.):

3.) I don't know what you mean by futures. I would call them assets. What you should do is you take your values $\hat{\mu}$ and $\hat{\Omega}$ and calculate the RE frontier. To do that you draw lots of samples from $\text{MVN}(\hat{\mu},\hat{\Omega})$, do MV optimization on each of them and compute the mean (both of the optimized pf and the frontiers).

4.) As you said, repeat the steps several times.

5.) You are right here. Just "score" all the portfolios (i think you meant $\Omega$ instead of $\Omega_n$).

6.) The plot shows the average in-sample and out-of-sample efficient frontiers. in-sample: You take all MV efficient frontiers for $\hat{\mu}$ and $\hat{\Omega}$ and build the average. That gives you the "Classical in-sample" curve. Computing the average over the REFs for the same parameters gives you the "REF in-sample" efficient frontiers. In the chart this means that if you consider the average of your sampled parameters as the true population parameter, the MV efficient frontier is always abvoe the resampled efficient frontier. out-of-sample: You take all calculated MV and REF weights but now you calculate the efficient frontier with respect to the true population parameters $\mu$ and $\Omega$. Of course (since they are optimal for another parameter set) this efficient frontiers are lower then their in-sample counterparts. Now comes the important point: The MV efficient frontier is now inferior to the REF. Thats what the whole point of the bootstrapping technique was.

Hope that enlightened you.

Before answering your questions directly i would like to briefly restate the idea of the resampled efficent frontier:

One of the problem with classical mean variance optimization is (even if the multivariate normal assumption holds) that you cant estimate $\mu$ and $\Omega$ (which is usually denoted as $\Sigma$) exactly. Thats why you incur estimation errors $\hat{\mu}-\mu$ and $\hat{\Omega}-\Omega$ when you deviate from the "population means". This error can change your portfolio dramatically (since MVO is not very robust in general). So the idea is to estimate $\mu$ and $\Omega$ to end up with $\hat{\mu}$ and $\hat{\Omega}$ and sample from the $\text{MVN}(\hat{\mu},\hat{\Omega})$ distribution. From all these samples, calculate the efficient frontier and the optimal portfolio. The RE frontier and RE optimal portfolio will be the mean of those.

This method is a bootstrapping method (where you strangely gain information for adding uncertainty).

Another nice way to think of it is (and here i quote the paper you mentioned): "Every simulated MV efficient frontier [...] is the right way to invest given a set of inputs. But the inputs are highly uncertain. How should an investor deal with portfolio optimality uncertainty?"

And the answer is: One should use the REF to obtain a sub-optimal (for one estimated set of parameters) portfolio which is more prone to estimation error and therefore should result in a better portfolio than from estimation alone. (look at Figure 4 in the paper you mentioned!)

As for your questions, you got everything correct up to step 3.):

3.) I don't know what you mean by futures. I would call them assets. What you should do is you take your values $\hat{\mu}$ and $\hat{\Omega}$ and calculate the RE frontier. To do that you draw lots of samples from $\text{MVN}(\hat{\mu},\hat{\Omega})$, do MV optimization on each of them and compute the mean (both of the optimized pf and the frontiers).

4.) As you said, repeat the steps several times.

5.) You are right here. Just "score" all the portfolios (i think you meant $\Omega$ instead of $\Omega_n$).

6.) The plot shows the average in-sample and out-of-sample efficient frontiers. in-sample: You take all MV efficient frontiers for $\hat{mu}$ and $\hat{\Omega}$ and build the average. That gives you the "Classical in-sample" curve. Computing the average over the REFs for the same parameters gives you the "REF in-sample" efficient frontiers. In the chart this means that if you consider the average of your sampled parameters as the true population parameter, the MV efficient frontier is always abvoe the resampled efficient frontier. out-of-sample: You take all calculated MV and REF weights but now you calculate the efficient frontier with respect to the true population parameters $\mu$ and $\sigma$. Of course (since they are optimal for another parameter set) this efficient frontiers are lower then their in-sample counterparts. Now comes the important point: The MV efficient frontier is now inferior to the REF. Thats what the whole point of the bootstrapping technique was.

Hope that enlightened you.

Before answering your questions directly i would like to briefly restate the idea of the resampled efficent frontier:

One of the problem with classical mean variance optimization is (even if the multivariate normal assumption holds) that you cant estimate $\mu$ and $\Omega$ (which is usually denoted as $\Sigma$) exactly. Thats why you incur estimation errors $\hat{\mu}-\mu$ and $\hat{\Omega}-\Omega$ when you deviate from the "population means". This error can change your portfolio dramatically (since MVO is not very robust in general). So the idea is to estimate $\mu$ and $\Omega$ to end up with $\hat{\mu}$ and $\hat{\Omega}$ and sample from the $\text{MVN}(\hat{\mu},\hat{\Omega})$ distribution. From all these samples, calculate the efficient frontier and the optimal portfolio. The RE frontier and RE optimal portfolio will be the mean of those.

This method is a bootstrapping method (where you strangely gain information for adding uncertainty).

Another nice way to think of it is (and here i quote the paper you mentioned): "Every simulated MV efficient frontier [...] is the right way to invest given a set of inputs. But the inputs are highly uncertain. How should an investor deal with portfolio optimality uncertainty?"

And the answer is: One should use the REF to obtain a sub-optimal (for one estimated set of parameters) portfolio which is more prone to estimation error and therefore should result in a better portfolio than from estimation alone. (look at Figure 4 in the paper you mentioned!)

As for your questions, you got everything correct up to step 3.):

3.) I don't know what you mean by futures. I would call them assets. What you should do is you take your values $\hat{\mu}$ and $\hat{\Omega}$ and calculate the RE frontier. To do that you draw lots of samples from $\text{MVN}(\hat{\mu},\hat{\Omega})$, do MV optimization on each of them and compute the mean (both of the optimized pf and the frontiers).

4.) As you said, repeat the steps several times.

5.) You are right here. Just "score" all the portfolios (i think you meant $\Omega$ instead of $\Omega_n$).

6.) The plot shows the average in-sample and out-of-sample efficient frontiers. in-sample: You take all MV efficient frontiers for $\hat{\mu}$ and $\hat{\Omega}$ and build the average. That gives you the "Classical in-sample" curve. Computing the average over the REFs for the same parameters gives you the "REF in-sample" efficient frontiers. In the chart this means that if you consider the average of your sampled parameters as the true population parameter, the MV efficient frontier is always abvoe the resampled efficient frontier. out-of-sample: You take all calculated MV and REF weights but now you calculate the efficient frontier with respect to the true population parameters $\mu$ and $\sigma$. Of course (since they are optimal for another parameter set) this efficient frontiers are lower then their in-sample counterparts. Now comes the important point: The MV efficient frontier is now inferior to the REF. Thats what the whole point of the bootstrapping technique was.

Hope that enlightened you.

Before answering your questions directly i would like to briefly restate the idea of the resampled efficent frontier:

One of the problem with classical mean variance optimization is (even if the multivariate normal assumption holds) that you cant estimate $\mu$ and $\Omega$ (which is usually denoted as $\Sigma$) exactly. Thats why you incur estimation errors $\hat{\mu}-\mu$ and $\hat{\Omega}-\Omega$ when you deviate from the "population means". This error can change your portfolio dramatically (since MVO is not very robust in general). So the idea is to estimate $\mu$ and $\Omega$ to end up with $\hat{\mu}$ and $\hat{\Omega}$ and sample from the $\text{MVN}(\hat{\mu},\hat{\Omega})$ distribution. From all these samples, calculate the efficient frontier and the optimal portfolio. The RE frontier and RE optimal portfolio will be the mean of those.

This method is a bootstrapping method (where you strangely gain information for adding uncertainty).

Another nice way to think of it is (and here i quote the paper you mentioned): "Every simulated MV efficient frontier [...] is the right way to invest given a set of inputs. But the inputs are highly uncertain. How should an investor deal with portfolio optimality uncertainty?"

And the answer is: One should use the REF to obtain a sub-optimal (for one estimated set of parameters) portfolio which is more prone to estimation error and therefore should result in a better portfolio than from estimation alone. (look at Figure 4 in the paper you mentioned!)

As for your questions, you got everything correct up to step 3.):

3.) I don't know what you mean by futures. I would call them assets. What you should do is you take your values $\hat{\mu}$ and $\hat{\Omega}$ and calculate the RE frontier. To do that you draw lots of samples from $\text{MVN}(\hat{\mu},\hat{\Omega})$, do MV optimization on each of them and compute the mean (both of the optimized pf and the frontiers).

4.) As you said, repeat the steps several times.

5.) You are right here. Just "score" all the portfolios (i think you meant $\Omega$ instead of $\Omega_n$).

6.) The plot shows the average in-sample and out-of-sample efficient frontiers. in-sample: You take all MV efficient frontiers for $\hat{mu}$ and $\hat{\Omega}$ and build the average. That gives you the "Classical in-sample" curve. Computing the average over the REFs for the same parameters gives you the "REF in-sample" efficient frontiers. In the chart this means that if you consider the average of your sampled parameters as the true population parameter, the MV efficient frontier is always abvoe the resampled efficient frontier. out-of-sample: You take all calculated MV and REF weights but now you calculate the efficient frontier with respect to the true population parameters $\mu$ and $\sigma$. Of course (since they are optimal for another parameter set) this efficient frontiers are lower then their in-sample counterparts. Now comes the important point: The MV efficient frontier is now inferior to the REF. Thats what the whole point of the bootstrapping technique was.

Hope that enlightened you. Cheers,

vanguard2k

Before answering your questions directly i would like to briefly restate the idea of the resampled efficent frontier:

One of the problem with classical mean variance optimization is (even if the multivariate normal assumption holds) that you cant estimate $\mu$ and $\Omega$ (which is usually denoted as $\Sigma$) exactly. Thats why you incur estimation errors $\hat{\mu}-\mu$ and $\hat{\Omega}-\Omega$ when you deviate from the "population means". This error can change your portfolio dramatically (since MVO is not very robust in general). So the idea is to estimate $\mu$ and $\Omega$ to end up with $\hat{\mu}$ and $\hat{\Omega}$ and sample from the $\text{MVN}(\hat{\mu},\hat{\Omega})$ distribution. From all these samples, calculate the efficient frontier and the optimal portfolio. The RE frontier and RE optimal portfolio will be the mean of those.

This method is a bootstrapping method (where you strangely gain information for adding uncertainty).

Another nice way to think of it is (and here i quote the paper you mentioned): "Every simulated MV efficient frontier [...] is the right way to invest given a set of inputs. But the inputs are highly uncertain. How should an investor deal with portfolio optimality uncertainty?"

And the answer is: One should use the REF to obtain a sub-optimal (for one estimated set of parameters) portfolio which is more prone to estimation error and therefore should result in a better portfolio than from estimation alone. (look at Figure 4 in the paper you mentioned!)

As for your questions, you got everything correct up to step 3.):

3.) I don't know what you mean by futures. I would call them assets. What you should do is you take your values $\hat{\mu}$ and $\hat{\Omega}$ and calculate the RE frontier. To do that you draw lots of samples from $\text{MVN}(\hat{\mu},\hat{\Omega})$, do MV optimization on each of them and compute the mean (both of the optimized pf and the frontiers).

4.) As you said, repeat the steps several times.

5.) You are right here. Just "score" all the portfolios (i think you meant $\Omega$ instead of $\Omega_n$).

6.) The plot shows the average in-sample and out-of-sample efficient frontiers. in-sample: You take all MV efficient frontiers for $\hat{mu}$ and $\hat{\Omega}$ and build the average. That gives you the "Classical in-sample" curve. Computing the average over the REFs for the same parameters gives you the "REF in-sample" efficient frontiers. In the chart this means that if you consider the average of your sampled parameters as the true population parameter, the MV efficient frontier is always abvoe the resampled efficient frontier. out-of-sample: You take all calculated MV and REF weights but now you calculate the efficient frontier with respect to the true population parameters $\mu$ and $\sigma$. Of course (since they are optimal for another parameter set) this efficient frontiers are lower then their in-sample counterparts. Now comes the important point: The MV efficient frontier is now inferior to the REF. Thats what the whole point of the bootstrapping technique was.

Hope that enlightened you.