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Chapter VII General Issues in Valuation and Arbitrage lnthischapterwecon idei general 1 39 1 391 1 e i t t 139 39 flowsbetapricingmodels and meanvariance efficiency These relationships are found to exist more generally than in the context of the CAPM only Much of the discussion in this chapter is my distillation of the material in Cochrane 1999 1 STOCHASTIC DISCOUNT FACTORS a Complete Markets and the Discount Factor rom equation 110 in Chapter V we know that we can write for the value of any asset in a complete F markets economy 1 p1 Emsxls where ms 0 gt 1 r 2 s Here xls represents the asset s payoff in state s The factor ms in each state s is the inverse of one plus the expected return of the ArrowDebreu security for state s and is thus positive in each state It is also unique since only ArrowDebreu prices are determined uniquely We can think of the expected return for the ArrowDebreu security in state s as the discount rate for a payoff to be received in state s Accordingly it is natural to think of m s as a discount factor that varies stochastically depending on the state Aside from the term stochastic discount factor ms depending on context is also called the pricing kernel or sometimes equivalentMartingale measure RadonNikodyme derivative marginalrate of39 p 1 1 39 39 or1 1 1r 39 39 variable lts significanceis thatthis one stochastic discount factor prices every asset in the economy Any asset can be priced by discounting payoffs by the same stochastic discount factor Note that we can write equivalently that 2 1 EmSRS where R s represents the gross return on asseti in state s 1 Examples of Stochastic DiscountFactors in Incomplete MarketEconomies Markets need not be complete in the context of the CAPM Under the assumption of norm a1 returns we found in Chapter 111 equation 312 that the firstorder conditions for investor k imply that 3 EukwkRl Rf o a mk a bukwk R BALVERS WEST VlRGIN39IA UNIVERSITY 130 FOUNDATIONS OF ASSET PRICING 501 CHAPTER VII GENERAL ISSUES IN VALUATION AND ARBITRAGE The implication follows by defining m consistent with equation 2 and since the first part of equation 3 holds for all 1 Thus the stochastic discount factor is related to the marginal utility of wealth and in this incomplete markets context may differ by individual but still prices each asset for that individual When we get into dynamic asset pricing theories in the next chapter we will derive the following standard first order condition 4 BEucfuciR Rf o a m wagonaxon The stochastic discount factor for individual khere equals the marginal rate of intertemporal substitution At least in the two cases describe above it is possible to find stochastic discount factors that prices all assets for a particular individual To what extent these factors may differ across individuals whether they most be positive as is clearly the case in equation 4 whether they are unique for an individual whether they can be mimicked by a portfolio of marketable assets and whether a discount factor must exist in all incompletemarket models is not immediately clear and will be considered next 2 INCOMPLETE MARKETS AND STOCHASTIC DISCOUNT FACTORS e will prove some quite general results that apply for stochastic discount factors in an incomplete R markets context a Value Additivily Consider the set of primary assets with imperfectly correlated returns arranged in the column vector x such that x T x1x2 xn with asset prices p T p1 pz 7 All primary assets are available to all investors If we define the paya space X as the set ofpayoffs available to all assets then we have x E X for all 1 E 1 n Assume now that markets are frictionless in the sense that portfolio combinations of the 11 primary assets are feasible for all individuals at no cost Or Assumption FrictionlessPanfaliaFarmalian For all 1 and j E 1 n and for all real numbers a and bxk E ax bxj EX Thus the payoff space consists of all linear combinations of the primary payoffs Further assume that Value Additivizy holds This implies that the price of a linear combination of payoffs is equal to the linear combination of the prices of the payoffs Cochrane 1999 calls this assumption the law of one price LOOP as two alternatives that are essentially equivalent are assumed to have the same price Cochrane calls this the happy meal assumption the price of a happy meal should just be equal to the price of a burger fries and a shake More in the realm of financial assets the price of say a government bond should be equal to the price of its stripped zerocoupon bond and the prices of the stripped coupons added together Form ally R BALVERS WEST VIRGINIA UNIVERSITY 131 FOUNDATIONS OF ASSET PRICING 501 SECTION 2 INCOMPLETE MARKETS AND STOCHASTIC DISCOUNT FACTORS Assumption 2 Value Addditivity For all real numbers a and b and for all x and j E X xk E ax bxj pxk apxl bpxj The following general result then applies in an environment where markets need not be complete RESULT 1 VALUE ADDITIVITY AND THE STOCHASTIC DISCOUNT FACTOR Provided that Assumption 1 holds Assumption2 a For all x EX Em xl p where m EX and unique withinX Proof First Em x1 p a Value Additivity Say xk E ax bx Then it is easy to verify the following pk Em xk Em axl bxj aEm x1 bEm xk apl bpj Second it is easy to derive that Value Additivity Em x1 p In vector notation we can write Em x p and we need to show that a stochastic discount factor exists such that this system of equations holds Assumption 1 in vector notation implies that X ch where c represents any nvector ofreal numbers Assume that m b TX so that it is a member ofX Set b x x T39 1p Substituting into E m x T p T shows that the equation holds as needed to be shown Note that the matrix x x Tis a matrix of second moments and is positive definite Hence its inverse exists As a result m as proposed exists and is determined uniquely D Fl I A m r I 1 I A min 1 111 t 391 1 WM nmimthatwithjn the set of attainable payoffs one and only one stochastic discount factor exists In other words not only does a stochastic discount factor exist it also can be constructed as a portfolio from the set of available payoffs and there is only one such portfolio in the payoff space Complete markets are not necessary for this result b Arbitrage The above result can be amended slightly if instead of assuming value additivity we assume absence of arbitrage opportunties AOAO While the noarbitrage assumption in a sense is similar to the law of one price assumption it is a little stronger The law of one price only assumes that two basically equivalent assets should have identical prices Absence of arbitrage implies that if one asset firstorder stochastically dominates another than it should have a higher price Or put differently if one asset in no state has a lower payoff than a particular other asset but in some states has a higher payoff then it should have a higher price Stated formally Assumption 3 Absence ofArbitrage Opportunities For every payoffx in X if xs 0 for all s and xs gt 0 for some s thenpx gt 0 A variant of the previous result then becomes R BALVERS WEST VIRGINIA UNIVERSITY 132 FOUNDATIONS OF ASSET PRICING 501 CHAPTER VII GENERAL ISSUES IN VALUATION AND ARBITRAGE RESULT 2 ABSENCE OF ARBITRAGE AND THE STOCHASTICDISCOUNT FACTOR Provided thatAssumption 1 holds Assumption3 a For all x EX Em x1 p with m EX andunique withinXandmgt 0 The slightly stronger assumption of no arbitrage thus implies the more specific result that the stochastic discount factor must also be positive in all possible states A complete proof of this result can be found in Cochrane 1999 Here we prove the reverse implication in general but the implication only for complete markets Proof First E m x p and m gt 0 4 Absence of Arbitrage Opportunities Clearly if mis strictly positive in all contingencies and x is never negative and strictly positive in some contingencies then 71 E m x gt 0 Second Absence of Arbitrage Opportunities Em x1 p and unique m gt 0 Although true for incomplete markets as well we prove this here only for complete markets Note that equation 1 proves this somewhat informally already Alternatively consider that Assumption 3 implies Assumption 2 Hence we know that a unique mquot exists that may be negative But say that for some state s m s lt 0 Then consider the ArrowDebreu security for state s its 71 E m x lt 0 which contradicts the assumption of no arbitrage opportunities D It should be pointed out that the above result is quite general It is possible of course to find a stochastic discount factor that is positive and uniquely determined in payoff space in the context of say the CAPM But this would require an assumption of norm ality or quadratic preferences together with other specific assumptions Here on the other hand a Hffir ient J quot than f1 39 39 1 f quot formation is absence of arbitrage opportunities Within the context of frictionless m arkets absence of arbitrage is implied by the very weak preference restriction of non satiation utility is strictly increasing in wealth or consumption c Multiple Stochastic Discount Factors In a complete markets economy m is unique The reason is that we know from Result 2 that only one m exists within the payoff space But with complete markets the payoff space includes all possibilities so no m can exist outside of the payoff space When markets are incomplete it is easy to construct additional stochastic discount factors Consider for instance m m 8 Then Emxl Em exl 7 Eexl In an incomplete markets economy it is always possible to select infinitely many random outcome variables with mean zero that are independent of all feasible payoffs in the incomplete markets payoff space so that Eexl 0 Many of these m s could be positive Here and in some of the further sections it is useful to use the idea of a projection A projection provides the least squares forecast of a particular variable Thus projyl l x 12 I x where 12 and I are the least squares estimates of a regression of y on x R BALVERS WEST VIRGINIA UNIVERSITY 133 FOUNDATIONS OF ASSET PRICING 501 SECTION 2 INCOMPLETE MARKETS AND STOCHASTIC DISCOUNT FACTORS Consider the projection of m on the space of payoffs X Then we can write that m projle e Eexl 0 for all i Thus 5 p1 Emxl Eprojle exl Eprgjm Xxl E0quot xxx The last equality follows since the projection of m on the payoff space will lie within the payoff space and must thus be the unique discount factor that lies in the payoff space under assumptions 1 and 2 or assumptions 1 and 3 If the stochastic discount rate generated by some particular asset pricing model with incomplete markets were m then it may not be found as a portfolio from the available asset opportunities However equation 5 shows that it is always possible to construct a mimicking portfolio mquot of m which has the exact same pricing implications as m Interestingly if one knew the true model and exact theoretical variable to represent m from the theoretical model but if this variable were measured with some error then sufficient data mining might produce the mimicking portfolio that as is the case with portfolio returns would likely be measured with less error and so might perform better than the true model d Systematic and Idiosyncratic Risk Given a stochastic discount factor and using projections it is possible to provide a natural definition of systematic and idiosyncratic risk Set x1 x elwhere x E projxllm Then 178 EON 0 1706 EON EON 1706 Thus the systematic risk is the risk that is perfectly correlated with the stochastic discount factor It is found by projecting the return of asset i on m without a constant so that 8 need not have zero mean The idiosyncratic risk is uncorrelated with m and is not priced as was shown 3 STOCHASTIC DISCOUNT FACTORS AND BETA PRICING MODELS e will explore here the connection between a stochastic discount factor representation a betapricing representation and a meanvariance efficiency representation and find that all three representations are equivalent a Stochastic Discount Factors and BetaPricing Models We know from equation 5 that one can always find a mimicking portfolio for the true underlying stochastic discount factor that prices assets equally well 6 p1 Emxl 7 Emxl a l EmR1 R BALVERS WEST VIRGINIA UNIVERSITY 134 FOUNDATIONS OF ASSET PRICING 501 CHAPTER VII GENERAL ISSUES IN VALUATION AND ARBITRAGE where m prajm lX From equation 6 we can derive two useful results First if a risk free asset exists its return would equal 7 l Em f EmRf a Rf lEm Second we know that by construction m is the payoff on a portfolio of generally available assets Now consider the return on the payoff m Rm E m pm From equation 6 for asset m we then have 8 pm Em2 Rm mEm2 Returning to equation 6 use the definition of covariance to write 1 C0vm R1 Em Em Varm Em 39 9 E R1 Clearly we already have a beta pricing formulation at this point However to obtain a more Standard formulation use equation 8 to relate the return on any asset to the return on the stochastic discount factor mimicking portfolio Apply equation 9 to the stochastic discount factor mimicking portfolio and use equation 8 to obtain Var R 10 ERm 4 A Em E RM Rewrite equation 9 to transform to returns on the mimicking portfolio and then combine equations 9 and 10 to produc e 11 ER 1 COVR quotR 1 ER Em VarRm Em quot 39 If a risk free asset exists then from equation 7 we can write R BALVERS WEST VIRGINIA UNIVERSITY 135 FOUNDATIONS OF ASSET PRICING 501 SECTION 3 STOCHAan DISCOUNT FACTORS AND BETA PRICING MODELS C ov Rm RI 12 ER1 Rf 1mERmRf where lm E W Or converting gross returns into returns 13 ll rfl31mquotllmquotrf Note that the steps that lead to equation 1 3 can also be reversed So the existence of a single beta pricing formulation with beta related to the covariance between asset return and the mimicking portfolio return implies the existence of an m that prices all assets In summary RESULT 3 STOCHASTIC DISCOUNT FACTORS AND BETA REPRESENTATIONS Iff a stochastic discount factor formulation applies a onebeta model exists with as its factor the return on the stochastic discount factor mimicking portfolio p1 Emx1 u rf 1mrf pm with lm E CovrmrlVarrm Thus assumptions 1 and 2 or 3 are seen to produce a CAPMtype model in which the covariance with one particular portfolio return is sufficient to price each asset The key portfolio here is the portfolio that mimics the stochastic discount factor m This portfolio of course in general is more difficult to obtain than the market portfolio used in the CAPM Additionally the portfolio has a negative risk premium as follows from equation 10 This makes sense since high returns when the future is discounted less are more valuable 1 Beta Pricing Models and Stochastic Discount Factors What if there are more betas or when the beta is not the beta with the mimicking portfolio We will see next that a linear factor model such as the APT is equivalent to a stochastic discount factor that is linear in the factors Define f as a demeaned column vector of the k factors and b as a column vector of k constants Note that demeaning the factors is without loss of generality as the factor means appropriately weighted can be added to the constant Additionally define L as the column vector of sensitivities of asset i to the k factors and A as the column vector of risk premia on the k factors Then RESULT 4 LINEAR STOCHASTIC DISCOUNT FACTORS AND GENERAL BETA REPRESENTATIONS The following two representations are equivalent A EmR1lmabe B ER1 06 9Tl3igt R BALVERS WEST VIRGINIA UNIVERSITY 136 FOUNDATIONS OF ASSET PRICING 501 CHAPTER VII GENERAL ISSUES IN VALUATION AND ARBITRAGE where u 111 and A ocEmf Proof From A and similar to equation 9 we get that 1 CovmRl Eon 39 ER1f Tb a 14 Em i The second equality follows from using A and since the means of f are equal to zero To put the Si into equation 14 recall that 15 3 2 EffT391Ele Substitute equation 15 into equation 14 to obtain 16 Em i Ii It is now easy to check with the appropriate substitutions that we obtain representation B Vice versa it is straightforward to go from equation 16 back to equation 14 and then to representation A D Note that the representations are not unique For instance we have seen that given incomplete markets you can add a random variable orthogonal to returns to m leaving pricing implications unchanged And in representation B adding risk factors with zero 3 or with zero A would leave pricing implications unchanged as well Note also that from a practical perspective Result 4 tells you how to discount cash flows if the APT is assumed to hold Further note that from Result 4 we can write AkRf pfk le the present value of the risk premium on factor k is equal to the price of the factor Lastly note that we know from Result 3 for instance that we could combine the different factors into just one factor m c Stochastic Discount Factors andMean Variance E iciency We will assume here for simplicity that a risk free asset exists The results in this section hold also for the more general case as shown in Cochrane 1999 We start again with the assumption that a stochastic discount factor representation holds for all assets We can then write using equations 6 and 7 and using the linearity property of the covariance operator R BALVERS WEST VIRGINIA UNIVERSITY 137 FOUNDATIONS OF ASSET PRICING 501 SECTION 3 STOCHASTIC DISCOUNT FACTORS AND BETA PRICING MODELS Cav m R 17 EmR1 1 a ER1 Rf M E m WPquot l if a risk free asset exists Using the definition of the correlation coefficient of the returns on asset i and the mimicking portfolio 0Rm r ERm 39 18 EmR1 1 a ER1 Rf pmoR By employing the fact that l 3 pm 3 1 it is possible to construct a meanvariance frontier 19 Ele 1 a lER1 Rfl g oR 0Rmquot ERm Figure 1 shows the wedgeshaped meanvariance frontier Clearly the risk free asset is on the frontier This is also true for the mimicking portfolio which follows from the fact that pm l Equation 18 shows that indeed the mimicking portfolio is on the meanvariance frontier It is however on the inefficient part of the frontier This provides the explanation for why the risk premium in equation 17 for instance is negative Note that the term in large parentheses equals the maximal Sharpe Ratio Combinations of the risk free asset and the mimicking portfolio trace out the meanvariance frontier This is clear from the fact that linear combinations of these two assets are either perfectly positively correlated or perfectly negatively correlated Thus for all portfolios i that are linear combinations of the risk free asset and the mimicking portfolio lpl ml l As a result it is possible to write either mquot or Rm m E m 2 as linearly related to any mean variance efficient return R So any frontier return is sufficient for pricing Thus MVF39 RESULT 5 STOCHASTIC DISCOUNT FACTORS AND THE MEAN VARIANCE FRONTIER The existence of a stochastic discauntfactar representation is equivalent to I RMVF a bm and t0 the betafarmulalian afbeta with any meanvariant e icient return 2 ER1 Rf 31 MW ERMVF Rf Our proof here assumed existence of a risk free asset but as stated previously the result holds also if no such asset exists Of course we already knew the result Roll 1976 proved that a single beta representation exists if and only if the benchmark return is on the meanvariance frontier Thus Result 5 follows from Roll s result plus Result 3 which relates the stochastic discount factor representation to a single beta pricing model Note that the expression in equation 1 9 is a key element in the HansenJagannathan Bound which relates excess returns to volatility in fundamentals Also note that in Figure 1 it is easy to indicate the idiosyncratic risk of any asset As discussed in section 2 the idiosyncratic risk is the part not correlated with m or mquot Hence it is causing the R BALVERS WEST VIRGINIA UNIVERSITY 138 FOUNDATIONS OF ASSET PRIC1NG 501 CHAPTER VII GENERAL ISSUES IN VALUATION AND ARBITRAGE ERi Figure 1 Stochastic Discount Factor and Portfolio Frontier return on an meanvariance ef cient port 0 390 must be perfectly correlated with Inquot variance in excess of the variance of the efficient asset with equal mean 20 x xe Where xX r039xm 02 0 02 V x F J 1 1 I I I s 39 Lastly Figure 1 Shows that Rm can be found as the minimum feasible second moment asset as Shown in Cochrane 2001 R BALVERS WEST VIRGINIA UNIVERSITY 139 FOUNDATIONS OF ASSET PRICING 501