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

1、-1In 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 mean-variance target All investors love FOSD,Risk aversion investors prefer SOSD-20.The St.Petersburg Paradox Game A:Flip a fa

2、ir coin,The payoff:If the head appears,you get$2.otherwise nothing.You would pay 1$for that game.-30.The St.Petersburg 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$2k How much would you be willing to pay for a chance to

3、play this game?Using an expected value,you should be willing to pay at least the expected value of the payoff from playing the game.What is the expected payoff?-40.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 .2k Expec

4、ted payoff=1+1+1+1 You would pay infinity to attend the game-51、utility function:function between consumption,or wealth and investors satisfied degree.:()U WU W-62、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 n

5、othing.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 (0)(1000)(1)(1000)UUU1000$(10)0(0)utilesutiles-72、How to construct utility function(binominal tree method)Step 3:repeat step2,and get different uti

6、lity of different losses and gains.such as:step4:plot wealth utility,and the utility curve is drawn.WealthUtility12(1000,2009:)(1000,3000:)UU:-83、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 Expect

7、ed Utility.Von Neumann-Morgenstern G EU(G)=U(100)+(1-)U(-100)In general expected utility function can be defined as followed:100$(10效 用)1:100$(10)dwwfwUPwUWEUii)()()()(-93、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,

8、0.5).(2)Whats the utility of the games expected value?(3)Which is bigger,EU(G)or UE(G)-104、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 uncertain outcome.A certain out

9、come brings greater utility to the risk aversioner than uncertain outcome,even though they have equal expected value.EE()=E()()(WEUWEU)(WEU)(WEUWWWW-114、Attitude toward risk:according to utility function Types of Attitude toward risk(1)Risk aversion:For bond,which gives you$2 interest,its price is 1

10、00.E()=2 For stocks,which gives you$3 dividend(P=50%),and$1 dividend(p=50%,its price is below 100.E()=2)()(WEUWEUWW-124、Attitude toward risk:according to utility function (2)Risk neutral:A certain outcome brings same utility to the risk neutral with uncertain outcome,if they have equal expected valu

11、e.EE()=E()()(WEUWEUWW-134、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()=E()()(WEUWEUWW-144、Attitude toward risk:according to utility function diagramFor almos

12、t every investor,he is a risk aversioner.U(w)0-155、Pricing of risk premium-Markowitz Risk Premium.(1)Certain Equivalent Wealth(CEW).If U(W*)is equal to W*is called CEW.(U(CEW)=)(Some certain value which has the same utility with an uncertain event is called CEW)(WEU)(WEU-16CEWCEW-175、What is Expecte

13、d Utility?Solve the St.Petersburg Paradox a=4-185、Pricing of risk premium-Markowitz Risk Premium.(2)Markowitz risk premium:the difference between EW and CEW；UE(W)EU(W)RPE(W)CEW-195、Pricing of risk premium-Markowitz Risk Premium.(2)Markowitz risk premium:the difference between E(W)and CEW.Example:inv

14、estors utility function is 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

15、(9.2)=9887.6,Risk Premium=E(W)-CEW=62.4$total RP CEW+50$-20Is insurance company risk lover?A little more different about risk Pure risk,loss 100%Speculative risk(Martin Halek&Joseph Eisenhauer)Speculative risk is binomial distributed for individual but normal for insurance company 对单个投资者而言，1/100可能的可

16、能遭受火灾，因此火灾是不确定事件、风险事件。设其期望收益与风险为(u,sigma)对保险公司而言，1万个投保人，基本上就是100次火灾，不确定性很少，发生火灾基本上就是必然的。其单位保单上的风险为sigma/100。1 million investors,std is almost 0.Binomial converges normal,and Sigma converges to 0.-21Is insurance company risk lover?你的家产为100万，去年火灾发生概率万分之一。你的期望收益为-100元。如果保费为90元，你都不愿支付，你是风险爱好者(投保后期望收益-90

17、元，风险为0元，不投保期望收益-100元，风险为1000元，因此投保的期望收益高但风险小）。绝大部分客户愿意支付100元以上的费用，人都是Risk aversion风险回避者。（或者说投资者确定性地花110元去避免一个预期亏100元的东西）.保险公司也是风险回避者，它们设计的保单110元，得毛利润10元。-22Behavior Finance:Sometimes people are risk lover Why does investor sell stock whose price goes up,and hold stock whose price goes down?Odean(199

18、8)。Suppose cost is 50，current is*，it may go up or down 5 with prob=0.5 If current is 55 Investor sells it,utility is U(5)If he holds it,utility is U(0)or U(10)If current is 45 Investor sells it,utility is U(-5)If he holds it,utility is U(0)or U(-10)When facing loss,he prefer uncertainty-236、Pricing

19、of risk premium-Pratt&Arrow Risk Premium(1)Deducing of Pratt&Arrow Risk Premium Suppose current wealth is w,denote future uncertain.usually probability of bad condition is very small.or.(is called risk neutral gamble)Pratt&Arrow define risk premium(RP)as followed:z%0Ezw%0Ez%z%)()()(RPWURPzEWUzWUE-24

20、6、Pricing of risk premium-Pratt&Arrow Risk PremiumCont.(1)Deducing of Pratt-Arrow Risk Premium right hand side(is expanded at W)=U(W)-RP*U(W)+terms of higher-order (A)left-hand side =(B)()()(RPWURPzEWUzWUERP221()higher-order term of 2E U WZ U WZ U WZ%221()()higher-order term of 2zzU wUwg-256、Pricing

21、 of risk premium-Pratt&Arrow Risk Premium by(A)and(B),for risk-aversion investors,U(w)0)()(212WUWURPz-266、Pricing of risk premium-Pratt&Arrow Risk Premium（2）Further measurement of risk aversion.（2.1）Absolute risk aversion:risk attitude when wealth changes(ARA),we define ARA=Given risk,for increasing

22、 function ARA of W,it means the investor will pay more risk premium when his wealth is increased.It is called Increasing ARAto keep RP constant,the second term of first equation will be lower.Even Wealth goes up,risky asset goes down Decreasing ARA，Constant ARA.)()(212WUWURPz)()(WUWU2z0dARAdw-276、Pr

23、icing of risk premium-Pratt&Arrow Risk Premium（2.2）Relative risk aversion We define RRA(Relative risk aversion)=Decreasing RRA when the ratio of risk to wealth is fixed,positive means risk premium increases when wealth increases.(Notice the condition).This type of people is called Increasing RRA.钱增加

24、后，股票持仓比例下降to keep RP constant,the second term of first equation will be lower.Decreasing RRA Constant RRA.)()(*212WUWUWWRPz)()(*WUWUWdWRRAd)(Wz2-286、Pricing of risk premium-Pratt&Arrow Risk Premium Example(The background is the same as before).investors utility function is U(W)=ln(W);current wealth$

25、10,000,he is faced potential losses(fire,earthquake),the loss is(-5000,0,0.01),how much would he pay for his risk under Pratt&Arrow?-296、Pricing of risk premium-Pratt&Arrow Risk Premium solution:z%0.01:50000.99:0()50E Z%2220.01 49500.99 50247500ZWWU1)(21)(WWU$275.12000,101*500,247*21)()(212WUWURPzHe

26、 would pay 62.28$-307,Property of Risk Aversion（a）Increasing ARA will be Increasing RRA Proof：（a）Suppose a investor with utility function，and he is a increasing ARA，then So （by RP=we know that ）WU,0)(dwARAd0)().(dwARAdwARAdwARAwd0212 ARAz0ARA-317,Property of Risk Aversion(b)Decreasing RRA belongs to

27、 Decreasing ARA Proof:Suppose a investor with utility function，and he is a decreasing RRA then Since So ,or,So the investor must decreasing ARA WU,0)(dwARAdwARA0ARA,0)(dwARAdw0)(dwARAd-328.Hyperbolic Absolute Risk Aversion(HARA)family-338.(HARA)family-349.Utility function and high moment-359.Utility

28、 function and high momentRight skew means r0,ln(W)investors love it-369.Utility function and high moment-3710.First-order stochastic dominance-3810.First-order stochastic dominance-3910.First-order stochastic dominance-4011.Second-order stochastic dominance F(.)second order stochastically dominates

29、G(.)if and only ifIf F second-order stochastically dominates G,and if F and G have the same mean,then F has a smaller variance than G.All risk-averse expected-utility maximizers prefer a second-order stochastically dominant lottery to a dominated lottery.00()(),for all yyG w dwF w dwy-4111.Second-or

30、der stochastic dominance-42FOSD&SOSD N(0,1)dominances N(0,2)?02sosd.xls-43测测你的风险容忍度（选择完后计算总分）你刚刚在电视游戏中获得一个大奖！你会选择哪一个？A元现金 B 的机会获得元 C，的机会获得元 D，的机会获得万元。有一个很好的投资机会，但是你得借钱。你会接受贷款吗？A绝对不会 B也许 C是的 你所在的公司是一家小型计算机公司。公司为融资要把股票卖给职工，公司管理层计划在三年后使公司上市，在上市之前，你不能出售手中的股票，也没有分红，但公司上市时，你的股票可能会翻倍，你会投资买多少钱买股票？A一点儿也不买 B半

31、年工资 C一年的工资 D两年工资。你住的单位公寓房要变成产权房。你可以花万元买下，或者单位给你万元了结。房子买后在市场上可卖万，但必须在个月以后，这其间你每个月得多开支元。A拿万元了事。B买后再卖。-445.5，你继承了一套万元的房子。房子很破旧了，但房子在市中心很可能升值。如果出租的话每个月净赚元，如果装修后再租的话每个月可赚元，但这得贷款。你会：A卖了。B出租。C装修后再租。6 你的朋友是地址勘探家，发现了一个油田，准备集资开发。成功的话利润可能是倍。成功的概率是。但如果是一个枯井的话大家将一无所获。A一点也不买。B半年工资 C一年的工资 D两年工资。7 你在一场赌博中输了元。A不再玩了。

32、B再拿元玩玩。C再拿元玩玩。D再拿元玩玩。8,你外出时：A，骑车宁愿多绕圈，也不愿意大拐弯。B，经常走左人行道。C，经常走右道？思考：设计一道题，通过人的处事态度推算对金融投资风险的容忍态度A-1,B-3,C-5,D-9;2A-1,B-2,C-3 3,A-1,B-2,C-3,D-5 4,A-1,B-2 5,A-1,B-2,C-3 6,A-1,B-2,C-3,D-5 7,A-1,B-2,C-4,D-6，8A-1，B-2，C-1-45Exercise Mr.Whites current wealth consist of his home,which is worth$50,000,and$20,

33、000 in savings which are earning 7%in savings accounts.His one year homeowners insurance is up for renewal,and he faces the following losses.Whites utility function is U(W)=ln(W).Which one should he select his insurance policy ,$40,000,or$30,000 or 50,000,no insurance?(hint:by maximize utility,inter

34、est of the insurance fee will also be considered).-46Exercise(AVL=expected value of insurer s loss).Assume that security returns are normally distributed.Compare portfolios A and B,using both first and second-order stochastic dominance:Case i:Var(A)Var(B),E(A)=E(B)Case ii:Var(A)=Var(B),E(A)E(B)Value of loss Prob.00.985,0000.0110,0000.00550,0000.005Amount of insurance Premium 30,00030+AVL1 40,00027+AVL250,00024+AVL3