<html xmlns:v="urn:schemas-microsoft-com:vml" xmlns:o="urn:schemas-microsoft-com:office:office" xmlns:w="urn:schemas-microsoft-com:office:word" xmlns:m="http://schemas.microsoft.com/office/2004/12/omml" xmlns="http://www.w3.org/TR/REC-html40"><head><meta http-equiv=Content-Type content="text/html; charset=us-ascii"><meta name=Generator content="Microsoft Word 14 (filtered medium)"><style><!--
/* Font Definitions */
@font-face
{font-family:Calibri;
panose-1:2 15 5 2 2 2 4 3 2 4;}
@font-face
{font-family:Tahoma;
panose-1:2 11 6 4 3 5 4 4 2 4;}
/* Style Definitions */
p.MsoNormal, li.MsoNormal, div.MsoNormal
{margin:0in;
margin-bottom:.0001pt;
font-size:11.0pt;
font-family:"Calibri","sans-serif";}
a:link, span.MsoHyperlink
{mso-style-priority:99;
color:blue;
text-decoration:underline;}
a:visited, span.MsoHyperlinkFollowed
{mso-style-priority:99;
color:purple;
text-decoration:underline;}
p.MsoPlainText, li.MsoPlainText, div.MsoPlainText
{mso-style-priority:99;
mso-style-link:"Plain Text Char";
margin:0in;
margin-bottom:.0001pt;
font-size:11.0pt;
font-family:"Calibri","sans-serif";}
p.MsoAcetate, li.MsoAcetate, div.MsoAcetate
{mso-style-priority:99;
mso-style-link:"Balloon Text Char";
margin:0in;
margin-bottom:.0001pt;
font-size:8.0pt;
font-family:"Tahoma","sans-serif";}
span.PlainTextChar
{mso-style-name:"Plain Text Char";
mso-style-priority:99;
mso-style-link:"Plain Text";
font-family:"Calibri","sans-serif";}
span.BalloonTextChar
{mso-style-name:"Balloon Text Char";
mso-style-priority:99;
mso-style-link:"Balloon Text";
font-family:"Tahoma","sans-serif";}
.MsoChpDefault
{mso-style-type:export-only;
font-family:"Calibri","sans-serif";}
@page WordSection1
{size:8.5in 11.0in;
margin:1.0in 1.0in 1.0in 1.0in;}
div.WordSection1
{page:WordSection1;}
--></style><!--[if gte mso 9]><xml>
<o:shapedefaults v:ext="edit" spidmax="1026" />
</xml><![endif]--><!--[if gte mso 9]><xml>
<o:shapelayout v:ext="edit">
<o:idmap v:ext="edit" data="1" />
</o:shapelayout></xml><![endif]--></head><body lang=EN-US link=blue vlink=purple><div class=WordSection1><p class=MsoPlainText><b><i>We are talking about convexity differently. You are talking about the mean-return versus risk efficient frontier. I am talking about the optimization problem (for any single point on the efficient frontier). By the way in mathematics the usual convention is that what you call convex is usually called concave Using that convention, the efficient frontier can be non-convex with some constraints even in the MVO case, e.g., with MIP for constraining the number of assets in the portfolio that happens (see Figure 3.1 in Scherer and Martin).<o:p></o:p></i></b></p><p class=MsoPlainText><b><i>See additional comments below.<o:p></o:p></i></b></p><p class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText>-----Original Message-----<br>From: gsoc-porta-bounces@lists.r-forge.r-project.org [mailto:gsoc-porta-bounces@lists.r-forge.r-project.org] On Behalf Of Brian G. Peterson<br>Sent: Sunday, October 06, 2013 6:05 AM<br>To: PortfolioAnalytics<br>Subject: Re: [GSoC-PortA] Mean-mETL objective?</p><p class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText>Can't we assume convexity, even if we're wrong?<o:p></o:p></p><p class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText>In many cases, the upper hull of the mean/mETL space will be convex, and the lack of convexity will manifest itself only in the lower hull.<o:p></o:p></p><p class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText>Right now, we don't do the mean/ETL tangency portfolio even when we have Gaussian ETL. It seems that we should be able to support this for the mean/ETL space, and extend it with appropriate caveats to the mean/mETL space.<o:p></o:p></p><p class=MsoPlainText><b><i><span style='color:black'>[Doug] It will be no problem to compute this by efficient search along a concave efficient frontier so long as the mean return of the global minimum risk portfolio is larger than the risk free-rate. Otherwise there is no tangency portfolio.</span></i></b><span style='color:black'><o:p></o:p></span></p><p class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText>Doug is correct that in the presence of skewness and kurtosis the upper hull may be sufficiently bumpy that the quadratic (or is it conical?) form won't hold. It seems that we could address this in the documentation.<o:p></o:p></p><p class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText>On 10/05/2013 10:10 AM, Doug Martin wrote:<o:p></o:p></p><p class=MsoPlainText>> Peter,<o:p></o:p></p><p class=MsoPlainText>><o:p> </o:p></p><p class=MsoPlainText>> While you have convexity with ETL, that would not hold with mETL due <o:p></o:p></p><p class=MsoPlainText>> to presence of skewness and kurtosis of portfolio returns that depend <o:p></o:p></p><p class=MsoPlainText>> upon the weights. I’ll have a look at the rest of this thread.<o:p></o:p></p><p class=MsoPlainText>><o:p> </o:p></p><p class=MsoPlainText>> Doug<o:p></o:p></p><p class=MsoPlainText>><o:p> </o:p></p><p class=MsoPlainText>> -----Original Message-----<o:p></o:p></p><p class=MsoPlainText>> From: Peter Carl<o:p></o:p></p><p class=MsoPlainText>> Sent: Friday, October 04, 2013 11:49 AM<o:p></o:p></p><p class=MsoPlainText>><o:p> </o:p></p><p class=MsoPlainText>> Hey Ross,<o:p></o:p></p><p class=MsoPlainText>><o:p> </o:p></p><p class=MsoPlainText>> I can't seem to get the Mean-mETL objective to select anything other <o:p></o:p></p><p class=MsoPlainText>> than the Min mETL portfolio using ROI. It looks like there should be <o:p></o:p></p><p class=MsoPlainText>> good convexity, but I think there's a substantial imbalance between <o:p></o:p></p><p class=MsoPlainText>> the size of the monthly mean return and the loss indicated by the ETL. <o:p></o:p></p><p class=MsoPlainText>> I've tried modifying the multiplier on the mean, but it doesn't seem <o:p></o:p></p><p class=MsoPlainText>> to have an effect.<o:p></o:p></p><p class=MsoPlainText>><o:p> </o:p></p><p class=MsoPlainText>> Any thoughts?<o:p></o:p></p><p class=MsoPlainText>><o:p> </o:p></p><p class=MsoPlainText>> pcc<o:p></o:p></p><p class=MsoPlainText>><o:p> </o:p></p><p class=MsoPlainText>> --<o:p></o:p></p><p class=MsoPlainText>_______________________________________________<o:p></o:p></p><p class=MsoPlainText>GSoC-PortA mailing list<o:p></o:p></p><p class=MsoPlainText><a href="mailto:GSoC-PortA@lists.r-forge.r-project.org"><span style='color:windowtext;text-decoration:none'>GSoC-PortA@lists.r-forge.r-project.org</span></a><o:p></o:p></p><p class=MsoPlainText><a href="http://lists.r-forge.r-project.org/cgi-bin/mailman/listinfo/gsoc-porta"><span style='color:windowtext;text-decoration:none'>http://lists.r-forge.r-project.org/cgi-bin/mailman/listinfo/gsoc-porta</span></a><o:p></o:p></p></div></body></html>