LectureUtilityFunctionandPriceofRisk资产组合金融系研究生课程课件

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1、Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,.,-,#,-,1,In this chapter.,How to construct utility function,Definition of risk aversion,Markowitzs risk premium=EW-CEW,Pratt&Arrows RP&RRA(-WU/U),Why investors love right skew,EU and

2、mean-variance target,All investors love FOSD,Risk aversion investors prefer SOSD,-1In this chapter.How to con,-,2,0.The St.Petersburg Paradox,Game A:Flip a fair coin,The payoff:If the head appears,you get$2.otherwise nothing.,You would pay 1$for that game.,-20.The St.Petersburg Parado,-,3,0.The St.P

3、etersburg Paradox,The game,：,Flip a fair coin until the first head appears,The payoff:If the first head appears on the k-th flip,you get$2,k,How much would you be willing to pay for a chance to play this game?,Using an expected value,you should be willing to pay at least the expected value of the pa

4、yoff from playing the game.,What is the expected payoff?,-30.The St.Petersburg Parado,-,4,0.The St.Petersburg Paradox,Outcomes=“Head appears in toss#”:1 2 3 k,Probability head occurs on given toss 1/8 1/k,Payoff =2 4 8 .,2,k,Expected payoff=1+1+1+1,You would pay infinity to attend the game,-40.The S

5、t.Petersburg Paradox,-,5,1,、,utility function:,function between consumption,or wealth and investors satisfied degree.,-51、utility function:functi,-,6,2,、,How to construct utility function(binominal tree method),step 1:,we arbitrarily assign a utility of 10 utiles to a loss of$1000,and 0 to nothing.,

6、Step 2,:investors must decide the probability,which makes indifferent between 0 and gamble(1000,-1000:),or,For an example:if =0.6 then U(1000)=6.7,-62、How to construct utility f,-,7,2,、,How to construct utility function(binominal tree method),Step 3,:repeat step2,and get different utility of differe

7、nt losses and gains.such as:,step4:,plot wealth utility,and the utility curve is drawn.,Wealth,Utility,-72、How to construct utility f,-,8,3,、,What is Expected Utility?-utility of uncertain,Just like that expected value is a parameter of R.V.,utility of uncertain outcome can be measured by Expected U

8、tility.Von Neumann-Morgenstern,G,EU(G)=U(100)+(1-)U(-100),In general expected utility function can be defined as followed:,-83、What is Expected Utility?,-,9,3,、,What is Expected Utility?,Example.For an investor,U(w)=ln(w),his initial wealth is 500.(1)Whats the utility of a game G(300,-300,0.5).(2)Wh

9、ats the utility of the games expected value?(3)Which is bigger,EU(G)or UE(G),-93、What is Expected Utility?E,-,10,4,、,Attitude toward risk:according to utility function,Types of Attitude toward risk,(1)Risk aversion:,utility of certain outcome.Notice:E()is a constant,is random.,expected utility of un

10、certain outcome.,A certain outcome brings greater utility to the risk aversioner than uncertain outcome,even though they have equal expected value.EE()=E(),-104、Attitude toward risk:a,-,11,4,、,Attitude toward risk:according to utility function,Types of Attitude toward risk,(1)Risk aversion:,For bond

11、,which gives you$2 interest,its price is 100.,E()=2,For stocks,which gives you$3 dividend(P=50%),and$1 dividend(p=50%,its price is below 100.,E()=2,-114、Attitude toward risk:a,-,12,4,、,Attitude toward risk:according to utility function,(2)Risk neutral:,A certain outcome brings same utility to the ri

12、sk neutral with uncertain outcome,if they have equal expected value.EE()=E(),-124、Attitude toward risk:a,-,13,4,、,Attitude toward risk:according to utility function,(3)Risk lover:,A certain outcome brings less utility to the risk lover with uncertain outcome,even if they have equal expected value.EE

13、()=E(),-134、Attitude toward risk:a,-,14,4,、,Attitude toward risk:according to utility function diagram,For almost every investor,he is a risk aversioner.,U,(w)0,-144、Attitude toward risk:a,-,15,5,、,Pricing of risk premium-Markowitz Risk Premium.,(1)Certain Equivalent Wealth(CEW).,If U(W*)is equal to

14、 W*is called CEW.(U(CEW)=),(Some certain value which has the same utility with an uncertain event is called CEW),-155、Pricing of risk premium-,-,16,CEW,CEW,-16CEWCEW,-,17,5,、,What is Expected Utility?Solve t,he St.Petersburg Paradox,a=4,-175、What is Expected Utility?,-,18,5,、,Pricing of risk premium

15、-Markowitz Risk Premium.,(2)Markowitz risk premium,:,the difference between EW and CEW,；,UE(W)EU(W),RP,E(W),CEW,-185、Pricing of risk premium-,-,19,5,、,Pricing of risk premium-Markowitz Risk Premium.,(2)Markowitz risk premium,:the difference between E(W)and CEW.,Example:investors utility function is

16、U(W)=ln(W);current wealth$10,000,he is faced potential losses(fire,earthquake),the loss is(-5000,0,0.01),how much would he pay for his risk.,Solution:Expected Value of future wealth(5000,10,000,0.01)=0.01*5000+0.99*10,000=9950,ln(CEW)=0.01*ln 5000+0.99*ln 10,000=9.2,CEW=Exp(9.2)=9887.6,Risk Premium=E(W)-CEW=62.4$total RP CEW+50$,-195、Pricing of risk premium-,-,20,Is insurance company risk lover?,A little more different about risk,Pure risk,loss 100%,Speculative risk(Martin Halek&Joseph Eisenhaue