Chapter1.Consumption-bad model and overview病句练习题
I start by thinking of an investor who thinks about how much to save and consume,and what portfolio of asts to hold.The most basic pricing equation comes from the¿rst-order conditions to that problem,and say that price should be the expected discounted payoff,using the investor’s marginal utility to discount the payoff.The marginal utility loss of consuming a little less today and investing the result should equal the marginal utility gain of lling the investment at some point in the future and eating the proceeds.If the price does not satisfy this relation,the investor should buy more of the ast.decision是什么意思
From this simple idea,I can discuss the classic issues in¿nance.The interest rate is related to the average future marginal utility,and hence to the expected path of consumption. High real interest rates should be associated with an expectation of growing consumption.In a time of high real interest rates,it makes n to save,buy bonds,and then consume more tomorrow.
Most importantly,risk corrections to ast prices should be driven by the covariance of ast payoffs with consumption or marginal utility.For a given expected payoff of an ast, an ast that does badly in states like a recession,in which the investor feels poor and is consuming little,is less desirable than an ast that does badly in states of nature like a boom when the investor feels wealthy and is consumi
ng a great deal.The former asts will ll for lower prices;their prices will reÀect a discount for their riskiness,and this riskiness depends on a co-variance.This is the fundamental point of the whole book.
Of cour,the fundamental measure of how you feel is marginal utility;given that asts must pay off well in some states and poorly in others,you want asts that pay off poorly in states of low marginal utility,when an extra dollar doesn’t really em all that important,and you’d rather that they pay off well in states of high marginal utility,when you’re hungry and really anxious to have an extra dollar.Most of the book is about how to go from marginal utility to obrvable indicators.Consumption is low when marginal utility is high,of cour, so consumption may be a uful indicator.Consumption is also low and marginal utility is high when the investor’s other asts have done poorly;thus we may expect that prices are
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C HAPTER1C ONSUMPTION-BASE
D MODEL AND OVERVIEW
low for asts that covary positively with a large index such as the market portfolio.This is the Capital Ast Pricing Model.The rest of the book comes down to uful indicators for marginal utility,things against which to compute a covariance in order to predict the risk-adjustment for prices.
1.1Basic pricing equation
An investor’s¿rst order conditions give the basic consumption-bad model,
p t=E t ·
u0(c t+1)
0(c t)x t+1
¸
.
Our basic objective is to¿gure out the value of any stream of uncertain cashÀows.I start with an apparently simple ca,which turns out to capture very general situations.
Let us¿nd the value at time t of a payoff x t+1.For example,if one buys a stock today, the payoff next period is the stock price plus dividend,x t+1=p t+1+d t+1.x t+1is a random variable:an investor does not know exactly how much he will get from his investment,but he can asss the probability of various possible outcomes.Don’t confu the payoff x t+1with the pro¿t or return;x t+1is the value of the investment at time t+1,without subtracting or dividing by the cost of the investment.
We¿nd the value of this payoff by asking what it is worth to a typical investor.To do this, we need a convenient mathematical formalism to capture what an investor wants.We model investors by a utility function de¿ned over current and future values of consumption,
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U(c t,c t+1)=u(c t)+ E t[u(c t+1)],
where c t denotes consumption at date t.We will often u a convenient power utility form,
u(c t)=
1
1 c
1
t
.
The limit as $1is
u(c)=ln(c).
The utility function captures the fundamental desire for more consumption,rather than posit a desire for intermediate objectives such as means and variance of portfolio returns. Consumption c t+1is also random;the investor does not know his wealth tomorrow,and hence how much he will decide to consume.The period utility function u(·)is increasing, reÀecting a desire for more consumption,and concave,reÀecting the declining marginal value of additional consumption.The last bite is never as satisfying as the¿rst.
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S ECTION1.1B ASIC PRICING EQUATION
This formalism captures investors’impatience and their aversion to risk,so we can quan-titatively corr
ect for the risk and delay of cashÀows.Discounting the future by captures impatience,and is called the subjective discount factor.The curvature of the utility func-tion also generates aversion to risk and to intertemporal substitution:The consumer prefers a consumption stream that is steady over time and across states of nature.
Now,assume that the investor can freely buy or ll as much of the payoff x t+1as he wishes,at a price p t.How much will he buy or ll?To¿nd the answer,denote by e the original consumption level(if the investor bought none of the ast),and denote by the amount of the ast he choos to buy.Then,his problem is,
max
{ }
u(c t)+E t u(c t+吸血鬼日记6
c t=e t p t
c t+1=e t+1+x t+1
Substituting the constraints into the objective,and tting the derivative with respect to equal to zero,we obtain the¿rst-order condition for an optimal consumption and portfolio choice,
p t u0(c t)=E t[ u0(c t+1)x t+1](1) or,
p t=E t ·
u0(c t+1)
u0(c t)
x t+1
¸
.(2)
The investor buys more or less of the ast until this¿rst order condition holds.
Equation(1.1)express the standard marginal condition for an optimum:p t u0(c t)is the loss in utility if the investor buys another unit of the ast;E t[ u0(c t+1)x t+1]is the increa in(discounted,expecte
d)utility he obtains from the extra payoff at t+1.The investor continues to buy or ll the ast until the marginal loss equals the marginal gain.
别对我撒谎Equation(1.2)is the central ast-pricing formula.Given the payoff x t+1and given the investor’s consumption choice c t,c t+1,it tells you what market price p t to expect.Its eco-nomic content is simply the¿rst order conditions for optimal consumption and portfolio for-mation.Most of the theory of ast pricing just consists of specializations and manipulations of this formula.
Notice that we have stopped short of a complete solution to the an expression with exogenous items on the right hand side.We relate one endogenous variable,price, to two other endogenous variables,consumption and payoffs.One can continue to solve this model and derive the optimal consumption choice c t,c t+1in terms of the givens of the model.In the model I have sketched so far,tho givens are the income quence e t,e t+1and a speci¿cation of the full t of asts that the investor may buy and ll.We will in fact study
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C HAPTER1C ONSUMPTION-BASE
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such fuller solutions below.However,for many purpos one can stop short of specifying (possibly wrongly)all this extra structure,and obtain very uful predictions about ast prices from(1.2),even though consumption is an endogenous variable.
1.2Marginal rate of substitution/stochastic discount factor
We break up the basic consumption-bad pricing equation into
p=E(mx)
no morem= u0(c t+1) u0(c t)
where m t+1is the stochastic discount factor.
A convenient way to break up the basic pricing equation(1.2)is to de¿ne the stochastic discount factor m t+1
m t+1 u0(c t+1)
u0(c t)
(3)
Then,the basic pricing formula(1.2)can simply be expresd as
p t=E t(m t+1x t+1).(4) When it isn’t necessary to be explicit about time subscripts or the difference between conditional and unconditional expectation,I’ll suppress the subscripts and just write p= E(mx).The price always comes at t,the payoff at t+1,and the expectation is conditional
on time t information.
十二月英文缩写The term stochastic discount factor refers to the way m generalizes standard discount factor ideas.If there is no uncertainty,we can express prices via the standard prent value formula
p t=1
zhonghex t+1(5)
where R f is the gross risk-free rate.1/R f is the discount factor.Since gross interest rates are typically greater than one,the payoff x t+1lls“at a discount.”Riskier asts have lower prices than equivalent risk-free asts,so they are often valued by using risk-adjusted
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S ECTION1.3P RICES,PAYOFFS AND NOTATION discount factors,
p i t=1
R
E t(x i t+1).
Here,I have added the i superscript to emphasize that each risky ast i must be discounted by an ast-speci¿c risk-adjusted discount factor1/R i.
In this context,equation(1.4)is obviously a generalization,and it says something deep: one can incorporate all risk-corrections by de¿ning a single stochastic discount factor–the same one for each ast–and putting it inside the expectation.m t+1is stochastic or random becau it is not known with certainty at time t.As we will e,the correlation between the random components of m and x i generate ast-speci¿c risk corrections.
m t+1is also often called the marginal rate of substitution after(1.3).In that equation, m t+1is the rate
at which the investor is willing to substitute consumption at time t+1for consumption at time t.m t+1is sometimes also called the pricing kernel.If you know what a kernel is and express the expectation as an integral,you can e where the name comes from. It is sometimes called a change of measure or a state-price density for reasons that we will e below.
For the moment,introducing the discount factor m and breaking the basic pricing equa-tion(1.2)into(1.3)and(1.4)is just a notational convenience.As we will e,however,it reprents a much deeper and more uful paration.For example,notice that p=E(mx) would still be valid if we changed the utility function,but we would have a different func-tion connecting m to data.As we will e,all ast pricing models amount to alternative models connecting the stochastic discount factor to data,while p=E(mx)is a convenient accounting identity with almost no content.At the same time,we will study lots of alter-native expressions of p=E(mx),and we can summarize many empirical approaches to p=E(mx).By parating our models into the two components,we don’t have to redo all that elaboration for each ast pricing model.
1.3Prices,payoffs and notation
The price p t gives rights to a payoff x t+1.In practice,this notation covers a variety of cas,including the following:
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C HAPTER1C ONSUMPTION-BASE影视编导培训班
D MODEL AND OVERVIEW
The price p t and payoff x t+1em like a very restrictive kind of curity.In fact,this notation is quite general and allows us easily to accommodate many different ast pricing questions.In particular,we can cover stocks,bonds and options and make clear that there is one theory for all ast pricing.
For stocks,the one period payoff is of cour the next price plus dividend,x t+1=p t+1+ d t+1.We frequently divide the payoff x t+1by the price p t to obtain a gross return
R t+1 x t+1 p t
We can think of a return as a payoff with price one.If you pay one dollar today,the return is how many dollars or units of consumption you get tomorrow.Thus,returns obey
1=E(mR)
which is by far the most important special ca of the basic formula p=E(mx).I u capital letters to denote gross returns R,which have a numerical value like1.05.I u lowerca letters to denote net returns r=R 1or log(continuously compounded)returns ln(R),both of which have numerical values like0.05.One may also quote percent returns100×r.
Returns are often ud in empirical work becau they are typically stationary over time. (Stationary in the statistical n;they don’t have trends and you can meaningfully take an average.“Stationary”does not mean constant.)However,thinking in terms of returns takes us away from the central task of¿nding ast prices.Dividing by dividends and creating a payoff
x t+1=μ
1+p t+1
d t+1
¶
d t+1
d t
corresponding to a price p t/d t is a way to look at prices but still to examine stationary vari-ables.
Not everything can be reduced to a return.If you borrow a dollar at the interest rate R f and invest it in an ast with return R,you pay no money out-of-pocket today,and get the
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