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Asset Pricing Stochastic Discount Factor Outline 1 Introduction 2 Traditional Linear Pricing Models 3 Empirical Asset Pricing 1 4 Uses of Pricing Models 5 Stochastic Discount Factor 6 Contingent Claims 7 Empirical asset pricing 2 46 / 171 Asset Pricing Stochastic Discount Factor Index Hypothesis and derivation asset prices: interpretation of prices and returns link with the discounted cash flow model linear pricing link with CAPM the mean-variance frontier discounted expected value model and market efficiency when return predictability does not imply inefficiency 47 / 171 Asset Pricing Stochastic Discount Factor SDF: strong points general hypothesis about preferences and time horizon it links pricing to some characteristic of the economy it prices assets whatever the return distribution is intertemporal model: it explains intertemporal diversification (with only one asset) implications for the dynamics of prices and returns over time. 48 / 171 Asset Pricing Stochastic Discount Factor The intertemporal problem of the representative investor two periods: savings and asset demand are equal to zero at time t + 1. 0 the investor is non satiated (U > 0) and risk averse 00 (U < 0) price-taking short selling and borrowing are allowed 49 / 171 Asset Pricing Stochastic Discount Factor max Et [U(Ct , Ct+1 )] = U(Ct ) + δEt [U(Ct+1 )] ω ( Ct = et − Pt ω Ct+1 = et+1 + xt+1 ω (26) where Ct consumption; et initial endowment; ω amount of asset bought; xt+1 asset payoff; Pt asset price; Et conditional expectation; δ subjective intertemporal discount factor, that ranges from 0 to 1. 50 / 171 Asset Pricing Stochastic Discount Factor Ct and ω are the choice variables and depend on parameters and realizations of random variables. ⇔ max U(et − Pt ω) + Et [δU(et+1 + xt+1 ω)] ω FOC: 0 0 Pt U (Ct ) = Et [δU (Ct+1 )xt+1 ] (27) The investor saves until the utility loss deriving from the 0 purchase of an additional unit of the asset, Pt U (Ct ), is equal to the expected utility growth deriving from the additional 0 payoff at time t + 1, Et [δU (Ct+1 )xt+1 ]. 51 / 171 Asset Pricing Stochastic Discount Factor Price as dependent variable 0 U (Ct+1 ) Pt = Et δ 0 xt+1 U (Ct ) (28) Given the payoff distribution xt+1 and consumption choices Ct , Ct+1 , we can determine the price. 0 t+1 ) If we define the factor mt+1 = δ UU(C , we can rewrite (28) 0 (Ct ) as follows: Pt = Et [mt+1 xt+1 ] (29) This is the SDF pricing formula. 52 / 171 Asset Pricing Stochastic Discount Factor Traditional pricing formulas Without uncertainty: Pt = 1 xt+1 Rf (30) With uncertainty, we use discount factor that are not stochastic, R1i , that account for asset i risk: Pti = 1 i ] Et [xt+1 i R with Ri > Rf (31) (29) includes corrections for the risk of the different assets in ONE COMMON discount factor, the same for every asset. i It is the correlation between mt+1 and xt+1 that generates the appropriate correction for asset i. 53 / 171 Asset Pricing Stochastic Discount Factor Expected return as dependent variable gross return= ratio between payoff and price. We divide (29) by P: P = E [mx] ⇒ 1 = E [mR] (32) In the case of a risk free asset we have: 1 = E [mR f ] ⇔ 1 = E [m]R f R f increases with decreasing 0 U (Ct+1 ) , U 0 (Ct ) i.e. with increasing (Ct+1 ) : (Ct ) it reflects that it is more expensive not to consume today when future consumption is larger, in terms of utility. Hence, savings should be remunerated more. 54 / 171 Asset Pricing Stochastic Discount Factor CCAPM and intertemporal diversification An asset is worth more than its payoff discounted at the risk free rate when it is a good hedge against future consumption risk. From (32), and as cov (m, x) = E [mx] − E [m]E [x]: P = E [mx] = E [m]E [x] + cov (m, x) = E [x] + cov (m, x) Rf 0 E [x] cov [δU (Ct+1 ), xt+1 ] P= f + R U 0 (Ct ) (33) If the second term is positive, the asset ensures a negative covariance between its payoff and future consumption. 55 / 171 Asset Pricing Stochastic Discount Factor Consumption is made more volatile by the asset whose return covaries positively with consumption (negatively with marginal future utility). Therefore, it has a return higher than the risk free asset. 1 = [mR] ⇔ 1 = E [m]E [R] + cov (m, R) ⇒ E [R] = 1 cov (m, R) − E [m] E [m] (34) Using the definition of m: 0 E [R] = R f − cov [U (Ct+1 ), Rt+1 ] E [U 0 (Ct+1 )] (35) 56 / 171 Asset Pricing Stochastic Discount Factor Static portfolio choice and the CAPM Static portfolio choice and the CAPM: the volatility of wealth is reduced by diversification among assets. CAPM: an asset risk premium is proportional to its amount of systematic risk. Intertemporal portfolio choice with one asset only and CCAPM: the volatility of the marginal utility of consumption is reduced by the intertemporal diversification. CCAPM: an asset risk premium is proportional to its contribution to consumption volatility. 57 / 171 Asset Pricing Stochastic Discount Factor Linearity and economic interpretation as in the CAPM From (34) E [R i ] = R f − cov (m,R i ) E [m] we have: var (m) cov (m, R i ) E [R ] = R + − var (m) E [m] i f (36) i.e. i) βi,m = − covvar(m,R : R i regression coefficient over the (m) discount factor m (with negative sign); (m) λm = var represents the price of risk. E [m] 58 / 171 Asset Pricing Stochastic Discount Factor With an appropriate parametrization of the utility function and an approximation, we can express: beta as a function of the cov between return and consumption growth the price of risk as a function of risk aversion and the consumption growth volatility. E [R i ] = R f + βi,∆c λ∆c with λ∆c = γvar (∆c) 59 / 171 Asset Pricing Stochastic Discount Factor Mean-variance frontier We know from (34) that: 1 = E [m]E [R i ] + cov (m, R i ) = E [m]E [R i ] + ρm,R i σ(R i )σ(m) ⇒ E [R i ] = R f − ρm,R i σ(m) σ(R i ) E [m] (37) with −1 ≤ ρ ≤ 1. Equation 37 implies |E [R i ] − R f | ≤ σ(m) σ(R i ) E [m] 60 / 171 Asset Pricing Stochastic Discount Factor every expected excess return is included inside the frontier the frontier is generated by: |ρm,R i | = 1. Returns on the upper part: perfectly negative correlated with m, positively with c: maximum risk. Returns on the lower part: perfectly negatively correlated with c: they are an insurance for investors. assets inside: have idiosyncratic risk, that is not correlated with consumption and that is not remunerated by any excess return. Asset B is riskier than asset A, but they have the same return. 61 / 171 Asset Pricing Stochastic Discount Factor Two−fund separation theorem holds! All returns on the frontier are perfectly correlated with m, hence they are perfectly correlated with each other. By selecting the market portfolio (with return R m ) on the frontier and the risk free asset (or two other efficient portfolios), it is possibile to generate every return belonging to the mean−variance frontier by combining the two: R mv = R f + a(R m − R f ) (38) 62 / 171 Asset Pricing Stochastic Discount Factor Market efficiency and return time−series EMH: excess returns are unpredictable if the market is using available information at best. SDF model: in an efficient market, from (35): i Et [Rt+1 − Rtf ] = − i cov (mt+1 , Rt+1 ) E [mt+1 ] (39) excess returns are unpredictable (constant) only if the second term of the equation is equal to zero (constant). Predictability − in an efficient market − depends on the dynamics of h 0 i δU (Ct+1 ) i cov , R i 0 t+1 cov (m, R ) U (C ) h 0t i = t+1 ) E [m] E U (C 0 U (Ct ) 63 / 171 Asset Pricing Stochastic Discount Factor Source of asset return predictability that does not contradict optimal exploitation of information: predictable changes in the risk premium associated to predictable changes in the asset exposure to risk, or predictable changes in the aggregate risk, i.e. in the market price of risk Return predictability does not necessarily imply informational inefficiency. 64 / 171 Asset Pricing Stochastic Discount Factor Implication of SDF on time-series of stock prices EMH: stock prices (adjusted by dividends) is a martingale: Pt = Et [Pt+1 + Dt+1 ] is equal to the flow of dividends, discounted by a constant rate: ∞ X Pt = Et (δ j Dt+j ) (40) j=1 65 / 171 Asset Pricing Stochastic Discount Factor Let us analyze the implication of SDF model: In the case of a stock, [xt+1 ] = [Pt+1 + Dt+1 ] Stock price is equal to: Pt = Et [mt+1 (Pt+1 + Dt+1 )] 0 (41) 0 Only if U (Ct ) = U (Ct+1 ), then Pt = δEt [Pt+1 + Dt+1 ] 66 / 171 Asset Pricing Stochastic Discount Factor In order to obtain the price as a function of the future dividend flow only, we can substitute in (41) future prices: Pt = Et [mt+1 (Pt+1 + Dt+1 )] Pt+1 = Et+1 [mt+2 (Pt+2 + Dt+2 )] ⇒ Pt = {Et mt+1 Et+1 [mt+2 (Pt+2 + Dt+2 )] + [mt+1 Dt+1 ]} ⇔ Pt = Et [mt+1 mt+2 Pt+2 + mt+1 Dt+1 + mt+1 mt+2 Dt+2 ] = Λ Pt = Et ∞ X mt+1,t+j Dt+j (42) j=1 where mt+1,t+j = mt+1 Λmt+j . 0 0 Only if U (Ct ) = U (Ct+1 ), Pt = Et ∞ X (δ j Dt+j ) (43) j=1 67 / 171 Asset Pricing Stochastic Discount Factor Summary SDF pricing accomodates any asset payoff, any well behaved investor preferences it nests DCF,CAPM-APT, CCAPM as special cases it shows that return predictability need not imply market inefficiency, contrary to the traditional EMH 68 / 171 Asset Pricing Contingent Claims Outline 1 Introduction 2 Traditional Linear Pricing Models 3 Empirical Asset Pricing 1 4 Uses of Pricing Models 5 Stochastic Discount Factor 6 Contingent Claims 7 Empirical asset pricing 2 69 / 171 Asset Pricing Contingent Claims Contingent Claims, Complete Markets and Risk Sharing 1 Another pricing method (contingent claims/risk neutral probabilities) that can be reconciled to the SDF. 2 Risk sharing among individuals 1 1. If we want to price an asset that is a perfect substitute, we apply the law of one price. APT: the same exposure to common risk factors, the same first and second moment HERE: perfect substitutes if they have the same payoff in the same future states of nature. 2. If we want to price an asset that is not a perfect substitute? APT: replicating its payoff with a portfolio of existing assets, HERE: replicating its payoff with a portfolio of basic imaginary assets. 70 / 171 Asset Pricing Contingent Claims 2 function of financial markets: sharing of risks among individuals it explains the development of financial markets (Allen and Gale, 1994) by trading assets, individuals eliminate idiosyncratic consumption fluctuations. in a complete market, full insurance is provided aggregate fluctuations, i.e. those that affect asset prices, are undiversifiable and must ultimately be borne by investors 71 / 171 Asset Pricing Contingent Claims Outline 1 The State Preference Approach 2 Contingent Claims 3 Pricing with contingent claims and SDF Risk neutral probabilities Equivalence between SDF and risk neutral probabilities. Application Almeida and Philippon: how do bankruptcy costs and optimal leverage decisions change if we use risk neutral bankruptcy probability instead of the historical one? 4 Pareto-efficient risk diversification. Brief digression on market incompleteness, financial innovation and pricing 72 / 171 Asset Pricing Contingent Claims State Preference Approach States of nature describe situation that investors cannot control. Relevant states could be good, poor or null orange harvest success or failure of a join−venture end or continuation of a war Two or more states cannot occur together. Once a state occurs, future wealth/consumption in that state becomes certain this guarantees that the simple mathematics used under certainty can be used also under uncertainty. 73 / 171 Asset Pricing Contingent Claims Individual problem rewritten: u(ct ) + βE [u(ct+1 )] = u(ct ) + βπ1 u(ct+1 , 1) + ... + βπs u(ct+1 , s) ct = et − pt ξ (44) ct+1,s = et+1,s + xt+1,s ξ, ∀s probability of state s is πs , with 0≤ πs ≤ 1, least one s. P πs = 1 and πs > 0 for at 74 / 171 Asset Pricing Contingent Claims Contingent Claims Asset: set of “contingent incomes” (conditional payoff). Matrix representation of the payoff of two assets (J = 2) in two states of the world (S = 2) is: x11 x12 XSxJ = (45) x21 x22 first column: payoff of asset 1 in states 1 and 2. 75 / 171 Asset Pricing Contingent Claims Pricing with Contingent Claims An Arrow−Debreu (imaginary) asset (AD): has a payoff of 1 in one state only and zero otherwise. Two AD assets in two states of the world can be represented as follows: 1 0 X = (46) 0 1 the first column indicates the payoff of AD asset 1 in states 1 and 2. AD asset: can be easly combined to replicate other assets. To replicate asset 1: buy x11 of the first AD and x21 of the second AD: 1 0 x11 x11 + x21 = (47) 0 1 x21 To replicate asset 2: x12 1 0 + x22 0 1 = x12 x22 (48) 76 / 171 Asset Pricing Contingent Claims Let the prices of the two assets be p1 and p2 respectively, the prices of AD assets pc1 and pc2 . If there are no arbitrage opportunities: ( p1 = x11 pc1 + x21 pc2 (49) p2 = x12 pc1 + x22 pc2 Pricing of a third, “redundant”: given the payoff and prices of the assets, we derive the (imaginary) prices of AD assets - provided the payoff matrix X is invertible, i.e. rank(X ) ≥ S given the payoff of a third asset x13 , x23 , we use state prices, (pc), to price it. 77 / 171 Asset Pricing Contingent Claims With S states of the world and J assets (J number of columns of matrix X ): 0 P = X PC , with (J × 1) = (J × S)(S × 1) (50) With J = S linearly independent payoffs, X is invertible. We invert the (sub) matrix of dimension S × S: PC = X 0 −1 P, with (S × 1) = (S × (S ∈ N))((S ∈ N) × 1) (51) The market is complete if rank(X ) ≥ S An asset is said to be redundant if the market is already complete before including it. 78 / 171 Asset Pricing Contingent Claims Pricing with Contingent Claims and SDF P Let us generalize (49): p(x) = s pc(s)x(s) Meaning: the value of a menu is equal to the sum of its components. Multiply and divide by the probability of each state: P p(x) = s π(s) pc(s) π(s) x(s). If we call m the ratio between state prices and probabilities: pc(s) π(s) P we can rewrite the equation as p = s π(s)m(s)x(s) = E [mx]. Note: if the market is complete, factor m in p = E [mx] exists. m is often called “state price density”. In the SDF section: the state price density coincides with consumption marginal utility ratio. m(s) = 79 / 171 Asset Pricing Contingent Claims Risk neutral probabilities Let us define π ∗ (s) = R f m(s)π(s) = R f pc(s) where R f = 1/ X pc(s) = 1/E [m] s (52) Note: π ∗ (s) are positive, smaller or equal to 1 (pc(s) ≥ 1/R f ), the price of a certain unit of consumption, and they add to 1. It is therefore straightforward to interpret π ∗ (s) as probabilities. Hence, we can rewrite the equation for the “menu”: X 1 X ∗ p(x) = pc(s)x(s) = f π (s)x(s) R s s E ∗ [x(s)] (53) Rf E ∗ indicates the use of “risk neutral probabilities”, rather than historical ones, to evaluate the expected value, ⇒ p(x) = 80 / 171 Asset Pricing Contingent Claims Risk neutral probabilities: put more weight on those state with higher marginal utility m. Risk aversion: is the same as paying more attention to unfavorable states of the world, w.r.t. their real probability. See Almeida on capital structure. 81 / 171 Asset Pricing Contingent Claims Equivalence between SDF and state price density u(ct ) + δE [u(ct+1 )] = u(ct ) + δπ1 (ct+1 , 1) + ... + δπs u(ct+1 , s) (∗)ct = et − pt ξ = wt (54) (∗∗)ct+1 = et+1,s + xt+1,s ξ = wt+1,s , ∀s Multiply both sides of (∗∗) by pc: (∗∗)pcs ct+1,s = pcs wt+1,s As thisP result holds for P every s, also the following equation holds: (∗ ∗ ∗) s pcs ct+1,s = s pcs wt+1,s . Now, adding (∗) and (∗ ∗ ∗), we obtain constraint: Pthe intertemporal budget P ct + s pcs ct+1,s = wt + s pcs wt+1 , s It shows that the individual can move resources from today to any state of the world tomorrow - as AD market is complete. 82 / 171 Asset Pricing Contingent Claims The solution of the problem stems from the maximization of the utility w.r.t. {c, c(s)}, accounting for the intertemporal budget constraint: X max u(c) + δπ(s)u[c(s)] c,c(s) s subject − to X X c+ pc(s)c(s) = w + pc(s)w (s) s L = u(c) + (55) s X s δπ(s)u[c(s)] − λ(c + X s pc(s)c(s) − w X pc(s)w (s)) s (56) 83 / 171 Asset Pricing Contingent Claims First order conditions: 0 u (c) = λ 0 δπu (c(s)) = λpc(s) 0 u [c(s)] pc(s) = δπ(s) u0 (c) 0 ⇒ pc(s) = δ u [c(s)] π(s) u 0 (c) (57) SDF and state price density coincide in complete markets: if (57) does not hold for some s, then it will be optimal to transfer resources between today and s by using assets. 84 / 171 Asset Pricing Contingent Claims Moreover, for every pair of states the following must hold: 0 0 δπ(s2 )u [c(s2 )] δπ(s1 )u [c(s1 )] = pc(s1 ) pc(s2 ) 0 pc(s1 ) π(s1 )u [c(s1 )] ⇒ = pc(s2 ) π(s2 )u 0 [c(s2 )] (58) 0 ⇔ m(s1 ) u [c(s1 )] = 0 m(s2 ) u [c(s2 )] Left−side: rate at which the investor can substitute consumption in state 2 with consumption in state 1, by buying and selling assets. Right−side: rate at which the investor is willing to substitute. If the two sides are not equal, the investor will trade assets and move resources. This clarifies the relation between SDF and state price density. 85 / 171 Asset Pricing Contingent Claims Risk sharing in a complete market if the market is complete, individuals can eliminate the idiosyncratic part of consumption risk by trading asset. each obtains full insurance: the same consumption in every state of the world, independently of individual bad/good luck NO equal distribution of consumption among individuals it explains why the only risk that matters in asset pricing is systematic risk Equation (57): marginal rate of substitution between c(s) and c for individual i is equal to pc(s). pc(s) is equal for individual j: hence, the marginal utility growth should be equal for everyone. 86 / 171 Asset Pricing Contingent Claims pc(s) is equal for individual j: hence, the marginal utility growth should be equal for everyone: 0 0 j i ) u (cs,t+1 u (cs,t+1 ) j δ = δ 0 0 j i u (ct ) u (ct ) i (59) where i and j refers to two different investors that can have different patience and utilities. If they have the same utility function, consumption of investor i and j should move in parallel between state s (at time t + 1) and time t: j i cs,t+1 cs,t+1 = cti ctj (60) shocks to consumption: perfectly correlated among individuals. In a complete market, all individuals share the risks. 87 / 171 Asset Pricing Contingent Claims Pareto efficiency This allocation is Pareto efficient, i.e. a benevolent planner - who maximizes consumer i expected utility without altering that of consumer j - would come to the same allocation of resources. S/he maximizes: max X δ it u(cti ) subject − to X δ jt u(ctj ) = k cti + (61) ctj = cta X X L= δ it u(cti ) + λ δ jt u(ctj ) subject − to cti + ctj = cta X X L= δ it u(cti ) + λ δ jt u(cta − cti ) where c a indicates the total available resources. 88 / 171 Asset Pricing Contingent Claims The FOC of this problem is: 0 0 δ it u (cti ) = λδ jt u (ctj ), hence the same risk shared as in the previous case, equation (59). This is the counterpart of the welfare theorem in the case of uncertainty, proved by Arrow and Debreu. Why don’t we observe full insurance in practice? 89 / 171 Asset Pricing Contingent Claims Are Markets Complete Individuals cannot obtain full insurance against idiosyncratic risks at fair prices insurance with deductible absence of insurance against longevity risk, or against the obsolescence of one’s human capital. Shiller (1993): some new assets can make these risks insurable. However, it is difficult to insure every risk, as individuals’ incentives change with risk hedging. Moral hazard and adverse selection do not allow for full insurance. 90 / 171 Asset Pricing Contingent Claims Assume to the contrary that an individual has full insurance at fair prices, that reflect the observed probability of a car theft S/he does not close her/his car properly, as this produces effort that reduces her/his utility: this is an example of “moral hazard” the probability of a car theft increases above the observed average and the insurance company goes bankrupt keeping this in mind, insurance companies do not offer full insurance at fair prices. Jane smokes, Elsa does not. Jane is willing to buy an health insurance at fair prices, whereas Elsa is not - this is an example of adverse selection. Then the probability of being ill among the pool of insured people is higher than the observed one and the company goes bankrupt. Clauses as bonus/malus or deductible allow to reach an equilibrium (Rothschild and Stiglitz, QJE, 1976) 91 / 171 Asset Pricing Contingent Claims The same situations may happen in financial markets the majority shareholder of a company has a loan with interest rate reflecting the average risk of investment projects S/he is willing to increase the risk, as stock prices will increase. This is a case of moral hazard, that leads to credit rationing (Stiglitz and Weiss, 1981). Assume a market maker is ready to buy at bid price that reflects the probability of a profit decrease, and to sell at ask price that reflects the probability of a profit increase, conditional on publicly available information. An insider is willing to sell when knowing the profits will decrease. A trader that is not informed is not willing to trade stocks. Adverse selection of the counterparts make the market maker reduce the bid price conditional to traders’ sell orders, as those orders are associated to a positive probability of privileged negative information. This is one of the sources of market illiquidity (Glosten and Milgrom, 1985). 92 / 171 Asset Pricing Contingent Claims A similar intuition holds also in auction markets: liquidity is reduced, i.e. the price impact of orders increase (Kyle, Econometrica 85; Grossman and Stiglitz, AER 80). In every example above, information asymmetry between insured and insurance underwriter, funded and funder, and among traders, exists and is associated to market incompleteness and incomplete risk diversification. The degree of asymmetry among traders depends on market micro−structure, particularly on negotiation mechanism (O’ Hara, 95) and on the abidance to insider trading and transparency regulation (Daouk and Bhattarcharya, 2002). Financial innovation is trying to complete markets, thus improving on risk diversification. 93 / 171