equal lo the product of the к single-period returns from I - к + 1 to I, i.<

l + R,(k) s (I + !(,) (I + , ,) (! + 1<<-ш)

A. fjll 4l / - + = JL IU Л !! /Vs /V. Л-*

0-4.2)

and its net return over the most recent к periods, written R,(k), is simply

equal lo its A-period gross return minus one. These multiperiod returns are

called compound returns.

Although returns are scale-free, it should be emphasized that they arc not unitless.but are always defined with respect lo some time interval, e.g., one period. In fact, R, is more properly called a rate of return, which is mon cumbersome terminology but more accurate in referring to R, as a rate or, in1 economic jargon, a flow variable. Therefore, a return of 20% is noi a complete description of ihe investment opportunity without specification of the return horizon. In the academic literature, the return horizon is generally given explicitly, often as part of the data description, e.g., The CRSP monthly returns file was used.

ftowever, among practitioners and in the financial press, a reiurn- horizon of one year is usually assumed implicitly; hence, unless stated otherwise, a return of 20% is generally taken to mean an annual return of 20%. Moreover, multiyear returns are often annualized to make investments with

.different horizons comparable, thus:

Annualized[rt,(A)] =

(1.4.3)

Since Isingle-period returns are generally small in magnitude, the following approximation based on a first-order Taylor expansion is often used to annualize multiyear returns:

AnnuaIized[rt,(A)]

(1.4.4)

Whether such an approximation is adequate depends on the particular application al hand; it may suffice for a quick and coarse comparison of investment performance across many assets, but for liner calculations in which the volatility of returns plays an important role, i.e., when the higher-order terms in the Taylor expansion arc not negligible, the approximation (1.4.4) may break down. The only advantage of such an approximation is convenience-it is easier to calculate an arithmetic rather than a geometric average-however, this advantage has diminished considerably with the advent of cheap and convenient computing power.

Continuous Compounding

! ;c difficulty of manipulating geometric averages such as (1.4.3) motivates a.;,- .er approach lo compound returns, one which is not approximate and also Iras important implications for modeling asset returns; this is the notion of continuous compounding. The continuously compounded return or logreturn r, of an asset is defined to be the natural logarithm of ils gross return (1 +R,):

r, s logd + й,) = log- = (1.4.5)

i-i

where = log/1;. When we wish to emphasize the distinction between rt( and r we shall refer to R, as a simple return. Our notation here deviates slightly from our convention that lowercase letters denote the logs of uppercase letters, since here we have r, = log(l + rt,) rather than log(rt,); we do this lo maintain consistency with standard conventions.

The advantages of continuously compounded returns become clear when we consider multiperiod returns, since

r,(k) = log(l + R,(k)) = log((l + rt,) (1 + rt,~,) (1 + rt, .+t = log(l + R,) + log(l + rt,-,) + + log(l + , *+,) = r, + r, i +--- + r, n+1, (1.4.6)

and hence the continuously compounded multiperiod return is simply the sum of continuously compounded single-period returns. Compounding, a multiplicative operation, is converted lo an additive operation by taking logarithms. However, the simplification is not merely in reducing multiplication to addition (since we argued above thai wilh modern calculators and computers, this is trivial), but more in the modeling of the statistical behavior of asset returns over time-it is far easier to derive the time-series properties of additive processes than of multiplicative processes, as we shall see in Chapter 2.

Continuously compounded returns do have one disadvantage. The simple return on a portfolio of assets is a weighted average of the simple returns on ihe assets themselves, where the weight on each asset is the share of the portfolios value invested in that asset. If portfolio ji places weight uty in asset i, then the return on the portfolio at lime /, /<) , is related to the returns on individual assets, rt , i = 1 ...N, by R/ = £)i=t ntyrt - Unfortunately continuously compounded returns do not share this convenient property. Since the log of a sum is not the same as the sum ol logs, fy, does not equal

L,=i Vr -

In empirical applications this problem is usually minor. When returns are measured over shorl intervals of time, and are therefore close to zero, ihe continuously compounded return on a portfolio is close to the weighted

I), 1>п

1+ i

Figure I.I. Dividend Itrtmrnl Timing (jinventinn

average of die continuously compounded returns on the individual assets:

/ ~ <C;=i V W ,lst ms nppioxiniaiitin in Chapter .5. Nonetheless it is common lo use simple returns when a cross-section of assets is being studied, as in Chapters 4-0, and continuously compounded returns when the temporal behavior of returns is the focus of interest, as in Chapters 2 and 7.

Dividend Payments

For assets which make periodic dividend payments, we must modify our definitions of returns and compounding. Denote by D, the assets dividend payment al date / and assume, purely as a matter оГ convention, that this dividend is paid just before the date-/ price / is recorded; hence / is taken lo be the ex-dividend price al dale /. Alternatively, one might describe as an end-ol-period asset price, as shown in Figure 1.1. Then the net simple return at date / may be defined as

I, + 1),

R, =-- - 1. (1/1.7)

/-I

Multipcriod and continuously compounded returns may be obtained in the same way as in the no-dividends case. Note that the continuously compounded return on a dividend-paying asset, r, = \og(P, + D,) - log(/ i), is a nonlinear function of log prices and log dividends. When the ratio of prices to dividends is not too variable, however, this function can be approximated by a linear function of log prices and dividends, as discussed in detail in Chapter 7.

Excess Returns

It is often convenient lo work with an assets excess return, defined as the difference between the assets return and the return on some reference asset. The reference asset is often assumed to be riskless and in practice is usually a shorl-lei in Treasury bill return. Working with simple returns, the

-lllttu- Until vvlutv lime i\( (mutinous, tins t ,riiitiui,(tiseltsse<l ill Section .1.1.2 o(( .li.t]itci < ;ltt be llsffl to tchtte simple ;ui(l I otililtilousll < oni>oiltl(te(t returns.

simple excess return on asset i is

A, = R -liit, (1-4.8)

where R , is the reference return. Alternatively one can define a log excess return as

Mi = rit - r, . (1.4.9)

The excess return can also be thought of as the payoff on an arbitrage portfolio that goes long in asset г and short in the reference asset, with no net investment at the initial date. Since the initial net investment is zero, the return on the arbitrage portfolio is undefined but its dollar payoff is proportional to the excess return as defined above.

1.4.2 The Marginal, Conditional, and Joint Distribution of Returns

Having defined asset returns carefully, we can now begin to study their behavior across assets and over time. Perhaps the most important characteristic of asset returns is their randomness. The return of IBM stock over the next month is unknown today, and it is largely the explicit modeling of the sources and nature of this uncertainty that distinguishes financial economics from other social sciences. Although other branches of economics and sociology do have models of stochastic phenomena, in none of them does uncertainty play so central a role as in the pricing of financial assets-without uncertainty, much of the financial economics literature, both theoretical and empirical, would be superfluous. Therefore, we m jst articulate at the very start the types of uncertainty that asset returns might exhibit.

77i?Joint Distribution Consider a collection of N assets at date (, each with return Рц, at date where ( = 1,..., T. Perhaps the most general model of the collection returns (/{,() is its joint distribution function:

J, of

G(Rn.....Rni\Rii.....Rm\...\RlT.....Rnt, x 6 ), (1.4.1J0)

where x is a vector of state variables, variables that summarize the economic environment in which asset returns are determined, and в is a vector [of fixed parameters that uniquely determines G. For notational convenience, we shall suppress the dependence of G on the parameters в unless it is needed. j

The probability law С governs the stochastic behavior of asset returns and x, and represents the sum total of all knowable information about them. We may then view financial econometrics as the statistical inference of в, given С and realizations of \R ). Of course, (1.4.10) is far too general to

he of any use for statistical inference, and we shall have to place further restrictions on G in the coining sections and chapters. However, (1.1.10) docs serve as a convenient way to organize the many models of asset returns to be developed here and in later chapters. For example, Chapters 2 through 6 deal exclusively with the joint distribution of [R ], leaving additional state variables x to be considered in Chapters 7 and 8. We write this joint distribution as Сц.

1 Many asset pricing models, such as the Capital Asset Pricing Model (CAPM) orSharpe (1064), Uintncr (H)65a,b), and Mossin (1066) considered in Chapter 5, describe the joint distribution of the cross section of returns /t,.....Rfj,} at a single dale /. To reduce (1.4.10) to this essentially

stajic structure, we shall have to asserl that returns are statistically independent through time and that the joint distribution of the cross-section of rellirns is identical across time. Although such assumptions seem extreme, they yield a rich set of implications for pricing financial assets. The CAPM, forjexample, delivers an explicit formula for the trade-off between risk and expected return, the celebrated security market line. 1

The\ Conditional Distribution

In thapter 2, we place another set of restrictions on Сц which will allow us to focus on the dynamics of individual asset returns while abstracting from cross-sectional relations between the assets. In particular, consider the joint

distribution Fof (/f,i.....RlT) for a given asset i, and observe that we may

always rewrite Fas the following product:

FlRn.....R,r) - i ,!)/ № i Лй. ,)

lv,iR,r I K.r-1.....Rn). (1.4.11)

From (1.4.11), the temporal dependencies implicit in {Rlt} are apparent. Issues of predictability in asset returns involve aspects of their conditioned distributions and, in particular, how the conditional distributions evolve through lime.

By placing further restrictions on the conditional distributions / ( ), we shall be able to estimate the parameters в implicit in (1.4.11) and examine the predictability of asset returns explicitly. For example, one version of the random-walk hypothesis is obtained by die restriction that the conditional distribution of return R is equal to its marginal distribution, i.e., ln(R,i I ) = F,i{Rji). If this is the case, then returns are temporally independent and therefore unpredictable using past returns. Weaker versions of the random walk are obtained by imposing weaker restrictions on / (/( ).

The Unconditional Distribution

In cases where an asset returns conditional distribution differs from its marginal or unconditional distribution, it is clearly the conditional distribu-

lion that is relevant for issues involving predictability. However, the proper-lies of die unconditional distribution of returns may still be of some interest, especially in cases where we expect predictability to be minimal.

One of the most common models for asset returns is the temporally independently and identically distributed (IID) normal model, in which returns are assumed to be independent over time (although perhaps cross-sectionally correlated), identically distributed over lime, and normally distributed. The original formulation of the CAPM employed this assumption of normality, although returns were only implicitly assumed to be temporally IID (since ii was a static two-period model). More recently, models of asymmetric information such as Grossman (1989) and Grossman and Stiglitz (1980) also use normality.

While the temporally IID normal model may be tractable, it suffers from at least two important drawbacks. First, most financial assets exhibit limited liability, so that the largest loss an investor can realize is his total investment and no more. This implies that the smallest net return achievable is -1 or -100%. But since the normal distributions support is the entire real H11, this lower bound of -1 is clearly violated by normality. Of course, it may be argued that by choosing the mean and variance appropriately, the probability of realizations below -1 can be made arbitrarily small; however it will never be zero, as limited liability requires.

Second, if single-period returns are assumed to be normal, then multi-period returns cannot also be normal since they are the products of the single-period returns. Now the sums of normal single-period returns are indeed normal, but the sum of single-period simple returns does not have any economically meaningful interpretation. However, as we saw in Section 1.4.1, the sum of single-period continuously compounded returns does have a meaningful interpretation as a multiperiod continuously compounded return.

The Lognormal Distribution

A sensible alternative is to assume that continuously compounded single-period returns r are IID normal, which implies that single-period gross simple returns are distributed as IID lognormal variatcs, since r = log(l + Kj,). We may express the lognormal model then as

TU ~ Mill a f). (1.4.12)

Under the lognormal model, if the mean and variance of r are /z, and aj, respectively, then the mean and variance of simple returns are given by

E[/< ] = , i+t - 1 (1.4.13)

Varf/r ] F ra -f -V - П. (1.4.14)