Summary of the Chief Features of Alternative Asset Pricing Theories

Transcription

1 Summary o the Chie Features o Alternative Asset Pricing Theories CAP and its extensions The undamental equation o CAP ertains to the exected rate o return time eriod into the uture o any security r r β r r, where r is the rate o return on the market ortolio o all risky securities and Cov r, r β The equation imlies that a security is "risky" and will command a ositive risk remium over the risk-ree rate only i its return has a ositive covariance with the return on the market ortolio That is, r r as Cov r, r Let J denote the exected ayout at date rom liquidating selling security, and let J denote the current date rice o this security By deinition r, and we can substitute in Equation to obtain the ollowing: + r J r r Equation is the asset ricing imlication o CAP t says that the current rice or resent value o a risky ayment to be made time eriod into the uture is the exected value o the ayout, discounted by the risk-adusted discount rate aearing in the denominator o the equation The risk adusted discount rate is r r and this is greater than the risk-ree rate or any "risky" security ie or any security with a return that is ositively correlated with the return on the market ortolio Shortcomings o CAP i CAP is only alicable or time eriod into the uture Asset ricing Equation cannot be extended to determine the current rice or resent value o any asset that will make ayouts more than time eriod into the uture This is a very serious shortcoming ii CAP is not emirically viable No one knows how to identiy the market ortolio o all risky securities, let alone how to measure its rate o return

2 a The Single ndex odel The Single ndex odel is airly simle extension o CAP that eliminates Shortcoming ii and can be used to emirically estimate values or β's and make redictions o exected returns t is motivated by the ollowing ex-ost reresentation o Equation : 3 r r r + ε, where r and r are the ex ost realized returns at date on any security and the market ortolio, resectively ε is a random comonent o the return on security that is unique to that security E[ ε ] and Cov ε, The random term ε is the source o what is called "non-systematic risk" associated with security Non-systematic risk can be comletely eliminated by diversiication: When security is combined in a ortolio with a large number o other securities, theε 's tend to cancel one another A well-diversiied ortolio is deined as a ortolio that is large enough that theε 's ully cancel one another The Single ndex odel asserts that large ortolios o stocks, such as used to deine stock market indices like the S&P5 in the US or the TSX/SP3 in Canada reresent well-diversiied ortolios Let ortolio be some well diversiied market index Then Equation 3 imlies that r r r and β The index ortolio relicates a ortolio with a Cov r, r share β invested in the market ortolio and a share β invested in the risk-ree security This ortolio must lie on the Caital arket Line CL Because the index ortolio lies on the CL and 'll ski the derivation, the exected return on any security is related to the exected return on the index ortolio via the ollowing: 4 r r β r r, Cov r, r where β The equation imlies that a security is "risky"and will command a ositive risk remium over the risk-ree rate only i its return has a ositive covariance with the return on the market index That is, r r as Cov r, r Equation 4 is the Single ndex odel and because r is an observable variable, this model can be used to make emirical redictions we relace r with r and r β

3 with β, both Equations and 3 are valid or the Single ndex odel The odel eliminates Shortcoming ii o CAP but still suers Shortcoming i in that it is valid or only one time eriod into the uture b Factor odels Factor odels start with CAP Equation 3 and then argue that luctuations in the return to the market ortolio must be the result o some set o underlying macroeconomic Factors, which will illustrate with a -actor model where the underlying Factors are the rate o growth o real GDP and the rate o growth o the money suly The hyothesis is that the rate o return on the market ortolio behaves as ollows: 5 r r af + bf, where a and b are constants The actor variables are deined as ollows: F rate o growth o GDP minus the exected rate o growth o GDP, F rate o growth o the money suly minus the exected rate o growth o the money suly Observe that E[F ] E[F ] And, while it may not be obvious, we can always choose to measure the Factors in such a way that Cov F, F, which we will assume to be the case here Equation 5 is substituted or r in Equation 3, the result may be written as 6 r r F F + ε Cov r, F Cov r, F Here β and β Also Cov ε, F Cov ε, F [ F β and β are sometimes reerred to as the actor loadings or security ] F Equation 6 identiies F and F as two searate sources o "systematic risk" mlicitly, the equation imlies that a security is "risky" and will command a ositive risk remium over the risk-ree rate only i its return has a ositive covariance with one or both o the Factors Observe that Equation 6 is a standard regression equation This means that estimates or the values o β and β can be obtained rom multile regression analysis using ast 3

4 observations o the values o the variables aearing in the equation But even i we have estimates o the values o β and β, Equation 6 is not very emirically viable t ermits us to redict how the return on security will resond to unexected movements in one or both o the Factors, but it does not tell us what determines the value o r n addition to this rather limited emirical viability, Factor odels o the sort described by Equation 6 share Shortcoming i with CAP and the Single ndex odel c Arbitrage Pricing Theory APT APT is simly a reinement o Factor odels develoed or the urose o enhancing their ability to make redictions about uture rates o return t utilizes something known as a actor ortolio To see what means look at Equation 6 and consider utting together a ortolio with a large number o securities such that a the ortolio is well diversiied theε 's ully cancel one another and b the ortolio has a value or β and a value or β This is the actor ortolio associated with Factor F and we will designate its rate o return as r Similarly, we will use r to identiy the return on the actor ortolio that is associated with Factor F, is well diversiied and has a value or β and a value or β Formally we have r r + F and r r + F Now consider some other well diversiied ortolio whose actor loadings are β and β Comare the return on this ortolio with the return on a ortolio z that has a share β invested in actor ortolio, a share β invested in actor ortolio, and a share - β - β invested in the risk-ree security Return on ortolio : r F F Return on ortolio z: [ β r r + β β r ] F F Since the two ortolios have exactly the same systematic risk, they must have the same exected returns That is, r must equal [ βr r + β β r ] ; otherwise there exists a roitable arbitrage oortunity Exactly the same argument alies in the case o an individual security That is, the exected return on any security must equal the exected return on a ortolio that has a share β invested in actor ortolio, a share β invested in actor ortolio, and a share - β - β invested in the risk-ree security Formally, this imlies 7 r β r r 4

5 The β's in Equation 7 are the same as deined on age 3 or the -Factor model The equation imlies that a security is "risky" and will command a ositive risk remium over the risk-ree rate only i its return has a ositive covariance with one or both o the Factors That this is identical to the concet o risk in the Factor odel should not be surrising since APT is simly a reinement o that model Equation 7 is the undamental equation o APT, and observe that it ermits one to emirically identiy the determinants o the exected return on any security The equation also ermits one determine the date rice or resent value o any -eriod asset As beore on age, let J denote the exected ayout at date rom liquidating selling security, and let denote the current date rice o this security Then, by analogy with Equation, we can derive J 8 + r r r r r APT is useul or emirical analysis but still suers Shortcoming i State Preerence Theory State reerence theory is an equilibrium asset ricing model that may be generalized to any number o uture time eriods Thus it does not suer Shortcoming i However, the model requires that we identiy or each uture time eriod some discrete number o "states o the world" And the odel must be imbedded in some careully seciied macroeconomic environment in which dierent households have either diering reerences utility unctions or ace dierent budget constraints As the numerical illustration eaturing "Tom" and "Judy" in a -time eriod, -state world demonstrated, State Preerence Theory quickly becomes exceedingly comlicated i we try to increase the number o dierent tye households, the number o time eriods, or the number o uture states o the world beyond those o this simle examle Thus, the chie Shortcoming o this odel is that it is not tractable in realistic situations Nonetheless, State Preerence Theory has some very imortant things to teach us Foremost is that a maor beneit o inancial markets is that they ermit economic agents acing diering ossible uture outcomes to engage in risk sharing by exchanging securities n the argon o Economics, this means that households are able to reduce the uncertainties o uture consumtion and increase exected lietime utility by exchanging securities with other households aximum risk sharing can only occur when security markets are "comlete"; that is, when there exist as many linearly indeendent securities as there are uture states o the world State Preerence Theory also rovides us with some very useul concets and tools or determining the rices o securities that have not yet been introduced into actual securities markets These include hedge ortolios, ure or Arrow-Debreu securities, 5

6 and risk neutral robabilities all o which are widely emloyed in determining the rices o derivative securities in the real world For urose o comaring this theory with other asset ricing models, a security is considered "risky" in State Preerence Theory and will command a ositive risk remium over the risk-ree rate i the security has ayos that are not identical over all uture states o the world 3 CCAP The consumtion based caital asset ricing model CCAP is based on the assumtion that all households are ininitely lived and have identical reerences and oortunities Unlike State Preerence Theory, households never exchange securities with one another in CCAP, though they may urchase securities rom or sell securities to other economic entities like cororations or governments The exected lietime utility o the "reresentative" household is given by the ollowing: 8 t E [ U ] u c + E β u ct, t where β is a arameter relecting time reerence and < β Equation 8 may be embedded in any kind o macroeconomic environment And given this environment, it is assumed that the reresentative household has chosen current consumtion c and made contingent lans or uture consumtion c, c, ct, etc in such a way as to maximize exected lietime utility Then, i aced with the rosect at date o urchasing or rice an incremental unit o any security that will deliver a one-time ayment o J ~ t at date t in the uture, we know that household will wish to urchase exactly units o this security Why? Because the household has already chosen the ortolio that maximizes its exected lietime utility and will wish to neith buy nor sell any additional securities The irst order condition associated with the household's decision to urchase exactly incremental units o security may be written t ~ ct 9 β E[ J t ] c Equation 9 is the main asset ricing equation o CCAP t tells us the current rice or resent value o a risk ayment made at any uture date t Unlike CAP and its extensions, CCAP is truly a multi-eriod asset ricing model Observe, however, that to evaluate the right-hand-side o Equation 9 requires taking the exected value o the roduct o random variables -- J ~ t and c t From the ersective o date, current consumtion c is known, but the uture consumtion variable c t is a random variable and so also is the marginal utility o this variable This comlicating eature o CCAP is its main Shortcoming, but one that can oten be made less severe by making a seciic assumtion about the orm o the utility unction [Economists and researchers in the 6

7 securities industry who use CCAP oten assume log utility, or which the term ct c u ] c c t Equation 9 tells us the resent value o any uture ayment a security will make a series o uture ayments, then its current rice is simly the sum o the resent values o the individual ayments For examle, the current rice S o a share o stock that will deliver dividend ayments over the indeinite uture starting with date is S E t t ct β Divt c n order to comare CCAP with CAP and its extensions, set t in Equation 9 and ~ interret J as the liquidation value at date o any security that is urchased at date J ~ or rice Then, emloy the deinition o the -eriod rate o return + r, and rewrite Equation 9 as ct β E[ + r ] c Observe that or the -eriod risk-ree rate revailing at date this equation becomes ct ' β + r E[ ] c With some maniulation omitted here, Equation minus Equation ' can be shown to imly that or CCAP a security is "risky" and will command a ositive risk remium over the risk-ree rate only i its return has a ositive covariance with uture consumtion c That is, r as Cov r, c This makes sense Households buy assets in order to smooth consumtion over time a articular asset has a high/low return whenever consumtion is high/low, the asset does little to hel smooth consumtion and is deemed to be "risky" 7