Lecture 9 Cochrane Chapter 8 Conditioning information

Transcription

1 Lecture 9 Cochrane Chapter 8 Condtonng normaton β u'( c t+ Pt = Et xt+ or Pt = Et mt+ xt+ or Pt = E mt+ xt+ It u'( ct normaton at tme t I x t and m t are d Vt, then uncondtonal expectatons are the same as condtonal expectatons. But ths s not always the case. We could make explct assumptons about I t (dcult. Instead o modelng condtonal dstrbutons, we would lke to use uncondtonal moments. Ths chapter dentes what condtonal dstrbutons mply or uncondtonal moments. Condtonng down Pt = Et mt+ xt+ Pt = Et mt+ xt+ Ω E( Pt = E mt+ xt+ E( Pt IcΩ = E mt+ xt+ I Law o terated expectatons: I you take an expected value (EV usng less normaton o an EV usng more normaton, you get back the EV usng less normaton. E E ( x = E( x { t } ( = E E x E x t t t+ t t+ E Ex ( Ω IcΩ = E xi

2 Multply the payos by an nstrument z t zp= E m x z t t t t+ t+ t New payo E( zp = E( m x z E( z = E( m R z New prce t t t+ t+ t t t+ t+ t Invest n asset accordng to z t managed portolo e.g. SMB,HML Checkng the (uncondtonal expected prce/return o all managed portolos s sucent to check all the mplcatons o condtonng normaton. So to deal wth condtonal dstrbutons, add managed portolos and use uncondtonal moments. A condtonal actor model does not mply an uncondtonal actor model. m = b b such that m = b ' ' t+ t t+ t+ t+ ' ' t+ = βtλt β ERt+ = β λ ER ( such that ( I senstvtes are tme-varyng, t s not OK to assume they are constant. I a portolo s MV ecent wth respect to a condtonal dstrbuton, t does not mply that the portolo s MV ecent uncondtonally.

3 Cochrane Chapter 9 Factor models m = a+ b' E( R = α + β ' λ t+ t+ t+ What characterstcs should actors have? ( based on economc oundaton e.g. related to consumpton ( orecastng varables predct returns or macro varables (3 hghly (maybe not completely unpredctable Classc dervatons o CAPM ( Consumpton CAPM, sngle perod ( Quadratc utlty, arbtrary returns, sngle perod (3 Negatve exponental utlty (general utlty, normally dstrbuted returns, sngle perod (4 Quadratc value uncton/bellman equaton, arbtrary returns, multperod (dynamc programmng (5 Intertemporal CAPM. Consumpton CAPM We dd ths at the end o Penat/Pennacch s notes enttled State Preerence Theory t Assume U '( C + = γr m perectly negatvely correlated 3

4 . Quadratc utlty, arbtrary returns t * Uc ( t = β ( ct c * * Uc ( t, ct+ = ( ct c β E ( ct+ c m * βu'( ct+ ( ct+ c t+ = = β * u'( ct ( ct c c = W t+ t+ W = R ( W c w t+ t t t N N w t t = = R = wr ; w = So m w * * Rt+ ( wt ct c βc β ( wt ct w t + = β = + R * * * t+ ct c ct c ct c Ths s a sngle actor model. c 3. Negatve exponental utlty, normally dstrbuted returns uc ( = e α, α > 0 coecent o absolute rsk averson We looked at some characterstcs o portolos chosen by an nvestor wth constant ARA n Penat/Pennacch s notes enttled Rsk Averson and Portolo Choce. α α Ec ( + σ ( c t I c η E(, c σ ( c E u( c = e wealth W, R, rsky assets return R, var-cov y = wealth ($ n each asset c= y R + y' R end o perod consumpton W = y + y' budget constrant Eu(c = e α α y R + y' E( R + y' y Max y, y E u(c gves y = ER ( R α 4

5 Amount nvested n rsky assets ndependent o wealth. E( R R = α y = αcov( R, R where R = y R + y ' R m m ER ( R m σ ( R m = α 4. Quadratc Bellman equaton, multperod U = u( c + β E V( W t t t+ η ( Wt + W * * I you have a quadratc utlty uncton uc ( t = ( ct c, you get a quadratc Bellman, so the problem looks lke the usual sngle perod quadratc utlty case. 5. Intertemporal CAPM We wll do ths later ater we go through contnuous tme mathematcs. Concerns about these models ( condtonal or uncondtonal? In general, these are condtonal models unless the structure s constant through tme (e.g. nvestment opportunty set s d, utlty uncton s the same, etc.. ( Does CAPM prce optons? Generally no apples to assets wth normally dstrbuted payos. Optons on stocks do not have normally dstrbuted returns. Yes quadratc utlty. 5

6 Fnance 400 A. Penat - G. Pennacch Arbtrage Prcng Theory The noton o arbtrage s smple. It nvolves the possblty o gettng somethng or nothng whle havng no possblty o loss. More speccally, suppose that an asset portolo can be constructed wthout requrng any ntal wealth. I ths zero-net-nvestment portolo can sometmes produce a postve return, but can never produce a negatve return, then t represents an arbtrage: startng rom zero wealth, a prot can sometmes be made but a loss can never occur. Aspecal case o arbtrage would be ths zero-net-nvestment portolo produces a rskless return. I ths certan return s postve (negatve, an arbtrage s to buy (sell the portolo and reap a rskless prot or ree lunch. Only the return was zero would there be no arbtrage. An arbtrage opportunty can also be dened n a slghtly derent context. I a portolo that requres a non-zero ntal net nvestment s created such that t earns a certan rate o return, then ths rate o return must equal the current (compettve maket rsk ree nterest rate. Otherwse, there would also be an arbtrage opportunty. For example, the portolo requred a postve ntal nvestment but earned less than the rsk ree rate, an arbtrage would be to (short sell the portolo and nvest the proceeds at the rsk ree rate, thereby earnng a rskless prot equal to the derence between the rsk ree rate and the portolo s certan (lower rate o return. In ecent, compettve, asset markets, t seems reasonable to thnk that easy prots dervng rom arbtrage opportuntes are rare and leetng. Should an arbtrage opportunty temporarly exst, then tradng by nvestors to earn ths rskless prot wll tend to move asset prces n a drecton that elmnates the arbtrage. For example, a zero-net-nvestment portolo produces Ths would lkely nvolve some borrowng or short sellng o assets as well as long postons n assets. Arbtrage dened n ths context s really equvalent to the prevous denton o arbtrage. For example, a portolo requrng a postve ntal nvestment produces a certan rate o return n excess o the rskless rate, then an nvestor should be able to borrow the ntal unds needed to create ths portolo and pay an nterest rate on ths loan that equals the rsk-ree nterest rate. That the nvestor should be able to borrow at the rskless nterest rate can be seen rom the act that the portolo produces a return that s always sucent to repay the loan n ull, makng the borrowng rsk-ree. Hence, combnng ths ntal borrowng wth the non-zero portolo nvestment results n an arbtrage opportunty that requres zero ntal wealth.

7 a rskless postve return, as nvestors create (buy ths portolo, the prces o the assets n the portolo wll be bd up. The cost o creatng the portolo wll then exceed zero. The portolo s cost wll rse untl t equals the present value o the portolo s rskless return, thereby elmnatng the arbtrage opportunty. Hence, n compettve asset markets, t may be reasonable to assume that equlbrum asset prces are such that no arbtrage opportuntes exst. As wll be shown, by assumng the absence o arbtrage, powerul asset prcng results can oten be derved. An early use o the arbtrage prncple s the covered nterest party condton n oregn exchange markets. To llustrate, let F 0t be the current (date 0 t-perod orward prce o one unt o oregn exchange. What ths orward prce represents s the dollar prce to be pad t perods n the uture or delvery o one unt o oregn exchange t perods n the uture. Let S 0 be the spot prce o oregn exchange, that s, the current (date 0 dollar prce o one unt o oregn currency to be delvered mmedately. Also let r 0t be the rsk ree borrowng or lendng rate or dollars over the perod 0 to t, and denote as r the rsk ree borrowng or lendng rate 0t or the oregn currency over the perod 0 to t. 3 Next consder settng up the ollowng portolo whch requres zero net wealth. Frst, let us sell orward (take a short orward poston n one unt o oregn exchange at prce F 0t. 4 Snce we are now commtted to delver one unt o oregn exchange at date t, let us also purchase the present value o one unt o oregn currency, /( + r, and nvest t at the oregn rsk 0t t ree rate, r0t. In terms o the domestc currency, ths purchase costs S 0 /( + r0t t, whch we nance by borrowng dollars at the rate r 0t. At date t, our oregn currency nvestment yelds ( + r /( + 0t t r = unt o the oregn 0t t currency whch we then delver to satsy our short poston n the orward oregn exchange contract. For delverng the currency, we receve F 0t dollars. But we also now owe rom our dollar borrowng a sum o ( + r 0t t S 0 /( + r. Thus, our net proceeds are 0t t F 0t ( + r0t t S0/( + r 0t t 3 For example, the oregn currency s the Japanese yen, r0t would be the nterest rate or a yen-denomnated rsk-ree nvestment or loan. 4 Takng a long or short poston n a orward contract requres zero ntal wealth, as payment and delvery all occur at the uture date t.

8 Note that these net proceeds are a certan return, that s, ths amount s known at date 0 snce t depends only on prces and rskless rates quoted at date 0. I ths amount was postve, then we should ndeed create ths portolo as t represents an arbtrage. I, nstead, ths amount was negatve, then an arbtrage would be or us to sell ths portolo, that s, we reverse each trade dscussed above (take a long orward poston, nvest n the domestc currency nanced by borrowng n oregn currency markets. Thus, the only nstance n whch arbtrage would not occur would be the net proceeds were zero, that s, F 0t = S 0 ( + r 0t t / ( + r 0t t whch s reerred to as the covered nterest party condton. Note that the orward exchange rate, F 0t, s determned wthout knowledge o the utlty unctons o ndvduals or ther expectatons regardng uture values o oregn currency. For ths reason, prcng (valung assets or contracts by usng arguments that rule out the exstence o arbtrage opportuntes can be very appealng. To motvate how arbtrage prcng mght apply to a very smple verson o the CAPM, suppose that there s a rsk ree asset that returns R and multple rsky assets. However, t s assumed that only a sngle source o (market rsk determnes all rsky asset returns and that these returns can be expressed by the lnear relatonshp R = a + b ( where R s the return on the th asset, a s ths asset s expected return, that s, E R =a. Further, s the sngle rsk actor generatng all asset returns, where E = 0, and b s the senstvty o asset to ths rsk actor. b can be vewed as asset s beta coecent. Note that ths s a hghly smpled example n that all rsky assets are perectly correlated wth each other. Now suppose that a portolo o two assets s constructed, where a proporton o wealth o w s nvested n asset and the remanng proporton o ( w s nvested n asset j. Ths portolo s return s gven by 3

9 R p = wa +( wa j +wb +( wb j ( = w(a a j +a j +w(b b j +b j I the portolo weghts are chosen such that w = b j b j b (3 then the uncertan (random component o the portolo s return s elmnated. The absence o arbtrage then requres that R p = R,sothat R p =w (a a j +a j =R or b j (a a j + a b j j = R b whch mples a R b = a j R b j λ (4 Ths condton states that the expected return n excess o the rsk ree rate, per unt o rsk, must be equal or all assets, and we dene ths rato as λ. λ s the rsk premum per unt o the actor rsk. The denomnator, b, can be nterpreted as asset s quantty o rsk rom the sngle rsk actor, whle a R can be thought o as asset s compensaton or premum n terms o excess expected return gven to nvestors or holdng asset. Thus, ths no-arbtrage condton s lke a law o one prce n that the prce o rsk, λ, whch s the premum dvded by the quantty, must be the same or all assets. Suppose that asset m has the same degree o rsk as the actor, that s, b m =. Thus, rom the above equlbrum condton λ = a m R, or we nterpret asset m as the beta = 4

10 market portolo, then λ = R m R. In terms o asset, the equlbrum condton can then be wrtten as a = R + b λ (5 = R + b ( R m R whch s the CAPM relaton. Thus, the CAPM can be derved by assumng there s only a sngle lnear rsk actor and that ths rsk actor has the same rsk as the market portolo. Let us now generalze the arbtrage prcng prncple to the case o multple rsk actors and allow ndvdual asset returns to have dosyncratc components. Thus, let there be k rsk actors and N assets n the economy, where k<n. Let b z be the senstvty o the th asset to the z th rsk actor, gven by z. Also let ε be the dosyncratc rsk component specc to asset, whch by denton s ndependent o the k rsk actors,,..., k, and the specc rsk component o any other asset j, εj. ε must be ndependent o the rsk actors or else t would aect all assets, thus not beng truly a specc source o rsk to just asset. I a s the expected return on asset, then the return generatng process or asset s gven by the lnear model where E ε = E z E z x = E ε ε j = E R = a + ε z k b z z + ε (6 z= = 0. For smplcty, we wll also assume that = 0, that s, the rsk actors are mutually ndependent. As t turns out, ths last assumpton s not mportant, as a lnear transormaton o correlated rsk actors can always be ound such that they can be redened as ndependent rsk actors. Another assumpton s that the dosyncratc rsk (varance or each asset be nte, that s E ε s <S (7 where S s some nte number. varance equal to one, so that we wll assume E ( cov R, z = cov (b z z, z Fnally, one can always normalze each rsk actor to have z =. Under these assumptons, note that ( = b z cov z, z = b z. Thus, b z s the covarance between the 5

11 return on asset and actor z. Let us now dene an asymptotc arbtrage opportunty. Denton: Let a portolo contanng n assets be descrbed by the vector o nvestment amounts n each o the n assets, w n w n wn... wn n. Consder a sequence o these portolos where n s ncreasng, n =,,.... Let σj be the covarance between the returns on assets and j. Then an asymptotc arbtrage exsts the ollowng condtons hold: (A The portolo requres zero net nvestment: w n =0 = (B The portolo return becomes certan as n gets large: lm n = j= w n wn j σ j 0 (C The portolo return s always bounded above zero j= w n a δ>0 We can now state the Arbtrage Prcng Theorem (APT: Theorem: I no asymptotc arbtrage opportuntes exst, then the expected return o asset, =,..., n, wll be descrbed by the ollowng lnear relaton a = λ0 + k z= bz λ z + ν where λ0 s a constant, λz can be nterpreted as the rsk premum or actor z, z =,..., k, and the expected return devatons, ν, satsy = ν =0 bz ν =0, z =,...,k ( = ( ( 6

12 lm n n = ν =0 ( Note that condton ( says that the average squared error (devaton rom the prcng rule ( goes to zero as nbecomes large. Thus, as the number o assets ncrease relatve to the rsk actors, expected returns wll, on average, become closely approxmated by the relaton a = λ0 + k z= b z λ z. Also note that the economy contans a rsk ree asset (mplyng b z =0, z, the rsk ree return wll be approxmated by λ 0. Proo: For a gven number o n assets >k, run a regresson o the a s on the bz s. In other words, project the dependent varable vector a =a a... an on the k explanatory varable vectors bz =bz bz... bnz, z=,..., k. Dene ν as the regresson resdual or observaton, =,...,n. Denote λ0 as the regresson ntercept and λz, z =,...,k, as the estmated coecent on explanatory varable z. The regresson estmates and resduals must then satsy k a = λ0 + bz λ z + ν (8 z= where by the propertes o an orthogonal projecton (Ordnary Least Squares regresson the resduals sum to zero, n = ν = 0, and are orthogonal to the regressors, n = bz ν = 0, z =,...,k. Thus, we have shown that (, (, and (, can be satsed. The last, but most mportant part o the proo, s to show that ( must hold n the absence o asymptotc arbtrage. Thus, let us construct an arbtrage portolo wth the ollowng weghts w = ν n= ν n (9 so that greater weghts are gven to assets havng the greatest expected return devaton. The total arbtrage portolo return s gven by R p = n = νn n ν R = = n = νn n = ν (a + k z= b z z + ε (0 Snce n = b zν =0,z=,...,k, ths equals 7

13 R p = n = νn n ν (a + ε = Let us calculate ths portolo s mean and varance. Takng expectatons, we obtan n E Rp = n ν = ν a n = ( ( snce E ε = 0. Substtutng n or a = λ0 + k z= b zλ z + ν,wehave E Rp = n = νn λ0 = ν + ( k z= λ z ν b z + = = ν (3 and snce n = ν = 0 and n = ν b z = 0, ths smples to E Rp = n = νn ν = = n ν (4 = To calculate the portolo s varance, start by subtractng ( rom ( Rp E Rp = n n= ν ν ε n = (5 Then because E ε εj =0or jand E ε =s, the portolo varance s E ( Rp E Rp = n= ν s n n = ν < n= ν S n n = ν = S n (6 Thus, as n becomes large (n, the varance o the portolo goes to zero, that s, the expected return on the portolo becomes certan. Ths mples that n the lmt the actual return equals the expected return n (4 lm n R p = E Rp = n ν (7 = and so there are no asymptotc arbtrage opportuntes, ths certan return on the portolo must equal zero, that s, 8

14 n ν = 0 (8 = whch s condton (. Q.E.D. Note that the APT can be vewed as a mult-beta generalzaton o CAPM. However, whereas CAPM says that ts sngle beta should be the senstvty o an asset s return to that o the market portolo, APT gves no gudance as to what are the economy s multple underlyng rsk-actors. Emprcal researchers have tended to select rsk actors based on those actors that provde the best t to hstorcal asset returns. We wll see another mult-beta asset prcng model, namely Merton s Intertemporal CAPM, whch s derved rom an ntertemporal consumer-nvestor optmzaton problem. However, that model predcts that the multple betas are not lkely to reman constant through tme, whch would cause sgncant dcultes when attemptng to estmate betas rom hstorcal data. 9

15 APT Cochrane Chapter 9 APT starts wth a statstcal characterzaton o outcomes/payos/returns. Ths eectvely places restrctons on the structure o the covarance matrx. e.g. cov( r, r = ββσ j j m Alternatvely, you can start wth the nvestor s utlty uncton and ask what varables/actors drve margnal utlty. The law o one prce (no arbtrage condton s very powerul when prcng redundant assets, but you want to prce a new non-redundant asset (.e. one that has senstvty to a new actor, the APT wll not help you. Exact actor model no resduals Approxmate actor model ncludes resduals I the resduals are small or dosyncratc, can the prce o an ndvdual asset be very derent rom the prce predcted by APT? In general, yes. It depends on cov( m, ε. See gure 7. How do you get rom the approxmate to the exact actor model? 6