ch7风险厌恶

风险厌恶第7章QiangLiu,SchoolofFinance,SWUFE,Chengdu,China7.27.1边际效用递减为凹的则称，，及，，如果对于函数定义)()()1()()]1[(1][0)(7.1uyuxuyxuyxuQiangLiu,SchoolofFinance,SWUFE,Chengdu,China7.3定理7.1如果凸的连续偏好由（6.4）的期望效用函数表示，那么相应的效用函数u是凹的提示：考虑[c0;c1]=[x;0]6.4：U(c)=u(x)凸性偏好U：xy,0a1u(ax+[1-a]y)au(x)+(1-a)u(y)QiangLiu,SchoolofFinance,SWUFE,Chengdu,China7.4定理7.2偏好的凸性）消费的边际效用递减（：足性）消费的边际效用（不满：消费的直接效用：二阶可微，则如果凹函数00)(0)(uuuuuQiangLiu,SchoolofFinance,SWUFE,Chengdu,China7.57.2风险厌恶的定义QiangLiu,SchoolofFinance,SWUFE,Chengdu,China7.6风险厌恶的经济意义确定w，不确定w+gE(w)=E(w+g)在期望值相同的确定与不确定性收支之间，一个风险厌恶者总是选择确定收支QiangLiu,SchoolofFinance,SWUFE,Chengdu,China7.7风险厌恶的定义（续）定理7.3当且仅当u是（严格）凹函数时，参与者（严格）风险厌恶当偏好可以由期望效用表示时，凸性意味着风险厌恶（偏好凸，u凹，风险厌恶）QiangLiu,SchoolofFinance,SWUFE,Chengdu,China7.8定理7.3证明)())~(()]~([Jensen)()]1[()()1()(:)()()1()]1[(:)}1,();,]1{([~)1,0(0)(2121wugwEugwuEuwppwuwupwpuwupxwupxpwpuppxpxpgpxu不等式：对凹函数应用风险厌恶是凹的充分性：或风险厌恶构造一个公平游戏，设是凹的必要性：风险厌恶QiangLiu,SchoolofFinance,SWUFE,Chengdu,China7.97.3风险厌恶的度量风险赌博的等价确定值：弃的财富（保险费）：为消除风险而愿意放为求的风险溢价参与一个公平游戏所要定义-)()]~([7.4wugwuEQiangLiu,SchoolofFinance,SWUFE,Chengdu,China7.10小风险方差度量小风险的风险系数代表风险厌恶程度小风险的风险溢价：为风险小的赌博的取值范围很小时，称当随机变量定义]~[])()([21)()(]~)(~)()([)]~([~~7.5221gVarwuwuwuwugwugwuwuEgwuEggQiangLiu,SchoolofFinance,SWUFE,Chengdu,China7.11风险厌恶的度量)(1)(ancerisktoler)()()(Pratt-ArrowwAwTwuwuwA系数：）风险容忍（：度量（绝对风险厌恶）风险厌恶的QiangLiu,SchoolofFinance,SWUFE,Chengdu,China7.12相对风险厌恶relativeriskaversion：对总财富的相对大小()()()()()wuwuwRwwuwuwQiangLiu,SchoolofFinance,SWUFE,Chengdu,China7.137.4线性或风险中性u(w)=wA(w)=R(w)=0QiangLiu,SchoolofFinance,SWUFE,Chengdu,China7.14负指数：恒定绝对风险厌恶CARAawwRawAaewuaw)(,)(0,)(QiangLiu,SchoolofFinance,SWUFE,Chengdu,China7.15平方awawwRawawAwaa1)(,1)(),0[,0,)(21221QiangLiu,SchoolofFinance,SWUFE,Chengdu,China7.16幂指数：恒定相对风险厌恶CRRAwRwwAwwu)(,)(1,0,11)(1QiangLiu,SchoolofFinance,SWUFE,Chengdu,China7.17对数：恒定相对风险厌恶CRRAwRwwAwwu1)(,1)(ln)(QiangLiu,SchoolofFinance,SWUFE,Chengdu,China7.18双曲线绝对风险厌恶（HARA）1,01,0,0,1,1,1)(11dddddwdwAaa对数：幂指数：负指数：平方：风险中性：QiangLiu,SchoolofFinance,SWUFE,Chengdu,China7.19风险厌恶与财富绝对风险厌恶递增、减：IARA,DARA相对风险厌恶递增、减：IRRA,DRRAQiangLiu,SchoolofFinance,SWUFE,Chengdu,China7.207.5风险厌恶的比较QiangLiu,SchoolofFinance,SWUFE,Chengdu,China7.21定理7.4证明更风险厌恶比参与者21:)()()())(()])~([())]~(([)]~([)(:43)())(()),(()(:32)(),()(,)()(:21212122221111212122121wAwAwuwufgwuEfgwufEgwuEwuwuwufzuuzfwuwzwuzzuwzQiangLiu,SchoolofFinance,SWUFE,Chengdu,China7.227.6一阶风险厌恶aaaawwawuaawwawwawuwugwuEgwuaa)()()()()()()()()]~([)},();,{(~,0),(),()(2121212121212121：假设财富正好是不可微）（