(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of 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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He currently has monthly income of 5,000 and has a 50 percent \ chance of incurring a loss of 500. How much would he be willing to pay to \ avoid this loss?\n\nFirst we define the CRRA utility function:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["u[wealth_,rho_] := (wealth^(1-rho))/(1-rho)"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Call the risk premium ", Evaluatable->False, AspectRatioFixed->True], StyleBox["pi", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[". To find it, we need to solve the following equation:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "rho=3\nloss=500\nFindRoot[u[5000-pi,rho] == .5*u[5000,rho] \n+ \ .5*u[5000-loss,rho],{pi,500}]"], "Input", AspectRatioFixed->True], Cell[TextData[ "Let's graph the risk premium as a function of the loss."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "rho=3\nplotrho3=Plot[5000 - (.5*5000^(1-rho) + \n\ .5*(5000-loss)^(1-rho))^(1/(1-rho)),\n{loss,0,2000}]"], "Input", AspectRatioFixed->True], Cell[TextData[ "If the consumer is more risk averse the risk premium increases more \ rapidly"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "rho=7\nplotrho7=Plot[5000 - (.5*5000^(1-rho) + \n\ .5*(5000-loss)^(1-rho))^(1/(1-rho)),\n{loss,0,2000}]\n\ Show[{plotrho3,plotrho7}]"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["A simple portfolio problem"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Let's calculate how much a consumer would invest in a risky asset.\nAgain, \ we assume a CRRA utility function and suppose that the risk asset has a 50 \ percent chance of a rate of return of 70% and a 50 percent chance of -30%. \ Utility as a function of the investment ", Evaluatable->False, AspectRatioFixed->True], StyleBox["a", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" is", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "eu[a_] := .5*u[wealth+.7*a,rho] + \n.5u[wealth-.3*a,rho]"], "Input", AspectRatioFixed->True], Cell[TextData[ "Fix wealth at 1000, say, and rho at 3, and plot this as a function of a."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["wealth=1000\nrho=5\nPlot[Evaluate[eu[a]],{a,1,600}]"], "Input", AspectRatioFixed->True], Cell[TextData[ "It looks like this consumer would like to invest about $200. Let's check:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["FindMinimum[-eu[a],{a,200}]"], "Input", AspectRatioFixed->True], Cell[TextData["So the optimal investment is about $175."], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Mean-Variance Utility"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Let's check the calculation in the text for Normally distributed wealth and \ constant absolute risk averse utility function. [To be completed...]"], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 640}, {0, 460}}, WindowToolbars->{}, CellGrouping->Manual, WindowSize->{520, 365}, WindowMargins->{{52, Automatic}, {30, Automatic}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, Magnification->1.5, MacintoshSystemPageSetup->"\<\ AVU/IFiQKFD000000V8nh09RAj0000000OXQ<097PXP0AP1Y06`0I@1^0642HSkP 0V97`0000001nR4@0TN2R000000000000000009R?^0000000000000000000000 00000000000000000000000000000000\>" ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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