Problem 45860. Algorithmic Trading - 2 (optimize Calmar Ratio)

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{"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","targetMode":"","relationshipId":"rId1","target":"/matlab/document.xml"},{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/output","targetMode":"","relationshipId":"rId2","target":"/matlab/output.xml"}],"parts":[{"partUri":"/matlab/document.xml","relationship":[],"contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n<w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>This is the second of a series of problems on quant trading. It builds on earlier problem 45776</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/45776-algorithmic-trading-1-optimize-trading-strategy-simple-case\"><w:r><w:t>&lt;https://www.mathworks.com/matlabcentral/cody/problems/45776-algorithmic-trading-1-optimize-trading-strategy-simple-case</w:t></w:r></w:hyperlink><w:r><w:t>&gt;</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Goal</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>Maximize the Calmar Ratio over the test period, calculated on daily returns, defined as</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>C = CAGR/MDD</w:t></w:r><w:r><w:t>, were MDD is the Maximum Drawdown of the return series, defined as the maximum relative drop from a peak (always positive).</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>Max Drawdown can be calculated by</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>MDD = -min((1+cR)./(1+cummax(cR))-1)</w:t></w:r><w:r><w:t>, where</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>cR=cumprod(R+1)-1</w:t></w:r><w:r><w:t> is the cumulative compound return over time,</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>mR=cummax(cR)</w:t></w:r><w:r><w:t> is the historical maximum cumulative compound return over time, and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>R = D.*(S(1:n-1)&gt;t)</w:t></w:r><w:r><w:t> is the daily return of the investment strategy</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>See</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Drawdown_(economics)\"><w:r><w:t>&lt;https://en.wikipedia.org/wiki/Drawdown_(economics</w:t></w:r></w:hyperlink><w:r><w:t>)&gt;</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>You are given</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>P</w:t></w:r><w:r><w:t>, a [n x 1] vector of daily prices of the traded security, at market open (always positive)</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>S</w:t></w:r><w:r><w:t>, a [n x 1] vector of trading signals, calculated just ahead of market open (can take any real value)</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>These will serve both as training and test set</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Your function should return</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>t</w:t></w:r><w:r><w:t>, a scalar threshold that determines this simple trading strategy: invest all the available funds in the security when S&gt;t, stay in cash otherwise</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Hints</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>The Calmar Ratio is a risk-weighted return metric. The optimal t also maximizes</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>log(C)</w:t></w:r><w:r><w:t> or, more simply maximizes</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>sum(log(1+D.*(S(1:n-1)&gt;t)))-cummax(cumsum(log(R+1)))</w:t></w:r></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray type=\"array\"/></embeddedOutputs>"}]}

This is the second of a series of problems on quant trading. It builds on earlier problem 45776 <a href = "https://www.mathworks.com/matlabcentral/cody/problems/45776-algorithmic-trading-1-optimize-trading-strategy-simple-case">https://www.mathworks.com/matlabcentral/cody/problems/45776-algorithmic-trading-1-optimize-trading-strategy-simple-case</a>Goal<ul><li>Maximize the Calmar Ratio over the test period, calculated on daily returns, defined as C = CAGR/MDD, were MDD is the Maximum Drawdown of the return series, defined as the maximum relative drop from a peak (always positive).</li><li>Max Drawdown can be calculated by <tt>MDD = -min((1+cR)./(1+cummax(cR))-1)</tt>, where <tt>cR=cumprod(R+1)-1</tt> is the cumulative compound return over time, <tt>mR=cummax(cR)</tt> is the historical maximum cumulative compound return over time, and <tt>R = D.*(S(1:n-1)&gt;t)</tt> is the daily return of the investment strategy</li></ul>See <a href = "https://en.wikipedia.org/wiki/Drawdown_(economics)">https://en.wikipedia.org/wiki/Drawdown_(economics)</a>You are given<ul><li>P, a [n x 1] vector of daily prices of the traded security, at market open (always positive)</li><li>S, a [n x 1] vector of trading signals, calculated just ahead of market open (can take any real value)</li><li>These will serve both as training and test set</li></ul>Your function should return<ul><li>t, a scalar threshold that determines this simple trading strategy: invest all the available funds in the security when S&gt;t, stay in cash otherwise</li></ul>Hints<ul><li>The Calmar Ratio is a risk-weighted return metric. The optimal t also maximizes <tt>log(C)</tt> or, more simply maximizes <tt>sum(log(1+D.*(S(1:n-1)&gt;t)))-cummax(cumsum(log(R+1)))</tt></li></ul>

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