A three mutual fund separation theorem

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1 A three mutual fund separaton theorem Fernando Alvarez and Andy Atkeson February 5, 08 Abstract We analyze a one perod economy wth CARA preferences, and normally dstrbuted aggregate rsk. We allow an arbtrary dstrbuton of dogmatc belefs on the expected value and varance of the aggregate rsk, as well as on the agent s rsk tolerance parameter. We show that there s a representatve agent whose rsk tolerance, and belefs over expected value and varance are approprately defned weghted averages of the same objects across all agents. We show a type of 3 fund separaton theorem: any effcent allocaton can be decentralzed wth three asset: an uncontngent bond, a Lucas s tree, and a smple varance swap whose prce s gven by the SVIX ndex, an ndex closely related to VIX, gven by the the prce of a portfolo of out of the money puts and calls. We gve smple expressons for the relatve prces of these securtes n terms of the representatve agent s parameters. We also gve smple expressons for each agent s holdng on the Lucas tree and the smple varance swap as functon of the agent s dfferences wth the representatve agent s values. The man novelty of the setup s the explct role for a smple varance swap or the portfolo of puts and calls n the decentralzaton of effcent allocatons. In equlbrum there are dfferent postons n the the varance swap or to the optons portfolo f and only f there s heterogenety n the belefs of the aggregate rsk of the economy, and hence these heterogenety maps nto trade volume. To understand the role of dfferent assumpton we also study the case of arbtrary belef and utlty functon for a small nose. Fnally, we extend the results to a multperod settng, for whch they hold condtonally on the hstory for each perod.

2 Introducton To be completed. Start wth an extended abstract wth summary of results.. Summary of the lterature revew on K mutual fund separaton results, to put our result n perspectve. 3. Summary of what a smple varance swap, portfolo of optons and SVIX. Relatonshp wth varance swap, and VIX, to put result n perspectve. 4. Dscusson of dfference of the results n the lterature: dfference on belefs, and role of a varance swap. In partcular, comparson wth other 3 fund separaton results. 5. Dscuss small nose approxmaton. 6. Dscuss extenson to dynamc set up. 7. Dscuss lmtatons of the results: dogmatc prors, CARA-Normal set up, dogmatc prors as opposed to ratonal expectatons, fast learnng of varance n dynamc case. Lterature Revew on K Mutual Fund Separaton Results The orgnal contrbutons s by Tobn 958, who show that mean-varance nvestors, whch have access to a rsk free return, can be restrcted to chose between two portfolos. The lterature has, roughly speakng, developed n two man branches to fnd suffcent condtons or to produce characterzatons for K mutual fund separaton. One branch studes the problem of fndng class of utlty functons whch produce separaton for arbtrary returns and levels of nvestors wealth, and the other branch studes the class of returns dstrbuton so that separaton occurs for arbtrary concave utlty functons. Among the orgnal contrbuton on suffcent condtons on the set of returns so that any rsk averse nvestor are ndfferent to select K mutual fund s Samuelson 967, and on the necessary and suffcent condtons s Ross 978. On the characterzaton of the class of utlty functon so that for any dstrbuton of returns and any level of wealth nvestors are ndfferent to select among K mutual funds there s the semnal paper by Cass and Stgltz 970 and subsequent work, such as results on Chen, Pelsser, and Vellekoop 0, and most notably the recent contrbutons by Dybvg and Lu 06. Other dstnctons are also present n the lterature. One s whether the set of asset for whch the nvestor can choose s complete or not. For nstance, Black 97 shows that wth mean varance preferences, two fund separaton obtans even n ncomplete markets. Yet, most of the lterature studes the problem wth complete markets. Another dstncton made n the lterature s whether a rsk free asset s ether avalable among the set of assets n the case of ncomplete market, or whether one of the K mutual funds s a rsk free assets. There s a handful of 3 mutual fund separaton results. One such result s due to Merton 973 n hs ntertemporal CAPM model, where the changng nature of the nvestment set are at the heart of the result. Two mportant cases of 3 mutual fund whch do not rely on a

3 changng nvestment set, are agents wth the SAHARA and GOBI utlty functon as proposed Chen, Pelsser, and Vellekoop 0 and Dybvg and Lu 06 respectvely. Fnally, and more closely related to our paper, are the results by Chab-Yo, Detmar, and Renault 04 and especally by Chab-Yo, Ghysels, and Erc 008. In both paper the authors consder general utlty functons, n the sense that they use analytcs expanson around small nose, and n both papers they consder heterogenety among nvestors. They gve a characterzaton of the equlbrum rsk-premum and a mutual fund separaton result, where n a specal case there are three funds. In partcular, Chab-Yo, Ghysels, and Erc 008 s even closer to ours n that the authors consder heterogenety n nvestor s belefs. Yet the type of heterogenety, and the features of dstrbuton of returns hghlghted ts skewness, are dfferent from the ones we emphasze n ths paper, as dscuss below. Also the mutual fund they consder do not nclude a varance swap. Thus, we vew the results as complementary to ours. Merton s 973 analyzes the dynamc consumpton and portfolo problem of an nvestor who faces assets whose prces follow dffusons, and whose nstantaneous expected return and varance also follow dffusons. He shows that n the case that the nterest rate follows a dffuson a 3 mutual fund separaton result for any strctly concave and ncreasng perod utlty functon. The mutual funds are the nstantaneous rsk-free asset, a mean varance effcent portfolo, and a hedgng portfolo n ths case one wth nnovatons perfectly negatvely correlated to the ones n short term nterest rates. The contnuos tme set up and the assumpton of dffusons are both central to the result. Ths s n contrast wth our result whch can be set up n a one perod model, and hence cannot be drven by hedgng demands. Dybvg and Lu 06 characterze the three mutual fund separaton that occurs n the one perod case wth SAHARA and GOBI utlty functons. In the SAHARA utlty functon, the three mutual funds are: a rsk-less bond, and two rsky assets, one equal to the dscount factor rase to a power, and the other equal to the recprocal of the frst. In the GOBI utlty functon, the three mutual funds are: a rskless bond, and two rsky assets, one s a power of the dscount factor, and the other s the square of the frst. The cases of SAHARA and GOBI preferences are dfferent from the preference we consder, and also, as explaned, do not produce a separatng mutual fund wth payoffs smlar to a varance swap. The set-up n both Chab-Yo, Ghysels, and Erc 008 and Chab-Yo, Detmar, and Renault 04 allows agent to have dfference utlty functons, and the ones n Chab-Yo, Ghysels, and Erc 008 also allow for dfferent belefs on the moments of the return to each asset. The authors fnd expressons for expansons as the functon of the nose, whch are carred out up to 3rd or 5th order. By the nature of the parameterzaton that the authors use on the nose, nvestor can only have dfferent belefs on the rsk premum,.e. the expected return of each asset relatve to the rsk-free asset. In ths set-up, skewness and hyper-skewness of the returns.e. lack of symmetry of returns s allowed, as well as dfferent fatness of tals.e. dfferent degree of excess kurtoss. The authors fnd that separatng portfolos are of three types: frst, as n the tradtonal theory, a market portfolo and a rsk-free bond. Second, a portfolo that weghts the co-skewness of the ndvdual portfolos so t depends on the lack of symmetry of the returns, and ts covaraton wth the market The nose s a parameter that multples the random component of the return, so when t s set to zero the model s determnstc. The authors work out expansons of equlbrum asset prces and portfolos as functons of the nose. In partcular the authors work out Taylor expanson of the nose, up to ffth order, for the excess expected return and for the separatng portfolos. 3

4 portfolo. Thrd a set of portfolos that span the dfference n belefs on the expected returns of the assets. The authors explan that n the gaussan case such as ours, due to the symmetry and proportonalty of ts even moments, many of the notable effects hghlghted n ther paper dsappear. In partcular, a varance swap s not one of the mutual funds that are requred n ther set up. Furthermore, heterogenety n varances s not allowed n ther set-up, andt s the central onenours. Fnally, they work out the case of ncomplete market.e. they take as gven a fnte number of assets, whle we consder a case where wth three funds one can decentralze the complete market allocaton. On the other hand, as explaned, ther set-up s otherwse more general n preferences and n the dstrbuton of assets. Thus, ths s the sense n whch we vew the result of our paper as complementary to thers snce we have normal aggregate uncertanty and allow for heterogenety on the belefs on varances. 3 Varance Swaps, VIX and SVIX In our one perod model, one the 3 mutual funds that suffce to support the complete market allocaton are a rsk-free bond, a market portfolo, and a smple varance swap on the market portfolo. A one perod smple varance swap n a securty s a contract where one part agrees to pay a prce before the realzaton of uncertanty and n exchange receves the square of the realzed smple return n that securty. The cash flow and the prce of the smple varance swap s equal to the cash flow and the prce of an equally weghted average of out of the money of calls and puts. Indeed the SVIX ndex s computed usng the prce of all avalable out of the money calls and puts. The SVIX ndex s closely related, but dfferent, from the VIX ndex, see Martn 0,03. Whle there s no trade n the VIX ndex per se, there are actvely traded futures markets on the VIX. Furthermore, the VIX ndex s the most commonly used market based measure of volatlty. We brefly outlne the dfference between the VIX, SVIX, as well as between ther assocated varance and smple varance swaps. Under the condtons stated below, the value of the VIX ndex s equal to the arbtrage free value of a varance swap. A varance swap of s contract where a seller at ts ncepton date promses to pay the square of the log.e. contnuously compounded daly returns of an underlyng asset from ncepton untl maturty, and n exchange the seller receves at maturty a fxed amount, whch we also refer to as the value or prce of the varance swap. Yet varance swaps, those whose prce s closely related to the VIX, dffer from smple varance swaps, those used n the model as a mutual fund and whose prce s closely related to the SVIX. In partcular, a varance swap uses the square of the daly log returns, whle the smple varance swap uses the square of the daly smple returns, where smple returns are defned as the dfference n prce of the underlyng securty over consecutve days, scaled, essentally, by the prce of the underlyng asset at ncepton. 3 The smple varance swap was proposed, ts propertes were characterzed, and ts synthetc We also allow heterogenety on rsk-tolerance and belefs on expected values, but they are not relevant for the demand of a varance swap, unless there s heterogenety on the belefs of the varance of the aggregate shocks. 3 It s actually dvded by the forward prce at ncepton wth maturty at the day returns are measured. For nstance, f nterest are zero and there are no dvdends the forward prce at ncepton s smply the prce of the underlyng asset at ncepton. Wth constant postve nterest rates t wll be proportonal 4

5 hstoral prces analyzed and compared wth the prce of standard varance swap n Martn 0,03. The prce of a varance swap s equal to the value of the VIX ndex under addtonal condtons, detaled below. The current value of VIX ndex s defned as, essentally, a weghted average of the current prces of the out of the money calls and puts, wth the SP500 as underlyng, all of them wth the same maturty, and weghted by the recprocal of the square of the strke prces of the optons. Takng the underlyng asset to be the SP500, and the current date as the ncepton date of the varance swap, the value of the VIX ndex should equal the arbtrage free prce of the varance swap under the followng assumptons: there are no arbtrage opportuntes and no transacton cost, the returns are measured at nfntesmal ntervals of tme as opposed to daly, there s a contnuum of optons wth all postve values for the strkes as opposed to a fnte number of strkes, v nterest rates and dvdend yelds are constant, and v the process for the underlyng follows a dffuson, perhaps wth random expected returns and volatlty, but wth no jumps. Lkewse, the smple varance swap can also be prced usng, essentally, an equally weghted average of the out of the money puts and calls for all postve strkes wth the same maturty as the varance swap and over the the same underlyng a quantty that Martn named and SVIX. Martn 0,03 uses prces of traded optons across strkes to approxmate the SVIX, n the same way that the VIX approxmated by such fnte sum. The condtons for ths equalty hold are weaker than the ones for the equalty between the VIX and the prce of the varance swap, n partcular t does not requre that prces has no jumps, see Martn 0,03. Fnally, the one perod model uses, naturally, the prce of a market portfolo wth dvdend on a sngle date, and also a one perod smple varance swap, whle n both traded varance swaps, and the VIX and SVIX ndces are defned usng mult-perod daly returns. In the general multperod verson of the model our 3 fund separaton theorem uses a one perod bond, a one perod dvdend strp on the market portfolo and a one perod varance swap n the one perod dvdend strp of the market. Yet, under certan crcumstances for nstance, when aggregate consumpton s a random walk, we extend our 3 fund separaton theorem to use a multperod setup were we use a one perod bond, the market return, and a multperod smple varance swap on the market return as separatng mutual funds. 4 Statc Model 4. Set up The economy has two perods, t = 0 and t =. In perod t = 0 agents trade fnancal assets whose payoffs are realzed n perod t =. There are I types of agents, each type denoted by, wth measure µ 0. Total output at tme t = takes values y n the set Y R. A type agent has belefs π : Y R, where π s the densty of the dstrbuton over values of y Y. A type agent has utlty functon u : X R, where X R. We assume that all u are strctly ncreasng and concave. Thus, agent s use expected utlty gven by: u c yπ ydy y Y 5

6 where c : X R s the consumpton of agent type, contngent on aggregate output to be y. We are not gong to work wth ths level of generalty on utlty functons and belefs, but t s useful to defned object at ths level of generalty to compare wth other results n the lterature. We wll specalze nto the case defne as CARA-Normal-Heterogenous economes. CARA-Normal-Heterogeneous Economes. Our results are for the specal case of economes whch have CARA utlty functon wth rsk tolerance parameter τ, so that u x = expx/τ, and belefs π about y Y = R to be normally dstrbuted wth expected value ȳ and varance σ > 0. Thus a CARA-Normal-Heterogeneous economy s parametrzed by θ R 4I contanng θ = τ,ȳ,σ µ R 4 + for all =,...,I and wth µ =. Feasble allocatons. An allocaton {c } I = s feasblty f the mpled aggregate consumpton at tme t = equals total output for each realzaton of aggregate output: y = I c yµ for all y Y. = Complete Market Equlbrum. We consder a complete set of tme t = 0 Arrow securtes,.e. agents can buy a unt of consumpton c y, contngent to the realzaton of y. Arrow prces are gven by p : Y R, so that py has the nterpretaton of the tme t = 0 prce of one unt of consumpton at tme t = contngent on y. We normalze prces so that the prce of a unt n all states of nature s,.e.: pydy =. 3 y Y We denote by the endowment of agent of a securty that pays e y when y s realzed. Thus agent s problem s: u c yπ ydy s.t pyc y e ydy = 0. 4 max c y Y We requre agent s endowments {e } to be feasble for ths economy,.e.: y Y I e yµ = y for all y Y. 5 = K-fund separaton. Fx an economy {u,π,µ } wth agent s endowment of securtes e {e }, and consder the allocaton of a compettve equlbrum c { c }. We say that there s a K mutual separaton f there are K funds: f k : Y R and K holdngs for each agent type αk such that K k=0 α kf k y = c y for all y Y and all =,...,I. 6

7 Moreover, K should be the smallest nteger for whch ths property holds. In words, the equlbrum allocatons can be obtaned by a lnear combnaton of K random varables, or portfolos. The holdngs α s can depend on characterstcs of the agent, but the random varables or portfolos f s can t. Ths property can also be defned for a set of economes {u,π,µ } or for a set of securty endowments e, or both. 4. Results We work out the condtonally effcent allocatons, shows that equlbrum prces can be obtaned by readng the margnal utlty of a representatve agent, and that the effcent allocaton can be decentralzed wth three assets: a rsk-free bond, a Lucas tree and a one perod varance swap whch we refer to as the VIX. In ths sense there s a three fund separaton theorem. Proposton. Take a CARA-Normal-Heterogeneous economy wth arbtrary endowments e = {e }. Let p be the Arrow-Debreu prce of contngent clam. Ths prce concdes wth one for an economy wth only one agent wth the average rsk tolerance, and belefs about the expected value and varance ȳ, σ gven by the followng weghted averages of the parameter for each agent: τ µ 6 µ τ µ j τ jµ j for all =,...,I 7 σ ȳ The Arrow prces are gven by: σ τ σ τ j j σj µ = ȳ = τ σ j µ and 8 τ j µ j σ j ȳ τ µ σ 9 py = e y ȳ σ / σ π σ for all y R, 0 and the equlbrum prces concde wth the ones of an economy where there s only one agent wth utlty functon ū and belefs π : ūc = expc/ and πy Nȳ, σ. Comments. The trplet,ȳ, σ are dfferent type of weghted averages.. The representatve agent rsk tolerance s the populaton average of the agent s rsk tolerance {τ }. 7

8 . The sem-elastcty of the prcng functon p y/py s equal to: p y py = y ȳ σ. 3. The representatve agent varance σ s the weghed harmonc mean of the agents varances {σ }. The weghts of each precson are drectly proportonal to populaton and rsk tolerant of each type. 4. The representatve agent varance σ s ndependent of a proportonal change n the rsk tolerance for all agents. 5. The representatve agent expected value ȳ s the weghed average of the agents expected values ȳ. The weghts are drectly proportonal to populaton dvded by the product of agent s type rsk averson and agent s varance. 6. The representatve agent expected value ȳ s ndependent of a proportonal change to rsk tolerance for all agents. The representatve agent expected value ȳ s also ndependent of a proportonal change to varance of all all agents. Market Portfolo Lucas s tree, Smple Varance Swap, and an Opton s portfolo. We defne as the market portfolo or Lucas tree an asset that pays y at t = when the aggregate output s y, wth prce F at tme t = 0. Wth our normalzaton, F has the unts of a forward prce, snce t s relatve to the prce of a rsk-less bond. We defne a smple varance swap as an asset that pays y F at t = when the aggregate output s y. We denote the tme t = 0 prce of ths asset as V, agan n terms of an asset that pays n each state of nature. In terms of the defnton of the notaton for K mutual fund theorem, we can set: f 0 y =, f y = y, and f y = y F. Note that the thrd fund, s defned n terms of the market prce of the second fund. Ths makes s closer to a smple varance swap, an asset traded n actual markets, but dfferent from typcal defntons of the mutual fund separaton theorem. Clearly, the funds are not unquely defned. We can take any other quadratc functon of y. We return now the prcng of the two key assets n our economy. Corollary. Usng the expresson from p n equaton 0 from Proposton we obtan that the prce of the Market and Varance Swap are: F V ypydy σ = ȳ pydy and y F pydy pydy = σ.. F, the forward prce of the Lucas tree, s gven by the representatve agent expected value µ by the rsk premum σ /. 8

9 . V, the prce of the varance swap, s gven by the representatve agent s varance σ, so t there s no drect rsk adjustment, except that the defnton of ts payoffs uses F, whch tself s mpacted by the rsk premum. 3. Any economy wth dfferent dstrbuton of τ,ȳ,σ,µ but wth the same,ȳ, σ has the same prces F,V. We now relate the one perod varance swap to the cash-flow of a portfolo of a portfolo of out of the money puts and calls, and ts prce, the SVIX. Let callk and putk be the prce at tme t = 0, n terms of one unt of consumpton at date t = of a call and put wth strke prce K,.e.: callk = We have: max{y K,0}pydy pydy and putk = max{k y,0}pydy pydy 3 Proposton. An equally weghted portfolo wth of each of the out of the money puts and calls.e. puts wth all the strkes K < F and calls wth all the strkes K > F, has the same payoff at t = than the smple varance swap y F, and thus has a prce at t = 0 equal to V: F V = putkdk + callkdk. 4 F We have few comments on ths proposton.. Ths result mples that agents can ether have access to a one perod smple varance swap, or equvalently to a portfolo of puts and calls on the market portfolo.. The Proposton s very closely related to Result 4 n Martn 0,03. There are mnor dfferences n the set-ups, such as that we have a only perod model and that y s unbounded from below, whch requres some modfcatons on hs proof. 3. Martn 0,03 uses an approxmaton of the expresson n Proposton usng fnte many strke prces and observed optons prces on the SP500 to compute emprcal counterparts of V, whch he named SVIX. He also compares t wth VIX. The next result gves smple expresson for the equlbrum portfolos. Proposton 3. Three Fund Separaton. The complete market allocaton can be decentralzed usng three assets: an uncontngent bond, the market portfolo n terms of rsk-less bonds a Lucas s tree, and a one-perod smple varance swap on the Lucas s tree or equvalently access to a portfolo of optons wth all strkes. We denote agent s holdngs of the Lucas s tree by α, and her holdngs of the smple varance swap by α. They are gven by: α = τ +ȳ ȳ τ σ α = σ σ τ σ σ σ σ + σ τ, for all =,...,I, 5, for all =,...,I. 6 9

10 The holdng of the uncontngent bond depend on the ntal α depend on the parameters as well as ndvdual endowments. Few comments are n order.. The representatve agent holds one unt of the Lucas s tree,.e. α = and no exposure to the smple varance swap of,.e. α = 0.. For all dstrbutons of the trplets τ,y,σ we have that averagng across agent s types: α µ = and α µ = If an agent beleves that y has hgher expected value than the representatve agent,.e. f ȳ > ȳ, she wll buy more unts of the Lucas s tree,.e. α wll be hgher. How much hgher wll depend on ths agent rsk tolerance relatve to perceved varance,.e. relatve to τ /σ. 4. If the agent s more rsk tolerant than the representatve agent,.e. f τ >, then she wll buy more unts of the Lucas s tree,.e. α wll be hgher. 5. If the agent beleves the economy s more volatle than the representatve agent,.e. f σ > σ, then ths agent wll adjust both her holdngs of the Lucas s tree and of the VIX. She wll hold fewer unts of the Lucas tree,.e. α wll be lower, but she wll buy the varance swap,.e. α > 0. The senstvty of these adjustment depends on the relatve varance, as well as on the agent s rsk tolerance. 6. If there sno heterogenety nthevarances σ, thefrst two funds suffce,.e. there sno need to dfferent postons on the varance swap, even f heterogenety n rsk-tolerance. 7. We have not wrtten an expresson for the portfolo of uncontngent bonds α because t depends on the ntal wealth of each agent. Agents wll hgher wealth wll hold more uncontngent bonds. On the other hand, the wealth of each agent does not affect the demand for Lucas trees nor the demands for varance swap. Trade volume. Under certan addtonal condtons Proposton 3 has sharp mplcatons on trade volume. Essentally, trade volume s gven by the mean absolute devaton on the cross sectonal heterogenety. In partcular, suppose that the ntal portfolos of agents are all the same, and hence they must be equal to the ones of the representatve agent. To be concrete, every agent start wth one unt of the Lucas tree, no rsk-free bonds, and no poston on the varance swaps. In ths case, the dfference between the optmal portfolo n Proposton 3 and the representatve agent s portfolo gves each agent s trade. Hence trade volume for the Lucas s tree denoted by TV and the trade volume n the varance swap denoted by TV 3 can be readly computed by takng the dfference n the absolute value 0

11 between the expresson n equaton 5 and equaton 6 respectvely: I TV = τ σ τ = σ +ȳ σ ȳ σ + σ ˆµ σ 7 TV 3 = I σ σ σ σ ˆµ where 8 ˆµ = σ τ σ = µ so that I ˆµ =. 9 = The expressons for both trade values use ˆµ as weghts on the dfferent types, whch are fractons of agents µ, reweghted by the rato τ /σ. Trade volume of the Lucas s tree TV has three components, correspondng to dfferences n rsk-tolerance, belefs on the expected value of the market portfolo, and belefs on the varance of the market portfolo. In the case n whch there s heterogenety n only one of these three dmensons, trade volume s proportonal to the mean absolute value n each of the parameters. The trade volume on the varance swap TV 3 depends only on the heterogenety on the belefs on the varance of the aggregate shocks, and t s proportonal to the mean absolute devaton of the varance belefs. The constant of proportonalty s the rato of the representatve agent s rsk-tolerance to the varance, so f agents are, n average, more rsk tolerant, the same dsperson on varance belefs generates more trade. Lkewse, an ncrease n the average varance belef, whle keepng the same dstrbuton say wth a multplcatve shft the varance belefs of each agent proportonally decreases trade volume n the varance swap. 5 Small Nose Approxmatons In ths secton nstead of usng a parametrc verson for preferences.e. CARA u and belef dstrbuton.e. normal y we develop a small nose approxmaton, or as termed by Samuelson 970 as compact dstrbutons. A smlar small nose approxmaton s also used, n a related but dfferent context, by Chab-Yo, Ghysels, and Erc 008 and Chab-Yo, Detmar, and Renault 04. We parameterze the belef dstrbutons by a common value σ > 0, and compute the lmt of the effcent allocatons as σ 0. The belefs are thus defned as: π y = σ π y ŷ a σ σ where > 0,σ > 0. 0 and where π s densty common to all agents. The nterpretaton of ths parametrzaton s that agent beleves that: y = ŷ +a σ +σ ǫ wth ǫ π forthesamerandomvarableǫ. Weassume that ǫπǫdǫ = 0and πǫdǫ = ǫ πǫdǫ =. Thus agent s belef for the expected value of y s ȳ = ŷ +a σ,

12 whch contans a common component ŷ, and an agent specfc component, a, whch scales wth σ. The varance of agent s belef for y s gven by σ = σ, where s the agent specfc component of ths varance. The parameterzaton of ȳ, as frst suggested by Samuelson s 970, ensures that as σ 0, the trade off of rsk and expected return for each agent are commensurate, so that t avods corner solutons. We study the shape of the effcent allocaton c y;σ as σ 0. In partcular, we are nterested n a second order approxmaton to c wth respect to y, evaluated at ŷ, as σ vanshes. For small σ we have: c y;σ c ŷ;σ+c ŷ;σy ŷ+ c ŷ;σy ŷ +o y ŷ We wll compare the values of c ŷ;σ and /c ŷ;σ for small σ wth those of α and α n the CARA-Normal case. We wll see that, f ether π 0 = 0 and π 0/π0 =, as t s for Normal varable, or f the product of the absolute rsk tolerance coeffcent to the absolute prudence coeffcent s one, as t s for a CARA utlty functon, then the expresson for α and /c ŷ;σ are the same. Also we wll see that, as n the case of CARA-Normal, dfferences n varances have a large mpact n c ŷ;σ, n the sense that c ŷ;σ /σ for small σ. For the next proposton we defne, abusng notaton, the absolute rsk tolerance and absolute prudence coeffcents: τ u c ŷ;0 u c ŷ;0 and κ u c ŷ;0 u c ŷ;0. 3 Usng the parameterzatons for σ = σ and ȳ = ŷ+σ a n the defntons of, ȳ and σ n equaton 6, equaton 9 and equaton 8 we have ȳ = ŷ +σ ȳ = ȳ +σ j j τ j µ j v j τ j µ j v j v = j τ jµ j a τ µ v 4 τ a µ v 5 τ µ, 6 We defne v, ˆv and ā as the harmonc mean of the normalzed varance and standard devaton, and the mean of the normalzed mean belef, whch are ndependent of σ, as follows: v k τ τ kµ k v µ 7 ˆv k τ τ µ and 8 kµ k ā k τ k µ k v k v a τ µ v 9

13 Thus, the followng expressons are ndependent of σ: ȳ ŷ σ = a τ µ v and ȳ ȳ σ = a ā v Wth these defntons at hand, we show that the result of the CARA-Normal case extend to symmetrc dstrbuton wth small nose. There are four cases of nterest, dependng on whether there s heterogenety on s and a s, and whether π 0 = 0 or not. We break t nto two propostons snce the most nterestng case for us s the one wth heterogenety on the belefs on varances. Proposton 4. Assume that v for some. If π 0/π0 = 0 : lm σ 0 c ŷ;σ = τ + τ lm σ 0 σ c ŷ;σ = τ v If π 0/π0 0 : lm σ 0 σc ŷ;σ = π 0 a ā π0 v v π 0 v π0 v π 0 τ π0 π lm σ 0 σ c ŷ;σ = 0 π0 π 0 π0 π 0 π ˆv v 33 ˆv { τ ˆv vk ˆv v κ k τ k κ τ }µ k τ v ˆv k v v v v k 34 We have several comments on the ths proposton. We frst comment on c.. For small σ, the expressons for c /σ. Ths s the same proportonalty factor than n the CARA-Normal case for α.. If the belefs have a symmetrc dstrbuton, then the densty evaluated a the mean value must be zero, π 0 = 0 and π 0 = If π 0 = 0, the expresson for c are the same regardless of the value of the ratos κ τ. 4. If the utlty functon s CARA then the product of prudence and rsk tolerance s one,.e. κ τ =. In ths case the expressons for c are the same regardless of the value of π If π s the densty of a Normal dstrbuton, then π 0/π0 = 0, and π 0/π0 =, so that the expressons for /c are dentcal to the values for α for the CARA- Normal case. For c we note that: 3

14 . For small TO BE FINISHED Now we state the results n the case of no heterogenety n the belefs about varances. Proposton 5. Assume that = v for all. If a ā for some : lm σ 0 c ŷ;σ = τ + τ v a ā lm ŷ;σ = v 3 σ 0 σc π 0 π0 If a = ā for all and σ : c ŷ;σ = τ c ŷ;σ = κ τ τ a ā π 0 π 0 π0 π0 τ τ π 0 π0 π 0 π0 π 0 π0 π 0 π κk τ k τk µk 38 k τ k 6 A Dynamc Verson In ths secton we extend the result to a multperod set-up. The result on the one perod model are clear, the asset used on t dffer from the actual market portfolo and varance swap n that n whle n the model they are based on one perod payoffs, the actual assets nvolve cash-flows on several perods. 6. Set up In ths secton we extend the results to a mult perod model. Tme s dscrete and runs t = 0,,...,T, where T can take the value +. Agents use dscounted expected utlty, wth possbly tme varyng dscount factor β t between perod t and t, common across agents, wth β 0 = and β t <. all t. The perod utlty for s state ndependent, but possbly heterogeneous, n partcular agents have specfc rsk tolerance τ. 4. As n the statc case, we consder a pure endowment case. We denote by y t the aggregate endowment a tme t. Hstores of aggregateendowment y t R t+ aredenoted by y t = y t,y t,...,y,y 0, whch solve the recurson y t = y t,y t. The economy starts at t = 0 where y 0 has been realzed. Each agent uses a densty π,t+ y t over hstores, wth the property that: π,t+ y t : Y R, wth π,t+ y t N ȳ,t+ y t, σ,t+ yt for all y t, and all t We can consder random rsk averson, whereda shocks to rsk averson have both aggregate and dosyncratc components, as done n Alvarez and Atkeson 07. In that paper we argue that n the presence of shocks to rsk averson t s more approprate to use non-expected utlty. 4

15 so the dstrbuton of y t+, condtonal on the hstory y t, s normal wth expected value and varance ȳ,t+ y t, σ,t+y t. Agent chose consumpton plans: c = {c,t y t } T t=0, for all wth y t R t, and for all t 0. Thus agent s objectve functon s: T t β s u c,t y t π,t dy t 40 y t Y t t=0 s=0 Feasblty s defned hstory by hstory as follows: I c,t y t µ = y t, for all y t R t and all t 0. 4 = To defne an Arrow-Debreu Compettve Equlbrum we ntroduce a prcng functon p = {p t y t } T t=0, for all wth yt R t, and for all t 0, wth the nterpretaton that p t y t s the prce of one unt of consumpton at date t, after hstory y t n terms of date t = 0 consumpton,.e. we have normalzed prces wth p 0 =. We wll use to three dfferent related prces. Frst we defne the prce of one unt of consumpton at date t+ contngent on hstory y t+,y t relatve to the one for the hstory y t. p t+ yt+ y t p t+y t,y t for all y t+ R t+ and all t, so 4 p t y t t p t+ y t+ = p s y s+ y s for all y t+ R t+ and all t 43 s=0 We also defne p t+ y t+ y t as the average condtonal prce,.e. the prce of one unt of consumpton at date t+ when y t+ s realzed, relatve to the prce of one unt at tme t+ for sure, both contngent on hstory y t, so that p t+ yt+ y t p t+y t,y t y p t+y,y t dy for all yt+ R t+ and all t, 44 Fnally, short term rsk-free bonds are defned as the prce of good at t+ across all realzaton of of y t+, n terms of unts of y t consumpton goods, both condton on hstory y t. b t y t y = p t+y,y t dy for all y t, and t 0 45 p t y t so that short term rates are R f ty t = /b t y t. Usng ths defnton we can relate the two condtonal prces as follows p t+ yt+ y t = b t y t p t+ y t+ y t 46 whch mples that: p t+ yt+ y t t = s=0 p s+ y s+ y s R f sy s 47 5

16 Another mportant prce s the one for an asset s a one perod market dvdend strp,.e. the prce of a clam to y t+ n terms of tme t consumpton, condtonal on hstory y t. We denote ths prce by S t y t and t s gven by: S t y t = yp t+ y y t dy 48 and F t y t s the one-perod forward prce of the one-perod market strp, so that F t y t = Rty f t S t y t = y p t+ y y t dy y We also let the market portfolo be an asset that at t promse payoffs {y s } T s=t+. We let the prce of ths asset be Q t y t, satsfyng: Q t y t = T s=t+ y p s ys y s...p t+ yt+ y t dy s...dy t+ 49 y s y t+ The prceof aoneperodvarance swap, onaone-perodmarket strps denotedby V t y t, and s gven by: 0 = y Ft y t Vt y t p t+ y y t dy 6. Results The results for the one perod economy extends mmedately condtonally to the dynamc case, approprately redefned. Proposton 6. Take a CARA-Normal-Heterogeneous economy wth arbtrary endowments e = {e }. Let p t+ y t be the average Arrow-Debreu condtonal prce of contngent clam at t+ condtonal on hstory y t. Ths prce concdes wth one for an economy wth only one agent wth the average rsk tolerance, and belefs about the expected value and varance ȳ t+ y t, σ t+ yt gven by the followng weghted averages of the parameter for each agent: τ µ 50 µ τ µ j τ jµ j for all =,...,I 5 σ t+y t ȳ t+ y t σ,t+ yt µ = τ σ,t+ yt j τ j σ j,t+ yt ȳ,t+ y t = τ σ,t+ yt µ and 5 j τ j µ j σ j,t+ yt ȳ,t+ y t τ µ σ,t+ yt 53 6

17 The Arrow average condtonal prces are gven by: p t+ y y t = e y ȳ t+ y t σ t+ yt / σ t+ yt π σ t+ y t for all y R, 54 and the equlbrum prces concde wth the ones of an economy where there s only one agent wth utlty functon ū and belefs π t+ y t : ūc = expc/ and π t+ y y t N ȳ t+ y t, σ t+y t. 55 Corollary. Usng the expresson from p t+ y t n equaton 54 from Proposton 6 we obtan that the forward prce of one-perod dvdend strp on the market portfolo and the one-perod Varance Swap are: ȳ t+ y t σ t+ yt F t y t = and Rty f t V t y t y Ft y t p t+ y y t dy = σ t+ yt. In ths case 3-mutual funds that suffce are also, condtonally, a rsk free bond, a dvdend strp of the market portfolo, and a one perod varance swap on the dvdend strp of the market portfolo. f 0,t+ yt+ y t =, f,t+ yt+ y t = y t+, and f,t+ yt+ y t = y t+ F t y t. 56 The 3 mutual funds are not unque. As we have seen n the statc model, and we wll repeat below, the varance swap can be replace by a portfolo of out of the money put and calls. Proposton 7. Condtonal three Fund Separaton. The complete market allocaton can be decentralzed usng three assets: an uncontngent rsk-less bond, a one forward contract on the a Lucas s tree, and a one-perod smple varance swap on the Lucas s tree. We denote agent s holdngs of the Lucas s tree by α, and her holdngs of the smple varance swap by. They are gven by: α α,t+ yt = τ + ȳ,t ȳ t+ y t τ σ,t+ yt + α,t+ yt = σ t+ y t σ,t+y t σ,t+ yt τ, for all =,...,I, 57 σ,t+ y t σ t+y t τ, for all =,...,I. 58 σ t+ yt σ,t+ yt Among the man dfferences between the one perod economy and the multperod economy are that wth at least two perods we can defne and analyze returns, and also that we 7

18 can consder securtes wth payoffs n multple perods. On the frst ssue, we can defne returns by combnng the prces descrbe n the one perod economy, whch are relatve to rsk-free securtes, and nterest rates. For ths purpose, we next defne and characterze the prce of one perod bonds b t y t at t that pays one unt of consumpton at t+ as follows b t y t ū y = β t ū y t π t+ y y t dy = β t e y t ȳ t+ yt e σ t+ yt Thus the log of gross nterest rates R t y t are gven by: y logr f ty t = logb t y t = logβ t + ȳt+y t y t σ t+ yt Let s denote by Put t,t+ K;y t and Call t,t+ K;y t the one perod call and put on the oneperod dvdend strp. These are prces tme t prces of puts and calls that have t+ payoffs. Put t,t+ K;y t = Call t,t+ K;y t = max{k y,0}p t+ y y t dy max{y K,0}p t+ y y t dy As n the statc case, we wll obtan that the cash-flow of one perod varance swap and a portfolo of puts and calls. Proposton 8. An equally weghted portfolo at tme t wth of each of the one perod out of the money puts and calls.e. puts wth all the strkes K < F t and calls wth all the strkes K > F t on the dvdend strp, has the same payoff at t+ than the smple varance swap wth payoffs y t+ F t, and thus has a prce at t equal to: Fty t V t y t = Rty f t Put t,t+ K;y t dk + Call t,t+ K;y t dk F ty t Note the small dfference wth the one perod model n that V t has an gross nterest rate. Ths s because for the dynamc verson we defne the puts and calls as traded, or wth prce at, tme t, and payoffs at tme t+. Random Walk Example. Let T =, and assume that all agents belefs π y t gven by y t+ = y t +m,t +σ,t ǫ t+ where ǫ t s..d normal 0,, and where m,t and σ,t are determnstc. Thus the representatve agent has belefs ȳ t+ = y t + m t and σ t+. The prce of the market portfolo s: Q t y t = φ t y t +κ t for two functons of tme {φ t,κ t }. We obtan: φ t = and κ t = φ t m t+ σ t+ +r /+ s s=t 8 s=t+ φ s m s+ σ s+/ s u=t +r u

19 In the case where m t = m, and σ t = σ are constant through tme, so aggregate consumpton s a random walk, then the net nterest r R f t = /βe m as well as φ and κ are constant throug tme: φ = r + r and κ = m σ / so that r Q t y t = y t + r r + m σ / r Note that n the case of a random walk, Q t s a lnear functon of y t, so a portfolo wth r number of share of the market portfolo and short n consol wth coupon r κ gve a payoff equal to {y t }. Thus, n ths dynamc economy, a multperod smple varance swap, and a smple portfolo on the aggregate portfolo wll gve the separatng mutual funds. Learnng Example. Suppose that each agent beleves that y t evolves accordng to: σ y t+ = m+y t +ω t+ where ω t as a tme nvarant but unknown varance σ. Suppose addtonally that each agent learnng s characterzed by two parameters σ,0 and k. The frst has the nterpretaton of agent s ntal expected value of her estmate of σ. The second has the nterpretaton of the precson of her ntal estmate. Then we assume that the agent update her estmate wth the observaton of y t as follows: σ,t+y t = t+k y t m y t + t+k t+k σ,ty t for all y t and t. 9

20 References Alvarez, Fernando and Andrew Atkeson. 07. Random Rsk Averson and Lqudty: a Model of Asset Prcng and Trade Volumes. Workng paper. Black, Fscher. 97. Captal Market Equlbrum wth Restrcted Borrowng. The Journal of Busness 45 3: Cass, Davd and Joseph E. Stgltz The structure of nvestor preferences and asset returns, and separablty n portfolo allocaton: A contrbuton to the pure theory of mutual funds. Journal of Economc Theory : 60. Chab-Yo, Foussen, Lesen Detmar, and Erc Renault. 04. Aggregaton of preferences for skewed asset returns. Journal of Economc Theory 54: Chab-Yo, Foussen, Erc Ghysels, and Renault Erc On Portfolo Separaton Theorems wth Heterogeneous Belefs and Atttudes towards Rsk. Workng paper. Chen, An, Antoon Pelsser, and Mchel Vellekoop. 0. Modelng non-monotone rsk averson usng SAHARA utlty functons. Journal of Economc Theory 46 5: Dybvg, Phlp and Fang Lu. 06. On nvestor preferences and mutual fund separaton. Tech. rep., Cornell Unversty, School of Hosptalty Admnstraton. Martn, Ian. 0,03. Smple Varance Swaps. Workng Paper 6884, Natonal Bureau of Economc Research. Merton, Robert C An Intertemporal Captal Asset Prcng Model. Econometrca 4 5: Ross, Stephen Mutual fund separaton n fnancal theory The separatng dstrbutons. Journal of Economc Theory 7 : Samuelson, Paul General Proof that Dversfcaton Pays. Journal of Fnancal and Quanttatve Analyss 0: 3. Samuelson, Paul A The Fundamental Approxmaton Theorem of Portfolo Analyss n terms of Means, Varances and Hgher Moments. The Revew of Economc Studes 37 4: Tobn, J Lqudty Preference as Behavor Towards Rsk. The Revew of Economc Studes 5 :

21 7 Appendx Lemma. Let { c } and p be a complete market compettve equlbrum gven endowments {e }, where prces are normalzed as n equaton 3. Then there s vector of weghts λ R I so that for each y Y: cy { c y} I = arg max x,...,x I I λ µ u x π y s.t.: = I x µ = y, 59 = and p s the Lagrange multpler of the feasblty constrant for y. Lkewse, a soluton of equaton 59 for a vector of weghts, corresponds to an equlbrum for some endowments {e }, Proof. of Lemma The economy satsfes local non-sataton, and hence the frst welfare theorem holds, and thus the equlbrum allocaton s Pareto Optmal. On the other hand, snce all u are strctly concave, there s a vector λ R I + so that the equlbrum allocaton solves I I λ µ u c yπ ydy + y µ c y pydy 60 = y Y On the other hand, gven the addtve separablty of expected utlty, the soluton must maxmzeequaton59. Forthesecondpart, weusethatthesolutonoftheproblemequaton59 for each y gves the suffcent condtons for the soluton of the problem equaton 60. Then, snce the preferences are strctly convex, by vrtue of the second welfare theorem we can fnd prces to decentralze the allocaton. Proof. of Proposton Consder the problem 59. The frst order condton for the planner s choce of c y s gven by: λ µ exp c y/τ exp y ȳ = py σ πσ where py s the multpler of the feasblty constrant correspondng to that value of y. Takng logs: y Y logλ +logµ c y/τ y ȳ σ = logπσ = logpy. Multplyng by τ µ, addng up across, and usng feasblty we have: logλ µ logπσ τ µ y µ y ȳ τ σ µ = logpy τ µ Defnng as τ µ

22 then py s: Note logpy = y ȳ = j σ j τ j µ j logλ µ τ µ logπσ y/ σ τ µ = y y = y ȳ σ ȳ σ + σ τ µ y y τ µ ȳ σ j σ σ j τ j µ j τ µ ȳ + σ τ µ ȳ + σ τ µ ȳ y ȳ σ σ τ µ τ µ ȳ where we defne µ µ τ j τ jµ j ȳ σ τ σµ ȳ µ j j τ j σ j µ σ Thus we can wrte: Thus logpy = logλ logπσ µ y/ y ȳ σ e y/ e y ȳ σ π σ σ py for all y R τ µ ȳ ȳ σ where use to mean that ts constant of proportonalty doe not depend on y. Note that ths s the margnal utlty of an agent wth,ȳ, σ that consumes the entre endowment,.e. t s the utlty of the representatve agent. We can rewrte ths expresson as: py e y/ e y ȳ σ π σ = e y yȳ σ /+ȳ σ π σ = e σ / ȳ σ / = e σ / ȳ σ / e y ȳ σ / σ π σ σ e y yȳ σ /+ȳ + σ / ȳ σ / σ σ e y ȳ σ / σ π σ π σ

23 so prces are proportonal to the pdf of a normal random varable wth expected value F = ȳ σ /, and wth varance σ. Proof. of Proposton Fx F and an arbtrary y. We frst show that an equally weghted portfolo of out of the money puts and calls as the same cash flow at t = that y F. We want to show that: F y F = max{k y,0}dk + max{y K,0}dK 6 To show ths we note that the value that the rght hand sde takes for two cases. If y F, then F F max{k y,0}dk = F y K ydk = y F and F max{y K,0}dK = 0 If y F, then max{y K,0}dK = y F F y KdK = y F and F max{k y,0}dk = 0 Thus the cash-flows n both sdes of equaton 6 are ndeed the same, and hence the market value y F at t = 0 s the same as an equally weghted portfolo of out of the money puts and calls,.e. the market value of the rght hand sde of equaton 6 s equal to F putkdk + callkdk. F Proof. of Proposton 3 Note that, by equatng the log of the frst order condton wth the expresson for the log of py, and multplyng t by τ, the optmal allocaton for c y can be wrtten as follows: logλ µ logπσ /τ c y y ȳ τ σ = a τ y τ y ȳ τ σ where a s a constant that does not depend on y. Solvng for c y: c y = logλ µ logπσ/ a τ +y τ + τ y ȳ σ y ȳ σ 3

24 and y ȳ σ y ȳ = y yȳ+ȳ y yȳ +ȳ σ σ σ = y σ ȳ ȳ y σ σ ȳ + σ σ ȳ σ = σ y F σ yf ȳ +F ȳ y + σ σ = σ σ σ σ ȳ y F y ȳ F + σ ȳ σ σ ȳ σ σ ȳ σ F σ F σ σ ȳ σ From the prevous expresson we can wrte the effcent consumpton as: c y = α + α y + α 3 y F. The drect exposure to y from the lnear term depends on the rsk tolerance, relatve to the average, just as n the case of common varance. To better match t wth the varance swap consder the one where the quadratc devaton s wth respect wth the market value of y, so that t reads y F, where F = ȳ σ /,.e., where the coeffcents satsfy: α = τ ȳ σ ȳ F σ σ F σ = τ ȳ F ȳ F τ σ σ = τ +ȳ ȳ τ σ σ τ + σ σ α3 = τ = σ σ σ σ σ τ Proof. of Proposton 4 and Proposton 5 The frst order condton for the planner problem s: λ u c y;σ y ŷ σ π a σ = py;σ 6 σ τ where λ s the planner weght on agent type. Evaluatng at ŷ, λ u c ŷ;σ a σ π = σpŷ;σ σ Takng σ 0: λ u c ŷ;0 π0 = lm σ 0 σpŷ;σ 4

25 Takn logs on equaton 6 log λ +logu c y;σ+logπ and dfferentatng wth respect to y: u c y;σ u c y;σ c y;σ+ σ y ŷ a σ = logσpy;σ σ y ŷ a σ σ y ŷ a σ π π σ = p y;σ py; σ Evaluatng the equaton for the frst dervatve at y = ŷ: π a σ u c ŷ;σ u c ŷ;σ c ŷ;σ+ σ π a σ = p ŷ;σ pŷ; σ 63 For future reference we also compute the second dervatve of c y;σ. We have: u c y;σ u c c y;σ y;σ+c y;σ d u c y;σ dy u c y;σ + π y ŷ a σ σ σ π y ŷ a σ σ y ŷ a π σ y ŷ a π σ σ σ = p y;σ p py;σ y;σ py; σ Evaluatng the equaton for the second dervatve at y = ŷ: u c ŷ;σ u c c ŷ;σ ŷ;σ+c ŷ;σ d u c ŷ;σ dy u c y;σ + π a σ σ π a σ a π σ a π σ = p ŷ;σ pŷ;σ p ŷ;σ pŷ; σ 64 Fnally we use feasblty to conclude that for all y and σ we have c y;σµ = and c y;σµ = 0 Case. Frst dervatve when π 0 = 0: 5

26 Takng the lmt as σ 0 n equaton 63 and usng L Hoptal, snce π 0 = 0 and π0 > 0, we get u c ŷ;0 u c ŷ;0 c ŷ;0 a π 0 π0 = lm p ŷ;σ σ 0 pŷ; σ Usng the defnton of τ then c a π 0 ŷ;0+τ π0 = τ p ŷ;σ lm σ 0 pŷ; σ Multplyng the prevous expresson by µ, addng up across and usng feasblty: or + π 0 π0 p ŷ;σ lm σ 0 pŷ;σ = π 0 π0 = π 0 π0 a p ŷ;σ τ µ = lm σ 0 pŷ; σ a τ µ ȳ ŷ τ µ σ = π 0 π0 Usng the defnton of σ and ȳ n equaton 8 and equaton 9: p ŷ;σ ŷ ȳ π 0 lm = σ 0 pŷ;σ σ π0 a σ σ Now, replacng on the expresson for c ŷ;0 we get: c ŷ;0 = τ +ŷ ȳ τ π 0 a π 0 τ σ π0 π0 = τ +ŷ ȳ τ π 0 + τ π a σ π0 σ σ 0 π0 = τ +ŷ ȳ τ π 0 + τ π 0 ȳ σ π0 σ ŷ π0 = τ + τ π 0 ȳ σ ȳ π0 τ µ Case. Frst dervatve when π 0 0, and v for some. Usng the defnton of τ nto frst dervatve, multplyng by µ, addng across, and multplyng both sdes by σ: c ŷ;σµ σ + τ σµ π π 6 a σ a σ = σ p ŷ;σ pŷ; σ τ σµ

27 Usng feasblty: σ + τ σµ π π a σ a σ = σ p ŷ;σ pŷ; σ τ σµ Takng the lmt as σ 0: p ŷ;σ lm σ 0 pŷ;σ σ = π 0 τ µ π0 v Replacng the expresson for lmp /p back nto the equaton for c : lmc ŷ;σσ + τ π 0 = τ τ µ π 0 σ 0 π0 π0 or or lm σ 0 c ŷ;σσ = lm σ 0 c ŷ;σσ = π 0 τ π0 π 0 τ ˆv = π0 j τ j µ j v j π 0 τ π0 Case 3. Frst dervatve when π 0 0, and = v for all. σ + τ σµ vσ σ ˆv v If = v for all then lm σ 0 c ŷ;σσ = 0, so t s consstent wth lm σ 0c ŷ;σ beng fnte. Indeed f = v we have: π a σ π a σ = p ŷ;σ pŷ; σ Insertng ths nto the equaton for c we have: c ŷ;σ+ τ π σ a σ vσ π a σ = τ σ p ŷ;σ pŷ;σ = τ σ σ + τ σ vσ v or Usng L Hoptal we get: c ŷ;σ = τ σ σ + τ σ vσ lm σ 0 c ŷ;σ = τ + τ v π a j a σ π a σ v j τ j µ j a j τ j σµ j σ π 0 π0 j ˆv τ j σµ j σ π a j σ v π a j σ v π 0 π0 π a j σ v π a j σ v 7

28 Hence, n the case where π 0 0 and v = for all the lmt for c ŷ;σ s fnte, and t depends on the dfferences on expected values. Case. Second dervatve when π 0 = 0 Multplyng both sdes of equaton 64 by σ we have: u c y;σ u c σ c y;σ y;σ+σ c y;σ d u c y;σ dy u c y;σ + π y ŷ a σ σ π y ŷ a σ σ π y ŷ a σ π y ŷ a σ σ σ = σ p y;σ p py;σ y;σ py; σ Evaluatng at y = ŷ: u c ŷ;σ u c ŷ;σ + σ c ŷ;σ+σ c ŷ;σ d u dy π a σ π a σ = σ p ŷ;σ py;σ π π p ŷ;σ pŷ; σ a σ a σ c ŷ;σ u c ŷ;σ Snce π 0 = 0 mples that lm σ 0 c ŷ;σ s fnte, then takng lmts we have: lm σ 0 σ c ŷ;σ = τ π 0 τ lmσ p ŷ;σ p π0 σ 0 py;σ ŷ;σ pŷ; σ Multplyng by µ addng up and usng feasblty: τ π 0 µ = lmσ π0 σ 0 or lm σ 0 σ p ŷ;σ py;σ p ŷ;σ = pŷ; σ p ŷ;σ py;σ π 0 π0 p ŷ;σ pŷ; σ τ µ = Replacng ths expresson nto the second dervatve of agent s we get: lm σ 0 σ c ŷ;σ = τ π 0 π 0 τ π0 π0 v π 0 = τ v π0 v π 0 τ v = v π0 v 8 v π 0 π0 v

29 Case. Second dervatve when π 0 0 and v for some. Multplyng equaton 64 by µ, addng up: + = c ŷ;σµ π a σ τ σ σ µ π p ŷ;σ pŷ;σ τ σµ c ŷ;σ d dy π a σ a π σ a σ p ŷ;σ pŷ; σ τ σ u c ŷ;σ u c y;σ Usng feasblty and multplyng both sdes by σ : c ŷ;στ d u c ŷ;σ σµ dy u c y;σ + τ σ v µ π a σ π a σ a π σ a π σ = p ŷ;σ pŷ;σ p ŷ;σ pŷ; σ σσ σ or Note that + = c ŷ;σ σ d u τ σµ c ŷ;σ dc u c y;σ τ σ µ π a σ π a σ a π σ a π σ p ŷ;σ pŷ;σ p ŷ;σ pŷ; σ τ σ d u c ŷ;σ dc u c y;σ σσ = κ στ σ τ σ where κ σ s the absolute prudence coeffcent,.e.: κ σ = u c ŷ;σ u c ŷ;σ. 9

30 so we can wrte: and from each : σ + τ j σ σ v j j = µ j c j ŷ;σ σ κj στ j σ τ j j σ π aj σ a v j π j σ v j aj σ aj σ π v j π v j p ŷ;σ pŷ;σ p ŷ;σ pŷ; σ σ µ j τ σ c ŷ;σσ c ŷ;σ σ κ στ σ τ σ + π a σ π a σ a π σ a π σ = p ŷ;σ pŷ;σ p ŷ;σ pŷ; σ σ Equatng both expressons we obtan: τ σ c ŷ;σσ = 65 c ŷ;σ σ κ στ σ τ σ + c j ŷ;σ σ κj στ j σ τ j j σ µ j + π a σ π a σ τ j σ π aj σ a v j a π σ a π σ σ v j j π j σ v j aj σ aj µ j σ π v j π v j Takng lmts as σ 0: lm σ 0 c j ŷ;σσ = lm σ 0 π 0 π0 c j ŷ;σσ τ κj τ j π 0 π0 τ j τ v τ j v v v lm σ 0 c ŷ;σσ κ τ τ µ j 30