Consider a consumer who decides on a consumption plan of pe- riods t (today) and t...

 

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Consider a consumer who decides on a consumption plan of pe- riods t (today) and t +1 (the next period). While the contingency on period t is certain, what would happen in period t + 1 is uncertain, which is expressed by the states $₁, $2,..., SK. Suppose that the consumer follows the axioms of the (subjective) expected utility and her utility function is given by -K U (c) = u(c₁) + BEKU (Ct+1(SK)) TT (SK) = u(c₂) + BE[u (C++1)], where c denotes the consumption plan, ß the (subjective) discount factor, c con- sumption in period t, C++1 (SK) consumption in period t + 1 under state sk, and TT(SK) the (subjective) probability of state Sk. Suppose that there is a financial market which trades an asset which is sold today at price pt and bares the payoffs x++1(SK) in period t + 1. The consumer's income in period t + 1 is also uncertain and is denoted by wr+1(sk). Her period-t income is certain and is given by w. Therefore, if she holds a units of the asset, her consumption plans can be expressed by the flow budget constraint C₁ + p₁a = w₁ C++1(S1) = Wt+1(S1) + X₁+1(S₁) a C++1 (52) = Wt+1(S2) + X₁+1($2)a Ct+1(SK) = Wt+1(SK)+x+1(SK)a (a) (Asset-pricing formula and stochastic discount factor) Solve the utility maximization problem for optimal asset holding a and ob- tain the "asset-pricing formula" Pt=P(C₁+1(SK)) x₁+1(SK) TT (SK) or P₁ = E[m₁+1 Xt+1], Bu' (C₁+1(SK)) where mi+1(SK) := is called "stochastic discount factor (SDF)." Intuitively, the SDF expresses the "value" of money for the individual in different states. (b) (Risk-free rate) Suppose that the asset is such that x++1 (Sk) = Rf for any state. Derive the value of E[m+1] (c) (Risk-neutral asset pricing) When u(c) = c, what is the asset-pricing formula? (d) (Subjective probability) Continue to assume u(c) = c. If the asset is such that x+1(s) = then what is the price of this asset? 1 ifs = Sk 0 otherwise (e) (Risk correction for asset pricing) For notational simplicity, drop the subscript for the formula: p = E[mx]. Recall that the covariance Cov (m, x) satisfies Cov(m, x) = E[mx] - E[m]E[x]. Show that, from the asset-pricing formula, we obtain E[x] P = + Cov(m, x). Rf This expression implies that the price of an asset depends on how the asset payoff and SDF covaries across states. Consider a consumer who decides on a consumption plan of pe- riods t (today) and t +1 (the next period). While the contingency on period t is certain, what would happen in period t + 1 is uncertain, which is expressed by the states $₁, $2,..., SK. Suppose that the consumer follows the axioms of the (subjective) expected utility and her utility function is given by -K U (c) = u(c₁) + BEKU (Ct+1(SK)) TT (SK) = u(c₂) + BE[u (C++1)], where c denotes the consumption plan, ß the (subjective) discount factor, c con- sumption in period t, C++1 (SK) consumption in period t + 1 under state sk, and TT(SK) the (subjective) probability of state Sk. Suppose that there is a financial market which trades an asset which is sold today at price pt and bares the payoffs x++1(SK) in period t + 1. The consumer's income in period t + 1 is also uncertain and is denoted by wr+1(sk). Her period-t income is certain and is given by w. Therefore, if she holds a units of the asset, her consumption plans can be expressed by the flow budget constraint C₁ + p₁a = w₁ C++1(S1) = Wt+1(S1) + X₁+1(S₁) a C++1 (52) = Wt+1(S2) + X₁+1($2)a Ct+1(SK) = Wt+1(SK)+x+1(SK)a (a) (Asset-pricing formula and stochastic discount factor) Solve the utility maximization problem for optimal asset holding a and ob- tain the "asset-pricing formula" Pt=P(C₁+1(SK)) x₁+1(SK) TT (SK) or P₁ = E[m₁+1 Xt+1], Bu' (C₁+1(SK)) where mi+1(SK) := is called "stochastic discount factor (SDF)." Intuitively, the SDF expresses the "value" of money for the individual in different states. (b) (Risk-free rate) Suppose that the asset is such that x++1 (Sk) = Rf for any state. Derive the value of E[m+1] (c) (Risk-neutral asset pricing) When u(c) = c, what is the asset-pricing formula? (d) (Subjective probability) Continue to assume u(c) = c. If the asset is such that x+1(s) = then what is the price of this asset? 1 ifs = Sk 0 otherwise (e) (Risk correction for asset pricing) For notational simplicity, drop the subscript for the formula: p = E[mx]. Recall that the covariance Cov (m, x) satisfies Cov(m, x) = E[mx] - E[m]E[x]. Show that, from the asset-pricing formula, we obtain E[x] P = + Cov(m, x). Rf This expression implies that the price of an asset depends on how the asset payoff and SDF covaries across states.

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Part a Assetpricing formula and stochastic discount factor To solve the utility maximization problem for optimal asset holding a we set up the Lagrangean La uct Euct1 wt pt a where is the Lagrange mul... View the full answer