(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 69050, 1903]*) (*NotebookOutlinePosition[ 69872, 1932]*) (* CellTagsIndexPosition[ 69828, 1928]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Putting", FontVariations->{"CompatibilityType"->0}], StyleBox[" Mathematica", FontSlant->"Italic"], " to work:\nIntegrated Value-at-Risk Modeling" }], "Title"], Cell[CellGroupData[{ Cell["Statement of the Problem", "Section"], Cell[TextData[{ "To demonstrate some useful features of ", StyleBox["Mathematica'", FontSlant->"Italic"], "s design, we turn to Benninga and Wiener's short paper on Value-at Risk \ (VaR), the modern risk-measuring index. VaR is defined as the lowest quantile \ of the potential losses that can occur within a given portfolio during a \ specific time period. The time period ", StyleBox["T", FontSlant->"Italic"], " and the confidence level ", StyleBox["q", FontSlant->"Italic"], " are the two major parameters, which are chosen carefully and are \ dependent upon the goal of the risk management (regulatory reporting, \ corporate risk management, etc.). For example, suppose that a portfolio \ manager has a daily VaR equal to $1 million at 1%. This means that, assuming \ normal market conditions, there is only one chance in a hundred that there \ will be a daily loss bigger than $1 million. While this problem can be easily \ stated, finding a solution can frequently be computationally difficult." }], "Text"], Cell[BoxData[{ \(Needs["\"]\), "\[IndentingNewLine]", \(Needs["\"]\)}], "Input", CellOpen->False, InitializationCell->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "The Single Asset Problem: Loading ", StyleBox["Mathematica", FontSlant->"Italic"], " Packages and Constructing a Model" }], "Section"], Cell[TextData[{ "First, we load the relevant add-on packages and use ", StyleBox["Mathematica", FontSlant->"Italic"], " to find the 5% VaR of $100 million invested in a single asset, \ lognormally distributed with expected return of 10% and variance of 30%. This \ is stated formally below (Hull, 2000).\n\n\t", Cell[BoxData[ \(TraditionalForm\`Log(v\_T) \[Tilde] Normal\ [\((\[Mu] - \[Sigma]\^2\/2)\)\ T + Log(v), \[Sigma]T]\)], FormatType->StandardForm], "\n" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(100 - Exp[Quantile[ NormalDistribution[ Log[100] + \(( .1 - .3\^2\/2)\), .3], .05]]\)], "Input"], Cell[BoxData[ \(35.49684754345434`\)], "Output"] }, Open ]], Cell["\<\ In other words, there is a 5% chance that $100 million invested in this \ single asset will lose more than $35,496,800.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(PDF = D[CDF[NormalDistribution[Log[100] + \(( .1 - .3\^2\/2)\), .3], Log[x]], x];\)\), "\[IndentingNewLine]", \(\(a = FilledPlot[{pdf}, {x, 0, 200}, PlotRange \[Rule] \ All, RGBColor[0.312505, \ 0.203128, \ 0.445319] AxesLabel \[Rule] {\*"\"\<\!\(Portofolio\_\(t + 1\)\)\>\"", \*"\ \"\\""}];\)\), "\[IndentingNewLine]", \(\(b = Graphics[ Line[{{100 - 35.4968, .0055}, {65, 0}}]];\)\), 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RWoo0P0033g_0P00XGoo000ROol00`00Oomoo`2;Ool20003?Nl01cW^?Nlmkcg_>Nhmk`000:=oo`00 8Woo00<007ooOol0SGoo2@00Y7oo000ROol00`00Oomoo`3oOolkOol0029oo`03001oogoo0?mooc]o o`008Woo00<007ooOol0ogoo>goo000ROol00`00Oomoo`3oOolkOol0029oo`03001oogoo0?mooc]o o`00ogooH7oo003oOomPOol00?moof1oo`00ogooH7oo0000\ \>"], ImageRangeCache->{{{0, 350.438}, {216.188, 0}} -> {-0.00091484, \ -5.00002*^-06, 0.0028588, 0.00462563}, {{0.3125, 350.063}, {216.188, 0}} -> \ {-22.3914, -0.00121228, 0.652057, 6.94527*^-05}, {{0.3125, 350.063}, \ {216.188, 0}} -> {-22.3914, -0.00121228, 0.652057, 6.94527*^-05}}] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Accounting for Correlations: Matrix Algebra with ", FontSlant->"Plain"], StyleBox["Mathematica", FontSlant->"Italic"] }], "Section"], Cell[TextData[{ "Of course, real portfolios have more than one asset. One key difficulty in \ modeling the behavior of portfolios with more than one asset is measuring, \ and then accounting for, correlations in the movement of asset prices. The \ above model is now generalized to accommodate three assets, the prices of \ which are lognormally distributed (meaning that returns are normally \ distributed).\n\n\tPortfolio size = $100 million\n\t\n\tPortfolio allocation \ = ", StyleBox["x", FontSlant->"Italic"], " = ", Cell[BoxData[ RowBox[{"(", GridBox[{ {\(x\_1\)}, {\(x\_2\)}, {\(x\_3\)} }], ")"}]]], "= ", Cell[BoxData[ RowBox[{"(", GridBox[{ {"0.30"}, {"0.25"}, {"0.45"} }], ")"}]]], ",\n\t\n\tMean returns =\[Mu] = ", Cell[BoxData[ RowBox[{"(", GridBox[{ {\(\[Mu]\_1\)}, {\(\[Mu]\_2\)}, {\(\[Mu]\_3\)} }], ")"}]]], ",\n\t\n\tVariance-covariance = ", StyleBox["S", FontSlant->"Italic"], " = ", Cell[BoxData[ RowBox[{"(", GridBox[{ {\(\[Sigma]\_11\), \(\[Sigma]\_12\), \(\[Sigma]\_13\)}, {\(\[Sigma]\_21\), \(\[Sigma]\_22\), \(\[Sigma]\_23\)}, {\(\[Sigma]\_31\), \(\[Sigma]\_32\), \(\[Sigma]\_33\)} }], ")"}]]], "." }], "Text"], Cell["\<\ This program will work in three pieces. First, some preliminary definitions \ are made.\ \>", "Text"], Cell[BoxData[{ \(\(X = Table[x\_i, {i, 3}];\)\), "\[IndentingNewLine]", \(\(S = Table[\[Sigma]\_\(i, j\), {i, 3}, {j, 3}];\)\), "\[IndentingNewLine]", \(\(means = Table[\[Mu]\_k, {k, 3}];\)\), "\[IndentingNewLine]", \(\(portfolioMean[X_, means_, initial_] := X . means*initial;\)\), "\[IndentingNewLine]", \(\(portfolioSigma[X_, S_, initial_] := Sqrt[X . S . X]*initial;\)\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(portfolioMean[X, means, initial]\), "\[IndentingNewLine]", \(portfolioSigma[X, S, initial]\)}], "Input"], Cell[BoxData[ \(initial\ \((x\_1\ \[Mu]\_1 + x\_2\ \[Mu]\_2 + x\_3\ \[Mu]\_3)\)\)], "Output"], Cell[BoxData[ \(initial\ \@\(x\_1\ \((x\_1\ \[Sigma]\_\(1, 1\) + x\_2\ \[Sigma]\_\(2, 1\ \) + x\_3\ \[Sigma]\_\(3, 1\))\) + x\_2\ \((x\_1\ \[Sigma]\_\(1, 2\) + x\_2\ \ \[Sigma]\_\(2, 2\) + x\_3\ \[Sigma]\_\(3, 2\))\) + x\_3\ \((x\_1\ \ \[Sigma]\_\(1, 3\) + x\_2\ \[Sigma]\_\(2, 3\) + x\_3\ \[Sigma]\_\(3, 3\))\)\)\ \)], "Output"] }, Open ]], Cell["\<\ Then, we can add in some numerical data and recalculate the portfolio's \ expected return and variance.\ \>", "Text"], Cell[BoxData[{\(X = {0.3, 0.25, 0.45};\), "\[IndentingNewLine]", \(means = {0.1, 0.12, 0.13};\), "\[IndentingNewLine]", \(initial = 100;\), "\[IndentingNewLine]", RowBox[{ RowBox[{"S", "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {"0.10", "0.04", "0.03"}, {"0.04", "0.20", \(-0.04\)}, {"0.03", \(-0.04\), "0.60"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}]}], ";"}]}], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(portfolioMean[X, means, initial]\), "\[IndentingNewLine]", \(portfolioSigma[X, S, initial]\)}], "Input"], Cell[BoxData[ \(11.85`\)], "Output"], Cell[BoxData[ \(38.48376280978771`\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Bringing Everything Together: Writing ", FontSlant->"Plain"], StyleBox["Mathematica", FontSlant->"Italic"], StyleBox[" Functions", FontSlant->"Plain"] }], "Section"], Cell["\<\ Finally, all of the pieces above are combined to write the new multivariate \ VaR for the portfolio.\ \>", "Text"], Cell[BoxData[ \(\(VaR[X_, s_, initial_, level_] := \[IndentingNewLine]initial - Quantile[\[IndentingNewLine]NormalDistribution[\[IndentingNewLine]\ portfolioMean[X, means, initial], \[IndentingNewLine]portfolioSigma[X, s, initial]], level];\)\)], "Input"], Cell["We can now look at VaR calculations for 1%, 5%, and 10%.", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(level = { .01, .05, .10};\)\), "\[IndentingNewLine]", \(VaR[X, S, initial, \ level]\)}], "Input"], Cell[BoxData[ \({177.6766197976416`, 151.45015683641952`, 137.4689264769305`}\)], "Output"] }, Open ]], Cell["\<\ Given the stated parameters, we can say that for a one-year horizon, there is \ a 1% chance of losing more than $177 million, a 5% chance of losing more than \ $151 million, and a 10% chance of losing more than $137 million.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["The Next Step: Databases, Monte Carlo Simulations, \ and Building a Real System", FontSlant->"Plain"]], "Section"], Cell[TextData[{ "While these two problems demonstrate the basic setup of a VaR calculation \ in ", StyleBox["Mathematica", FontSlant->"Italic"], ", modeling of real-world returns requires more-involved calculations of \ the asset correlations from empirical data. Benninga and Wiener's paper \ delves more deeply into this process, including the techniques of risk \ mapping, historical simulation, variance-covariance techniques, and the Monte \ Carlo approach. Interested readers are strongly encouraged to ", ButtonBox["download", ButtonData:>{ URL[ "http://finance.wharton.upenn.edu/~benninga/wiener.html"], None}, ButtonStyle->"Hyperlink"], " this and other white papers. To assist in the implementation of ", StyleBox["Mathematica", FontSlant->"Italic"], " in your firm, Wolfram Research Account Executives can recommend ", StyleBox["Mathematica", FontSlant->"Italic"], " consultants in your region, tools to link ", StyleBox["Mathematica", FontSlant->"Italic"], " to your database, and tools to value derivative instruments." }], "Text"] }, Closed]], Cell[TextData[{ StyleBox["Note:", FontWeight->"Bold", FontVariations->{"CompatibilityType"->0}], StyleBox[" All of the code used in this example was modified from the \ article \"Value-at-Risk (VaR)\" by Simon Benninga and Zvi Wiener. 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