(*^ ::[paletteColors = 128; automaticGrouping; magnification = 150; currentKernel; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e8, 24, "Times"; ; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e6, 18, "Times"; ; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, L1, e6, 14, "Times"; ; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, L1, a20, 18, "Times"; ; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, L1, a15, 14, "Times"; ; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, L1, a12, 12, "Times"; ; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = input, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L1, 12, "Courier"; ; fontset = output, output, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; ; fontset = message, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = print, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = info, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1, 12, "Courier"; ; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, L1, 10, "Times"; ; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = Left Header, nohscroll, cellOutline, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, L1, 12; fontset = Left Footer, cellOutline, blackBox, 12; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Courier"; ; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; next21StandardFontEncoding; ] :[font = title; inactive; preserveAspect; startGroup; ] Uncertainty :[font = subsubtitle; inactive; preserveAspect; ] Hal Varian January 1992 :[font = section; inactive; Cclosed; preserveAspect; startGroup; ] Risk Premium :[font = text; inactive; preserveAspect; ] A consumer has a constant relative risk aversion utility function with risk aversion of 3. 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? First we define the CRRA utility function: :[font = input; preserveAspect; ] u[wealth_,rho_] := (wealth^(1-rho))/(1-rho) :[font = text; inactive; preserveAspect; ] Call the risk premium pi. To find it, we need to solve the following equation: ;[s] 3:0,0;22,1;24,2;79,-1; 3:1,11,8,Times,0,12,0,0,0;1,10,8,Courier,1,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; ] rho=3 loss=500 FindRoot[u[5000-pi,rho] == .5*u[5000,rho] + .5*u[5000-loss,rho],{pi,500}] :[font = text; inactive; preserveAspect; ] Let's graph the risk premium as a function of the loss. :[font = input; preserveAspect; ] rho=3 plotrho3=Plot[5000 - (.5*5000^(1-rho) + .5*(5000-loss)^(1-rho))^(1/(1-rho)), {loss,0,2000}] :[font = text; inactive; preserveAspect; ] If the consumer is more risk averse the risk premium increases more rapidly :[font = input; preserveAspect; endGroup; ] rho=7 plotrho7=Plot[5000 - (.5*5000^(1-rho) + .5*(5000-loss)^(1-rho))^(1/(1-rho)), {loss,0,2000}] Show[{plotrho3,plotrho7}] :[font = section; inactive; Cclosed; preserveAspect; startGroup; ] A simple portfolio problem :[font = text; inactive; preserveAspect; ] Let's calculate how much a consumer would invest in a risky asset. Again, 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 a is ;[s] 3:0,0;266,1;267,2;270,-1; 3:1,11,8,Times,0,12,0,0,0;1,10,8,Courier,1,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; ] eu[a_] := .5*u[wealth+.7*a,rho] + .5u[wealth-.3*a,rho] :[font = text; inactive; preserveAspect; ] Fix wealth at 1000, say, and rho at 3, and plot this as a function of a. :[font = input; preserveAspect; ] wealth=1000 rho=5 Plot[Evaluate[eu[a]],{a,1,600}] :[font = text; inactive; preserveAspect; ] It looks like this consumer would like to invest about $200. Let's check: :[font = input; preserveAspect; ] FindMinimum[-eu[a],{a,200}] :[font = text; inactive; preserveAspect; endGroup; ] So the optimal investment is about $175. :[font = section; inactive; Cclosed; preserveAspect; startGroup; ] Mean-Variance Utility :[font = text; inactive; preserveAspect; endGroup; endGroup; ] Let's check the calculation in the text for Normally distributed wealth and constant absolute risk averse utility function. [To be completed...] ^*)