Monetary Economics: PDF Lecture Notes

Monetary Economics 1 / 181 Monetary Economics Giuliano Curatola Topic 1 Winter Semester 2024-25 Monetary Economics 2...

Monetary Economics 1 / 181 Monetary Economics Giuliano Curatola Topic 1 Winter Semester 2024-25 Monetary Economics 2 / 181 Introduction Main topics I We study theoretical models of financial markets and their empirical applications I Goal: understand i) determinants of expected returns, ii) determinants of stock return volatility, iii) determinants of price fluctuations... I Working framework 1. Real asset pricing (AP) models (Lucas pure-exchange economy) 2. AP model with money/inflation I We compare theoretical predictions with data I Discrete-time models and continuous-time models Monetary Economics 3 / 181 Introduction Introduction to Topic 1 Monetary Economics 4 / 181 Introduction Some stylized facts we wish to explain 1. Stocks have higher average returns than bonds (premium between 4-8%, see Link) I Why do people do not hold more stocks? (1 2 3)... I yes, stocks are risky but what makes stock market risk ”special”? I maybe stocks fall in value at a ”particularly inconvenient” time (state of nature)... 2. Expected returns, prices and equity premium are high (low) in recessions (booms) 3. The term structure of interest rates is upward sloping but may change slope before recessions (FED Cleveland ) 4. Stock market volatility is high in bad times... Monetary Economics 5 / 181 Introduction Asset returns: overview Figure: Asset return across countries (Munk, Chapter 1). For more recent data: Link Monetary Economics 6 / 181 Introduction Price-dividend ratio: overview Figure: Price-dividend ratio (Munk, Chapter 1) Monetary Economics 7 / 181 Introduction Interest rates: overview Figure: Interest rates (Munk, Chapter 1) Monetary Economics 8 / 181 Introduction Our goal I Our goal is to rationalize the previous (and some others) empirical regularities using equilibrium models. Precisely: 1. We look for prices such that demand equals supply for any asset 2. At the equilibrium prices individuals make optimal consumption-investment decisions (i.e. max utility function) I We consider endowment economies in which the agents’ endowments (initial wealth and random future income) are given I There is a single perishable consumption good (the numeraire). Prices, dividends etc. measured in units of the consumption good Monetary Economics 9 / 181 Asset pricing models without inflation: discrete-time Asset pricing models without inflation: discrete-time Monetary Economics 10 / 181 Asset pricing models without inflation: discrete-time Single period model for bond pricing A simple model for bond pricing I Consider a 2 periods (1 and 2) economy populated with N investors with different endowments, indexed by i. I Preferences, endowments and production possibilities are known with certainty. I Investment opportunity: I Single good for consumption and investment I A bank gives loans and accepts deposits without restriction at the interest rate r. I Investors choose consumption and saving to max their utility function Monetary Economics 11 / 181 Asset pricing models without inflation: discrete-time Single period model for bond pricing The investors’ problem I Investor i chooses consumption in both periods to maximize utility U i (c0i , c1i ) which is strictly concave in both arguments: max U i (c0i , c1i ) c0i ,c1i I subject to the budget constraint c0i = w0i − s0i c0i ≥ 0 c1i = w1i + (1 + r )s0i c1i ≥ 0 where w0i and w1i represent the endowment in period 1 and 2, respectively, and s0i is saving (or loan). Monetary Economics 12 / 181 Asset pricing models without inflation: discrete-time Single period model for bond pricing The investors’ problem (Cont) I Plugging c1i into the utility function max U i (c0i , w1i + (1 + r )(w0i − c0i )) c0i I First order condition ∂U i /∂c0i MRSc i ,c i = =1+r ∀i 1 0 ∂U i /∂c1i I That is, the marginal rate of substitution is equalized across investors Monetary Economics 13 / 181 Asset pricing models without inflation: discrete-time Single period model for bond pricing Equilibrium conditions I Market clearing condition: deposits equal loans N  X  w0i − c0i = 0 i which has to be solved for the equilibrium interest rate, say r ∗. I Once we have r ∗ , we can price bonds with the usual present value relation x1 x2 p(0) = x0 + ∗ + +... 1+r (1 + r ∗ )2 where x is a risk-free payoff. Monetary Economics 14 / 181 Asset pricing models without inflation: discrete-time Single period model for bond pricing Interpretation and examples I The interest rate r ∗ depends on the investors’ preferences and model’s parameters I With this model we can address the following questions: I Ho does risk aversion and/or subjective discount rates affect the price of the bond? I How does the investors’ endowment affect the price of the bond? I Consider the following example: I 2 Investors I U i = logc0i + βlogc1i , i = 1, 2 I Find the equilibrium r ∗. Monetary Economics 15 / 181 Asset pricing models without inflation: discrete-time Single period model for bond pricing Example: log utility I The two investors in the economy solve max log c01 + β log c11 {c01 ,c11 } max log c02 + β log c12 {c02 ,c12 } subject to c1i = w1i + (1 + r )[w0i − c0i ] i = 1, 2 and the last equations combines the two budget constraints together I We substitute the budget constraint into the utility and take FOC with respect to c0i Monetary Economics 16 / 181 Asset pricing models without inflation: discrete-time Single period model for bond pricing Example: log utility (Cont) I Solving the optimality conditions w11 + w01 (1 + r ) w12 + w02 (1 + r ) c01 = , c02 = (1 + β)(1 + r ) (1 + β)(1 + r ) I We impose the market clearing condition (w01 − c01 ) + (w02 − c02 ) = 0 and we derive the equilibrium interest rate 1 w11 + w12 1 + r∗ = β w01 + w02 Monetary Economics 17 / 181 Asset pricing models without inflation: discrete-time Single period model for bond pricing Example: log utility (Cont) I The interest rate is decreasing in β I If β ↑ investors give more importance to the future and would like to postpone consumption I As a result investors would like to increase saving I However saving must stay at zero and therefore r ↓ to keep saving at the same level I The interest rate is increasing (decreasing) in the total future (current) endowment I When future endowment increases, investors would like to borrow against future endowment and anticipate consumption I The interest rate rises to keep saving at zero I The opposite happens if current endowment increases Monetary Economics 18 / 181 Asset pricing models without inflation: discrete-time Single period model for bond pricing Example: log utility (Cont) I The model previously described has several limitations I A limited amount of heterogeneity Suggested exercise: solve the same model assuming different subjective discount rates (i.e. β 1 6= β 2 ) I Endowments and production possibilities are known with certainty Suggested exercise: solve the same model assuming that there is only 1 investor. w0 is known while at time 1 the endowment can increase to w1u with probability p or decrease to w1d with probability 1 − p I The model considers only the problem of bond pricing The next example considers the problem of stock pricing Monetary Economics 19 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets A 2-period model with a risky asset I We formalize the problem using the concept of representative agent: The representative agent is an agent whose preferences are assumed to be representative of the entire economy I Consider a representative agent with endowments wt and wt+1 who chooses consumption to maximize U(ct , ct+1 ) I There is only one risky asset, (supply = 1 share) with I Time t price pt I Random payoff given by xt with distribution F (x ) I The objective is to determine the equilibrium price of the risky asset pt. Monetary Economics 20 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets The problem of the representative agent I The investor’s problem max U(ct , ct+1 ) ct ,ct+1 subject to: ct = wt − pt st ct ≥ 0 ct+1 = wt+1 + xt+1 st ct+1 ≥ 0 where st is the quantity of the risky asset held at time t I Since the endowment is exogenous this framework is also called endowment economy or Lucas economy Monetary Economics 21 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Optimality condition I Assume that the utility function is the weighted sum of time-separable utility functions: U(ct , ct+1 ) = u(ct ) + βEt [u(ct+1 )] I We substitute ct and ct+1 into the utility function and take FOC with respect to st   −u 0 (wt − pt st )pt + βEt u 0 (wt+1 + xt+1 st )xt+1  = 0   | {z } | {z } ct ct+1 which is known as Euler equation Monetary Economics 22 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets How to use the Euler equation 1. Fix the price of the risky asset & the distribution of payoff and compute st (portfolio choice, no equilibrium) 2. Fix the distribution of payoff, impose market clearing conditions (demand = supply, that is st = 1 ) and compute the equilibrium price/returns 3. We follow approach 2 Monetary Economics 23 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Optimality condition and the pricing equation I Rearranging the Euler equation yields the basic equation of consumption-based asset pricing:  0 u (ct+1 )  pt = βEt xt+1 u 0 (ct ) I The asset price equals the current expectation of its discounted payoff I SDF (or pricing kernel): marginal valuation of future consumption in the relevant state, expressed in units of current consumption. Monetary Economics 24 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets The equilibrium pricing kernel I The pricing equation in the previous slide can be rewritten as: pt = Et [mt+1 xt+1 ] where the SDF is u 0 (ct+1 ) mt+1 = β u 0 (ct ) I The price pt is the expectation of the product of two random variables, both conditional on time−t information I We can compute the expectation analytically or numerically Monetary Economics 25 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Economic interpretation of the pricing kernel I How does the SDF depend on macro-factors (consumption growth in this case)? I Given that U(c) is concave, U 0 (c) is decreasing in c I Therefore mt+1 is high when ct+1 is small compared to ct I An asset that pays off when consumption is low is worth more I What matters for asset prices is how payoffs are correlated with consumption I Different asset pricing models produce different asset pricing implications through different specification of the pricing kernel Monetary Economics 26 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Introducing a risk-free asset I Once the pricing kernel is known we can price any asset I Assume we want compute the price (bt ) of a risk-free asset with known payoff x̄t+1 bt = Et [mt+1 x̄t+1 ] = Et [mt+1 ] x̄t+1 I Rearranging bt 1 = = Et [mt+1 ] x̄t+1 Rf where Rf is the risk-free rate Monetary Economics 27 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Introducing a risk-free asset (Cont) I Using the covariance decomposition pt = Et [mt+1 xt+1 ] = Et [mt+1 ] Et [xt+1 ] + Covt [mt+1 xt+1 ] I and substituting 1 Rf = Et [mt+1 ] Et [xt+1 ] pt = + Covt [mt+1 xt+1 ] Rf I In other words, asset are highly valued when their payoff is positively (negatively) correlated with the SDF (consumption) Monetary Economics 28 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Example: log utility I Assume that the representative agent is equipped with log utility ⇒ U = log c, U 0 = c1. Thus ct ct     pt = Et [mt+1 xt+1 ] = Et β Et [xt+1 ] + Covt β xt+1 ct+1 ct+1 1 ct   = βEt Rf ct+1 I The previous equation makes it clear the relation between consumption growth and asset prices I Assets that pay high dividends in states of low consumption growth have high price Monetary Economics 29 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Example: log utility (Cont) I A usual assumption to close the model is that consumption growth and stock returns are log-normally distributed ct+1   gt+1 = log ∼ N(µc , σc ), c  t  xt+1 rt+1 = log ∼ N(µr , σr ) pt with covariance σcr I Using the normality assumption we can compute expectations in closed form Monetary Economics 30 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Example: log utility (Cont) I We start by computing the expected value of the SDF "  −1 # ct+1 1 2 = βEt e −gt+1 = βe −µc + 2 σc
Et β ct I Which implies that the risk-free rate can be computed as follows 1 1 1 2 Rf = −g = e µc − 2 σc βEt (e t+1 ) β 1 ⇒ rf = log Rf = − log β + µc − σc2 2 I Suggested exercise: compute Rf when U(c) = c 1−γ 1−γ Monetary Economics 31 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets The equilibrium risk-free rate I The risk-free rate is decreasing in the subjective discount rate⇒ the same economic interpretation as before I The risk free rate is increasing in consumption growth I If µc ↑ investors expect higher future consumption I ⇒ Investors would like to borrow against future expected increase in consumption⇒ the risk free rate increases I The interest rate is decreasing in the volatility of consumption growth I The higher is σc the higher is the overall risk in the economy. Investors would like to save more today to hedge against possible drop in consumption tomorrow (precautionary saving)⇒ the risk free rate decreases Monetary Economics 32 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets The equilibrium stock return I We start by rewriting the Euler equation for the stock as follows "  −1 # "  −1 # ct+1 ct+1 pt = Et β xt+1 ⇒ 1 = Et β Rt+1 ct ct I Which implies that 1 2 1 2 1 = βEt e −gt+1 +rt+1 = βe −µc +µr + 2 σc + 2 σr −σcr
1 1 ⇒ µr = − log β + µc − σc2 − σr2 + σcr | {z 2 } 2 rf 1 E (Rt+1 )   ⇒ µr − rf + σr2 = log = σcr 2 Rf Monetary Economics 33 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets The equilibrium stock return I The equilibrium excess returns are function of σrc only I Investors require higher premium to hold assets that pay high payoff when consumption is already high I Asset whose payoff is negatively correlated with consumption growth are worth more ⇒ They have high initial price and low expected returns I Problems: in the data 1) σrc is close to zero (Equity premium puzzle) 2) the equity premium is time-varying Monetary Economics 34 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Limitations I The previous model implies 1 rf = − log β + µc − σc2 2 E (Rt+1 )   log = σcr Rf I Set β = 0.98, σc = 2%, µc = 2% and σcr = 0.0004 (consistent with US data) rf ' 4%   E (Rt+1 ) log = 0.0004 Rf I Mismatch between model and data Monetary Economics 35 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Limitations I With power utility we obtain the following trade-off: I Matching the equity premium needs a high risk aversion I Matching the risk-free rate puzzle needs a low risk aversion I The model-implied moments are constant while in the data are time-varying I In slide 44 (log utility) we show that the price of the risky asset is pt = Act for some constant A I Thus Rt+1 = pt+1p+ct+1 = Act+1 +ct+1 Act = A+1 ct+1 A ct t ⇒ Returns and consumption growth have the same vol Monetary Economics 36 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets A 2-period model with a risky asset: Comments I The key result of the model is the pricing equation pt = Et [mt+1 xt+1 ] I This equation would still be valid if we changed the utility function. This would just change the expression for the SDF mt+1 I All AP models amount to alternative specifications of SDF mt+1 Monetary Economics 37 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets A 2-period model with a risky asset: Comments I The pricing equation pt = Et [mt+1 xt+1 ] has several interesting implications I Assume free portfolio formation (if one can trade payoffs x1 and x2 one can also trade payoff ax1 + bx2 ). Then we have: I The price of a portfolio equals that of its constituent assets: p(ax1 + bx2 ) = ap(x1 ) + bp(x2 ) I Portfolios with the same payoff must have the same price (Law of one price) x1 = x2 −→ p(x1 ) = p(x2 ) Monetary Economics 38 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Infinite-horizon, 1-asset model I Consider a model with a representative investor who earns capital income only I The investor decides how many stocks to buy and consumes the payoff of the stock I Consumption and stock allocation are linked by the familiar budget constraint: ct = (dt + pt )st−1 − pt st ct+1 = (dt+1 + pt+1 )st − pt+1 st+1 I In other words, the endowment in period t is the payoff from investment in period t − 1 (dividend dt and price pt ). Monetary Economics 39 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Infinite-horizon, 1-asset problem (Cont) Assumptions: I Dividends (dt ) are a sequence of random variables that follow a Markov process I The following transversality condition lim Et β s U 0 (ct+s )pt+s = 0 s→∞ makes the equilibrium price compatible with market clearing conditions Monetary Economics 40 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Infinite-horizon, 1-asset problem: The investor’s problem I The investor’s problem takes the familiar form ∞ X max ∞ E β t U(ct ) {ct }t=0 t=0 subject to: ct = (dt + pt )st−1 − pt st , ∀t I As usual, take FOC with respect to st U 0 (ct )pt = βEt U 0 (ct+1 )(dt+1 + pt+1 )   Monetary Economics 41 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Infinite-horizon, 1-asset problem: market clearing I Assume there is 1 share of the risky asset (representing the entire market) I We want investors to hold the entire supply ⇒ st = 1, ∀t ct = (dt + pt )st−1 − pt st = ct = dt , ∀t ⇒ consumption = dividends I Extension: the risk-less asset =0 =0 z}|{ z }| { ct = (dt + pt )st−1 − pt st − bt + bt−1 (i + it ) = dt , ∀t Monetary Economics 42 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Infinite-horizon, 1-asset problem: The pricing equation I Starting from some period t and recursively substituting, one obtains: U 0 (ct+1 ) U 0 (ct+1 ) U 0 (ct+2 )      pt = Et β 0 dt+1 + Et β 0 Et+1 β 0 dt+2... U (ct ) U (ct ) U (ct+1 ) 0 0     U (ct+1 ) U (ct+2 ) = Et β 0 dt+1 + Et β 2 0 dt+2 +... U (ct ) U (ct ) U 0 (ct+3 )   + Et β 3 0 dt+3... U (ct ) "∞ # X U 0 (ct+i ) i = Et β dt+i i=1 U 0 (ct ) I The price of the risky asset is the expectation of the future dividends discounted at the marginal utility of future consumption, in terms of the marginal utility of current consumption. Monetary Economics 43 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Infinite-horizon, 1-asset problem: The equilibrium I In equilibrium st = 1∀t ⇒ ct = dt "∞ # X U 0 (dt+i ) i pt = Et β dt+i i=1 U 0 (dt ) I Example 1: Assume log utility ⇒ U = log c, U 0 = 1 c ∞ ∞ " # i X 1/dt+i X dt β pt = Et β dt+i = dt βi = i=1 1/dt i=1 1−β pt+1 − pt dt+1 − dt ct+1 − ct Rt+1 = = = pt+1 dt+1 ct+1 ⇒ PD ratio constant, returns and consumption growth have same moments Monetary Economics 44 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Infinite-horizon, 1-asset problem: The equilibrium I Example 2: power utility & log-normal i.i.d. consumption growth "∞ # " ∞ # −γ −γ ct+i ct+i ct+i X  X  −δ×i −δ×i pt = Et e ct+i = ct Et e ct ct ct i=1 i=1   1−γ   1−γ   ct+1 ct+2   = ct Et e −δ + Et e −δ +... ct ct   1−γ   1−γ 1−γ   ct+1 ct+1 ct+2    = ct Et e −δ + Et e −2δ × +... ct ct ct+1 h 2 2 i = ct e −δ+(1−γ)(µc +.5(1−γ)σc ) + e −2δ+2(1−γ)(µc +.5(1−γ)σc ) +... A = ct 1−A where e −δi = B i I same problem as before... Monetary Economics 45 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Power utility: option pricing I After having identified the SDF we can also price options I Consider an asset which pays consumption at time T̃ only I Asset price   C −γ  p̃(t) = Et β T̃ −t T̃ CT̃ Ct I Assume that gt+1 ∼ N(µc , σc ) for t = 0, 1..., T̃ − 1 then 2 2   log β + (1 − γ)µc + 0.5(1 − γ) σc (T̃ −t) | {z } p̃(t) = Ct e =δ I Consider a call option with maturity T ≤ T̃ and strike price K. Pricing formula:  −γ  CT  pc (t) = Et β T −t (p̃(T ) − K )+ Ct Monetary Economics 46 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Power utility: option pricing I We need to compute  1−γ   −γ  CT CT pc (t) = Ct β T −t Et e δ(T̃ −T ) 1p̃(T )>K − K β T −t Et 1p(T˜ )>K Ct Ct | {z } | {z } I1 I2 I Computation of I1 : slide 160 with y = CT Ct and n = 1 − γ  1−γ  CT I1 = Ct β T −t Et e δ(T̃ −T ) 1 C −δ(T̃ −T ) K Ct { CT >e Ct } t 2 2 = Ct β T −t e δ(T̃ −T ) e [(1−γ)µc +(1−γ) (σc ) ](T −t) N(d1 ) = p̃(t)N(d1 ) where log(Ct e δ(T̃ −T ) /K ) + [(1 − γ)σc2 + µc ](T − t) d1 = √ σc T − t Monetary Economics 47 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Power utility: option pricing I Massage d1

log(Ct e δ(T̃ −T ) /K )+ log e δ(T −t) − log e δ(T −t) + [(1 − γ)σc2 + µc ](T − t) d1 = √ σc T − t
log(p̃(t)/K )− log e δ(T −t) + [(1 − γ)σc2 + µc ](T − t) = √ σc T − t log(p̃(t)/K ) − [log β + (1 − γ)µc + 0.5(1 − γ)2 σc2 ](T − t) + [(1 − γ)σc2 + µc ](T − t) = √ σc T − t   log(p̃(t)/K ) + − log β + γµc + σc2 (1 − γ) (1 − 0.5 + 0.5γ) (T − t) = √ σc T − t  =r  z }| { 2 2 log(p̃(t)/K ) + − log β + γµc − 0.5γ σc +0.5σc2  (T − t) log(p̃(t)/K ) + (r + 0.5σc2 )(T − t) = √ = √ σc T − t σc T − t Monetary Economics 48 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Power utility: option pricing I Computation of I2 : slide 160 with y = CT Ct and n = 1
− γ)  −γ  CT 1p̃(T )>K = −K β T −t e −γµc (T −t)+0.5γ 2 σc2 (T −t) I2 = −K β T −t Et N(d2 ) Ct r z }|
{ − − log β + γµc − 0.5γ 2 σc2 (T −t) = −Ke N(d2 ) = −Ke −r (T −t) N(d2 ) I where d2 is given by log(Ct e δ(T̃ −T ) /K ) + [(1 − γ)σc2 + µc ](T − t) d2 = √  σc T − t
log(p̃(t)/K )− log e δ(T −t) + [(1 − γ)σc2 + µc ](T − t) = √  σc T − t 
 log(p̃(t)/K ) + − log β + γµc − σc2 γ + 0.5(1 − γ)2 (T − t) log(p̃(t)/K ) + (r −.5σc2 )(T − t) = √ = √ σc T − t σc T − t Monetary Economics 49 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Power utility: option pricing I Finally, in equilibrium stock return vol = vol of consumption growth and we obtain the usual B-S formula for the price of a call option c(t) = p̃(t)N(d1 ) − Ke −r (T −t) N(d2 ) I We observe than now the price of the option depends on preferences I the price of the underlying asset (p̃(t)) depends on γ and ρ I the interest rate r depends on γ and β Monetary Economics 50 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets 1.5 1 =.98 =.9 0.8 1 p/C rf 0.6 0.5 =.98 0.4 =.9 0 0.2 0 50 100 0 50 100 0.6 =.98 =.9 0.4 pc 0.2 0 0 10 20 30 40 50 60 70 80 90 100 Figure: µc =.03, σc = 0.3, K =.5, T = 30/250, T̃ = 1, t = 0. Monetary Economics 51 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Power utility: option pricing I GE models provide additional insights as compared to BS I For instance, in the customary BS model the price of a call option increases with r I In equilibrium we observe a negative relationship between r and the price of the call I When r ⇑ the present value of K decreases (positive effect on the option price) I When r ⇑ the price of the underlying asset ⇓ (negative effect on the option price) I This does not contradict BS: in GE the change in risk aversion affects interest rates and the price of the underlying asset simultaneously Monetary Economics 52 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Infinite-horizon, N-assets problem I Introducing multiple risky assets is useful to study the return differential or correlations between sectors. Motivation: I Sin premium (Link) and boycott strategy (Link) I Value and size effects: expected return tend to increase with the book-to-market ratio and the capitalization of the issuing companies I... I Asset pricing models with N risky assets allow to understand the reason why agents prefer a given asset and the implications for equilibrium returns Monetary Economics 53 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Infinite-horizon, N-assets problem: Assumptions I N ”trees”. Each tree represents a single share that entitles the holder to all the output produced I Output cannot be stored, is exogenous and stochastic, and its law of movement is defined by a Markov process I The representative agent maximizes ∞ X E β t U(ct ) t=0 I The agents chooses at each date t a consumption level ct and share holdings zj,t+1 to be held in the subsequent period, for each share j (for j = 1,... , N) Monetary Economics 54 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Infinite-horizon, N-assets problem (Cont) I The maximization problem is subject to a constraint that relates output, consumption and financial investments N X ct = (dj,t zj,t + pj,t zj,t − pj,t zj,t+1 ) ∀t j=1 where where pt is the price of a share in period t I Market clearing: I The representative agent has to hold all the N shares zj,t = 1, j = 1,..., N I The output (= dividends) of all ”trees” is consumed N X ct = dj,t = dt j=1 Monetary Economics 55 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Infinite-horizon, N-assets problem: Optimality conditions I We have N Euler equations for optimal portfolio choice, one for each share holding zj,t U 0 (ct )pj,t = βEt U 0 (ct+1 )(dj,t+1 + pj,t+1 )   I To find the equilibrium price one recursively substitutes this relation to get "∞ # X U 0 (ct+i ) i pj,t = Et β dj,t+i i=1 U 0 (ct ) and impose the market clearing condition ct = dt for all t Monetary Economics 56 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Infinite-horizon, N-assets problem: Example I Assume that the representative agent is equipped with quadratic utility ⇒ U(c) = act − 12 bct2 , U 0 (c) = a − bct I In this case the expression for excess returns becomes Rf E [Rj,t+1 ] − Rf = − β Cov [(a − bct+1 )Rj,t+1 ] a − bct Rf = βb Cov [ct+1 Rj,t+1 ] a − bct I Let z be the asset whose whose returns are mostly highly correlated with aggregate consumption Monetary Economics 57 / 181 Asset pricing models without inflation: discrete-time Single and multi-period models with risky assets Infinite-horizon, N-assets problem: Example (Cont) I Divide the previous expression for any asset j by the same expression for asset z Cov [ct+1 Rj,t+1 ] E [Rj,t+1 ] − Rf Cov [ct+1 Rj,t+1 ] Var [ct+1 ] βj,c = = Cov [ct+1 Rz,t+1 ] = E [Rz,t+1 ] − Rf Cov [ct+1 Rz,t+1 ] βj,z Var [ct+1 ] I Therefore, E [Rj,t+1 ] − Rf = βj,t (E [Rz,t+1 ] − Rf ) I where βj,t = ββj,c. If it is possible to construct a portfolio such j,z that βj,z = 1 one gets the direct analogue to the CAPM Monetary Economics 58 / 181 Asset pricing models without inflation: discrete-time Extensions: habit formation and disagreement Extensions I Alternative preferences and/dynamics of aggregate consumption I Habit formation in consumption I Behavioral preferences I Long run risk in consumption I Heterogeneity: preference, beliefs or endowment process I Production economy Monetary Economics 59 / 181 Asset pricing models without inflation: discrete-time Extensions: habit formation and disagreement Habit formation I We consider the model of Campbell and Cochrane (1999) I The main idea is that agents form consumption habits depending on past consumption I When current consumption is close to the habit level risk aversion increases and induces agents to demand higher risk premium I In addition agents save more today as a precaution against the possibility that consumption tomorrow get closet to the habit level I In this way habit formation helps solve the equity premium and the risk free rate puzzle I Alternative specification: Chan and Kogan (2022). See also The Joneses) Monetary Economics 60 / 181 Asset pricing models without inflation: discrete-time Extensions: habit formation and disagreement Habit formation: the model I Agents are equipped with utility "∞ (Ct+i − Xt+i )1−γ # X Et β t+i i=0 1−γ where X is the reference level of consumption I Let St be the surplus consumption ratio Ct − Xt St = Ct I Thus, the relative risk aversion is UCC γ RRAt = −Ct = UC St Monetary Economics 61 / 181 Asset pricing models without inflation: discrete-time Extensions: habit formation and disagreement Habit formation: the model (Cont) I The relative risk aversion is decreasing in the surplus consumption ratio St I The SDF St+1 −γ Ct+1 −γ     mt,t+1 = β St Ct I The aggregate consumption follows Ct+1   gt+1 = log = ḡ + σεt+1 , εt+1 ∼ N(0, 1) Ct I The surplus consumption ratio st = log(St ) ∼ AR(1) st+1 = (1 − φ)s̄ + φst + λ (gt+1 − ḡ) 0 < φ < 1, ḡ and s̄ are parameters Monetary Economics 62 / 181 Asset pricing models without inflation: discrete-time Extensions: habit formation and disagreement Habit formation: the risk-free rate I The risk-free rate can be computed in the usual way 2 σ 2 (λ2 +1+2λ) E[mt,t+1 ] = βe −γ[(1−φ)(s̄−st )+ḡ]+0.5γ γ 2 σ2 ⇒ log(Rf ) = − log(β) + γ ḡ−γ(1 − φ)(st − s̄) − (1 + 2λ + λ2 ) 2 I Subjective discount rate and expected consumption growth same effect as before (in red) I If the surplus consumption ratio ⇓, marginal utility ⇑, agents wish to borrow to increase consumption, the risk-free rate ⇑ I Fluctuations of St creates additional precautionary saving motives, the risk-free rate ⇓ Monetary Economics 63 / 181 Asset pricing models without inflation: discrete-time Extensions: habit formation and disagreement Habit formation: the risk-free rate I CC assumes λ is a function of st : λ is selected to satisfy 3 conditions I 1: constant risk-free rate γ 2 σ2 − γ(1 − φ)(st − s̄) − (1 + 2λ + λ2 ) = r0 2 γ 2 σ2 ⇒ (1 + λ)2 = −r0 − γ(1 − φ)(st − s̄) 2 I 2: habit predetermined (known in advance) at st = s̄: ∂xt+1 ∂ct+1 |st =s̄ = 0 Xt+1 = (1 − St+1 )Ct+1 ⇒ xt+1 = log[1 − e st+1 ] + ct+1 ∂xt+1 −λe st+1 λ ⇒ =1− s = 1 − −st+1 ∂ct+1 1−e t+1 e −1 ∂xt+1 1 |s =s̄ = 0 ⇔ λ(S̄) = − 1 ∂ct+1 t S̄ Monetary Economics 64 / 181 Asset pricing models without inflation: discrete-time Extensions: habit formation and disagreement Habit formation: the risk-free rate ∂x ∂ ∂ct+1 I habit is predetermined near s̄: t+1 ∂st+1 |st =s̄ = 0 ∂ ∂xt+1 ∂ct+1 0 λ (e −st+1 − 1) + λe −st+1 = ∂st+1 (...)2 ∂ ∂xt+1 ∂ct+1 0 1 ⇒ |st =s̄ = 0 ⇔ λ(S̄) = − ∂st+1 S̄ I Putting the 3 conditions together implies 1−φ r γ r0 = −γ , S̄ = σ 2 1−φ 1p λ= 1 − 2(st − s̄) − 1 if λ > 0, otherwise λ = 0 S̄ Monetary Economics 65 / 181 Asset pricing models without inflation: discrete-time Extensions: habit formation and disagreement Habit formation: the risk-free rate I Replacing λ into the expression for the risk free rate γ rf = log(Rf ) = − log(β) + γ ḡ − (1 − φ) 2 I The risk free rate becomes constant and the term − γ2 (1 − φ) allows to reduce the risk-free rate I when φ is high, the future (t + 1) surplus consumption ratio tends to be higher compared to the current one (t)...the agents prefer to anticipate consumption and the risk-free rate ⇑ Monetary Economics 66 / 181 Asset pricing models without inflation: discrete-time Extensions: habit formation and disagreement Habit formation: the equity premiums I When economic conditions deteriorate (i.e. St ⇓) I Risk aversion ⇑, thus the equity premium ⇑ I Marginal utility more sensitive to macroeconomic fluctuations ⇒ Higher rebalancing ⇒ stock market volatility ⇑ I This is how habit formation solves the problems of standard AP models Monetary Economics 67 / 181 Asset pricing models without inflation: discrete-time Extensions: habit formation and disagreement Habit formation: equity premium Figure: Source: Campbell and Cochrane (1999) Monetary Economics 68 / 181 Asset pricing models without inflation: discrete-time Extensions: habit formation and disagreement Habit formation: stock market vol Figure: Source: Campbell and Cochrane (1999) Monetary Economics 69 / 181 Asset pricing models without inflation: discrete-time Extensions: habit formation and disagreement Heterogeneity I Agents differ along many dimensions I Wealth I Preferences: RA, subjective discount rate... I Beliefs I How does these differences affect optimal choices? I We need to introduce heterogeneity in our framework... Monetary Economics 70 / 181 Asset pricing models without inflation: discrete-time Extensions: habit formation and disagreement Heterogeneous beliefs: motivations I Heterogeneity in beliefs is an appealing feature of general equilibrium asset pricing models I It is intuitive and realistic that people in the financial market have different expectations about the future link to SPF, ECB I We need to introduce heterogeneous beliefs into GE models... Monetary Economics 71 / 181 Asset pricing models without inflation: discrete-time Extensions: habit formation and disagreement Heterogeneous beliefs: SPF, Philadelphia Fed Monetary Economics 72 / 181 Asset pricing models without inflation: discrete-time Extensions: habit formation and disagreement Heterogeneous beliefs: SPF, Philadelphia Fed Optimists Pessimists Std Dev 70-30q Std Dev 70-30q GDP λ = 0.95 0.25 0.25 -0.04 -0.08 λ = 0.90 0.23 0.15 -0.03 -0.04 λ = 0.85 0.16 0.04 0.00 -0.02 λ = 0.80 0.16 0.06 -0.04 -0.06 IP λ = 0.95 0.33 0.27 -0.24 -0.17 λ = 0.90 0.34 0.27 -0.26 -0.22 λ = 0.85 0.31 0.29 -0.22 -0.15 λ = 0.80 0.27 0.20 -0.29 -0.28 Monetary Economics 73 / 181 Asset pricing models without inflation: discrete-time Extensions: habit formation and disagreement Correlations: disagreement, uncertainty and trad. volume 3-month horizon Opti Std Pessi Std Total std Opti 70q-30q Pessi 70q-30q Total 70q-30q Correlation with Macroeconomic Uncertainty 0.17 0.25 0.35 0.16 0.23 0.34 Correlation with Economic Policy Uncertainty -0.05 0.08 0.09 -0.08 0.10 -0.05 Correlation with Trading Volume -0.04 0.15 -0.06 -0.00 0.21 -0.04 Pess. (Opt.) disagreement reflects hete. beliefs (uncertainty) Monetary Economics 74 / 181 Asset pricing models without inflation: discrete-time Extensions: habit formation and disagreement Heterogeneous beliefs: facts I People disagree (to be expected) I More disagreement in bad times than in good times I More optimists (pessimists) in good (bad) times I Optimists (pessimists) disagree more in good (bad) times I Disagreement and trading volume? Monetary Economics 75 / 181 Asset pricing models without inflation: discrete-time Extensions: habit formation and disagreement Heterogeneous beliefs (Cont) I Consider the discrete time economy of slides 14-18 but with two agents I There are 2 periods t = 0, 1 and a risky asset that pays x1 at t=1 I The payoff of the risky asset is random and can take two values: x1h (high value) and x1l (low value) I The two agents assign different probabilities to the states of the world I Optimistic agent: prob good state=πho and prob bad state=1 − πho I Pessimistic agent: prob good state=πhp and prob bad state=1 − πhp with πho > πhp Monetary Economics 76 / 181 Asset pricing models without inflation: discrete-time Extensions: habit formation and disagreement Heterogeneous beliefs (Cont) I Each agent decides the optimal investment in the risky assets 1−γ 1−γ " # ci,t i ci,t+1 max + βEt ci,t ,ci,t+1 1 − γ 1−γ subject to: cti = wti − pt sti i i ct+1 = wt+1 + xt+1 st where st is the quantity of the risky asset held at time t and pt the price I Eit means that agents compute expected values using their own probability distribution Monetary Economics 77 / 181 Asset pricing models without inflation: discrete-time Extensions: habit formation and disagreement Heterogeneous beliefs (Cont) I The FOC of the optimistic agent   −(wto − pt sto )−γ pt + βEot  o −γ   (wt+1 + xt+1 st ) xt+1  = 0  | {z } | {z } cto o ct+1 which implies " −γ # co t+1 pt = βEot xt+1 cto I For the pessimistic agent  !−γ  cp pt = βEpt  t+1 xt+1  ctp Monetary Economics 78 / 181 Asset pricing models without inflation: discrete-time Extensions: habit formation and disagreement Heterogeneous beliefs (Cont) I Clearly there can exist only one price, therefore we impose  !−γ  cp " −γ # co βEot t+1 xt+1 = βEpt  t+1 xt+1  cto ctp I expanding the expected values o !−γ o !−γ ch,t+1 cl,t+1 πho h xt+1 + (1 − πho ) l xt+1 cto cto p !−γ p !−γ ch,t+1 cl,t+1 = πhp h xt+1 + (1 − πhp ) l xt+1 ctp ctp which has to hold state by state (check price of AD securities) Monetary Economics 79 / 181 Asset pricing models without inflation: discrete-time Extensions: habit formation and disagreement Heterogeneous beliefs (Cont) h I Equating terms multiplying xt+1 l and xt+1 we get p !1 cl,t+1 ctp πlp γ o = o cl,t+1 ct πlo for state l and !1 p ch,t+1 cp πhp γ o = to ch,t+1 ct πho for state h Monetary Economics 80 / 181 Asset pricing models without inflation: discrete-time Extensions: habit formation and disagreement Heterogeneous beliefs (Cont) πhp πlp I Note that πho > πhp ⇒ πho < 1 and πlo >1 I Assume that at t=0 the agents have the same consumption (ctp = cto ) =1 =1 p z}|{ !1 p z}|{ !1 cl,t+1 ctp πlp γ ch,t+1 cp πhp γ o = , = to cl,t+1 cto πlo o ch,t+1 ct πho | {z } | {z } >1 0 and chooses consumption at any time t. I The dividend (consumption) process Ct is a GBM dCt = µC Ct dt + σC Ct dBt for some µC > 0 and σC > 0 I The price of the risky asset St and the price of the risk-less asset Zt satisfy dZt = rt Zt dt dSt + Ct dt = µt St dt + σt St dBt rt , µt and σt are determined in equilibrium Monetary Economics 87 / 181 Asset pricing models without inflation: continuous-time The basic economy (Cont) I The SDF Ht (or state-price density in continuous time models) follows a GBM (see also slide 178) dHt = −rt Ht dt − θt Ht dBt where µt − rt θt = σt is the price of risk and represents the remuneration (in excess of the risk-free rate) for unit of risk σ. I The consumption plan of the agent is said to be feasible if satisfies the budget constraint Z T x ≥E Ht ct dt 0 Monetary Economics 88 / 181 Asset pricing models without inflation: continuous-time The Equilibrium I The equilibrium is defined by a triple (r , µ, σ) such that: I The utility of the representative agent is maximized for some optimal consumption ĉt and optimal fraction of wealth invested in the risky asset π̂. I The demand and supply of consumption are equal, that is ĉ = Ct , ∀t I Demand = supply for each asset π̂Wt = St , (1 − π̂)Wt = 0 where Wt is the wealth of the representative agent at time t Monetary Economics 89 / 181 Asset pricing models without inflation: continuous-time Solution method I We solve the problem using the familiar Lagrangian function Z T " Z T # −ρt E e U(ct )dt + λ x − E Ht ct dt 0 0 where λ is the Lagrange multiplier associated to the budget constraint. I Taking F.O.C. with respect to ct (complete markets !!) e −ρt Uc (ct ) = λHt I If we normalize H0 = 1 −→ λ = Uc (c0 ). Thus, Uc (ct ) Ht = e −ρt (1) Uc (c0 ) Monetary Economics 90 / 181 Asset pricing models without inflation: continuous-time Solution method (Cont) I We know that the SDF satisfies dHt = −rt Ht dt − θt Ht dBt I Applying Ito’s lemma on Eq 1 and matching terms dHt cUcc 1 cUccc 2    = − −ρ − µc + σ dt Ht Uc 2 Ucc c | {z } rt cUcc   − − σc dBt Uc | {z } θt I... the equilibrium interest rate (rt ) and the equilibrium price of risk (θt ) depend on the utility function Monetary Economics 91 / 181 Asset pricing models without inflation: continuous-time Solution method (Cont) I The equilibrium price of the risky asset is given by Z T Z T Hs Hs St = Et Cs ds = Et ĉs ds = Wt t Ht t Ht Where the second equality follows from the equilibrium condition of the consumption market and the last equality from the equilibrium condition of the stock market I Finally, for the risk-less asset Rt rs ds Zt = e 0 Monetary Economics 92 / 181 Asset pricing models without inflation: continuous-time The Consumption CAPM I We recall that the price of risk is defined by µt − rt θt = σt I In equilibrium, CUcc (ct , t) θt = − σc Uc (ct , t) I Using the previous equations the stock return can be written as CUcc (ct , t) µt − rt = − × σ σ Uc (ct , t) | c{z }t | {z } Cov [ dC , dS C S ] Relative risk aversion Monetary Economics 93 / 181 Asset pricing models without inflation: continuous-time Example: Power utility I The representative agent is equipped with power utility c 1−γ U(c) = 1−γ where γ is the parameter of relative risk aversion. I The max. problem is therefore defined as ct1−γ max e −ρt − λHt ct c 1−γ where λ is the Lagrange mult. associated to the budget constraint. I Taking F.O.C. with respect to c e −ρt ct−γ = λHt Monetary Economics 94 / 181 Asset pricing models without inflation: continuous-time Example: Power utility (Cont) I The market clearing condition implies λHt = e −ρt ct−γ = e −ρt Ct−γ I Applying Ito’s lemma on the previous expression dλH dH 1   = = − ρ + γµc − γ(1 + γ)σc2 dt − γσc dBt λH H 2 I As a result 1 rt = ρ + γµc − γ(1 + γ)σc2 2 θt = γσc Monetary Economics 95 / 181 Asset pricing models without inflation: continuous-time The basic economy (Cont) I The price of risk is increasing in γ and σc. I The risk free rate is I Increasing in ρ: the more impatient is the agent the more she desires to borrow in order to be able to anticipate consumption. I Increasing in µc : when expected consumption growth is high, future marginal utility is low compared to the present. The agent desires to borrow in order to anticipate consumption. I Decreasing in σc : when volatility is high consumption is risky and the agent desires to save more today as a precaution against drop in consumption. Monetary Economics 96 / 181 Asset pricing models without inflation: continuous-time Example: Power utility (Cont) I The price of the risky asset is given by Z T Z T Hs St =Et Cs ds = Ctγ Et e −ρ(s−t) Cs1−γ ds t Ht t Z T 2 = Ct e −ρ(s−t) Et e (1−γ)(µc −.5σc )(s−t)+(1−γ)σc (Bs −Bt ) ds t Ct   = 1 − e −η(T −t) η where η = ρ + (γ − 1)(µc − γ2 σc2 ) I Finally, applying Ito’s lemma on the stock price σt = σc µt − rt = σt θt = γσc2 Monetary Economics 97 / 181 Asset pricing models without inflation: continuous-time Power utility: limitations I The model with power utility is simple and provides closed form solutions for asset prices I However it delivers poor empirical predictions: I The volatility of US consumption growth is about 2% per annum. Stock return vol σ = 16% I Empirical/experimental estimations suggest γ = 2 I Therefore µ − rt = 2 × 0.022 = 0.08% I This number is extremely low if compared with the mean stock return over the last century (7%) [Equity premium puzzle] I In order to match the observed equity premium we would need 0.07 γ= = 175. 0.02 × 0.02 Monetary Economics 98 / 181 Asset pricing models without inflation: continuous-time Power utility: limitations (Cont) I The basic model predicts that the risk free rate is 1 rt = ρ + γµc − γ(1 + γ)σc2 2 I Let σc = µc = 0.02 and ρ = 0.02 and γ = 10 (to get a sizable equity premium) −→ r = 21% I This number is extremely high if compared with market data (1.48% over the last 10 years) [Risk-free rate puzzle] I The equity premium puzzle needs a high risk aversion while the risk free rate puzzle needs a low risk aversion!! I The model predicts that consumption growth and stock returns have the same volatility. Empirically stock returns have much larger volatility (16% vs 2%) [Volatility puzzle] Monetary Economics 99 / 181 Asset pricing models without inflation: continuous-time Power utility: option pricing I After having identified the SDF we can also price options I Consider an asset which pays consumption at time T only HT   S̃(t) = Et CT = Ct e −η(T −t) Ht I Consider a call option with maturity T and strike price K. Pricing formula: HT h i Sc (t) = Et (S(T ) − K )+ Ht  1−γ   −γ  CT CT = Ct e −ρ(T −t) Et 1S̃(T )>K − Ke −ρ(T −t) Et 1S(T˜ )>K Ct Ct | {z } | {z } I1 I2 Monetary Economics 100 / 181 Asset pricing models without inflation: continuous-time Power utility: option pricing I As before, we work with I1 and I2 separately  1−γ   1−γ  CT CT I1 = Ct e −ρ(T −t) Et 1S̃(T )>K = Ct e −ρ(T −t) Et 1 CT > K Ct Ct Ct Ct I Computation of I1 : slide 160 with y = CT Ct and n = 1 − γ  1−γ  CT Ct e −ρ(T −t) Et 1 CT > K Ct Ct Ct (1−γ)(µc −0.5σc2 )(T −t)+0.5(1−γ)2 σc2 (T −t) = Ct e −ρ(T −t) e N(d1 ) =η z }| { = Ct e − (ρ + (γ − 1)(µc − 0.5γσc2 )(T −t) N(d ) = S̃ N(d ) 1 t 1 where log(Ct /K ) + [(1 − γ)σc2 + µc − 0.5σc2 ](T − t) d1 = √ σc T − t Monetary Economics 101 / 181 Asset pricing models without inflation: continuous-time Power utility: option pricing I Massage d1

log(Ct /K )+ log e η(T −t) − log e η(T −t) + [(1 − γ)σc2 + µc − 0.5σc2 ](T − t) d1 = √ σc T − t
log(S̃t /K )+ log e η(T −t) + [(1 − γ)σc2 + µc − 0.5σc2 ](T − t) = √ σc T − t γ log(S̃t /K ) + [ρ + (γ − 1)(µc − 2 σc2 )](T − t) + [(1 − γ)σc2 + µc − 0.5σc2 ](T − t) = √ σc T − t   log(S̃t /K ) + ρ + γµc + σc2 (1 − γ − 0.5 + 0.5(1 − γ)γ) (T − t) = √