One-Period Valuation Theory

One-Period Valuation Theory Part 2: Chris Telmer March, 2013 1 / 44

1. Pricing kernel and financial risk 2. Linking state prices to portfolio choice Euler equation 3. Application: Corporate financial leverage What is alpha? 2 / 44

Reference Point Recall our fundamental pricing equation, q = Dψ: Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 q j = s ψ s d js 3 / 44

Reference Point Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Recall our fundamental pricing equation, q = Dψ: Reference point: EPDV q j = s q j = E(D j) 1+i f = ψ s d js s P sd js 1+i f 3 / 44

Reference Point Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Recall our fundamental pricing equation, q = Dψ: Reference point: EPDV q j = s q j = E(D j) 1+i f = ψ s d js This ignores risk. What can we do? s P sd js 1+i f Adjust the probabilities (numerator) Adjust the discounting (denominator) 3 / 44

Risk Neutral Valuation Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 q j = s ψ s d js 4 / 44

Risk Neutral Valuation Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 q j = s = s ψ s d js ψ s s ψ s d js s ψ s 4 / 44

Risk Neutral Valuation Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 q j = s = s = ψ s d js ψ s s ψ s s Q sd js 1+i f d js s ψ s 4 / 44

Risk Neutral Valuation Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 q j = s = s = ψ s d js ψ s s ψ s s Q sd js 1+i f = EQ D j 1+i f d js s ψ s 4 / 44

Risk Neutral Valuation Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 q j = s = s = ψ s d js ψ s s ψ s s Q sd js 1+i f = EQ D j 1+i f d js s ψ s Why the language risk neutral? 4 / 44

Risk Neutral Valuation Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 q j = s = s = ψ s d js ψ s s ψ s s Q sd js 1+i f = EQ D j 1+i f d js s ψ s Why the language risk neutral? Important: if the state prices aren t unique, then neither are the risk-neutral probabilities 4 / 44

Pricing Kernel Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 q j = s ψ s d js 5 / 44

Pricing Kernel Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 q j = s = s ψ s d js P s ψ s P s d js 5 / 44

Pricing Kernel Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 q j = s = s = s ψ s d js P s ψ s P s d js P s m s d js 5 / 44

Pricing Kernel Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 q j = s = s ψ s d js P s ψ s P s d js = P s m s d js s = E(mD j ) 5 / 44

Pricing Kernel Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 q j = s = s ψ s d js P s ψ s P s d js = P s m s d js s = E(mD j ) We call m the pricing kernel. It is a stochastic discount factor. Expectation is taken w.r.t. objective probabilities 5 / 44

Pricing Kernel Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 q j = s = s ψ s d js P s ψ s P s d js = P s m s d js s = E(mD j ) We call m the pricing kernel. It is a stochastic discount factor. Expectation is taken w.r.t. objective probabilities Main link between theory and arbitrage-free pricing q j = E(mD j ) is called the Euler equation 5 / 44

Risk-Free Rate Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Again, if d js = 1 for all s, then asset j is a risk-free bond. 6 / 44

Risk-Free Rate Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Again, if d js = 1 for all s, then asset j is a risk-free bond. It s price? b = E(m1) 6 / 44

Risk-Free Rate Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Again, if d js = 1 for all s, then asset j is a risk-free bond. It s price? b = E(m1) = E(m) = 1 1+i f 6 / 44

Risk-Free Rate Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Again, if d js = 1 for all s, then asset j is a risk-free bond. It s price? So, b = E(m1) = E(m) = 1 1+i f Risk-free bond price equals sum of state prices Risk-free bond price equals mean of pricing kernel 6 / 44

Three Different Representations Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Capital budgeting: q j = E (md j ) 7 / 44

Three Different Representations Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Capital budgeting: q j = E (md j ) Asset allocation: 1 = E (md j /q j ) 1 = E (mr j ) 7 / 44

Three Different Representations Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Capital budgeting: q j = E (md j ) Asset allocation: 1 = E (md j /q j ) 1 = E (mr j ) Derivatives and risk premiums 0 = E (m(r j R k )) 7 / 44

Why is the Kernel a Useful Normalization? Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Link to theory and observables By using expectations under P, not under Q, we can estimate our model s parameters using historical data Economic insights regarding risk Fundamental distinction between systematic and idiosyncratic risk 8 / 44

Risk = Covariance with m Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Risk is covariance with the kernel: 9 / 44

Risk = Covariance with m Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Risk is covariance with the kernel: q j = E (md j ) 9 / 44

Risk = Covariance with m Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Risk is covariance with the kernel: q j = E (md j ) = E(m)E(D j )+Cov (m, D j ) 9 / 44

Risk = Covariance with m Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Risk is covariance with the kernel: q j = E (md j ) = E(m)E(D j )+Cov (m, D j ) = E(D j) 1+i f +Cov (m, D j) 9 / 44

Risk = Covariance with m Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Risk is covariance with the kernel: q j = E (md j ) = E(m)E(D j )+Cov (m, D j ) = E(D j) 1+i f +Cov (m, D j) If Cov(m,D j ) < 0 then this asset pays a risk premium: it s price is less than its EPDV. 9 / 44

...in terms of returns Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Using the excess returns formulation (let asset k be riskless): 0 = E (m(r j R k )) 10 / 44

...in terms of returns Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Using the excess returns formulation (let asset k be riskless): 0 = E (m(r j R k )) = E ( m(r j i f ) ) 10 / 44

...in terms of returns Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Using the excess returns formulation (let asset k be riskless): 0 = E (m(r j R k )) = E ( m(r j i f ) ) = E (m) E(r j i f )+Cov (m,r j ) 10 / 44

...in terms of returns Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Using the excess returns formulation (let asset k be riskless): 0 = E (m(r j R k )) = E ( m(r j i f ) ) = E (m) E(r j i f )+Cov (m,r j ) = E(r j ) = i f Cov (m,r j )/E(m) 10 / 44

...in terms of returns Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Using the excess returns formulation (let asset k be riskless): 0 = E (m(r j R k )) = E ( m(r j i f ) ) = E (m) E(r j i f )+Cov (m,r j ) = E(r j ) = i f Cov (m,r j )/E(m) For risk adjusted EPDV, put this into the definition of return: 10 / 44

...in terms of returns Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Using the excess returns formulation (let asset k be riskless): 0 = E (m(r j R k )) = E ( m(r j i f ) ) = E (m) E(r j i f )+Cov (m,r j ) = E(r j ) = i f Cov (m,r j )/E(m) For risk adjusted EPDV, put this into the definition of return: 1+r j D j q j 10 / 44

...in terms of returns Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Using the excess returns formulation (let asset k be riskless): 0 = E (m(r j R k )) = E ( m(r j i f ) ) = E (m) E(r j i f )+Cov (m,r j ) = E(r j ) = i f Cov (m,r j )/E(m) For risk adjusted EPDV, put this into the definition of return: 1+r j D j q j = E(1+r j ) = ED j q j 10 / 44

...in terms of returns Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Using the excess returns formulation (let asset k be riskless): 0 = E (m(r j R k )) = E ( m(r j i f ) ) = E (m) E(r j i f )+Cov (m,r j ) = E(r j ) = i f Cov (m,r j )/E(m) For risk adjusted EPDV, put this into the definition of return: 1+r j D j q j = E(1+r j ) = ED j = q j = q j ED j E(1+r j ) 10 / 44

...in terms of returns Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Using the excess returns formulation (let asset k be riskless): 0 = E (m(r j R k )) = E ( m(r j i f ) ) = E (m) E(r j i f )+Cov (m,r j ) = E(r j ) = i f Cov (m,r j )/E(m) For risk adjusted EPDV, put this into the definition of return: 1+r j D j q j = E(1+r j ) = ED j = q j = = q j = q j ED j E(1+r j ) E(D j ) 1+i f Cov(r j,m)/e(m) 10 / 44

Example: Factor Analysis Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Factor structure: m t+1 = 1 β (f t+1 µ) 11 / 44

Example: Factor Analysis Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Factor structure: m t+1 = 1 β (f t+1 µ) If f t+1 = market return, this is CAPM 11 / 44

Example: Factor Analysis Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Factor structure: m t+1 = 1 β (f t+1 µ) If f t+1 = market return, this is CAPM CAPM works badly for the cross-section 11 / 44

Example: Factor Analysis Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Factor structure: m t+1 = 1 β (f t+1 µ) If f t+1 = market return, this is CAPM CAPM works badly for the cross-section Pricing kernel: framework for building, estimating and implementing dynamic multi-factor models 11 / 44

Example: Factor Analysis Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Factor structure: m t+1 = 1 β (f t+1 µ) If f t+1 = market return, this is CAPM CAPM works badly for the cross-section Pricing kernel: framework for building, estimating and implementing dynamic multi-factor models Alpha: Pricing kernel framework designed to ask is it risk? or is it an anomaly? 11 / 44

Example: Term Structure Modeling Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Price of n-period zero coupon bond at date t is denoted b n t. Pricing kernel equation says that b n+1 t = E t m t+1 b n t+1 12 / 44

Example: Term Structure Modeling Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Price of n-period zero coupon bond at date t is denoted b n t. Pricing kernel equation says that Algorithm: b n+1 t = E t m t+1 b n t+1 1. Write down a model of m t 12 / 44

Example: Term Structure Modeling Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Price of n-period zero coupon bond at date t is denoted b n t. Pricing kernel equation says that Algorithm: b n+1 t = E t m t+1 b n t+1 1. Write down a model of m t 2. Price the one-period bond: b 1 t = E tm t+1 1 12 / 44

Example: Term Structure Modeling Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 Price of n-period zero coupon bond at date t is denoted b n t. Pricing kernel equation says that Algorithm: b n+1 t = E t m t+1 b n t+1 1. Write down a model of m t 2. Price the one-period bond: b 1 t = E tm t+1 1 3. Price the two-period bond... and so on b 2 t = E tm t+1 b 1 t+1 12 / 44

(continued) Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 where logm t+1 = δ +z t +λε t+1. z t+1 = (1 ϕ)θ +ϕz t +σε t+1, with ε t+1 N(0,1). 13 / 44

Reference Risk Neutral Kernel Risk-Free Rate Representations Usefulness Risk Example 1 Example 2 There are three alternative ways to write the pricing-kernel-based valuation equation: 1. Payoffs: 2. Returns: 3. Excess Returns: 14 / 44

P & Q of Risk Applications 15 / 44

Price and Quantity of Risk P & Q of Risk Applications Consider some benchmark asset with return r. This can be the market portfolio, or something more investment-manager specific. Or, it could be an asset that mimics your consumption. Its relation with the kernel must be: Er i f = Cov ( m,r ) /E(m) 16 / 44

Price and Quantity of Risk P & Q of Risk Applications Consider some benchmark asset with return r. This can be the market portfolio, or something more investment-manager specific. Or, it could be an asset that mimics your consumption. Its relation with the kernel must be: Er i f = Cov ( m,r ) /E(m) Define this asset such that it is risky: Er i f > 0 This necessarily means that Cov ( m,r ) < 0. 16 / 44

Price and Quantity of Risk P & Q of Risk Applications Consider some benchmark asset with return r. This can be the market portfolio, or something more investment-manager specific. Or, it could be an asset that mimics your consumption. Its relation with the kernel must be: Er i f = Cov ( m,r ) /E(m) Define this asset such that it is risky: Er i f > 0 This necessarily means that Cov ( m,r ) < 0. Combine equation (1) with its counterpart for any asset, i: Er i i f Er = Cov ( ) m,ri i f Cov ( m,r ) = Er i i f = Cov( ) m,r i Cov ( ( m,r ) Er ) i f = β i }{{} quantity ( Er i f ) } {{ } price 16 / 44

(continued) P & Q of Risk Applications The quantity of risk is covariance with the pricing kernel, relative to the kernel s covariance with the benchmark asset return. The price of risk is how much compensation the benchmark receives. An asset with lots of risk is one that has a large, negative covariance with the kernel. Or, a large, negative covariance with marginal utility. It is an asset that pays little when marginal utility is high. 17 / 44

(continued) P & Q of Risk Applications The quantity of risk is covariance with the pricing kernel, relative to the kernel s covariance with the benchmark asset return. The price of risk is how much compensation the benchmark receives. An asset with lots of risk is one that has a large, negative covariance with the kernel. Or, a large, negative covariance with marginal utility. It is an asset that pays little when marginal utility is high. Note: Suppose that r = m. Then things look more beta-like: β i = Cov( ) m,r i Cov ( m,r ) = Cov ( ) m,ri Var ( m ) 17 / 44

(continued) P & Q of Risk Applications The quantity of risk is covariance with the pricing kernel, relative to the kernel s covariance with the benchmark asset return. The price of risk is how much compensation the benchmark receives. An asset with lots of risk is one that has a large, negative covariance with the kernel. Or, a large, negative covariance with marginal utility. It is an asset that pays little when marginal utility is high. Note: Suppose that r = m. Then things look more beta-like: β i = Cov( ) m,r i Cov ( m,r ) = Cov ( ) m,ri Var ( m ) Suppose that m = r m. We get CAPM: β i = Cov( ) r m,r i Var ( ) r m 17 / 44

(continued) Intuition is the same as Assignment 1: P & Q of Risk Applications 18 / 44

(continued) Intuition is the same as Assignment 1: Assignment 1: Cov(MU,r i ) < 0 means risk P & Q of Risk Applications 18 / 44

(continued) P & Q of Risk Applications Intuition is the same as Assignment 1: Assignment 1: Cov(MU,r i ) < 0 means risk Here: Cov ( ) r m, r i > 0 means risk Suppose that r m 1/MU Makes sense? (yes) 18 / 44

(continued) P & Q of Risk Applications Intuition is the same as Assignment 1: Assignment 1: Cov(MU,r i ) < 0 means risk Here: Cov ( r m, r i ) > 0 means risk Suppose that r m 1/MU Makes sense? (yes) Additional insight: systematic vs. non-systematic risk (same as diversifiable vs. non-diversifiable risk). Variation in r m is systematic. Can t be diversified away. We all cannot avoid it. 18 / 44

(continued) P & Q of Risk Applications Intuition is the same as Assignment 1: Assignment 1: Cov(MU,r i ) < 0 means risk Here: Cov ( r m, r i ) > 0 means risk Suppose that r m 1/MU Makes sense? (yes) Additional insight: systematic vs. non-systematic risk (same as diversifiable vs. non-diversifiable risk). Variation in r m is systematic. Can t be diversified away. We all cannot avoid it. Cov(r m, r i ) > 0 = r i contains systematic risk. Expected return positive. 18 / 44

(continued) P & Q of Risk Applications Intuition is the same as Assignment 1: Assignment 1: Cov(MU,r i ) < 0 means risk Here: Cov ( r m, r i ) > 0 means risk Suppose that r m 1/MU Makes sense? (yes) Additional insight: systematic vs. non-systematic risk (same as diversifiable vs. non-diversifiable risk). Variation in r m is systematic. Can t be diversified away. We all cannot avoid it. Cov(r m, r i ) > 0 = r i contains systematic risk. Expected return positive. Cov(r m, r i ) = 0 = r i contains only non-systematic risk. Expected return = risk-free rate. 18 / 44

(continued) P & Q of Risk Applications Intuition is the same as Assignment 1: Assignment 1: Cov(MU,r i ) < 0 means risk Here: Cov ( r m, r i ) > 0 means risk Suppose that r m 1/MU Makes sense? (yes) Additional insight: systematic vs. non-systematic risk (same as diversifiable vs. non-diversifiable risk). Variation in r m is systematic. Can t be diversified away. We all cannot avoid it. Cov(r m, r i ) > 0 = r i contains systematic risk. Expected return positive. Cov(r m, r i ) = 0 = r i contains only non-systematic risk. Expected return = risk-free rate. i.e., r i = a+br m +ε i 18 / 44

Applications P & Q of Risk Applications Assignment 1, final question 19 / 44

Question Contingent Choice Solution Implications State-Contingent Choice Problem 20 / 44

Question Question Contingent Choice Solution Implications State prices, ψ and the pricing kernel m are basically the same thing. Where do they come from? Easy answer: ψ = D 1 q Hard (often unavoidable) answer: Supply and demand for primitive securities They arise, simultaneously, in a market equilibrium in which asset prices, consumption, investment, etc., are determined. 21 / 44

Simplest Problem Question Contingent Choice Solution Implications Same basic setting: S future states of nature Expected utility: U(c) = P s u(c s ) = E [ u(c) ] s Budget constraints: θ s ψ s = W (1) s c s = y s +θ s (2) Choice problem: { max P s u(c s ) } subject to (1) and (2) θ s s 22 / 44

Solution Question Contingent Choice Solution Implications Lagrangian function: L = ( P s u(y s +θ s )+λ W s s (S + 1) first-order conditions: P s u (y s +θ s }{{} c s ) λψ s = 0 W s ψ s θ s = 0 ψ s θ s ) 23 / 44

Solution Question Contingent Choice Solution Implications Lagrangian function: L = ( P s u(y s +θ s )+λ W s s (S + 1) first-order conditions: P s u (y s +θ s }{{} c s ) λψ s = 0 W s ψ s θ s = 0 ψ s θ s ) Implication: state prices linked to marginal utility ψ s = P s u (c s )/λ 23 / 44

Implications Question Contingent Choice Solution Implications Riskfree (real) interest rate: ψ s = P s u (c s )/λ = ψ s = P s u (c s )/λ s s = b = Eu (c)/λ 24 / 44

Implications Question Contingent Choice Solution Implications Riskfree (real) interest rate: ψ s = P s u (c s )/λ = ψ s = P s u (c s )/λ s s = b = Eu (c)/λ Pricing kernel: = s ψ s = P s u (c s )/λ ψ s d js = P s u (c s )d js /λ s ] ] = q j = E [u (c)d j /λ = E [md j 24 / 44

General Setup Savings Problem Uncertainty Euler Equation Risk Premiums 25 / 44

General Setup General Setup Savings Problem Uncertainty Euler Equation Risk Premiums where, maxu(c 0,c) s.t. θ,c,c 0 c 0 +q θ y 0 +q e c y +D θ e R N θ R N y R S ++ y 0 R ++ 26 / 44

Savings Problem S = 1 and N = 1. Budget constraints: General Setup Savings Problem Uncertainty Euler Equation Risk Premiums c 0 +q b θ b y 0 c y +θ b 27 / 44

Savings Problem S = 1 and N = 1. Budget constraints: General Setup Savings Problem Uncertainty Euler Equation Risk Premiums c 0 +q b θ b y 0 c y +θ b Substitution with q b = b = 1/(1+i f ) c 0 +c/(1+i f ) y 0 +y/(1+i f ), or c y +(1+i f )(y 0 c 0 ) 27 / 44

(continued) Let U be additively separable across time. General Setup Savings Problem Uncertainty Euler Equation Risk Premiums U(c 0,c) = u(c 0 )+βu(c) 28 / 44

(continued) Let U be additively separable across time. General Setup Savings Problem Uncertainty Euler Equation Risk Premiums Problem: subject to: U(c 0,c) = u(c 0 )+βu(c) max c 0 {u(c 0 )+βu(c)} c y +(1+i f )(y 0 c 0 ) 28 / 44

(continued) General Setup Savings Problem Uncertainty Euler Equation Risk Premiums Sub-in FOC: max c 0 { u(c0 )+βu(y +(1+i f )[y 0 c 0 ]) } u (c 0 ) = β(1+i f )u (c) Simplest example of Euler equation 29 / 44

with Uncertainty Problem: General Setup Savings Problem Uncertainty Euler Equation Risk Premiums max{u(c 0 )+βe[u(c)]} s.t. θ,c,c 0 c 0 +q θ y 0 +q e c y +D θ, 30 / 44

with Uncertainty Problem: General Setup Savings Problem Uncertainty Euler Equation Risk Premiums max{u(c 0 )+βe[u(c)]} s.t. θ,c,c 0 c 0 +q θ y 0 +q e c y +D θ, Lagrangian: L = u(c 0 )+β s P s u ( y s + j d js θ j ) + λ(y 0 +q e c 0 q θ) 30 / 44

(continued) Lagrangian: L = u(c 0 )+β s P s u y s + j d js θ j General Setup Savings Problem Uncertainty Euler Equation Risk Premiums FOC: β s + λ(y 0 +q e c 0 q θ) P s u ( y s +(D ) θ) s djs λq j = 0, j = 1,2,...,N u (c 0 ) λ = 0 c 0 +q θ y 0 q e = 0 N +2 (non-linear) equations in N +2 unknowns: θ, c 0 and λ. Gives Asset Demands = θ(y 0,y,e ; q,d) 31 / 44

Euler Equation FOC combined give β s P s u (c s )d js = u (c 0 )q j General Setup Savings Problem Uncertainty Euler Equation Risk Premiums Which can be written: βe[u (c)d j ] = u (c 0 )q j = q j = β E[u (c)d j ] u (c 0 ) Stochastic, one-period Euler equation 32 / 44

General Setup Savings Problem Uncertainty Euler Equation Risk Premiums 1. State Prices. Let Then, ψ s = P s βu (c s ) u (c 0 ) q j = s ψ s d js 33 / 44

General Setup Savings Problem Uncertainty Euler Equation Risk Premiums 1. State Prices. Let Then, 2. Pricing Kernel. Let Then, ψ s = P s βu (c s ) u (c 0 ) q j = s ψ s d js m s = βu (c s ) u (c 0 ) q j = s P s m s d js 33 / 44

(continued) General Setup Savings Problem Uncertainty Euler Equation Risk Premiums Risk-Neutral Valuation. Define, and Q s = P s βu (c s ) u (c 0 ) q b = s 1 q b P s βu (c s ) u (c 0 ) Then, q j = q b Q s d js s = EQ D j 1+i f 34 / 44

Risk Premiums Euler equation: E(m(1+r j )) = 1 for all j General Setup Savings Problem Uncertainty Euler Equation Risk Premiums 35 / 44

Risk Premiums Euler equation: E(m(1+r j )) = 1 for all j General Setup Savings Problem Uncertainty Euler Equation Risk Premiums Implies E(r j ) = i f 1 E(m) Cov(m,r j) 35 / 44

Risk Premiums Euler equation: E(m(1+r j )) = 1 for all j General Setup Savings Problem Uncertainty Euler Equation Risk Premiums Implies Implies E(r j ) = i f 1 E(m) Cov(m,r j) E(r j ) = i f 1 E(u (c)) Cov(u (c),r j ) 35 / 44

Risk Premiums Euler equation: E(m(1+r j )) = 1 for all j General Setup Savings Problem Uncertainty Euler Equation Risk Premiums Implies Implies E(r j ) = i f 1 E(m) Cov(m,r j) E(r j ) = i f 1 E(u (c)) Cov(u (c),r j ) Valuable intuition. 35 / 44

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1. State prices, ψ s, given by no-arbitrage 2. The state price per unit probability is the pricing kernel. It is also marginal utility. 3. If the pricing kernel varies across states risk premiums are non-zero. If it doesn t we have risk neutrality. 4. Essential intuition: assets that have low payoffs in high marginal utility states are risky. 37 / 44

Risk Aversion (continued) Consumption (continued) 38 / 44

Applications What is alpha? How does one find it? Article by R. Rajan Risk Aversion (continued) Consumption (continued) 39 / 44

Applications Risk Aversion (continued) Consumption (continued) What is alpha? How does one find it? Article by R. Rajan Possible answers: Arbitrage? Find negative implied state prices? 39 / 44

Applications Risk Aversion (continued) Consumption (continued) What is alpha? How does one find it? Article by R. Rajan Possible answers: Arbitrage? Find negative implied state prices? A good deal? Find positive state prices that represent compensation for risks that you don t care so much about. Necessarily implies incomplete markets. 39 / 44

Applications Risk Aversion (continued) Consumption (continued) What is alpha? How does one find it? Article by R. Rajan Possible answers: Arbitrage? Find negative implied state prices? A good deal? Find positive state prices that represent compensation for risks that you don t care so much about. Necessarily implies incomplete markets. Changing the payoffs? e.g., a takeover changes the payoff matrix. 39 / 44

Applications Risk Aversion (continued) Consumption (continued) What is alpha? How does one find it? Article by R. Rajan Possible answers: Arbitrage? Find negative implied state prices? A good deal? Find positive state prices that represent compensation for risks that you don t care so much about. Necessarily implies incomplete markets. Changing the payoffs? e.g., a takeover changes the payoff matrix. Clientele effects? 39 / 44

Applications Risk Aversion (continued) Consumption (continued) What is alpha? How does one find it? Article by R. Rajan Possible answers: Arbitrage? Find negative implied state prices? A good deal? Find positive state prices that represent compensation for risks that you don t care so much about. Necessarily implies incomplete markets. Changing the payoffs? e.g., a takeover changes the payoff matrix. Clientele effects? An illusion? Tail risk is beta, not alpha. 39 / 44

Risk Aversion Risk Aversion (continued) Consumption (continued) Decision-making under uncertainty Simplest model: Von-Neuman Morganstern expected utility maximization Useful but lots of limitations and/or inconsistencies with observed behavior For example, no sense of uncertainty about uncertainty. Example: valuing mortgages under various macro scenarios and then they trade at the minimum? Risk aversion Aversion to mean preserving spreads. People will pay to avoid them. This is the nature of insurance company profits. Tradeoff: risk vs. return in consumption units implied by no tradeoff in utility units. All this captured by concave utility over sure things. 40 / 44

Market Sentiment? q j = E(D j ) 1+i f Cov(r j,m)/e(m) Risk Aversion (continued) Consumption (continued) Sources of variation: Numerator: cash flow news Denominator: discount factor news (variation) 41 / 44

(continued) Risk Aversion (continued) Consumption (continued) Key distinction Was a big move in prices due to bad news or due to increased risk aversion? Answer is critical to money management, alpha... Why? Because increased risk aversion implies increased expected returns. Not necessarily so for bad news. Key tradeoff. Diminishing probability of increasingly ugly events vs. increasing marginal utility, conditional on those events occurring. Superficially similar to increased variance increases the downside, but also the upside. But substantively way different. 42 / 44

Consumption-Based Asset Pricing Risk Aversion (continued) Consumption (continued) State prices proportional to marginal utility What s random? Difference between known state-contingent variables and unknown state-contingent realizations? Key implication of state-contingent decision making. Recall the skiing story. Consider a state-contingent inventory problem: how much will I value a line-of-credit in the event that a supplier defaults on me? Risk inherent in the marginal utility of consumption. Euler equation: tradeoff marginal utility units today for expected (discounted) marginal utility units tomorrow. Applications of the Euler equation: Understanding portfolio choice Asset pricing: what are the risk factors? Those in the Euler equation! 43 / 44

(continued) Risk Aversion (continued) Consumption (continued) What determines interest rates? Inflation and real interest rates. What determines real interest rates? Puzzles Patience Productivity and consumption smoothing Precautionary savings and uncertainty Growth is high relative to observed real interest rates? Why? Why don t people try to smooth more, thereby driving up real rates? Is it the precautionary demand? Precaution against what? The black swan? Is stock market risk macroeconomic risk? No. Or, maybe there are macroeconomic black swans and by holding the stock market you risk getting hammered by one. How much do you care? Do you buy the calibration? 44 / 44