Asset Pricing Solutions

Asset Pricing SolutionsStill missing 1624th November 20061Problem 1This is the most basic of asset pricing questions.1.1Part AYou can just remember the asset pricing equation from the lecture notes:σb,s (rs rf ) σs2 (rb rf )wb wsσb,s (rb rf ) σb2 (rs rf )and plug in the values given:0 1(.1)wb ws0 .5(.1)T 2wb 2wsWeights must sum to one, therefore:wb ws 12ws ws 11ws 3Or you can do it the long way if you forgot the formula. Start with the return of the portfoliorp (1 w)rb wrs (1 w)(.1) w(.10) .1And the variation of the portfolio:σp2 var((1 w)b ws) (1 w)2 σb2 w2 σs2 2σs,b (1 w)2 (.5) w2 (1) .5 w .5w2 w2 .5 w 1.5w2Now, minimize the variation subject to the above return:min σp2 1.5w2 w .5 s.t. rp .10L 1.5w2 w .5 λ(rp .1)1

The FOC with respect to w is:3w 1 0w 13Which is the same answer that we had from above. Note that the constraint doesn't enter in theFOC because regardless of the weights the return will always be .10.1.2Part BIn order to draw the picture in return-variance space, we must calculate the variance of the optimalmutual fund.313 11 13691 σp2 w2 w ( )2 222 33 218 18 1831 1Therefore, if we invest in 100% of the mutual fund, then we will be at a point (r, σ 2 ) ( 10, 3 ). If12we invest 100% in the riskless asset, then we will be at a point (r, σ ) ( 20 , 0). Draw a straightline between the two, which represents the return/variance tradeo for combinations of the mutualfund and the riskless asset. If there is an interior solution, the solution will occur at wherever thetangency of the line between the two assets and the indi erence curve. Otherwise, the optimalportfolio will be a corner solution of 100% mutual fund, or 100% riskless asset.2Problem 2Remember that in general, with two risky assets we have:rp (1 ws wb )rf ws rs wb rbσp2 wb2 σb2 ws2 σs2 2ws wb σs,bIn this case we have σs2 σb2 σ 2 and σs,b 0. So if we knowwbws α wb αws then:σpσp2 (1 α2 )ws2 σ 2 ws p(1 α2 )σAnd so:σpασp(rs rf ) p(rb rf )2(1 α )σ(1 α2 )σrp rf pIf we have positive cash balances then:wb0.42 ws0.63so: wb 23 ws and rp 0.03 σpq2(1 23 )σ(0.06) q2σ3 p2(1 32 )σ(0.04) 0.03 0.26σp 13σIf we were to spend all our income in bonds and stocks we would get ws σp 513 σ 0.72σ and rp 0.082. If we have negative cash balances then:wb0.21 ws0.4235 0.03 and wb 250.072σp.σand so:

so: wb 21 ws and rp 0.05 σpq2(1 12 )σ1σ2 p(0.04) q2(0.02) 0.05 (1 12 )σ0.1σp 5σ 0.05 0.045σp.σIf we were to spend all our income in bonds and stocks we would get ws 32 and wb 13 and so:σp 35 σ 0.74σ and rp 0.083.So what have we found out? Well, the slope of the rp line is larger when you are a lender thana borrower. Furthermore, when you stop lending at an interest rate lower and with less risk thanwhen you start borrow (the rst line is tangent to the CAPM e cient frontier at a lower point thanthe second line of portfolio, so there must be some points you are on the CAPM e cient frontier).3Problem 3We want to show thatcov(ki , p) nXwj cov(ki , kj )j 1where p nj 1 wj kj , or the whole portfolio; kj indexes assets, and wj are weights correspondingto the fraction of the portfolio each assets holds.Notice thatPcov(ki , p) cov(ki ,nXwj kj ) E(kinXwj ki kj ) E(ki )E(j 1nXwj kj ) j 1 nXwj kj ) E(ki )E(nXwj E(ki kj ) j 1wj [E(ki kj ) E(ki )E(kj )] j 14nXwj kj ) j 1j 1j 1 E(nXnXwj E(ki )E(kj )j 1nXwj cov(ki , kj )j 1Problem 4Let the payo to a portfolio beP nX1i 1nxi ,where xi is the payo to asset i. Thereforevar(P ) varnX1i 1n!xinX1 2 varxini 1 4.1! nn Xn1 X1 Xvar(x) cov(xi , xj ) in2 i 1n2 j 1 i6 j1n2 n2·nσ cov(xi , xj )n2n2Part AIf cov(xi , xj ) 0, thenvar(P ) σ2 lim var(P ) 0n n

4.2Part BIf cov(xi , xj ) σ 2 , thenvar(P ) 55.1n 2 n2 n 2 n2 2σ σ 2 σ σ 2 lim var(P ) σ 2n n2n2nProblem 5U ln(C)Case 1:Portfolio 1E(U ) 111ln(100) ln(200) ln(300) 5.202333Portfolio 2E(U ) 111ln(85) ln(248.5) ln(266.5) 5.181333There is a di erence of 0.405% between the two numbers, with the utility being higher the rstportfolio.5.2U C 0.001C 2Case 2:Portfolio 1E(U ) 111(100 .001(10, 000)) (200 .001(40, 000)) (300 .001(90, 000)) 333 1111(90) (160) (210) 1533333Portfolio 2E(U ) 1 1 1 85 .001 · 852 248.5 .001 · 248.52 266.5 .001 · 266.52 333111(85 7.225) (248.5 61.752) (266.5 71.022) 153.3335333There is a di erence of 0.0001%, with the utility being higher for second portfolio. However,as we will show in the next problem, the di erence only occurs because of the slight variation invariance between the two portfolios. If they had the exact same variance, then the agent would gainthe same level of utility from both bundles.6Problem 6Let the utility be quadratic, orU C αC 2Let there be n posible outcomes of consumption, where outcome Ci occurs with probability πi .ThennE(U ) Xi 1πi (Ci αC 2 ) E(C) αE(C 2 )

Since var(C) E(C 2 ) E(C)2 , we haveE(U ) E(C) αvar(C) αE(C)2Notice that E(U ) α 0, var(C)which implies that an increase in variance reduces utility. Also E(U ) 1 2αE(C) E(C) E(U )1If E(C) 2α, then E(C) 0, and the expected utility depends positively on the return. Remember,the assumption we made when we postulated quadratic utility was that it valid only for a rangewhere utility was increasing with C , or where U 1 2αC 0, C11or C 2α. If E(C) 2α, then at least one outcome of consumption has C invalid. Therefore, the result holds for all appropriate values of C .712α ,and therefore isProblem 7When ever people are identical, the equilibrium is one in which people just stick with their endowments. If one person wanted to trade, then everyone would want to trade in the same direction, andthere would be no one with whom to trade. Other than that, this is very much like a consumptionquestion with uncertainty.This is a one period model, so consumption endowment - assets purchased return on assetsPeople are endowed with one unit of A and one unit of B , so that: e pA pB . Individuals'budget constraint in the good state is: c pA A pB B e A B , and in the bad state isc p A A p B B e A.Since A pays one with certainty, we can make it the numeraire with price 1. In this case thebudget constraint in the good state can be re-written as c 1 pB (1 pB )B and in the badstate c 1 pB pB BExpected utility is then:1 (1 pB (1 pB )B)1 σ1 (1 pB pB B)1 σE(u) [] []21 σ21 σMaximize WRT B:11(1 pB (1 pB )B) σ (1 pB ) (1 pB pB B) σ ( pB ) 022All people are identical, the price must be set so that everyone demands one unit of B, the oneunit that they are endowed with:(1 pB (1 pB )) σ (1 pB ) (1 pB pB ) σ ( pB ) 02 σ (1 pB ) pB 0

(2 σ 1)pB 2 σ2 σ2 σ 1pB So the derivative of pB with respect to σ is: pB 2 σ ln 2(2 σ 1) 2 σ ( 2 σ ln 2) 0 σ(2 σ 1)2if σ 1 thenpB 123213The easiest way to see the e ect of raising σ , is to try σ 2,pB 1454 15Intuition: σ is a measure of a person's risk aversion. As σ increases, this person is more riskaverse and would be willing to pay more for a certainty asset, asset A. As a result the amountthat a person can sell their B for must decrease so that they cannot purchase any A. When σ 1,selling 3 B could get your one A. When σ 2, you have to sell 5 B to get one A.88.1Problem 7.5Part ASince everyone is identical, no one will trade in the bond and we will have ct 1 yt 1 . Supposethat an individual buys units of the asset in period t. The price is p, so she gives up p units ofconsumption. The marginal utility of consumption is U e αct e α cSo the price in utility is:M C pe αThe gain is times the marginal utility in period t 1:M B E e αct 1 eE( αct 1 ) 2 Var( αct 1 ) e α 1 Setting the marginal bene t and cost equal, we have pe α e α α2 σ 22 p eα2 σ 22α2 σ 22

8.2Part BNotice that α 2σ 0 p 1. The reason this occurs is because even though expected consumptionin period t 1 is the same as in t, the expected marginal utility is higher. To see this:2 2E(Ut0 ) e α0E(Ut 1) E(e αct 1 ) e αE(ct ) σ22 e α σ22 e αThis is just like precautionary saving. The bigger is σ 2 , the more uncertain is ct 1 , and thus higherexpected marginal utility. Higher α also raises p because it means more risk aversion and moresensitivity of U 00 to c.9Problem 7.6In this problem, we have1 U CctSo if you buy units, the cost in utility isMC pctThe bene t is ct 1 units of extra consumption, the value of which isct 1 E ct 1 U (ct 1 ) Ect 10 Setting cost and bene t equal10 p ctp ctProblem 8The optimal consumption is provided for us. This consumption satis es the FOC. We can use thisfact to solve for the price. People will buy the security when the marginal utility forgone when theypurchase it equals the expected marginal utility gain when they purchase it.u0 (100)p .5u0 (50) .5u0 (150) 2100 4 p .5(50 4 ) (150 4 )p 10041004 2 504 150424 504 24 504 42 5043 50416 8 81

11Problem 9Similar to problem 7, people are identical so the equilibrium will be where no one wants to trade,or in this case not save.For an individualc1 1 sbut there is uncertainty about the second period consumption:c2 1 s(1 r)2with probability 1/2, andc2 3/2 s(1 r)with probability 1/2.The utility maximization ismax E(u) ln(1 s) u1113ln( s(1 r)) ln( s(1 r))2222FOC:(1 r) 1(1 r) 3 0 11 s 2( 2 s(1 r)) 2( 2 s(1 r))In equilibrium there is no saving b/c aggregate consumption in one period equals aggregateoutput in that period. Therefore, 1 (1 r) 1 r 3 02( 12 )2( 2 ) 1 (1 r) 1 r 034(1 r) 13(1 r) 3/4r 14A negative interest rate. This makes sense as without the constraint that s 0, people wouldwant to save. A negative interest rate forces people to pay to save, making it a less attractivealternative, and forcing s 0.12Problem 10Both the Red and Green people will optimize their consumption treating the interest rate as anotherparameter.Red:FOC:U 0 (ct )1 r 0U (ct 1 )1 θ

c2 1 rc1c2 c1 (1 r)BC:c1 c2y2 y1 1 r1 r11 r2 r2c1 1 r2 rc1R 2(1 r)c1 c1 1 therefore,c2R 2 r2 r (1 r) 2(1 r)2Green:FOC:c2 1 rc1BC:c1 y2c2 y1 1 r1 rc1 c1 12c1 11c1G 2and1c2G (1 r)2Notice that the consumption choice of Greens does not depend on the interest rate. The interestrate must be set so that Greens can consume 1/2 in the rst period. Because is no storage, thetotal consumption in period one must be less than or equal to the total income. This is macro sowe ignore the less than part, and say that total consumption in period one total income in periodone. Since the populations are equal, the exact sizes are irrelevant.c1R c1G y1R y1G2 r1 1 12(1 r) 22 r 31 r2 r 3 3r

r 1/2This negative interest rate will force Red to borrow from Green, allowing green to consume inperiod 2.c1R 3/22(1/2) 3/2Red borrows 1/2 from Green (s1R 1 3/2 1/2).c1G 1/2c2R 3/22 3/4c2G 1/2(1/2) 1/413Problem 11Identical people, therefore in equilibrium everyone consumes their endowment.The returns of B and C are uncorrelated; there is just as likely a chance of them both returning 0, both returning 0, B returns while C doesn't, and C returns while B doesn't. If they wereperfectly correlated then either both would return 0 or neither would. If they were perfectlynegatively correlated, then one would return when the other didn't.This is an uncertainty problem with four di erent states of the world; it's expected utility time.L ln c1 1111ln A ln(A B) ln(A 2C) ln(A B 2C) 4444 λ(c1 pA A pB B pc C 1 pA pB pC ){E(u) -λ(expenditures-income-endowment)}Where capital letters denote the number of a given asset held.FOCs:WRT c11 λ 0c1WRT A:1111 λpA 04A 4(A B) 4(A 2C) 4(A B 2C)WRT B:

11 λpB 04(A B) 4(A B 2C)WRT C:22 λpc 04(A 2C) 4(A B 2C)Identical people consume endowment:c1 y1 1A B C 1Plug these values into the FOCsλ 11111 pA 04 4(2) 4(3) 4(4)pA 25/4811 pB 04(2) 4(4)pB 31611 pc 02(3) 2(4)pc 7/24And as expected pA pc pB .14Problem 12(A)Uncertainty, RBC, and asset pricing. The triple wammy!De ne: a amount of asset held; f amount of risk free asset held.She will choose the portfolio that maximizes her expected utility subject to her budget constraints.11max E(u) (ln cs ln ns ) (ln cr ln nr )22subject to