(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 661482, 15230] NotebookOptionsPosition[ 357999, 10128] NotebookOutlinePosition[ 649098, 14847] CellTagsIndexPosition[ 649017, 14842] WindowFrame->Normal ContainsDynamic->True *) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Risk Measures in Finance", "Title", CellMargins->0, CellFrameMargins->{{60, 9}, {9, 12}}, CellChangeTimes->{{3.41252692941323*^9, 3.412526931828525*^9}, { 3.412532832874496*^9, 3.412532837988109*^9}, {3.432536749375*^9, 3.432536768296875*^9}}, FontColor->GrayLevel[1], Background->GrayLevel[0]], Cell["", "Subtitle", CellChangeTimes->{{3.4331391540695267`*^9, 3.4331391557101517`*^9}}], Cell["Jan Hurt", "Author", CellChangeTimes->{{3.432536777953125*^9, 3.4325367835625*^9}}], Cell["Charles University of Prague", "Affiliation", CellChangeTimes->{{3.4325367878125*^9, 3.432536804421875*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"], ButtonFrame->"None"], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"], ButtonFrame->"None"] }], "PreviousNext"] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Introduction", FontColor->RGBColor[ 0.6502021820401312, 0.7862973983367666, 0.08026245517662318]]], "Section", CellChangeTimes->{{3.432909640984375*^9, 3.432909651625*^9}}], Cell[CellGroupData[{ Cell["Preliminaries", "Subsection", CellChangeTimes->{{3.433150255203125*^9, 3.433150260515625*^9}, { 3.433768916859375*^9, 3.433768917515625*^9}}], Cell["", "Text", CellChangeTimes->{{3.43376883159375*^9, 3.433768837578125*^9}}], Cell[CellGroupData[{ Cell["Notation", "Subsubsection", CellChangeTimes->{{3.43376888265625*^9, 3.433768885265625*^9}}], Cell[TextData[{ "Let ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["S", "0"], ",", SubscriptBox["S", RowBox[{"1", ","}]], ",", "..."}], TraditionalForm]]], " be the prices of securities at times ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"t", "=", "0"}], ",", "1", ",", "..."}], TraditionalForm]]], ". 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One of the \ disadvantages of \[Sigma] is that it measure the global volatility of either \ the return or the loss. While left-hand tail of the loss distribution is of \ interest for shareholders in a pleasant way, the right-hand tail is a \ nightmare not only for the shareholders. Some decades ago this observation \ led financial analysts to the introduction of a new measure known as quantile \ to statisticians distinctively called Value-at-Risk. 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If the distribution function is increasing on the support of the \ respective distribution, then ", Cell[BoxData[ FormBox[ SuperscriptBox["F", RowBox[{"-", "1"}]], TraditionalForm]]], " is simply the inverse function of ", Cell[BoxData[ FormBox["F", TraditionalForm]]], ". For a specific value of ", Cell[BoxData[ FormBox["y", TraditionalForm]]], ", the value ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["F", RowBox[{"-", "1"}]], "(", "y", ")"}], TraditionalForm]]], " is called the ", Cell[BoxData[ FormBox["y", TraditionalForm]]], "-", StyleBox["quantile", FontSlant->"Italic"], ". The ", Cell[BoxData[ FormBox["\[Alpha]", TraditionalForm]]], "-quantile of ", Cell[BoxData[ FormBox["L", TraditionalForm]]], " " }], "Text", CellChangeTimes->{{3.433149243203125*^9, 3.43314933571875*^9}, { 3.4331494058125*^9, 3.433149476109375*^9}, {3.433149525234375*^9, 3.4331496399375*^9}, {3.4331496731875*^9, 3.43314969953125*^9}, { 3.433149934328125*^9, 3.43314999575*^9}, 3.43315003053125*^9}], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["VaR", "\[Alpha]"], "(", "L", ")"}], ":=", RowBox[{ SubscriptBox["VaR", "\[Alpha]"], "=", RowBox[{ SuperscriptBox["F", RowBox[{"-", "1"}]], "(", "\[Alpha]", ")"}]}]}], TraditionalForm]], "DisplayFormulaNumbered", CellChangeTimes->{{3.433156364640625*^9, 3.433156369109375*^9}}, TextAlignment->Center], Cell[TextData[{ "is called ", StyleBox["Value-at-Risk at level ", FontSlant->"Italic"], Cell[BoxData[ FormBox[ StyleBox["\[Alpha]", FontSlant->"Plain"], TraditionalForm]], FontSlant->"Italic"], ", shortly VaR. The probability that the loss exceeds ", Cell[BoxData[ FormBox[ SubscriptBox["VaR", "\[Alpha]"], TraditionalForm]]], " is ", Cell[BoxData[ FormBox[ RowBox[{"1", "-", "\[Alpha]"}], TraditionalForm]]], ". The typical values for ", Cell[BoxData[ FormBox["\[Alpha]", TraditionalForm]]], " are ", Cell[BoxData[ FormBox["0.99", TraditionalForm]]], " (Bankers Trust), ", Cell[BoxData[ FormBox["0.95", TraditionalForm]]], " (J P Morgan) or ", Cell[BoxData[ FormBox["0.954", TraditionalForm]]], " for Citibank. VaR is still today the most popular risk measure for market \ risk quantification. Mathematicians were thinking on the properties of VaR, \ however, and discovered that a good risk measure should fulfill some \ postulates which are summarized below." }], "Text", CellChangeTimes->{{3.43315003765625*^9, 3.433150081828125*^9}, { 3.4331501210625*^9, 3.43315012759375*^9}, 3.4331503525625*^9, { 3.43315089046875*^9, 3.433151115609375*^9}, {3.433151220515625*^9, 3.433151290046875*^9}, {3.43315132571875*^9, 3.43315132853125*^9}, { 3.433153670546875*^9, 3.433153879671875*^9}, {3.433153915671875*^9, 3.433153987453125*^9}, {3.433156376078125*^9, 3.43315637671875*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Coherent Risk Measure", "Subsubsection", CellChangeTimes->{{3.433151338625*^9, 3.43315135096875*^9}}], Cell[TextData[{ "A coherent risk measure is a real function \[ScriptR] on a space of random \ variables \[ScriptCapitalP] that for ", Cell[BoxData[ FormBox[ RowBox[{"X", ",", RowBox[{"Y", "\[Element]", "\[ScriptCapitalP]"}]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"a", "\[Element]", "\[DoubleStruckCapitalR]"}], TraditionalForm]]], " possesses the following properties: " }], "Text", CellChangeTimes->{{3.433151439421875*^9, 3.433151514*^9}, { 3.43315157440625*^9, 3.433151617375*^9}, {3.4331518960625*^9, 3.4331520369375*^9}, {3.433152214953125*^9, 3.43315247103125*^9}, { 3.433152530078125*^9, 3.433152719*^9}, {3.433152764453125*^9, 3.43315280434375*^9}, 3.43368506396875*^9}], Cell[BoxData[ RowBox[{ StyleBox["A1", FontSlant->"Italic"], " ", FormBox[ RowBox[{"X", "\[LessEqual]", RowBox[{"Y", " ", "almost", " ", RowBox[{"surely", " ", "\[DoubleLongRightArrow]", " ", RowBox[{"\[ScriptR]", "(", "X", ")"}]}]}], "\[LessSlantEqual]", RowBox[{ RowBox[{"\[ScriptR]", "(", "Y", ")"}], RowBox[{"(", StyleBox["Monotonicity", FontSlant->"Italic"], ")"}]}]}], TraditionalForm]}]], "DisplayFormula", CellChangeTimes->{{3.433152158453125*^9, 3.433152162109375*^9}, { 3.43315249284375*^9, 3.433152512109375*^9}}], Cell[BoxData[ RowBox[{ StyleBox["A2", FontSlant->"Italic"], " ", FormBox[ RowBox[{ RowBox[{"\[ScriptR]", "(", RowBox[{"X", "+", "Y"}], ")"}], "\[LessSlantEqual]", RowBox[{ RowBox[{"\[ScriptR]", "(", "X", ")"}], "+", RowBox[{ RowBox[{"\[ScriptR]", "(", "Y", ")"}], RowBox[{"(", StyleBox["Subaditivity", FontSlant->"Italic"], ")"}]}]}]}], TraditionalForm]}]], "DisplayFormula", CellChangeTimes->{{3.433152158453125*^9, 3.433152190421875*^9}, { 3.433152576109375*^9, 3.43315257809375*^9}}], Cell[BoxData[ RowBox[{ StyleBox["A3", FontSlant->"Italic"], " ", FormBox[ RowBox[{ RowBox[{"\[ScriptR]", "(", RowBox[{"a", " ", "X"}], ")"}], "=", RowBox[{"a", " ", RowBox[{"\[ScriptR]", "(", "X", ")"}], " ", RowBox[{"(", StyleBox[ RowBox[{"Positive", " ", "homogeneity"}], FontSlant->"Italic"], ")"}]}]}], TraditionalForm]}]], "DisplayFormula", CellChangeTimes->{{3.433152158453125*^9, 3.433152162109375*^9}, { 3.433152192296875*^9, 3.43315219296875*^9}, {3.433152740359375*^9, 3.433152742203125*^9}}], Cell[BoxData[ RowBox[{ StyleBox["A4", FontSlant->"Italic"], " ", RowBox[{ FormBox[ RowBox[{ RowBox[{"\[ScriptR]", "(", RowBox[{"a", "+", "X"}], ")"}], "=", RowBox[{"a", " ", "+", " ", RowBox[{ RowBox[{"\[ScriptR]", "(", "X", ")"}], " ", RowBox[{"(", RowBox[{ StyleBox[ RowBox[{"T", StyleBox["ranslation", FontSlant->"Italic"]}]], StyleBox[" ", FontSlant->"Italic"], StyleBox["equivariance", FontSlant->"Italic"]}], ")"}]}]}]}], TraditionalForm], "."}]}]], "DisplayFormula", CellChangeTimes->{{3.433152158453125*^9, 3.433152162109375*^9}, { 3.433152194953125*^9, 3.433152195765625*^9}, {3.433152820265625*^9, 3.433152822890625*^9}, {3.433152907765625*^9, 3.433152915375*^9}, 3.433152997890625*^9}], Cell[TextData[{ "The Var does not meet ", StyleBox["A2", FontSlant->"Italic"], " in general. Moreover, VaR does not say how large losses above this value \ may be. " }], "Text", CellChangeTimes->{{3.433153018203125*^9, 3.433153054859375*^9}, { 3.433155329703125*^9, 3.433155336578125*^9}, {3.433155378875*^9, 3.433155477453125*^9}, {3.433155536765625*^9, 3.433155543375*^9}, 3.43315581921875*^9, {3.433155851765625*^9, 3.43315590121875*^9}, { 3.43315601615625*^9, 3.433156073921875*^9}, {3.433156187078125*^9, 3.433156189625*^9}, {3.43315626334375*^9, 3.4331562660625*^9}, { 3.433156773765625*^9, 3.4331568244375*^9}, {3.433685100296875*^9, 3.433685103578125*^9}, {3.433685157140625*^9, 3.433685165875*^9}, { 3.43377486821875*^9, 3.433774870296875*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Conditional Value-at-Risk", "Subsubsection", CellChangeTimes->{{3.4331562275*^9, 3.433156238125*^9}}], Cell[TextData[{ "More intuitive measure of risk and even a coherent measure is the \ Conditional Value-at-Risk, also called Tail Conditional Expectation or \ Expected Shortfall. Note that if the loss distribution is not absolutely \ continuous, these definitions slightly differ. ", StyleBox["Conditional Value-at-Risk at level \[Alpha]", FontSlant->"Italic"], " is defined as" }], "Text", CellChangeTimes->{{3.4331563065*^9, 3.433156334765625*^9}, { 3.433156693265625*^9, 3.43315669859375*^9}, {3.433156735390625*^9, 3.433156758625*^9}, {3.433156849515625*^9, 3.43315698609375*^9}, { 3.433157064625*^9, 3.4331570730625*^9}, {3.433685173984375*^9, 3.433685184671875*^9}}], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["CVaR", "\[Alpha]"], "(", "L", ")"}], "=", RowBox[{ SubscriptBox["CVaR", "\[Alpha]"], "=", RowBox[{"E", RowBox[{ RowBox[{"{", RowBox[{"L", "|", RowBox[{"L", "\[GreaterEqual]", RowBox[{ SubscriptBox["VaR", "\[Alpha]"], "(", "L", ")"}]}]}], "}"}], "."}]}]}]}], TraditionalForm]], "DisplayFormulaNumbered", CellChangeTimes->{3.433157056046875*^9}, TextAlignment->Center], Cell[TextData[{ "Another advantage of this measure is that it says how large average loss \ may be after exceeding VaR. Note that even this measure is not a novelty. 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