Empirical Asset Pricing Winter 2009 Final exam review problems.

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1 Empirical Asse Pricing Winer 009 John Cochrane Final exam review problems. These are a collecion of problems from old exams, which should guide your sudy and give you some sense of my syle. No, I m no going o provide wrien answers! In general, you do no need o memorize formulas. You should know he or 3 main poins of each paper we read. Sae he main poin of x and how i was documened is legiimae. You should also be able o recognize facs documened in papers we read (and when we alked abou hose in class). See he Fama - French quesions for good examples. 1. Consider a VAR represenaion of reurns and dividend growh wih wo righ hand variables, r +1 = a r z + b r dp + ε r,+1 (5) d +1 = a d z + b d dp + ε d,+1 dp = φ dp,dp dp 1 + φ dp,z z 1 + ε dp,+1 z = φ z,dp dp 1 + φ z,z z 1 + ε z,+1 (6) where as usual r represens log reurns, d log dividend growh, dp = d p log dividend yield, z is he exra forecasing variable, and all variables are de-meaned. (Assume ha r and d as well as furher lags of dp and z have zero coefficiens.) (a) If z is no presen, wha rough values do you expec o see for b r,b d and φ dp,dp in annual daa? (b) Wha resricions on he coefficiens a, b, φ of his VAR flow from ideniies? (c) Suppose he exra variable helps o forecas dividend growh, i.e. a d 6= 0. Wehavesome inuiion ha if a variable helps o forecas dividend growh, i should also help o forecas reurns. Is his rue in your sysem? If a d 6=0,doeshaimplyhaa r 6=0? (d) To invesigae his issue a bi furher, wrie a srucural sysem raher han he reduced form VAR, in which z only exiss o forecas dividend growh x +1 = φ x x + ε x+1 (7) z +1 = φ z z + ε z+1 E (r +1 )=x E ( d +1 )=z. (8) People in he economy observe boh x and z; weobservez, as well as prices, dividends, and reurns. Find he coefficiens in he VAR represenaion of he form (5)-(6) ha resuls from his srucural sysem. (e) If he VAR resuling from his sysem displays a d 6=0mus i also display a r 6=0? (f) Your VAR represenaion from par c may have some zeros or oher resricions no presen in (5)-(6). How would you modify he seup of (7)-(8) o remove hose resricions? (Jus wrie down he sysem you hink you need o use, don solve i.) Hin: In case you forgo, he reurn linearizaion (ignoring consans as usual) and associaed formulas are r +1 ρ (p +1 d +1 ) (p d )+ d +1.

2 X X p d E ρ j 1 d +j E ρ j 1 r +j ; ρ = j=1 j=1 j=1 X X r E 1 r (E E 1 ) d + ρ j 1 d +j j=1 1 1+D/P 0.96 ρ j 1 r +j. (a) (5) Wha is a reasonable approximae value for he regression coefficien of marke reurns on he dividend price raio in annual daa? Answer boh for log reurns, log dp, and for percen reurns on percen dp (b) (5) We said ha roughly speaking 100% of he variance of marke dividend yields comes from reurns and 0% from dividend growh, bu only roughly 60% of he variance of marke reurns comes from expeced reurns, wih 0% from dividend growh. How is his possible do dividends maer, or don hey? (Hin: Wha happens if d is iid?) (c) (5) Cochrane claims ha long run coefficiens b r /(1 ρφ) are more powerful ess of reurn predicabiliy. Bu you can bea maximum likelihood, and he ML esimae and es of b r is jus he OLS esimae. Which is righ? 3. (5) We sudied he following able, updaing Fama and Bliss s resuls r (n) +1 y(1) ³ a + b f (n 1 n) y (1) f (n 1 n) y (1) + ε +1 n a b σ(a) σ(b) R a b σ(a) σ(b) R = ³ y (1) + ε +1 a + b +n 1 y(1) = forecasing one year reurns forecasing one year raes on n-year bonds n years from now I highlighed he number Wha, exacly, is herefore equal o 0.15?(No need o prove, jus sae he answer. Be very careful where you pu your ns and your s. Feel free o draw a picure oo. Clarify any noaion you inven by defining i in erms of bond prices. ). (10) (a) Wha do Cochrane and Piazzesi mean by a one facor model of expeced reurns? (A few equaions are appropriae here.) How does his concep of facor relae o usual eigenvalue decomposiions? (b) Do Cochrane and Piazzesi find ha, saisically, bond expeced reurns follow a one-facor model? Explain how one migh es i and wheher i does or does no pass he es. (If you can remember wha we did, inven a new es. I is enough o say wha regression you would run or momen condiion you would look a you do no have o derive he covariance marix or es saisic.) 5

3 5. (5) Brand, Cochrane and Sana Clara say ha discoun facors are highly correlaed across counries. Bu people exhibi a lo of home counry bias, and sock markes are no ha well correlaed across counries. How do we resolve his apparen conradicion? (Hin: i migh be useful o hink abou wo counries wih uncorrelaed sock markes and a consan exchange rae) 6. (5) If you run cross-secional Fama-MacBeh regressions of average reurns on full-sample beas and loglinear funcions of he characerisics size and book/marke raio, which se of variables drives he oher ou? 7. (10) Sar wih he CAPM, R ei = α i + β i R em Now, le s consider adding anoher facor F, which is also an excess reurn (hml or smb for example) R ei = α i + β i R em + ε i + γ i F + ε i (a) Suppose ha he saisic for γ i is significan, for all i, he R of he regression improves, and E(F ) > 0 and also is saisically significan. Does ha mean we should adop his mulifacor model, i.e. ha we should describe average reurns by 8. (10) E R ei = αi + β i E (R em )+γ i E (F )? (b) Suppose he GRS es rejecs he second model, bu does no rejec he firs model. Does ha mean ha he pricing errors of he second model are larger? (a) Which ges beer reurns going forward, socks ha had grea pas growh in sales, or socks ha had poor pas growh in sales? Is his paern consisen wih some paern of beas? (b) If you sor socks ino winners ha wen up from year -5 o one year ago, and losers ha wen down from year -5 o one year ago, which ones do beer for he nex year? Is his consisen wih some paern of beas? (c) If we form a momenum porfolio, from socks ha did well las year, are he reurns on ha porfolio correlaed wih he reurns on value socks over he nex year? If value socks go up, do momenum socks end o go up, down, or remain he same? 9. (15) (a) Here s an idea: Companies should issue sock and inves when he cos of capial is low, meaning expeced reurns are low. Thus, porfolios of companies ha are repurchasing sock should have a lo higher reurns going forward han porfolios of companies ha are issuing sock. Does his idea work? (b) Wai a minue hose issuing companies have high sock prices and he repurchasers low sock prices. Surely he big issuers are growh socks and he repurchasers are value socks, so we are jus finding ha value socks have high average reurns? (c) If he answer o b is no, does his fac mean ha we need o form a new issues facor, a porfolio of all high issues firms minus low issues firms, and hen run facor models ha include his facor R ei = α i + b i rmrf + h i hml + a i iss + ε i? Explain exacly how an exra facor migh no be necessary, even if he answer o b is no, and wha regressions you would run o check. 6

4 10. (5) Your assignmen is o evaluae he CAPM using he FF 5 porfolios on poswar daa. One group member uses a pure ime-series regression. She repors ha he CAPM is lousy; he marke premium is posiive, bu he alphas are huge; some alphas are even bigger han he average excess reurns. The oher group member uses a cross-secional approach wih a free inercep. He repors ha no, he CAPM is doing fine. The alphas are reasonable, hough in his sample i seems he marke premium came ou negaive. Can boh of hese resuls happen, or did one of hem make a misake? If a misake, who made he misake? (Illusrae your answer wih an appropriae graph. Label he axes. 11. (10) (a) Lamon and Thaler hink Palm invesors are behaving irraionally. Wha should hese invesors have done wih heir money raher han buy Palm? (b) Name a leas wo piece of evidence ha Cochrane cies for he convenience yield" heory as opposed o he morons heory or he heory ha shor sales consrains means ha pessimiss can express heir views in explaining he high price of Palm over 3com. 1. (10) Wha does his picure represen? Be explici, wih equaions. 0 Unresriced Resriced (a) 7

5 forward rae mauriy Rank forward curves 1- by which provides he sronges signal of one year excess reurns on 5 year bonds i) according o Fama and Bliss regressions ii) according o Cochrane and Piazzesi s regressions. (There may be ies.) (A: FB look for slope, CP look for en shapes.) 13. The curren log yield on 1, and 3 year bonds is 0%, 15%, 10% an invered yield curve (a) Find curren log prices and forward raes. (b) Find he expeced one year reurn on and 3 year bonds, and he expeced one and wo year yields one year from now, 1. According o he expecaions hypohesis. According o Fama and Bliss. Simplify heir regression coefficiens o 1 or 0, as appropriae.( Hin: you can figure ou he expeced wo year yield one year from now from he expeced reurn on he hree year bond.) (c) Plo he expeced bond prices hrough ime in each case. (Your plo has ime on he x axis and bond price on he y axis. You do no have o find he FB pah for he 3 year bond pas ime 1). 1. Show ha a discoun facor linear in he marke reurn implies he CAPM m +1 = a br m +1 E(R ei )=β i E(R em ) (A: Sar wih 0=E(mR e ), and use he definiion of covariance. 0=E(m)E(R e )+cov(m, R e ),E(R e )= cov(m, R e )/E(m)...) 15. Do you expec ineres raes o be higher in good imes or bad imes? Back up your view wih an equaion and an explanaion. 8

6 16. An invesor lives for wo periods, ime 0 and ime 1. He has a uiliy funcion over consumpion c 0 in period zero and random consumpion c 1 in period 1 given by 1 h E (c c 0 ) (c c 1 ) i c is a parameer (number), c 0 and c 1 are consumpion in he firs and second periods of life. We learn from a deailed saisical analysis ha his consumpion follows a random walk, c 1 = c ; he random shock 1 is normally disribued wih mean 0 and variance σ. (A useful preliminary: As of ime zero, i.e., knowing c 0, wha is he mean and variance of c 1?) We observe consumpion a period 0, c 0.Iislesshanc ; c 0 <c. Your answers o he following quesions can conain c 0.Find he price a ime 0 (i.e. knowing c 0 ) of he following securiies. (a) A one period zero coupon bond. (You may assume zero inflaion if his worries you.) (b) A Sock, which pays a random dividend equal o c 1 and nohing hereafer. (c) 1. Is he sock price greaer or less han he price of a bond wih c 0 face value?. How and why does he price depend σ? 3. How and why does he price depend on c? In paricular, explain wha happens as c 0 ges closer and closer o c? 17. Verdelhan, Lusig and Roussanov formed 6 carry rade porfolios. The mean annual reurns on hese 6 porfolios (percen per year) are They performed an eigenvalue decomposiion of he covariance marix of hese reurns, QΛQ 0 = cov(r e,r e0 ) Here are heir resuls: Q = diag(λ) P λ i = Wha were heir porfolios? Wha do hese resuls mean? 18. (30) (This is from 3590, a bi harder han I am heading owards on his final, bu we did alk abou nonseparable uiliy, so i s useful) Le s hink abou how our asse pricing formulas would change if 9

7 we recognize ha he consumpion series we re using is durable. Assume a single durable good, so he represenaive invesor objecive is X E β j u(k +j )s.. k =(1 δ) k 1 + c j=0 c now represens durable good purchases. General hin: This problem does no require los of algebra. I used no more han 3 lines for each par. I srongly advise you o work i ou on he scrach paper a he end before answering i here! (a) (5) Sae he invesor s firs order condiions for buying an asse wih price p and payoff x +1. How is his equaion differen from he sandard nondurable case pu 0 (c )=E [βu 0 (c +1 )x +1 ]? (b) (10) Now assume a consan riskfree rae R f =1/β. Use he equaion for pricing he risk free rae o collapse he new erms, so you have an asse pricing equaion p = E(mx) expressed in erms of u 0 (k ) and u 0 (k +1 ). (Hins: 1) Do he δ =1case firs, hen show his soluion works for he δ<1 case. You do no have o prove his is he only soluion. ) If you re having rouble, sar wih quadraic uiliy, and hen generalize o arbirary u(k). ) (c) (7.5) In he case of power uiliy, express he discoun facor in erms of c +1 /c and a purchases/sock c /k raio. Suppose as in he Campbell/Cochrane model ha he variance of purchases growh σ(c +1 /c )is consan over ime, When does his model generae high risk premia in booms when purchases are high relaive o he sock of durables or in recessions when purchases are low relaive o he sock? (d) (7.5) Express he model in coninuous ime, Z max E 0 e ρ u(k )d dk = δk d + c d 0 assume c follows a diffusionprocessandpoweruiliyu 0 (k) =k γ. Assume your resuls from par a, b go hrough so Λ = e ρ u 0 (k ). By characerizing his discoun facor, do risk premia increase or decrease in his model relaive o he nondurable model? How migh you modify his coninuous-ime seup o generae he opposie resul (one senence)? 19. Show ha he asse pricing predicions of inernal vs. exernal habi models are he same for power uiliy, an AR(1) habi and linear echnology. 0. (A very simple version of Cochrane/Piazzesi) Le s modify he basic Vasicek erm srucure model, and see if we can accoun for Fama-Bliss regressions. The basic model has a consan marke price of risk. We need o have a ime-varying price of risk. The obvious way o do ha is jus o make he price of risk depend on he single facor. So, le s pursue he obvious exension, in which raher han jus λ we have a ime-varying λ = λ 0 + λ 1 x, x +1 δ = ρ(x δ)+ε +1 log m +1 = x 1 (λ 0 + λ 1 x ) σ ε (λ 0 + λ 1 x ) ε +1 (a) Find p (1), p (), hence y (1),f (), rx () +1, E rx () +1 in his model. Hin: hey are sill linear funcions (suff)+ (suff) x! You have o use Ee x = e Ex+ 1 σx, exacly as we did in lecure. 30

8 (b) Find he prediced value of he Fama-Bliss coefficiens, i.e. wrie E rx () () +1 =( )+( )(f y (1) ). (All you re doing here is subsiuing ou he previous resuls. You had E rx () +1 = a + bx and (f () y (1) )=c + dx,soifyoujuswrie ³ (f () E rx () +1 = a + b y (1) ) c d E rx () +1 = a bc d + b () (f y (1) ) d your re done. ) Forge he mess in he consan, we re only ineresed in he coefficien,b/d. Can we find λ 0,λ 1 so ha his model capures he Fama-Bliss slope coefficien of approximaely 1? 1. More Paper quesions (a) Wha is Goyal and Welch s main complain abou reurn predicabiliy regressions? (b) Does he dog ha did no bark show anyhing wrong wih Goyal and Welch s calculaions, or does i admi hem bu couner in some oher way? (c) Cochrane and Piazzesi AER decisively rejec heir single-facor model. Ye hey ignore his rejecion and rumpe he single facor model as a grea success. Why? (d) Expeced reurns are always earned for covariance of reurns wih shocks. According o Cochrane and Piazzesi s Decomposing he yield curve wha is he imporan shock, covariance wih which drives expeced bond reurns? Do Lusig, Verdelhan and Roussanov find ha he same srucure works across counries? (e) Campbell and Cochrane claim ha hey produce imperfec correlaion beween consumpion growh and sock reurns. Ye heir model has a single shock doesn his mean every variable has o be perfecly correlaed? 31