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the discount factor
To value an asset, we have to account for the delay and for the risk of its payments. The effects of time are not too difficult to work out. However, corrections for risk are much more important determinants of an many assets values. For example, over the last 50 years U.S. stocks have given a real return of about 9% on average. Of this, only about 1% is due to interest rates; the remaining 8% is a premium earned for holding risk. Uncertainty, or corrections for risk make asset pricing interesting and challenging.
Asset pricing theory shares the positive vs. normative tension present in the rest of economics. Does it describe the way the world does work or the way the world should work? We observe the prices or returns of many assets. We can use the theory positively, to try to understand why prices or returns are what they are. If the world does not obey a models predictions, we can decide that the model needs improvement. However, we can also decide that the world is wrong, that some assets are mis-priced and present trading opportunities for the shrewd investor. This latter use of asset pricing theory accounts for much of its popularity and practical application. Also, and perhaps most importantly, the prices of many assets or claims to uncertain cash flows are not observed, such as potential public or private investment projects, new financial securities, buyout prospects, and complex derivatives. We can apply the theory to establish what the prices of these claims should be as well; the answers are important guides to public and private decisions.
Asset pricing theory all stems from one simple concept, derived in the first page of the first Chapter of this book: price equals expected discounted payoff. The rest is elaboration, special cases, and a closet full of tricks that make the central equation useful for one or another application.
There are two polar approaches to this elaboration. I will call them absolute pricing and relative pricing.In absolute pricing, we price each asset by reference to its exposure to fundamental sources of macroeconomic risk. The consumption-based and general equilibrium models described below are the purest examples of this approach. The absolute approach is most common in academic settings, in which we use asset pricing theory positively to give an economic explanation for why prices are what they are, or in order to predict how prices might change if policy or economic structure changed.
In relative pricing, we ask a less ambitious question. We ask what we can learn about an assets value given the prices of some other assets. We do not ask where the price of the other set of assets came from, and we use as little information about fundamental risk factors as possible. Black-Scholes option pricing is the classic example of this approach. While limited in scope, this approach offers precision in many applications.
Asset pricing problems are solved by judiciously choosing how much absolute and how much relative pricing one will do, depending on the assets in question and the purpose of the calculation. Almost no problems are solved by the pure extremes. For example, the CAPM and its successor factor models are paradigms of the absolute approach. Yet in applications, they price assets relative to the market or other risk factors, without answering what determines the market or factor risk premia and betas. The latter are treated as free parameters. On the other end of the spectrum, most practical financial engineering questions involve assumptions beyond pure lack of arbitrage, assumptions about equilibrium market prices of risk.
The central and unfinished task of absolute asset pricing is to understand and measure the sources of aggregate or macroeconomic risk that drive asset prices. Of course, this is also the central question of macroeconomics, and this is a particularly exciting time for researchers who want to answer these fundamental questions in macroeconomics and finance. A lot of empirical work has documented tantalizing stylized facts and links between macroeconomics and finance. For example, expected returns vary across time and across assets in ways that are linked to macroeconomic variables, or variables that also forecast macroeconomic events; a wide class of models suggests that a recession or financial distress factor lies behind many asset prices. Yet theory lags behind; we do not yet have a well-described model that explains these interesting correlations.
In turn, I think that what we are learning about finance must feed back on macroeconomics. To take a simple example, we have learned that the risk premium on stocks - the expected stock return less interest rates - is much larger than the interest rate, and varies a good deal more than interest rates. This means that attempts to line investment up with interest rates are pretty hopeless - most variation in the cost ofcapital comes from the varying risk premium. Similarly, we have learned that some measure of risk aversion must be quite high, or people would all borrow like crazy to buy stocks. Most macroeconomics pursues small deviations about perfect foresight equilibria, but the large equity premium means that volatility is a first-order effect, not a second-order effect. Standard macroeconomic models predict that people really dont care much about business cycles (Lucas 1987). Asset prices are be-ginningtorevealthattheydo-thattheyforegosubstantialreturnpremiatoavoidassetsthat fall in recessions. This fact ought to tell us something about recessions!
This book advocates a discount factor / generalized method of moments view of asset pricing theory and associated empirical procedures. I summarize asset pricing by two equations:
pt = E(mt+i xt+i)
mt+i = f (data, parameters). where pt = asset price, xt+i = asset payoff, mt+i = stochastic discount factor.
The major advantage of the discount factor / moment condition approach are its simplicity and universality. Where once there were three apparently different theories for stocks, bonds, and options, now we see each as just special cases of the same theory. The common language also allows us to use insights from each field of application in other fields.
This approach also allows us to conveniently separate the step of specifying economic assumptions of the model (second equation) from the step of deciding which kind of empirical representation to pursue or understand. For a given model - choice of f ( ) - we will see how the first equation can lead to predictions stated in terms of returns, price-dividend ratios, expected return-beta representations, moment conditions, continuous vs. discrete time implications and so forth. The ability to translate between such representations is also very helpful in digesting the results of empirical work, which uses a number of apparently distinct but fundamentally connected representations.
Thinking in terms of discount factors often turns out to be much simpler than thinking in terms of portfolios. For example, it is easier to insist that there is a positive discount factor than to check that every possible portfolio that dominates every other portfolio has a larger price, and the long arguments over the APT stated in terms of portfolios are easy to digest when stated in terms of discount factors.
The discount factor approach is also associated with a state-space geometry in place of the usual mean-variance geometry, and this book emphasizes the state-space intuition behind many classic results.
For these reasons, the discount factor language and the associated state-space geometry is common in academic research and high-tech practice. It is not yet common in textbooks, and that is the niche that this book tries to fill.
I also diverge from the usual order of presentation. Most books are structured following the history of thought: portfolio theory, mean-variance frontiers, spanning theorems, CAPM, ICAPM,APT,optionpricing, and finally consumption-based model. Contingent claims are an esoteric extension of option-pricing theory. I go the other way around: contingent claims and the consumption-based model are the basic and simplest models around; the others are specializations. Just because they were discovered in the opposite order is no reason to present them that way.
I also try to unify the treatment of empirical methods. A wide variety of methods are popular, including time-series and cross-sectional regressions, and methods based on generalized method of moments (GMM) and maximum likelihood. However, in the end all of these apparently different approaches do the same thing: they pick free parameters of the model to make it fit best, which usually means to minimize pricing errors; and they evaluate the model by examining how big those pricing errors are.
As with the theory, I do not attempt an encyclopedic compilation of empirical procedures. The literature on econometric methods contains lots of methods and special cases (likelihood ratio analogues of common Wald tests; cases with and without riskfree assets and when factors do and dont span the mean variance frontier, etc.) that are seldom used in practice. I
try to focus on the basic ideas and on methods that are actually used in practice.
The accent in this book is on understanding statements of theory, and working with that theory to applications, rather than rigorous or general proofs. Also, I skip very lightly over many parts of asset pricing theory that have faded from current applications, although they occupied large amounts of the attention in the past. Some examples are portfolio separation theorems, properties of various distributions, or asymptotic APT. While portfolio theory is still interesting and useful, it is no longer a cornerstone of pricing. Rather than use portfolio theory to find a demand curve for assets, which intersected with a supply curve gives prices, we now go to prices directly. One can then find optimal portfolios, but it is a side issue for the asset pricing question.
My presentation is consciously informal. I like to see an idea in its simplest form and learn to use it before going back and understanding all the foundations of the ideas. I have organized the book for similarly minded readers. If you are hungry for more formal definitions and background, keep going, they usually show up later on in the chapter.
Again, my organizing principle is that everything can be traced back to specializations of the basic pricing equation p = E(mx). Therefore, after reading the first chapter, one can pretty much skip around and read topics in as much depth or order as one likes. Each major subject always starts back at the same pricing equation.
The target audience for this book is economics and finance Ph.D. students, advanced MBA students or professionals with similar background. I hope the book will also be useful to fellow researchers and finance professionals, by clarifying, relating and simplifying the set of tools we have all learned in a hodgepodge manner. I presume some exposure to undergraduate economics and statistics. A reader should have seen a utility function, a random variable, a standard error and a time series, should have some basic linear algebra and calculus and should have solved a maximum problem by setting derivatives to zero. The hurdles in asset pricing are really conceptual rather than mathematical.
PART I Asset pricing theory
Chapter 1. Consumption-based model and overview
I start by thinking of an investor who thinks about how much to save and consume, and what portfolio of assets to hold. The most basic pricing equation comes from the first-order conditions to that problem, and say that price should be the expected discounted payoff, using the investors marginal utility to discount the payoff. The marginal utility loss ofconsuming a little less today and investing the result should equal the marginal utility gain of selling 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 asset.
From this simple idea, I can discuss the classic issues in finance. 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 sense to save, buy bonds, and then consume more tomorrow.
Most importantly, risk corrections to asset prices should be driven by the covariance of asset payoffs with consumption or marginal utility. For a given expected payoff of an asset, an asset that does badly in states like a recession, in which the investor feels poor and is consuming little, is less desirable than an asset that does badly in states of nature like a boom when the investor feels wealthy and is consuming a great deal. The former assets will sell for lower prices; their prices will reflect a discount for their riskiness, and this riskiness depends on a co-variance. This is the fundamental point of the whole book.
Of course, the fundamental measure of how you feel is marginal utility; given that assets must pay off well in some states and poorly in others, you want assets that pay off poorly in states of low marginal utility, when an extra dollar doesnt really seem all that important, and youd rather that they pay off well in states of high marginal utility, when youre hungry and really anxious to have an extra dollar. Most of the book is about how to go from marginal utility to observable indicators. Consumption is low when marginal utility is high, of course, so consumption may be a useful indicator. Consumption is also low and marginal utility is high when the investors other assets have done poorly; thus we may expect that prices are
low for assets that covary positively with a large index such as the market portfolio. This is the Capital Asset Pricing Model. The rest of the book comes down to useful indicators for marginal utility, things against which to compute a covariance in order to predict the risk-adjustment for prices.
1.1 Basic pricing equation
An investors first order conditions give the basic consumption-based model,
p E Lu,(ct+i) x
Pt Et p---- Xt+i .
Our basic objective is to figure out the value of any stream of uncertain cash flows. I start with an apparently simple case, which turns out to capture very general situations.
Let us find the value at time t of a payoff xt+1. For example, if one buys a stock today, the payoff next period is the stock price plus dividend, xt+1 pt+1 + dt+1. xt+1 is a random variable: an investor does not know exactly how much he will get from his investment, but he can assess the probability of various possible outcomes. Dont confuse the payoff xt+1 with the profit or return; xt+1 is the value of the investment at time t + 1, without subtracting or dividing by the cost of the investment.
We find 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 defined over current and future values of consumption,
U(ct,Ct+1) u(ct)+ PEt [u(ct+1)],
where ct denotes consumption at date t. We will often use a convenient power utility form,
The limit as 7 - 1 is
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 ct+1 is 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, reflecting a desire for more consumption, and concave, reflecting the declining marginal value of additional consumption. The last bite is never as satisfying as the first.
Section 1.1 Basic pricing equation
This formalism captures investors impatience and their aversion to risk, so we can quantitatively correct for the risk and delay of cash flows. Discounting the future by в captures impatience, and в is called the subjective discount factor. The curvature of the utility function 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 sell as much of the payoff xt+\ as he wishes, at a price pt. How much will he buy or sell? To find the answer, denote by e the original consumption level (if the investor bought none of the asset), and denote by £ the amount of the asset he chooses to buy. Then, his problem is,
max u(ct) + Et0u(ct+i) s.t.
ct = et - pt£ ct+i = et+i + xt+i£
Substituting the constraints into the objective, and setting the derivative with respect to £ equal to zero, we obtain the first-order condition for an optimal consumption and portfolio choice,
ptu(ct)= Et [ви1 (ct+i)xt+i] (1)
r u(ct+i) 1 pt = Et в )f+yxt+i . (2)
The investor buys more or less of the asset until this first order condition holds.
Equation (1.1) expresses the standard marginal condition for an optimum: ptu(ct) is the loss in utility if the investor buys another unit of the asset; Et [pu(ct+i)xt+i] is the increase in (discounted, expected) utility he obtains from the extra payoff at t+1. The investor continues to buy or sell the asset until the marginal loss equals the marginal gain.
Equation (1.2) is the central asset-pricing formula. Given the payoff xt+i and given the investors consumption choice ct,ct+i, ittells you what marketprice pt to expect. Its economic content is simply the first order conditions for optimal consumption and portfolio formation. Most of the theory of asset pricing just consists of specializations and manipulations of this formula.
Notice that we have stopped short of a complete solution to the model, i.e. 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 ct,ct+i in terms of the givens of the model. In the model I have sketched so far, those givens are the income sequence et,et+i and aspecification of the full set of assets that the investor may buy and sell. We will in fact study
such fuller solutions below. However, for many purposes one can stop short of specifying (possibly wrongly) all this extra structure, and obtain very useful predictions about asset prices from (1.2), even though consumption is an endogenous variable.
1.2 Marginal rate of substitution/stochastic discount factor
We break up the basic consumption-based pricing equation into
where mt+1 is the stochastic discount factor.
A convenient way to break up the basic pricing equation (1.2) is to define the stochastic discount factor mt+1
Then, the basic pricing formula (1.2) can simply be expressed as
pt Et(mt+1xt+1). (4)
When it isnt necessary to be explicit about time subscripts or the difference between conditional and unconditional expectation, ill 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.
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 present value formula
Pt Rf xt+1 (5)
where Rf is the gross risk-free rate. 1/Rf is the discount factor. Since gross interest rates are typically greater than one, the payoff xt+1 sells at a discount. Riskier assets have lower prices than equivalent risk-free assets, so they are often valued by using risk-adjusted
Section 1.3 Prices, payoffs and notation
Here, I have added the i superscript to emphasize that each risky asset i must be discounted by an asset-specific risk-adjusted discount factor 1/Д\
In this context, equation (1.4) is obviously a generalization, and it says something deep: one can incorporate all risk-corrections by defining a single stochastic discount factor - the same one for each asset - and putting it inside the expectation. mt+i is stochastic or random because it is not known with certainty at time t. As we will see, the correlation between the random components of m and xl generate asset-specific risk corrections.
mt+i is also often called the marginal rate of substitution after (1.3). In that equation, mt+i is the rate at which the investor is willing to substitute consumption at time t +1 for consumption at time t. mt+iis sometimes also called the pricing kernel. If you know what a kernel is and express the expectation as an integral, you can see where the name comes from. it is sometimes called a change ofmeasure or a state-price density for reasons that we will see below.
For the moment, introducing the discount factor m and breaking the basic pricing equation (1.2) into (1.3) and (1.4) is just a notational convenience. As we will see, however, it represents a much deeper and more useful separation. For example, notice that p = E(mx) would still be valid if we changed the utility function, but we would have a different function connecting m to data. As we will see, all asset 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 alternative expressions of p = E(mx), and we can summarize many empirical approaches to p = E(mx). By separating our models into these two components, we dont have to redo all that elaboration for each asset pricing model.
1.3 Prices, payoffs and notation
The price pt givesrightstoa payoff xt+i. In practice, this notation covers a variety of cases, including the following:
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