The term structure of interest rates(Campell 1986) Consider the Lucas asset-pric

Question

The term structure of interest rates(Campell 1986) Consider the Lucas asset-pricing model, with only one asset. Assume that the dividend on the asset follows

log(dt) = g + log(dt-1) + et

where et is normally distributed, white noise, with mean zero and variance 2 . Assume that utility is constant relative risk averse (CRRA), with a coefficient of relative risk aversion .

a.) Consider an N-period pure discount bond at time t, a bond that pays one unit of the good at time t+i and has a price pNt .Drive the equilibrium value of pNt .

b.) Define the yield to maturity on an N-period pure discount bond as rNt , such that (1+rNt)-N = pNt . Characterize the term structure of yields to maturity at time t. Is it upward or downward sloping? Explain?

Can someone help me with part a and b?

Explanation / Answer

We use (a particular instantiation of) the Lucas asset pricing model with heterogeneous agents that is simple enough to implement in the laboratory and yet complex enough to generate a rich set of predictions about prices and allocations. As we shall see, testable predictions emerge under very weak assumptions (allowing complete heterogeneity of endowments and preferences across agents); stronger predictions emerge under stronger assumptions (identical preferences). Because we wish to take the model to the laboratory setting, a crucial feature of our design is that it generates a great deal of trade; indeed Pareto optimality (hence equilibrium) requires that trading takes place every period. This is important in the laboratory setting because subjects do not know the “correct” equilibrium prices (nor do we) and can only learn them through trade, which would seem problematic (to say the least) if theory predicted that trade would take place infrequently. We therefore follow Bossaerts and Zame (2006) and insist that individual endowments not be stationary (where by “stationary” we mean “to be a time-invariant function of dividends”) – although aggregate endowments are 5 stationary, which is a key assumption of the Lucas model.3 As Crockett and Duffy (2010) confirm, not giving subjects a reason to trade in every period – or at least frequently – is a recipe for producing price bubbles in the laboratory, perhaps because subjects are motivated to trade out of boredom rather than for financial gain.)

We consider an infinite horizon economy with a single perishable consumption good in each time period. In the experiment, the consumption good is cash so we use the terms ‘consumption’ and ‘cash’ interchangeably here. How we make cash “perishable” and how we make the laboratory economy infinite-lived will be explained later. In each period there are two possible states of nature H (high), L (low), which occur with probabilities , 1 independently of time and past history. Two long-lived assets are available for trade: (i) a Tree that pays a stochastic dividend d H T when the state is H, d L T when the state is L and (ii) a (consol) Bond that pays a constant dividend d H B = d L B = dB each period.13 We assume d H T > dL T 0 and normalize so that the Bond and Tree have the same expected dividend: dB = dH T + (1 )d L T . Note that the dividend processes are stationary in levels. (In the experiment proper, we choose = 1/2; d H T = 1, d L T = 0; dB = 0.50, with all payoffs in dollars.) There are n agents, where n is even (in the experiments n will be between 12 and 30). Each agent i has an initial endowment bi of Bonds and i of Trees, and also receives an additional private flow of income ei,t (possibly random) in each period t. Write b = Pbi , = Pi and e = Pei for the social (aggregate) endowments of bonds, trees and additional income flow. We assume that the social income flow e is stationary – i.e., a time-invariant function of dividends (in the experiment proper it will be constant) – so that aggregate consumption bd B + d T + e is also stationary ( indexes the state), but we impose no restriction on individual endowments. (As noted earlier, we wish to ensure that in the experimental setting subjects have a reason to trade each period.) We induce the following preferences. Agent i maximizes discounted expected lifetime utility for infinite (stochastic) consumption streams Ui({ct}) = E "X t=1 t1ui(ct) # where ct is (stochastic) consumption at time t. We assume that the period utility functions ui are smooth, strictly increasing, strictly concave and have infinite derivative at 0 (so that optimal consumption choices are interior), but make no assumptions as to functional forms.

Note that agent endowments and utility functions may be heterogeneous but that all agents use the same constant discount factor , which we induce to equal 5/6. In each period t agents receive dividends from the Bonds and Trees they hold, trade their holdings of Bonds and Trees at current prices, use the proceeds together with their endowments to buy a new portfolio of Bonds and Trees, and consume the remaining cash. How exactly agents buy and sell in our laboratory economy will be explained later on. Here, we follow the theory and assume that agents take as given the current prices of the Bond pB,t and of the Tree pT ,t (both of which depend on the current state), make forecasts of (stochastic) future asset prices pB,t0 , pT ,t0 for each t 0 > t and optimize subject to their current budget constraint and their forecast of future asset prices. (More directly: agents optimize subject to their forecast of future consumption conditional on current portfolio choices.) At a Radner equilibrium (Radner, 1972) markets for consumption and assets clear at every date and state and all price forecasts are correct (“perfect foresight”). This is not quite enough for equilibrium to be well-defined because it does not rule out the possibility that agents acquire more and more debt, delaying repayment further and further into the future – and never in fact repaying it. In order that equilibrium be well-defined, such schemes must be ruled out. Levine and Zame (1996), Magill and Quinzii (1994) and Hernandez and Santos (1996) show that this can be done in a number of different ways. Levine and Zame (1996) show that all ‘reasonable’ ways lead to the same equilibria; the simplest is to require that debt not become unbounded.14 (In the experimental setting, we forbid short sales so debt is necessarily bounded.)

Because markets are dynamically complete, agents can trade to Pareto optimal allocations. Pricing in a Pareto-optimal allocation can be derived using the representative agent approach. Indeed, a representative agent exists, and prices should be such that the representative agent is willing to hold the supply of assets and consume no more or less than the aggregate dividend. In an economy with heterogeneous agents, the preferences of the representative agent are hard to derive. These preferences may not even look like those of any individual agent. However, since each period we only have two possible states (the state is either high H or low L), things simplify dramatically. To see this, fix an individual agent i; write {ci} for i’s equilibrium consumption stream. Write i’s first-order condition for optimality: p A,t = u 0 i (c H i ) u 0 i (c i ) (d H A + p H A,t+1) + (1 ) u 0 i (c L i ) u 0 i (c i ) (d L A + p L A,t+1) where superscripts index states and subscripts index assets, time, agents in the obvious way. We can write this in more compact form as p A,t = E u 0 i (ci) u 0 i (c i ) (dA + pA,t+1) (1) for = H, L and A = B, T. Equality of the ratios of marginal utilities across all agents, which is a consequence of Pareto optimality, implies that (1) is independent of the choice of agent i, and hence that we could write (1) in terms of the utility function of a representative agent. Let µ = u 0 i (c L i )/u0 i (c H i ) be the marginal rate of substitution in the Low state for consumption in the High state. Pareto optimality guarantees that µ is independent of which 12 agent i we use, so it is the marginal rate of substitution of any individual and that of the representative agent. (Note that risk aversion implies µ > 1.) Consequently: p H A = h (d H A + p H A ) + (1 )(d L A + p L A)µ i p L A = h (d H A + p H A )(1/µ) + (1 )(d L A + p L A) i Solving yields: p H A = 1 h dH A + (1 )d L A µ i p L A = 1 h dH A (1/µ) + (1 )d L A i (2) Specializing to the parameters of the experiment d H T = 1, dL T = 0; d H B = d L B = 0.5; = 5/6 yields: p H B = (2.5)(1 + µ)/2 p L B = (2.5)(1 + µ)/2µ p H T = 2.5 p L T = 2.5/µ (3) The important thing to note here is that: p B > p T , in each state . That is, the Bond is always priced above the Tree. Intuitively, this is because the “consumption beta” of the Tree is higher, and hence, is discounted more (relative to expected future dividends). The consumption beta of a security is the covariance of its future dividends with aggregate future consumption. Bond dividends are deterministic, while those of the Tree increase with aggregate consumption. Hence, the consumption beta of the Tree is higher than that of the Bond. (Notice also that, under our parametrization, p H T = 2.5; the price of the tree in the High state is independent of risk attitudes. In addition, p H B /pL B = p H T /pL T ; the ratios of asset prices in the two states are the same.)

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