The numeraire portfolio

Journal

of Financial

Economics

The numeraire

26 (1990) 29-69.

North-Holland

portfolio

John B. Long, Jr.* Unicersiry of Rochester, Received

November

Rochester,

NY l-1627, WA

1988, final version

received

March

1990

A portfolio formed from a given list of assets is defined as a numeraire portfolio for the list if (a) it is self-financing, (b) its value is always positive. and (c) zero is always the best conditional forecast of the numeraire-dominated rate of return of every asset on the list. The numeraire portfolio exists if and only if there are no profit opportunities from trading assets on the list. For a sample list of heterogeneous assets (NYSE size-quintile portfolios, corporate bonds, and short-term bills). numeraire-dominated returns are similar to market-model forecast errors and, as abnormal return measures, clearly dominate market-adjusted returns.

1. Introduction

I will say that a list of assets presents a profit opportunity if, trading only assets on the list, you can get something for nothing in one of the following ways: (i) for a zero or negative current price you can buy claim to a nonnegative future payoff that is positive with positive probability or (ii) you can sell claim to a certain zero future payoff for a positive current price.’ Absence of such opportunities is a minimal implication of hypotheses that the asset market is efficient or in equilibrium. My central theoretical conclusion is a refinement of the idea that the absence of profit opportunities involves some sort of ex ante equality of asset *I have received helpful comments from Avi Bick, Eugene Fama (the editor), Wayne Ferson, Marshall Freimer, Michael Jensen, Richard Roll (the referee), Clifford Smith, Reni Stulz, Jerold Warner, and from G. William Schwert, who also provided data. I also benefited from reactions to earlier versions of this work at the University of Chicago, New York University, Ohio State University, and Vanderbilt University. Financial support for this research was provided by the Bradley Policy Research Center and the Managerial Economics Research Center. ‘arbitrage opportunities of the first type’ and ‘Ingersoll (1987) calls these. respectively. ‘arbitrage opportunities of the second type’. In the finance literature [for example, Ross (1978)], ‘arbitrage opportunities’ are often defined as only the first type although the second type better fits the dictionary definition of arbitrage. .

0304-405X/90/503.50

J.F E.-B

D 1990-Elsevier

Science

Publishers

B.V. (North-Holland)

30

J.B. Long, Jr.. The numeraire portfolio

rates of return. Most asset-pricing models express this idea as an C-Yante equality of risk-adjusted expected rates of return, My conclusion, in contrast, does not directly involve any notion of risk. Given frictionless trading and two restrictions on the joint distribution of returns on individual assets (that per-unit prices and dividends are bounded over finite time intervals and that there is a self-financing portfolio’ whose value is always positive), the conclusion is: An asset list offers no profit opportunities if and only if a numeraire portfolio can be formed from the list. A numeraire portfolio is defined to be a self-financing portfolio such that, if current and future asset prices and dividends are denominated in units of the numeraire (that is, divided by the contemporaneous value of the numeraire portfolio), the expected rate of return of every asset on the list is always equal to zero. This conclusion is quite general with respect to how trading and asset returns are modeled. It is true, for example, with continuousor discrete-time trading, with continuous or discrete return distributions and with complete or incomplete markets. The essential assumptions are those already mentioned: no transaction costs, no restrictions on short sales, bounded prices and dividends of individual assets, and the availability of a self-financing portfolio whose value is always positive. The numeraire portfolio is related to several ideas that have been much discussed in the literature of asset pricing, portfolio theory, and event-study methodology. They include (9 the behavior of asset returns in efficient capital markets, (ii) pricing by risk-neutral valuation, (iii) portfolio strategies that are optimal (by some criterion) in the long run, and (iv) empirical definitions of abnormal returns. The relations between these ideas and the numeraire portfolio are summarized briefly below. In the academic finance literature on Asset returns in efficient markets. efficient markets3 the dominant question has been: ‘How should asset prices behave if there are no profit opportunities?’ In this context, ‘no profit opportunities’ usually refers not only to what I call profit opportunities, but also to other deviations of asset prices or expected returns from their equilibrium values. Thus, ‘no profit opportunities’ implies at least that there are no profit opportunities as I have defined them. A specific answer to the question of how asset prices should behave is then that, based on all available information, zero should always be the best forecast of every asset’s numeraire-denominated rate of return. This is an exact sense in which efficient ‘After its initial purchase, a self-financing (net dividends minus net new purchases). ‘See Fama (1976)

portfolio

is managed

to have zero net cash-flows

for an extensive discussion df the notion of efficient

markets.

1. B. Long, Jr.. The numrram

31

portfolio

markets imply zero serial correlation in asset rates of return and zero profits to trading strategies based on available information. It is also an appropriate basis for variance-bound tests of whether asset prices in fact reflect best forecasts of future dividends.” Risk-neutral caluation. Cox and Ross (1976) argue that if a claim’s payoffs can be exactly replicated by trading in securities and riskless loans, the equilibrium relation between the price of the claim, the interest rate, and security prices should be what it would be if all investors were risk-neutral. Since, in equilibrium, risk-neutral investors must expect the same rate of return on all securities, pricing the claim by this argument involves (a) substituting for the actual probability measure an equivalent measure5 under which the expected rate of return on all securities equals the riskless interest rate, (b) computing the expected payoff(s) of the claim using the equivalent probability measure, and cc> discounting the result to the present at the riskless rate. If there is a constant riskless interest rate and if present and future asset prices are divided by the contemporaneous price of a long-term riskless pure discount bond, then (assuming no dividends) the bond-denominated prices of all assets are martingales under the Cox-Ross equivalent probability measure. Harrison and Kreps (1979) thus call the equivalent measure an ‘equivalent martingale measure’ and show, in a fairly general model, that existence of such a measure is necessary and sufficient for the absence of profit opportunities. My contribution is to show that if prices are denominated in the numeraire portfolio, then the actual probability measure is a martingale measure. No substitution of a different measure is necessary in valuation. Long-run growth. The numeraire portfolio for a given asset list is the portfolio that would be chosen by a log utility investor if his trading were restricted to that list. With a probability approaching one as the investment horizon recedes, this portfolio grows faster than any other self-financing portfolio from the list. Thus, when the list includes all available assets, the numeraire portfolio is the ‘growth-optimal’ portfolio first suggested by Kelly (1956), Latane (1959), Markowitz (19591, and Breiman (1960). The long-run growth property of the portfolio is of no interest here, but it has fascinated dozens of authors who have written a large literature focused on the mathematical properties of the portfolio.6 There is also a large literature that ‘See, for example. 5Two probability events.

Leroy and Porter measures

(1981) and Schiller

are equivalent

if they

(1981).

assign

positive

probabilities

to the same

‘This literature seems to be concentrated in the 1960s and early 1970s. Hakansson (1971) is an example that also compares the growth-optimal portfolio with mean-variance-efficient portfolios. Hakansson’s paper contains an extensive bibliography.

32

J.B. Long, Jr., Thr numemrr

portfolio

develops an equilibrium asset pricin g model based on the assumption that actual or representative consumer-investors have log utility.‘Although the log utility assumption is never made here. it does have a distinctive equilibrium implication: that the numeraire portfolio for the entire market is the market portfolio of all assets. Thus tests of the log utility model by Roll (19731, Fama and MacBeth (19741, Kraus and Litzenberger (1975), and Grauer (1978) can be interpreted as tests of whether common empirical proxies for the market portfolio also proxy for the stock market numeraire portfolio (even though the theory of the numeraire portfolio does not require that it be the market portfolio).’ The evidence is consistent with the proposition that expected proxy-denominated stock returns are zero. Roll (1973, p. 555) even worries that the evidence supports the proposition ‘too strongly’!

Measuring abnormal returns. Numeraire-denominated returns are natural measures of abnormal returns. If there are no profit opportunities, the best forecast of future numeraire-denominated returns is zero. An asset’s numeraire-denominated gross return (one plus its rate of return) for a period is defined as its nominal gross return divided by the numeraire’s nominal gross return. Thus, numeraire-denominated returns are nominal returns adjusted to reflect the contemporaneous return on the market (as measured by the nominal return on the numeraire portfolio), and the numeraire-denominated rate of return on itself is zero by construction. In this sense, numerairedenominated returns measure asset-specific returns in the same sense as market-model residuals. Given a time series of nominal returns on a numeraire proxy, however, numeraire-denominated returns have at least two advantages over market-model forecast errors as measures of abnormal returns. First, because the multivariate process of numeraire-denominated returns depends only on relative gross returns, it is stationary under a broader range of circumstances than the multivariate process of nominal returns. Shifts in the nominal inflation process, for example, may cause shifts in assets’ market-model parameters even though they don’t affect relative gross returns (and hence numeraire-denominated returns). Second, given a time series of nominal returns on a numeraire proxy, numeraire-denominated returns for individual assets are computed by simple division. This eliminates the requirement that observations be available for each asset to estimate its market-model parameters. ‘See, for example.

Roll (1973), Kraus

and Litzenbeger

(1975), and Rubinstein

(1976a.b).

‘More recent test of the log utility model [for example. Litzenberger and Ronn (1986) and references therein] have focused on the relation berween asset prices and consumption. These tests do not have direct implications for the quality of alternative numeraire proxies. Brown and Gibbons (1985) do not use consumption, but report only the log of one plus the market proxy-denominated return on one-month T-bills.

33

J.B. Long, Jr., The numeraire portfolio

To be able to use numeraire-denominated returns, one must first have an empirical numeraire proxy. Section 4 of this paper examines some proxies suggested by theory and evidence. For some of the proxies (for example, the value-weighted NYSE portfolio and an approximately equally-weighted portfolio of stocks, bonds, and bills), proxy-denominated stock returns are very similar to market-model forecast errors and, as abnormal return measures, clearly dominate market-adjusted returns. The remainder of the paper discusses the numeraire portfolio, its properties, and its applications. Basic definitions and assumptions are presented in a discrete-time setting in section 2. Section 3 defines the numeraire portfolio in this setting, outlines the proof of the main conclusion, and describes the composition of the numeraire portfolio and its relation to mean-varianceefficient portfolios. Section 4 addresses the problem of empirically identifying the numeraire portfolio and offers sample statistics for alternative proxies. Concluding remarks are offered in section 5. An appendix derives the main conclusion in a continuous-time setting and shows the special features of the numeraire portfolio in that case.

2. Basic definitions

and assumptions

The numeraire portfolio is illustrated here in the context of a standard discrete-time mode1 of an asset market. Special features associated with continuous-time trading are described in appendix B.

2.1. The market

Assets may be bought and sold on a discrete sequence of trading dates, costs or restrictions on short sales.

t = 0, . . . ) T. There are no transaction

2.2. Assets There are K assets. The dividend per unit of asset j at time t is Djt. It goes to the investors who hold units of asset j on arrival at time t. Purchases and sales of asset j at time t occur at the ex-dividend price Pj,. If P,,(_ i f 0, then R, denotes the rate of return on asset j over the (t - 1, t] period. The K x 1 vectors of per-unit dividends, ex-dividend prices, and rates of return at time t are denoted by D,, P,, and R,. It is assumed that all prices and dividends are bounded in the sense that

Pr{lD,,lsMand

lpi,l
j=l,...,

K,

t=O ,...,

T}=l,

J.B. Long, Jr.. The numrrawe portfolio

34

for some finite number M. (The formal modeling of probabilities and random variables is described in appendix A.) 2.3. Portfolios A portfolio X is a vector-valued

stochastic process. Its composition at time Xjr, specifies the number of units of asset j in the portfolio immediately after trading at time t. In other words, X, is the composition of the portfolio on exit from time t. X, can be regarded as a decision rule. To be feasible, the rule X, must always specify finite quantities that depend only on information available at time t (for example, the history of asset prices and dividends through time r>. t, X,, is a K x 1 vector whose jth element,

2.4. Portfolio value and cash flow

If not modified by some term like ‘arrival’ or ‘with-dividend’, the ‘value of portfolio X at time t’, denoted by I/xr, refers to the market value of the portfolio on exit from time t. Thus,

vx*= X,,P,

,

t = 0,. . . , T.

The cash flow from portfolio X at time t, CXr, is simply the difference between the with-dividend value of the portfolio held on arrival at time t and the value of the portfolio held on exit from time t. This in turn is equal to net dividends received minus net asset purchases at time t. Thus, C,,=X,‘_,(D,+P,)-X,‘P,=X,‘_,D,-AX,’P,,

t=l

,...t

T,

where AX, =X, -X,_,.

2.5. Self-financing portfolios

A self-financing portfolio is a portfolio whose cash flows at future dates (t 2 1) are zero with probability one. If X is a self-financing portfolio, then, at each t 2 1, the with-dividend value of the arrival portfolio X,_, exactly finances purchase of the exit portfolio X,. 2.6. Self--financing portfolios with always positice calue I assume that there is at least one self-financing portfolio whose value is always positive in the strong sense that its value is always greater than some

35

1. B. Long, Jr., The numeratre portfolio

positive number. If 2 is such a portfolio, there is an E, > 0 such that Pr(Z;P,>&,,O~t~T]

= 1.

The purpose of this assumption is to provide at least one portfolio that can serve as a numeraire in the sense that, when dividing prices and dividends by the value of this portfolio, one is always dividing by a strictly positive number. Moreover, if nominal prices and dividends are bounded with probability one, so are the numeraire-denominated prices and dividends. 2.7. Profit opportunities A profit opportunity is a portfolio that satisfies three conditions: (a) its initial value is nonpositive, (b) its intermediate cash flows and terminal with-dividend value are nonnegative with probability one, and (c) either its initial value is negative or there is a positive probability of a positive future cash flow or with-dividend terminal value. Thus, S is a profit opportunity if and only if (a) V,, 5 0, (b) Pr(C,, L 0, 1 5 t I T - 1, and C,, + VST101 = 1, (c> Pr( - VsO+ XrCSr + V,, > 01 > 0. 3. The numeraire

portfolio

A numeraire portfolio N is defined as a self-financing portfolio with always positive value such that, for each asset j and each t, 0 I t < T,

with probability one, where E,{ .} denotes expected value conditional on all information at time t. If a numeraire portfolio exists and if 1+

Rj,,+,=

e,r+l+bjj,t+l pjt

available

1 +Rj.r+l =

1 + R,vv,t+,’

then the numeraire’s definition and the law of iterated expectations

imply

with probability one for all j and t. In other words, when prices and dihfends

36

J. B. Long. Jr.. The numertnre portfolio

are denominated in units of a numeraire portfolio, asset prices equal rhe undiscounted sum of e,rpected future payoffs and zero is always the best conditional forecast of any asset’s future rate of return.

The following basic theorem provides a necessary and sufficient condition for the existence of a numeraire portfolio in a market and shows that its nominal rate of return is unique. Theorem 1. Among all feasible portfolios, there is a numeraire portfolio if and only if there are no profit opportunities. Moreor*er, if A and B are two numeraire portfolios and R,4, and R,, are their rates of return oL‘er the (t - 1, t] interr.al, then R,_,, = R,, with probability one for all t 2 1. Proof.

It is easy to show that the existence of a numeraire portfolio is inconsistent with profit opportunities. If a numeraire portfolio N exists, its definition and the law of iterated expectations imply that

where X is any portfolio, d,Y, = C,,/I/,,, and pX, = VX,/qv,. But if X is a profit opportunity, then (since the value of the numeraire portfolio is always positive)

and at least one of these two inequalities is strict. But that is a contradiction of the formula for FYo implied by the existence of a numeraire portfolio. Thus, if a numeraire portfolio exists, there are no profit opportunities. It is more difficult to show that the absence of profit opportunities implies the existence of a numeraire portfolio. The key to a relatively easy and general demonstration, however, is to consider the problem of maximizing E&In VXTj subject to VXO= 1 and the constraint that X be a self-financing portfolio with always nonnegative value. The conditions that define a numeraire portfolio are first-order conditions for this maximization problem. Under the assumptions of section 2 (bounded prices and dividends, the existence of a self-financing portfolio with always positive value) a solution exists if there are no profit opportunities. Thus, the absence of profit opportunities implies the existence of a numeraire portfolio. As a solution to maxE(ln VX7.),the numeraire portfolio is also the growth-optimal portfolio of Kelly (1956), LatanC (19.591, and Breiman (1960). A more formal elaboration of this argument is contained in appendix A.

J.B. Long, Jr.. The numeraire portfolio

37

Finally, if there are redundant assets in the market (assets whose returns can be duplicated by portfolios of other assets), the composition of the numeraire portfolio is not unique. 9 Nevertheless, the rate of return on the numeraire portfolio is unique. If A and B are portfolios that both satisfy the definition of the numeraire portfolio, then, for every t,

These simultaneous conditions probability one. End ofproof.

are satisfied if and only if R,, = R,,

with

Perhaps the most striking feature of the numeraire portfolio is the condition E,(R,.,, i} = 0, j = 1,. . . , K. Zero is always the best conditional forecast of the future numeraire-denominated rates of return on any asset. This condition is perfectly consistent with cross-sectional heterogeneity and secular variation in expected dollar rates of return as long as these are not indicative of profit opportunities. For a given marginal distribution of the dollar rate of return on the numeraire, R, ,+,, the conditional mean of the ratio

1 +R;.t+t

= 1 + dj+(+,

1 + R,v.r+, depends positively on E,(Rj,, + , 1, but it also depends negatively on cov,(Rj,!+iTRN,r+i ) . Thus, for all assets to have the same (zero) expected numeraire-denominated return, the composition of the numeraire portfolio for the time interval (t, t + 11 must be such that assets with high mean dollar returns also have high covariances with the dollar return to the numeraire portfolio. In this respect, the relation between individual assets and the numeraire portfolio is similar to the relation between individual assets and the market portfolio of the Capital Asset Pricing Model (CAPM). This and more general features of the numeraire portfolio’s composition are discussed in more detail below. 3.1. Composition of the numeraire portfolio As observed in the proof of Theorem 1, the value proportions in the numeraire portfolio are those that would be chosen by an investor with log utility. Thus, the numeraire portfolio is the growth-optimal portfolio of 9See Vasicek (1977) for’an extreme example of redundancy and nonuniqueness of the numeraire-portfolio composition. In his model bond market, any portfolio of long-term bonds can be combined with the shortest-term bond to form a numeraire portfolio.

38

J. B. Long, Jr.. The numrram

portfolio

Kelly (1956), Latane (1959), and Breiman (1960). Although the focus here is not on investors’ preferences or optimal portfolios, the fact that the numeraire portfolio is optimal for log utility investors provides some well-known characterizations of its composition. As noted above, there are also similarities between the numeraire portfolio and the market portfolio of the CAPM. Specific characteristics of the numeraire portfolio (familiar as characteristics of the log utility investor’s portfolio) are given below in Theorem 2. Theorem 3 then provides a mean-variance approximation of the portfolio. (The approximation is perfect in a continuous-time setting.) Theorem 2. The nrtmeraire portfolio is ‘myopic’, that is. its r,alue proportions on exit from time t depend only on the time t conditional distribution of the one-step-ahead rates of return R,,,, ,, j = 1,. . . , K. Specifically, the c,alue proportions depend only on the conditional distribution of ratios of the onestep-ahead gross returns (1 + R, I f ,I, j = 1, . . . , K. If the market is complete and if A denotes an erent that is l*erifiable at time t + 1, then on exit from time t a fraction qA of the ualue of the numeraire portfolio is inr,ested in a portfolio whose with-dirsidend L,alue at time t + 1 is positice if ec’ent A occurs and zero otherwise. The fraction q,_, is equal to the time t conditional probability of A. Proof. Let 71 be the K x 1 vector of value proportions in the numeraire portfolio on exit from time t. 77, can depend on any information available at time t (for example, the history of prices and dividends through time t). It is the solution to the simultaneous equations

1,

j=l

,...,

K,

where

1 +R,v.,+, =$(I

+R,+,).

Thus v1 depends only on the time t conditional distribution of the relative valuesof(l+Rj,,i,), j=l,..., K. If the market at time t contains a complete menu of claims to value at time t + 1, then the securities in the market at time t (denoted up to now by j=l ,a.., K) can be taken to be pure state-contingent claims to payoffs at time t + 1. Let rrt, be the finest partition of the sample space (possible histories of the world from 0 to T) that is discernible by investors at time t + 1. This means that the events in r,,, are verifiable at time t + 1 and

J. B. Long, Jr., The numeraire portfolio

39

every other event that is verifiable at t + 1 is a union of events in I’,+,. Assume there is a state claim for each event (state) e in Tr+i and that the claim for state e has a time r + 1 with-dividend value of one in event e and zero otherwise. For an e in T(+i with positive time t probability, let R,.,, L denote the ex post rate of return on the state claim that pays off in state e and let ne denote the fraction of the value of the numeraire invested in this claim on exit from time r. (Since there are no profit opportunities by assumption, the claim for state e has a positive price at time t if and only if the time r probability of state e is positive.) Then one plus the ex post dollardenominated rate of return on the numeraire portfolio is 71~.(1 + R,,,, ,> if state e occurs. Substituting this into the condition that expected numerairedenominated returns be zero yields

ve = r
for every e in r, + I,

where 7, is the time t conditional probability that state e will occur at time t + 1. If A is an event that is verifiable at time t + 1, then, since A is a union of events in I-,+, and since the events in r,, , are disjoint, the fraction of the time t value of the numeraire invested in claims that pay off in event A is TV, the time t conditional probability of A.‘O End ofproof. The myopia that characterizes the numeraire portfolio’s value proportions is potentially important in empirical applications. It implies that numerairedenominated rates of return can be stationary stochastic processes even if nominal rates of return are not. Numeraire-denominated rates of return depend only on the ex ante distribution and the ex post realizations of the relative values of gross nominal returns. Thus, for example, numerairedenominated returns are independent of expectations, and realizations of pure price-level inflation (variation in asset prices and dividends that doesn’t affect their relative contemporaneous values). For the special case of complete markets, Theorem 2 gives a simple exact characterization of the numeraire portfolio’s value proportions. Unfortunately, the characterization is uninformative in the general incomplete market case (even though it is also true in that case for a set of state prices that support the prices of traded assets). Given the familiarity of mean-variance portfolio theory, more information is conveyed by an approximation of the numeraire portfolio’s location in the mean-variance opportunity set. “This property of the numeraire portfolio was first noted by authors of the early literature on the growth-optimal portfolio, for example, Kelly (1956) and Brieman (1960).

10

J.B. Long, Jr., The numera,re porrfoko

3.2. Relation portfolios,

between the numeraire portfolio, and the CAPM market portfolio

mean-variance-e%ficient

portfolio [for example, From the literature on the growth-optimal Hakansson (197111, it is well known that the portfolio is not generally mean-variance-efficient period by period. The numeraire portfolio on exit from time t, however, does maximize the expected log of one plus the (t, t + 11 rate of return; this is the basis for a mean-variance approximation. Specifically,

E,(Nl +R,v,,+,)j = E,(Rx,,+ll - i~ar,(R.~.,-,}~ for portfolios X whose value is sure to be positive. The approximation is better the smaller are E,(R, r+lj and var,(R.,,_,j. As shown in appendix B, the approximation becomes, ;n effect, perfect when periods become arbitrarily short allowing continuous trading opportunities.” Maximizing the approximate objective function yields a mean-variance-efficient portfolio that approximates the numeraire portfolio.” Theorem 3. If riskless borrowing and lending are ar,ailable for the (t, t + l] period at the rate rF ,+ ,, the approximate numeraire portfolio on exit from time t is a ler.ered position in the mean-variance-target portfolio,13 Q, for the (t, t + l] period. The nominal rate of return on the approximate numeraire is gicen by R .v.,+~=h(t)R~.,+,=(l-A(t))r,,,_,, where A(t)

=

E,&.,+J -rF.t+l var,(RQ,,, J

’

With this degree of leverage, the numeraire equal to the variance of its rate of return. “Though it is frequently multivariate normal returns that assumption.

portfolio’s

expected excess return is

used to justify mean-variance-objective functions, an assumption of wouldn’t make sense here, since the expected log is undefined under

“See Markowitz (1959, ch. 6) for two slightly different mean-variance approximations. See also Hansen and Richard (1987) and Hansen and Jagannathan (1990) for the mean-variance characterization of a different benchmark portfolio. Roughly, the gross return on the benchmark in these hvo papers behaves like the reciprocal of the gross return on the numeraire. 13See Sharpe

(1964).

J.B. Long, Jr., Thr numrmrr

portfolio

31

The mean-variance-tangent portfolio, Q. has the property that each individual asset’s expected excess return is proportional to the covariance between the asset’s return and the return on the tangent portfolio. Any levered position in the tangent portfolio will also have this property. Leverage only affects the constant of proportionality. The value of A(t) above, however, implies that each individual asset’s expected excess return is equal to the covariance between its return and the return of the (approximate) numeraire portfolio, that is, E,(R,.,+,} - rf.,+, = cov,(R,.,+,, Rs,,,,}, j = l,..., K. If the mean-variance-tangent portfolio is the market portfolio (that is, if the standard CAPM describes expected asset returns), then the numeraire portfolio is a levered position in the market portfolio. The market in which the CAPM market portfolio is defined, however, is the market for all individual assets in the economy. A numeraire portfolio, on the other hand, exists for any market defined by any given list of assets that present no profit opportunities when trading is restricted to assets in the list. Moreover, the numeraire portfolio for a given market contains only assets from that market. Thus, in empirical applications, the numeraire portfolio can be defined as the numeraire for the market consisting of a limited list of assets. 4. Empirical

proxies for numeraire

returns

In this section, I examine actual returns for a group of assets that make up a market in the sense described above. I assume there are no profit opportunities in this market,14 so there is, by assumption, a numeraire portfolio for the market. The presence of an actual numeraire portfolio in this market is thus not the empirical issue. The empirical issue and the practical problem is to identify the market’s numeraire portfolio, or at least a proxy that closely mimics its principal return features. The potential uses of a good numeraire proxy are roughly the same as the uses of the various market-portfolio proxies. Just as market-portfolio proxies are used to estimate abnormal returns for individual assets, a numeraire proxy is used to estimate the numeraire-denominated returns of individual assets. Numeraire-denominated returns have the same qualitative interpretation as conventional abnormal return measures. Ignoring measurement and estimation errors, both measure asset performance relative to the average contemporaneous performance of other assets in the market. Both statistics are constructed so that, in equilibrium, the best one-step-ahead forecast of the statistic is zero. In practice, both numeraire-denominated returns and conventional abnormal return measures involve estimation errors. For measures based on “More accurately, I assume there would be no profit opportunities if transaction restrictions on short sales were removed but asset returns remained unchanged.

costs and

J.B. Long, Jr., Thhr numern~re portfolio

12

market-portfolio proxies errors come from

(for example,

market-model

forecast

errors)

these

(al use of a proxy for the CAPM market portfolio (or at least for a portfolio that is mean-variance-efficient), and (bl estimation of market-model parameters for each asset being studied. For numeraire-denominated returns, estimation errors come from construction of the numeraire proxy, that is, from estimation of the composition (weights) of the numeraire portfolio for the market being studied. The potential advantage of the numeraire proxy, especially in event studies, is that market-model regressions for each asset are not necessary. In addition to,saving some computation, this allows the use of firms for which market-model parameter estimates are unavailable. Evidence examined below indicates that numeraire(proxy)-denominated returns dominate marketadjusted returns as a way of measuring abnormal returns when market-model parameters are not available. When market-model parameter estimates are available, numeraire-denominated returns for stocks have distributional properties (means, standard deviations, autocorrelations, etc.1 that are very similar to those of the corresponding market-model forecast errors. Numeraire-denominated returns on bills and bonds are more variable than the corresponding market-model forecast errors. 4.1. The data base

The data base consists of monthly rates December 1983 for the following assets:

of return

from January

1926 to

Five firm-size ranked portfolios of common stocks. These are the stock portfolios used in Keim and Stambaugh (19861. The stocks in portfolio j for month t are those of the firms in the jth size quintile of NYSE firms at the beginning of the year. The month r return for portfolio j is the second-month return on an equally-weighted buy-and-hold portfolio formed at the beginning of month t - 1. The nominal return in month t for the jth quintile portfolio is denoted by RQj,. (ii) The value-weighted portfolio of NYSE stocks. For month t, its nominal is denoted by RMV,. (iii) Long-term corporate bonds. This is the Ibbotson-Sinquefield (1982) series. The month t nominal return on corporate bonds is denoted by CB, . yield series. This is the riskless rate series from French, (iv> A Treasury-bill Schwert, and Stambaugh (1987). Its month t value is denoted by TBY,. (i>

Statistics

for these series are given in panel

A of table

1.

13

J. B. Long, Jr., The numeraireportfolio

4.2. Market-adjusted

returns and market-model

forecast

errors

If a satisfactory numeraire proxy can be identified, numeraire-denominated returns are an alternative to conventional measures of abnormal returns. This section summarizes the behavior of the conventional abnormal return measures for the sample described above. I then use the summary as a benchmark for evaluating alternative numeraire proxies. In event studies, market-adjusted returns offer the same kind of advantage offered by numeraire-denominated returns: no estimates of market-model parameters for the firms in the study are required. As defined by Brown and Warner (1980), the market-adjusted return of asset j in month t is A, = R,, - R,wt, where R,, and R,, are the month t nominal returns on asset j and the market-portfolio proxy. The obvious disadvantage of market-adjusted returns is that the cross-sectional average of their expected values is not zero for arbitrary sets of stocks. (According to Brown and Warner, this problem can be serious if the value-weighted portfolio is used as the market proxy.) If a good proxy for the numeraire portfolio is available, numerairedenominated returns do not have this disadvantage. Panel B of table 1 shows the behavior of market-adjusted returns. The cross-sectional spread of mean adjusted returns is necessarily identical to the cross-sectional spread of mean unadjusted returns as shown in panel A. This is the principal limitation on the use of market-adjusted returns as abnormal return measures. Assets with different expected raw returns cannot both have zero expected market-adjusted returns. Thus nonzero market-adjusted returns are not necessarily ‘abnormal’. For assets with a beta of one (for example, the large firm portfolio Q5>, however, the mean market-adjusted return is near zero and its standard deviation is much smaller than the standard deviation of the asset’s unadjusted return. Panel C of table 1 shows the behavior of market-model forecast errors. The forecasts are fitted values from regressions of asset returns on f&40’, the return on the value-weighted NYSE portfolio. For each month and each asset, the parameters of the regression are estimated using the first 60 of the latest 72 months of observations. To compare market-adjusted returns and market-model forecast errors, it helps to think of the market-adjusted returns as forecast errors from a version of the market model in which the intercept is constrained to equal zero and the slope is constrained to equal one. These constraints are removed in constructing the forecast errors in panel C. Separate nonzero intercepts are allowed for each asset and thus the mean forecast errors in panel C are much smaller than the mean market-adjusted returns in panel B (except for portfolio Q-5, which closely mimics the market portfolio). Similarly, removing the slope constraint allows the standard deviations of forecast errors for bonds and bills to be much lower than the standard deviations of

Table

I

4.55 4.Yl 2.h3 2.46 4.7h -0.2x - 0.42

11.073I 0.0434 0.02Y7 0.01 Y6 O.WXh 0.0562 1J.OS7S

o.w74 o.w3s 0.ocl26 O.ol)IS o.ooK! - O.~N)S4 - o.tH)hs

0.000x 0.olJI0 o.owi 0.ool~3 - 0.woI O.~N)O.7 O.oot)J

0.0h92 0.0405 0.0274 o.oIxo O.(NlXO o.olHx OAH)I4

‘12 lesl for zero mans p-value = 0.000

RQIM RL)2M RQ.jM RC)JM RQ.5M C’IIM ‘1‘11 YM

4.12 5.5X 2.70 2.S) 4.3Y 0.7’) I .54

3.07 2.03 I .3’) I.05 I .w I.1 I I.41 0.3x

0. I 102 O.OXhX 0.0773 0.060’) 0.06 I2 O.OlYl 0.0026 0.0574

Slicw.”

0.0 I65 0.0 I25 O.OIIh 0.0105 o.ow2 O.W3h 0.002h o.cnNo

Slit. dcv.

7‘2 lest for zero means p-value = 0.07h

RQlA RL)bt RQ3A RL)JA RQ.5A CRA WY/1

Variahlc” tcpj

O.YS

0.9 I 0.0x

0.x7 O.YS 0.08

rt’tt~rf~s

RQJ

0.X1 0.00 0.05 O.YX

ICY.5

3O.YJ 72.77 24.hl IY.OS 4X.52 X.1,.% s.44

O.XX

0.80 0.00

O.hh 0.7X 0.x7

0.25 0.27 0.43 0.50

u.07 0.0’) 0.07 0. I I 0. IS

0.20 0.23 0.22 0.24 0.24

C’lI

Cross-sectiomd correl;rtions

I’NIWI A: Rw

Icy2

I’ctrd C: Mdrkrr-rfwtle~ fiwfxwf erronE

IO.41 2.13 x.04

I222

23.2h IX.14 12.1~7 Il.37

Kur(.’

0.42 0.47 0.53 0.54 0.30 O.Y4

0.04 0.04 0.04 0.03 0.04 0.04

- 0.08 -0.10 -O.IO -O.IO ~ 0.02 - 0.00

-

-

7SY

December 10X3 (hYh months). Identical btalislics for suhperiotls are in appendix C.

0.03 0. IO 0.05 0.13 0.0 I 0.01 0.x’)

0.01 O.Oh 0.03 0.0x - O.OI 0.0 I - 0.ol)

0.07 O.I2 0. I.5 0.16 0. I7 0. IO 0. I2

0.03 0. IO 0.17 0.22 0.2 I 0.15 0.Y3

0.0 I 0.02 O.(H) 0.01 - 0.02 - 0.01 0.00 - 0.0I

r2

0. I7 0.20 0.17 0. IS 0.12 0.14 O.YX 0. I I

rl

-

0.02 O.Oh 0.00 0.01 0.04 0. IO 0.X4

- 0.03 - 0.07 -0.0X - 0.06 -0.0X -0.14 - 0.12

-0.12 -0.15 -0.11 0.05 -0.1

-0.0x -0.IO -0.12

r3

Ihv and transformed monlhly rates of return on NYSE quintile portfolios. corpora~r hods, and one-monOI T-bills for the period Jlcnunry iY20 10

? 2 3 $, 2 : 3 2 0

6 & _>

:

t

- 0.0032

TBYN

0.0580

0.06h9 0.04OtJ 0.0277 0.0 I X4 o.tttt7x 0.0570

o.ooo9

RQ3N

1.10

3.53 3.44 I.51 I .33 2.54 I.21

x.1 I

23.93 37.hl 11.44 4.44 24.cJ9 X.30

O.X9

0.82 0.92

prmy 0.2x 0.32 0.43 0,s I

- 0.35 -0.42 - 0.47 - 0.47 - 0.26 -

= NYSE rdrrcwvri~l~~I

____._~ 0.70 0.82 KU9

(tIlo~renrirrporrli,lio

___ 0.38 0.4h 0.50 0.s I 0.3 I 0.05

- 0.56 - I.1 I 0.0x - 0.09

- 0.67 x.54 X.16 6.59 7.14

Xhl

13.31

I S.26 0.76

0.47 0.x0

OS17 0.50 0.79

- 0.41 - 0.05 0.35 0.7 I - 0.09

- 0.58

-0.x2 - 0.9 I - 0.84

-0.x2 - 0.9 I - 0.83 - 0.57 - 0.07 0.94

0. to 0.15 0.03 -0.0x - 0.06 0.15 0.15

0.w 0.W WJO 0.00 0.04 0.02

o.ut

osJ7 0.03 0.0X - 0.02 0.03 1J.01

0.01

0.03 0.06 0.07 0.05 0.08 0. I I

-0.0x - 0.06 -0.11 - 0.09 - 0.0X

- 0.03 - O.Oh

-0.10

-

‘/~@I is the monthly rate of return on the portfdio of firms in the 11th NYSE size quintile. The smallest firms are in quinlile I. Firms are reclassilied a~ the heginning of each year. CR is the monthly holding-period return on the tbbotson-Sinqueliel~ Corporate Bond Index. TBY is the one-month T-bill rate. RMV is the return on the value-weighted NYSE portfolio. htf returns are normally distributed, estimated skewness has mean 0 and st;mdard error 0.09. ‘If returns are normutty distributed, estimntrd kurtosis has man 0 and standard error 0.19. “Markel-;l‘ljustccI,xle‘l returns are raw returns minus RMV. the return on the NYSE portfolio. ‘For each month ;IINI each ;ISSL’I, market-model parameters for the forec;lsr are computed using the returns from the tirst 60 of the talesI 72 IIWI~I~IS. This nlr;ms ttwt the market-model forecast-error series begins in January 1932 and extends for 624 months. ;Ipproximates the maximum-likelihood estimate of the ‘The nl;lxirllunl-tiketillcn,tl numerairc. proxy is IIIC lixcd-weight portfolio IIWI IWI unobserved linlc scrics of rcluriis 011 lhr ‘Iruc nunlcrairc pulldio for lhis nurkcl. ‘t‘t~c wcighls on /
= 0.260

0.0 19’) 0.0209 0.0527 0.0544

0.0222

2.44 O.h4

0.10

0.06 0.11 0.1 1 0. tit 0. IO 0.09

porf/blio)

f’~‘cr,dE: N~rrttrrcrir~,-ck~,r~(~~~~it~(~/~,~/ rv/wtu (rrlottcrrrire-/Jl~r/fl~l;~) prmy = roux.-likulil~ood pmry )’

0.0501 0.02X7

I‘2 lest for zero meims p-vdur

UN 7BYN

RQ5N

o.tKut2 - 0.0005 - 0.0033 - o.lNt4o

osto I3

RL)ZN

RQjN

0.0043

RUIN

.~

72 tesl for zero means p-value = 0.238

0.0018

O.Wt9 tt.lWNWt - 0.0024

RQQN RL)SN CUN

0.0023

RL)3N

O.W58

R@lN

RQZN

I’utwl D: N~o~~rn~ir~-clr~~ort~irrtrrrd MII~IF

their market-adjusted returns. For stocks, however, the standard deviations of forecast errors are not significantly lower than the standard deviations of market-adjusted returns. 4.3.

The distribution

of numeraire-denominated

The numeraire-denominated (t - l,t] is

rate

of return

retwxs

on

asset

; for the

period

where R,, is the nominal (S) rate of return on asset J’ for the period and R,,., is the contemporaneous nominal rate of return on the numeraire portfolio. Theoretically, the numeraire-denominated rate of return of every asset in the numeraire’s market is a mean zero, serially uncorrelated time series. This distinctive property of the true numeraire portfolio will be used here as the standard by which to judge several proxies. For each proxy, the time series of monthly numeraire(proxy)-denominated returns is computed for each asset in the data base. The behavior of these time series can then be compared in table 1 with the behavior expected if the pro.xy is the true numeraire and with the behavior of the abnormal return measures discussed above. For empirical application purposes (for example, using numerairedenominated returns as estimates of abnormal returns), the ideal numeraire pro,xy would be such that all of the numbers in table 1 for proxy-denominated returns (except the p-values of the Hotelling T’ tests) are small in absolute value. In comparisons of alternative proxies, smaller means. standard deviations, cross-sectional and serial correlations, etc. of proxy-denominated returns are better. 4.-C. Numeraire

proxy #I:

Market-portfolio

proxy

Motivated by the prediction of the log utility equilibrium model (that, in equilibrium, the market portfolio is the numeraire portfolio), Roll (1973) and Fama and MacBeth (1974) compute numeraire-denominated returns for individual stocks (Roll) and for portfolios (Fama and hlacBeth) using market-portfolio proxies as the numeraire. (Roll uses the S&P 500 and Fama and MacBeth use the NYSE equally-weighted return.) They then compute Hotelling T’ statistics for the hypothesis that the expected numerairedenominated return is equal to zero for all assets in the test. (Roll tests that the expected numeraire-denominated returns are the same for all stocks.) In

J.B. Long, Jr., The numrrave portfolio

41

many replications of the test, the p-values for the statistics are uniformly consistent with the hypothesis. (Roll repeats the test for 68 groups of individual stocks in the same time period. Fama and MacBeth repeat the test for five-, ten-, twenty-, and thirty-year time periods using 20 beta-ranked portfolios.) For the seven-asset universe made up of the five quintile portfolios, corporate bonds, and T-bills, similar tests and other sample statistics are given in panel D of table 1. In panel D, the value-weighted NYSE portfolio is used as the numeraire proxy. Viewed as an abnormal return measure, the numeraire-denominated returns reported in panel D weakly dominate the corresponding marketadjusted returns reported in panel B. (In comparison with the returns in panel B, the returns in panel D have means closer to zero and, for stocks, less skewness and kurtosis. In other respects, the panels are essentially identical.) For stocks, numeraire-denominated returns are also very similar to the forecast errors reported in panel C in all dimensions except mean value. (For bonds and bills, numeraire-denominated returns are inherently more variable than raw nominal returns, and nominal returns, in turn, are more variable than forecast errors.)

4.5. Numeraire proxy #2: Lecered positions in the market-portfolio proxy The analysis in section 3.2 concluded (subject to the approximation of discrete-time trading ‘with the continuous-time model) that the numeraire portfolio is a mixture of investment in T-bills and investment in the mean-variance-tangent portfolio. For month t, the value weight of the tangent portfolio in the numeraire portfolio is h(t). the ratio of the beginning-of-month estimates of the tangent portfolio’s mean excess return and its variance. Even if the tangent portfolio is the market portfolio (that is, if the standard CAPM holds) and the market-portfolio proxy is accurate, there is no theoretical reason to expect A(t) to be constant, much less constant and equal to one. Moreover, available empirical evidence suggests that, on average (over time), A(t) is greater than one and it varies over time. These generalizations are apparent in Fama and MacBeth (1974) [who compute for various time periods the constant value of A that maximizes the sample mean of In(1 + R,)], in the direct estimates of A(t) provided by Merton (19801, and in the relation between the market risk premium and volatility estimated by French, Schwert, and Stambaugh (1987). For the seven-asset universe of the quintile portfolios, corporate bonds, and T-bills, I construct a variety of numeraire proxies consisting of constant

48

J.B. Long, Jr., The mtmrruire portfolio

and variable

leverage

positions

in the market-portfolio

proxy. These

include:

(a) a proxy with A constant and equal to the sample mean of f&VTBY for the entire 1926-1983 period divided by the sample variance of IiiMv for the period, (b) a proxy with A different in each of four sample subperiods, where, in subperiod n, A is the sample mean of RMV- TBY for the nth subperiod divided by the sample variance of RMV for the nth subperiod, (c) a pro,xy with subperiod values of A chosen to ma.ximize the ex posr geometric mean numeraire rate of return, and (d) a proxy where A varies monthly, where, for month f, A is equal to the one-month-ahead forecast (made at the beginning of the month) of R&IV- TBY divided by the one-month-ahead forecast of the variance of R,UV [the forecasts are those constructed in French, Schwert, and Stambaugh (198711. Without reporting the details, the principal implication of this exercise is that high leverage (in the numeraire-proxy portfolio) causes high standard deviations and high cross-sectional correlations in numeraire-denominated returns. In proxy (b), the value of A is high enough (4.54) in the second subperiod to cause the numeraire portfolio to go bankrupt. The values of A in pro.xy (d) (based on the French, Schwert, and Stambaugh forecasts) are often extreme. The mean is 3.54, the standard deviation is 2.6, and the maximum is 20.3. (The information content of the French, Schwert, and Stambaugh forecasts is illustrated by the fact that this numeraire proxy avoids bankruptcy in spite of its often extreme leverage.) The means and autocorrelations of returns denominated in these proxies are not substantially closer to zero than those in panel D of table 1, where the unlevered value-weighted portfolio is used as the numeraire proxy. This is in spite of various levels of optimization in construction of the Ievered proxies. The simplest explanation for the failure of the levered proxies to mimic the theoretical numeraire is that the value-weighted CRSP portfolio is not a good pro,xy for the tangent portfolio of the seven-asset universe considered here. 4.6. Numeraire proxy #3: Quasi-maximum-likelihood

estimates

As proxies for the numeraire portfolio, market-portfolio proxies ered positions in market portfolio proxies have at least three limitations: (i)

the market portfolio may not be the standard CAPM may not hold),

tangent

portfolio

(that

and lev-

a priori

is, the

19

1. B. Long, Jr.. The numrraire portfolio

(ii) the market-portfolio proxy may not accurately represent the true market portfolio, and (iii) estimation error in A, even if it has zero mean, can affect both the mean and the autocorrelations of (proxy) numeraire-denominated returns. Quasi-maximum-likelihood estimation of the numeraire portfolio avoids the first two limitations, if not the third. Let (1 + R,) be the 7 X 1 vector whose ith element is one plus the month t nominal rate of return on the jth asset in the data base (the seven assets being the quintile portfolios, corporate bonds, and T-bills). The following quasi-maximum-likelihood procedure is based on the fact that E,_,{(l+R,)/(l+R,,)}

=l

(avectorofones),

t=l

,...,T.

Let Q be the covariance matrix of the (r - l&conditional distribution of (I + R,)/(l + R,“,) and assume that Q is constant. Then, if the conditional distribution is normal, maximizing the log-likelihood of the observed sample with respect to the (unobserved) time series of numeraire-portfolio weight vectors (ql} is equivalent to minimizing

with respect to subject to

nr,

t=O I....

1 + R,v, = q;_ ,( 1 + R,)

T-

1, and

n;_ ,l = 1,. . . , T.

Solutions to this minimization are called quasi-maximum-likelihood estimates if the normal distribution assumption is used only to specify the estimation objective function, and not to deduce the small-sample distribution of the estimates. An alternative rationale for the objective function is simply that it measures a distance between the sample counterparts of E,_ ,((l + R,)/(l + RN,)}, t=l ,. . ., T, and their theoretical -value, 1. The weighting matrix Q-r is chosen to use the sample information efficiently and make the estimation error variance small. Since a numeraire return is estimated for each month in the sample, however, theorems about the asymptotic features of minimumdistance estimators [for example, Malinvaud (197O)I do not apply. For the same reason, the incorporation of instrumental variables as in Hansen (1982) and Hansen and Singleton (1982) would be superfluous and have no effect on the estimated numeraire returns.

50

J. B. Long, Jr.. The nwnerarre portfolio

Minimizing the objective yields

lrR

_

.L.r-

U+WQ-'U+R,) (1 + R,)'Q-'1

’

t = l,...,T,

where R,L.ris the estimate of R,v,. Since Q is unknown, the time series RNr is first computed for Q = I. This first stage estimate of the numeraire return is then used to compute first-stage numeraire-denominated returns for the seven assets. The sample covariance matrix of these seven first-stage numeraire-denominated returns is then used as the matrix Q to produce second-stage estimates of the numeraire portfolio return. The final estimates of the numeraire portfolio returns are defined as the fitted values from regression of the second-stage estimates on the contemporaneous nominal returns to the seven assets with the intercept constrained to be zero and the coefficients constrained to sum to one. Thus the final estimates are returns to a fixed-weight portfolio. This final step is somewhat arbitrary. Its purpose is to simplify interpretation of the final estimates and to provide an easy way to use sample information from one subperiod to define a numeraire for another subperiod. Theoretically, constraining the portfolio weights to be constant is consistent with an assumption that the conditional distribution of relative gross returns is constant. Statistics from this estimation procedure are given in table 2. The covariante matrix of first-stage numeraire-denominated returns is based on the entire 19X5-1953 sample period. It is used in all subperiods to compute the second-stage estimates of the nominal returns to the numeraire portfolio. (If subperiod-specific covariance matrices are used to compute second-stage numeraire returns, the coefficients in the second-stage regressions vary a little more across subperiods than in table 2, but the statistics in panel E of table 1 are essentially unaffected.) The constrained regressions of the second-stage estimates on the nominal returns of the seven assets are shown in the last panel of table 2. Since the coefficients are nearly constant across subperiods, the coefficients from the overall 1926-1983 period regression are used in all subperiods as the numeraire portfolio weights in computing the numeraire-denominated returns reported in panel E of table 1. The statistics reported in panel E of table 1 are very similar on most dimensions to the statistics in panel D (value-weighted portfolio as numeraire proxy). The notable differences are in the cross-sectional correlations and in the standard deviations. The cross-sectional correlations in panel E between stocks and bonds/bills are much more negative than in panel D. This is to be expected, given the negative correlations in panel D between stocks and bonds/bills and the fact that the numeraire in panel E is a mixture of stocks, bonds, and bills.

Table 7 Statistics from maximum-likelihood estimation of the numeraire-portfolioweights in the market consistingof the NYSE quintile portfolios,corporate bonds, and one-month T-billsfor the periodJanuary 1926to December 1983 (696 months). Firsf-stage

Mean

of

esrimnre

numeraire-portfolio

Std. dev. Skew. Kurt. RQf

rerum:

(1 + R,,)

= (1 +

RYfl + R)/l’(l

+ R)

Coefficients from constrained regression of R,v, on R (standarderrorsinparentheses)

RQ2

RQ3

RQ4

l/1926-12/1983 =6%

RQ5

CB

TBY

R=

months

0.0111 0.0594 2.61 20.60 0.1986 0.1488 0.1270 0.0964 0.1521 0.1609 0.1161 0.99 (0.0053)(0.1028)(0.0186)(0.0200)(0.0142)fO.0091) l/1926-12/1938= 156months 0.0127 0.0993 2.23

9.47 0.1962 0.1512 0.1311 0.1254 0.1215 0.3050 -0.0305 0.99 (0.0146)(0.0302)(0.0446)(0.0475)(0.0367)(0.0538) l/1939-12/1953 = 180months

0.0108 0.0479 1.43 12.51 0.2211 0.0093 0.2077 0.1769 0.0810 0.2125 0.0916 0.99 (0.0060)(0.0238)(0.0310)(0.0360)(0.0254)(0.0277) l/1954-12/1968 = 180months 0.0112 0.0279 0.10

1.51 0.1490 0.1535 0.1395 0.1333 0.1438 0.1479 0.1330 0.99 fO.0042)(0.0074)(0.0092)(0.0098)(0.0064)(0.0037)

0.0099 0.0458 0.62

3.00 0.1917 0.1306 0.0615 0.1931 0.1362 0.1435 0.1434 0.99 (0.0074)(0.0165)(0.0205)(0.0215)(0.0129)(0.0047)

l/1969-12/1983 = 180months

Cocoriance

matrix

offirst-stage

numerarredenominated

RQIN

0.002s

RQW

0.0011

0.0008

RQJN

0.0005

O.OCQ5

0.0005

RQ4N

0.0001

o.cQO3

0.00X

RQ5N

-O.OGIM

CBN

-0.0021 -0.0022

TBYN Second-stage

estimate

_-O.OCOO

-0.0013 -0.0014

0.0003

-0.ooO9 -0.0010

-0.0006 -0.0006

of numeraire-portfolio

Std. dev. Skew. Kurt.

I/

1926-12/ 1983

O.OC04

0.0002

return:

o.OaO4

-0.0001 -0.0001

(1 + RN2)

where Q is the incerse of the&t-stage

Mean

returns,

=

0.0027 0.0026 0.0029

(1 + RYQtl + R)/l'QfI + R),

cocorionce

motrti

Coefficients from constrained regression of R,vZ on R (standarderrorsinparentheses) RQI

RQ2

RQ3

RQ4

RQS

CB

TBY

R=

l/1926-12/1983=696months 0.0098 0.0579 1.89 15.32 0.1939 0.1352 0.1246 0.1077 0.1441 0.1632 0.1313 0.99 f0.0002)fO.CGO6)fO.CH308) fO.0009)f0.0006)(0.0004) l/1926-12/1938 = 156months 0.0092 0.0954 1.74

7.18 0.1932 0.1361 0.1244 0.1098 0.1421 0.1703 0.1241 0.99 fO.WO6) (0.0012)(0.0018)(0.0019)(0.0015)(0.0022)

0.0099 0.0476 0.93

9.90 0.1954 0.1287 0.1282 0.1102 0.1420 0.1646 0.1309 0.99 (O.ooo2)(O.cc09)(0.0012)(0.0014)fO.cKllO) (0.0011)

0.0109 0.0277 0.40

1.54 0.1923 0.1349 0.1261 0.1072 0.1446 0.1625 0.1324 0.99 fO.ooO3)fO.0005)fO.OC06)(0.0006)fO.0004)(O.ooO2)

0.0091 0.0459 0.47

2.64 0.1934 0.1329 0.1250 0.1094 0.1446 0.1625 0.1322 0.99 fO.OUO6)(0.0013)(0.0016)(0.0016)fO.0010)(0.0004)

l/1939-12/1953 = 180months

l/1954-12/1968 = 180months

l/1969-12/1983 = 180 months

52

J. B. Long, Jr.. The numeraire portfolio

The most important difference between panels D and E is in the standard deviations. Except for quintile portfolios 4 and 5 (large-firm stocks), the standard deviations of numeraire-denominated returns are uniformly smaller in panel E than in panel D. For application purposes, this makes the maximum-likelihood numeraire used in panel E the best numeraire proxy of all those considered. In comparison with the best nominal abnormal return measures (marketmodel forecast errors reported in panel 0, the maximum-likelihood numeraire-denominated stock returns have similar means and standard deviations and lower autocorrelation. The numeraire-denominated returns, however. do not involve estimation of market-model parameters for each asset. Thus, in some applications, they will be a useful alternative to marketmodel forecast errors. Although numeraire-denominated returns on individual securities have not been examined here, they have been computed with daily data for individual firms in a recent large-scale study by Loderer and Martin (1989) of over 10.000 corporate acquisitions. Usin g the NYSE value-weighted portfolio as the numeraire proxy, Loderer and Martin compute numeraire-denominated returns, market-model residuals, size-adjusted returns, and industry-adjusted returns for days surrounding the corporate events in their sample. In their reported statistics, results based on numeraire-denominated returns are very similar to results based on market-model forecast errors. 5. Conclusion

The theory developed here says that if there are no profit opportunities in a market, a numeraire portfolio can be formed by trading in assets from that market. The numeraire portfolio is self-financing, its value is always positive, and zero is always the best forecast of future numeraire-denominated rates of return on every asset in the market. Though not developed much here, applications of the numeraire in theoretical modeling stem from the very simple representation of asset prices it allows. When prices and dividends are denominated in the numeraire, an asset’s price equals the sum of its expected future payoffs. Pricing does not involve risk adjustment, discounting, or substitution of a fictitious risk-neutral probability law. Especially for theoretical questions that focus on relative asset prices (for example, option pricing) and not on the relation of asset prices to commodity prices, numeraire denomination simplifies mathematical modeling without obscuring anything essential. Even for questions that do involve the relation of asset and commodity prices, such as consumptioninvestment choices, numeraire denomination of all prices, including commodity prices, introduces at most one additional state variable (for example, the numeraire-denominated price of $1) and yet it (a) provides a nominally riskless asset (the numeraire portfolio itself) and (b) makes the conditional

J. B. Long, Jr., The numeraire portfolio

53

mean of every asset’s nominal rate of return always zero. It then becomes clear, for example, that hedging is the only reason for a risk-averse consumer not to choose the numeraire as his investment portfolio. The potential empirical applications of the numeraire portfolio are qualitatively the same as the existing applications of market-portfolio proxies. Empirical applications of the numeraire portfolio, however, do not require market-model regressions for each asset. Given a time series of nominal asset returns and a contemporaneous time series of nominal returns on the numeraire, simple division is all that is required to compute numerairedenominated returns. As with applications of the CAPM (where the market portfolio must be approximated), a practical problem faced by applications of the numeraire is estimation of the time series of nominal returns on the numeraire portfolio. To have all of its unique properties, the numeraire portfolio may have to change in composition over time. The theory says that the portfolio is operational (its composition on exit from any date depends at most on information available at that date), but if the relative values of gross returns on individual assets are nonstationary, or even stationary but autocorrelated, then the exact composition of the numeraire portfolio can be a complicated function of history. In that case, there would be little hope of identifying good empirical proxies for the numeraire. Fortunately, the statistics reported in section 4 suggest that relative gross returns are sufficiently stationary that a simple constant-value-weights portfolio (not far from equal weights on everything, including bonds and bills) can serve as a good proxy for the numeraire portfolio. As an abnormal return measure, the numeraire(proxy)-denominated returns dominate marketadjusted returns and (for stocks) were very similar to market-model forecast errors. Appendix

A: Existence

of the numeraire

portfolio

Consider the problem of maximizing E&lnV,,} subject to VXO= 1 and the constraint that X be a self-financing portfolio with always nonnegative value. If there are no profit opportunities, this problem has a solution and the solution satisfies the conditions that define the numeraire portfolio. Thus the absence of profit opportunities implies the existence of the numeraire portfolio. A formal detailed version of this argument involves formal representations of probabilities, random variables, and ‘information’. A. 1. Probabilities, random cariables, and information

Probabilities and random variables in the model are defined in terms of a probability space {f2,/, n) for the model. R is the set of outcomes (the sample space) of the model and J? is the field of all events (subsets of 0) to

J. B. Long. Jr.. The nnmemtre portfolio

5-I

which probabilities are assigned by the probability measure f7. All random variables in the model are functions defined on R. Thus an outcome of the model, w E R, uniquely specifies the values of all random variables in the model (e.g., the realized sample paths of prices and dividends). Since prices and dividends are bounded integer multiples of pennies and since the number of assets and trading dates is finite. a finite-sample space is adequate and will be assumed here. A model of continuous trading in appendix B requires an infinite-sample space. The information structure of the model is defined in terms of the times at which events in +,F are verifiable. Event A is verifiable at time f if investors can perceive at time t whether A contains the outcome w. The set of all events that are verifiable at time r is a subfield, ft, of / and. since it is assumed that investors do not forget information, the sequence of subfields, (A), is increasing in the sense that A CA+, for each t. Loosely speaking, A defines the kinds of information that will be available at investors at time t. If investors will know the realized value of a random variable at time t, the random variable is said to be observable at time t or A-measurable. More precisely, if knowledge of which events in A contain the outcome w implies knowledge of the value of a random variable, the random variable is A-measurable. If, for every t = 0,. . . , T, the time t value of a stochastic process is observable at time I, the process is said to be adapted to {/‘,I. the model’s information structure. It is assumed, for example, that the price and dividend processes are adapted to C/J. A.2

Existence proof

If there are no profit opportunities, then, with bounded prices and dividends, the only way to generate the possibility of an arbitrarily large positive terminal value Vxr is to incur the risk of a negative (and arbitrarily large in absolute value) value of VXT. But such a portfolio violates the nonnegativity constraint of the problem. Thus, feasible V&.‘S are bounded. The set of feasible VXr’s is also convex and closed. Thus, a solution to the constrained maximization problem e?tists. If N is a solution to the problem of maximizing E&In Vr), then no other feasible solution Q yields a higher value of E&In I’,-). Consider, in particular, other feasible portfolios of the form Q,=N,,

t
P1.7+1

v

+ N.r+l

Dj.rtl

f>;+l,

55

J.B. Long. Jr.. The numerairr porrfolio

where (Y is a scalar with ICTI > 0 but small enough that Q is feasible,” ,Y,_ is the random variable that equals one if event AT E,L( occurs and zero otherwise, and e, is the K x 1 vector with 1 in the jth slot and zeros elsewhere. This Q is the same as N unless event A, occurs. If A, occurs, then Q differs from N only by the following transactions: (i) at time T, (Y units of asset j are purchased and the purchase is financed by simultaneously selling CYP,JI(~; units of portfolio N, and (ii) at time T + 1, the (Y units of asset j are sold (with dividend) and the proceeds are used to purchase (UC P,. ~+ , + D,, TL, )/b(,.. + L units of portfolio N. These transactions increase the proportion of asset j in the portfolio during the interval (7 ,T + 11 while leaving asset proportions in all other intervals unchanged. At.time T, the market value of Q is

vp7. =

+D,.r+I

q.:+,

VN.r+

VNT.

I

Since Q is feasible for CYin a neighborhood of zero and since E&In VQr) is maximized at (Y= 0, the derivative of E&In I&.} with respect to CKmust be zero at (Y= 0, i.e.,

Since this must hold for any choice of j, T, and AT E/?,

with probability one for every choice of j and t. But this is exactly the condition that defines the numeraire portfolio. Appendix

B: The continuous-trading

case

If continuous trading is allowed, then prices, dividends, portfolios, cash flows, etc. must be defined as continuous-time stochastic processes and some I5If Pr&., = 0) > 0, then E&In Vsr) = - =. But, by assumption, there is a feasible portfolio Z with always positive value and E&In VZT)> - 2. Thus, Pr(k’v, > 0) = 1 and, given that prices and dividends are bounded, there is a 6 > 0 such that Q is feasible for IQI< S.

J.B. Long, Jr., The numeratre portfolio

56

assumptions

must be modified.

The conclusion,

however,

remains

the same:

There are no profit opportunities if and only if a numeraire portfolio is allailable. B. 1. Special assumptions and definitions in the continuous-trading case Let P(t) be the K x 1 vector of per-unit ex-dividend asset prices at time t. Let D(t) be the K x 1 vector of cumulative dividends per unit paid in the interval (0, t]. Finally, let Q(t) be the 2K x 1 vector of prices and cumulative dividends formed by stacking f’(t) on top of D(t). It is assumed that Q(t) is an It6 process of the form

Q(t) - Q(O) =

L’m(sJ ds + jo’S(s)

dw(s),

where m(s) is a 2 K x 1 stochastic process, S(s) is a 2K x :! K stochastic process, and w(s) is a standard 2K x 1 Wiener process [the elements of w(s) are iid normal variates with mean zero and variance ~1.‘~ The first integral above is a vector of Lebesgue integrals and the second integral is a vector of It6 stochastic integrals.” Defined this way, prices and cumulative dividends have continuous sample paths in the interval [0, T], and thus are finite, with probability one. They need not be jointly Markov, however, and thus conditional distributions at time t may depend on the entire history of prices, dividends. and any other information available at time t.” As in the discrete-time case, a portfolio X is a K x 1 right-continuous stochastic process [i.e., X(t) specifies the portfolio composition on exit from time t] that is operational in the sense that, for every t, X(t) depends only on information available at time t. The market value of portfolio X at time t, denoted by V,(t), is also defined as before: Vu(t)

OltlT.

=X(r)‘P(t),

In continuous time, portfolio cash flows, like dividend flows, are best defined in cumulative terms. Thus, let C,(t) denote the cumulative cash flows from portfolio X in the interval (0, t]. Since the cumulation begins after “Both m(s) and S(s) are observable at time s and nonanticipating with respect to increments in W(S), i.e., for any t, their behavior from time 0 through time t is independent of increments in w(s) subsequent to time t. “For these integrals to be well detined for 0
<=.

See Friedman (1975) and Malliaris and Brock (1982) for more on Ito integrals. ‘*For the special case where Q is a multivariate diffusion process, see Merton (1971).

57

J.B. Long, Jr., The numeraire portfolio

time 0, C,(t) does not include the initial value of the portfolio. defined to be zero and, for t > 0,

c,(t) = jbWs)‘[dP(s) +dD(

C,(O) is

s)] - [ vx( t) - v,(o)],

where dP(s) and dD(s) are the first and last K rows, respectively, of m(s) dr + S(s) dw(s). The integral in this expression for cumulative cash flow is the sum of all capital gains and dividends for the portfolio in the interval (0, t]. (This includes capital gains and dividends on reinvested dividends and net contributions in the interval.) Portfolio X is self-financing if, for every t in the interval (0, Tl, Pr(C,(t) = O] = 1. From the cash-flow equation above, the value of a self-financing portfolio is therefore an It6 process (the sum of a Lebesgue and an It6 integral, like the elements of Q). Thus, in the interval [0, T], its same path is continuous and hence finite with probability one. As in the discrete-time case, it is assumed that there is a self-financing portfolio with always positive value. With continuous trading, a profir opportunity is defined as a self-financing portfolio that satisfies (i) its value at time 0 is nonpositive, (ii) its terminal (time T) value that is nonnegative with probability one, and (iii) its initial value is negative and/or its terminal value is positive with positive probability. This is a slightly less inclusive definition than the definition for the discrete-time case, but given the presence of a self-financing portfolio with always positive value,.any more general profit opportunity (e.g., a portfolio with zero initial and terminal values and positive cash flows) can be converted into a profit opportunity as defined here. B.2. Tile numeraire portfolio

in the continuous-trading

A numeraire portfolio is a self-financing value that satisfies, with probability one,

portfolio

case

with always positive

where V,(t) is the market value of the numeraire portfolio at time t, and P/V and dD/(V + dV) are, respectively, vectors of numeraire-denominated prices and dividends. Since the numeraire portfolio is self-financing, dV, is given by dyv(t) = N(tY(dP(t) + dD(t)). The integral in the definition is the sum of numeraire-denominated dividends in the (t, Tl interval. It is technically defined by /

Pgjv=/(7)(l-(Gy

J.B. Long, Jr.. The numerarrr portfolio

58

together with the It6 multiplication (ds)‘=

0,

(dw(s))(ds)

rules =0

(vector of zeros)

and (dw(s))(dw(s))‘=lds. As in the discrete-time case, a numeraire portfolio can be characterized as a solution to the problem of maximizing E&In L’,(T)) (with respect to self-financing portfolios with a given positive initial value and always positive future value). A solution exists if and only if there are no profit opportunities. The first-order conditions of the problem are (equivalent to) the conditions above that define a numeraire portfolio.” B.3.

The composition of the numeraire portfolio

The problem of maximizing E&In V,(T)} is actually much more tractable in the continuous-time setting than in discrete time. From the cash-flow equation above, if X is self-financing, then

=

Y&V + ~‘X(S)‘A~S) ds + !3’XW-(3)dwb),

where /.L(s)~s + r(s)dw(s) = dP(s) + dD(s). b’u(t) is an It6 process in this case and Ito’s formula applied to In P’y:,(t>implies that E,{ln VX( T)) = In V,(O) +

where Y(S) = X(s>/V,(s>

(~T[Y(s)‘~(s) - ;Y(s)'2(s)Y(s)]

En

ds),

and X(s) = T(s)T(s)‘.

“Let V&(r) be the time t value of the portfolio that. among all self-financing portfolios with initial value one and always positive future value, maximizes E,,(ln V,y(T)l. Then, an alternative feasible portfolio can be formed by adding at time r a small investment in asset i financed by selling units of the original portfolio [at a price per unit k’\(r)]. Any dividends paid between time I and T on the investment in asset i are used to purchase additional units of the original portfolio. At time T. this alternative portfolio contains a units of asset i and 1 -~P,(r)/~~(r)+ojl’~[l/(r/,(r)+dV,(s))]dD,(s) units of the original portfolio, where u is the number of units of asset i purchased at time 1. The quantity CI can be chosen based on any information available at time t and the alternative portfolio will be feasible for Ial sufficiently small. Since the original portfolio is optimal, if X(cr) is the alternative portfolio, the derivative of E,{ln Vxca, CT)) with respect to CI must be zero at cr = 0. This yields the condition that defines the numeraire portfolio.

J.B. Long, Jr., The numeruire portfolio

59

Since Y(s) can depend on any information available at time s, maximizing E,(V,.JT)) subject to L’,.(O)= 1 and the self-financing condition (that all dividends and capital gains be reinvested) is equivalent to:

maxhe

Y(s)‘P(s) - +Y(s)‘~(s)~(s),

subject to

Y(s)‘P(s)

= 1,

for all s. But, given that Y(sYP(s> = 1. Y(s)‘pu(s) is the expected instantaneous rate of return on the portfolio at time s [conditional on information available at time s, e.g., p(s)] and Y(s)“(s)Y(s) is the instantaneous variance rate of the portfolio rate of return at time s.“’ Thus, the solution to the problem, the numeraire portfolio, is always mean-l*ariance-efficient in terms of the conditional return.

distribution

of its short-step-ahead

(‘instantaneous’)

rate of

If there is an instantaneously riskless asset at time t [if 2(t) is singular] and its instantaneous rate of return is r(t), then the numeraire portfolio at time t will be the mean-variance-efficient portfolio whose expected excess rate of return, Y(tl’p(t) -r(t), is equal to the variance, Y(t)‘C(t)Y(r), of its rate of return. This equality of mean excess return and variance is achieved by investing a fraction A(t) of portfolio value in the tangent portfolio” and the remainder in the riskless asset, where h(t) is equal to the mean excess return on the tangent portfolio divided by the variance of the tangent portfolio return. Coincidentally, the numeraire portfolio is such that the expected excess rate of return on any portfolio whose value is not zero is equal to the covariance between its rate of return and the rate of return on the numeraire portfolio. If there is no riskless asset at time t [if S(t) is nonsingular], then the numeraire portfolio is the portfolio that would be the tangent portfolio if a riskless asset were available and its rate of return were equal to the mean rate of return on the minimum-variance portfolio minus the variance of the rate on the minimum-variance portfolio.” This is the riskless rate at which A(t) is equal to one. If the market portfolio is the mean-variance-tangent portfolio (i.e., if the standard CAPM describes expected asset returns), then the numeraire portfolio is a levered position in the market portfolio. If the market portfolio is mean-variance-efficient but no riskless asset is available, the zero-beta form of the CAPM [Black (1972)] holds and, like every other mean-variance-efficient portfolio, the numeraire portfolio is a mixture of the market portfolio and the zero-beta portfolio. 2oFormally, lim, _,(E,[R(s,

let r)l/(t

R(s, t) = [Vx(r) - V,(S)]/C’,~(S) for -sH and Y(s)‘ZX)Y(J) = lim, _,(var,[R(s.

“See

Sharpe

“The

minimum-variance

I 2 s. Then I&%

Y(s)‘p(s)

=

-s)).

(1964). portfolio

is the portfolio

that minimizes

Y’CY subject

to Y’P = 1.

60

J. B. Long, Jr.. The nrmrraire portfolio

J.B. Long, Jr., Thr numeraire porrfollo

WE.-

C

61

J.B. Long, Jr

The nrcmrrnire portfolio

zP3rJ-PP =. =. 3 = = I

=, =

=

r2

PI

p

s

==

f

=

I-

5

Ic.

-

=.

z. 7 =: =’ _ = =

o.ooott - O.WSh - O.W7 I

RC)SN C/IN ‘IXYN

0.0 I67 0.045 1 0.0404

0.6 I 0.27 0.X2 0.x4

- 0.0”) 0.5’) 0.70

0.3x

- 0.61 - O.OI

0.14

- 0.46

- 0.73

- 0.83 - 0.Y I

0. I2 0.0’)

- 0.47

- 0.74

- 0.82 - 0.00

- 0.01 - 0.02 - 0.0s

- 0.20

- 0. I5

0.02 - 0.07

0.03 0.03 - 0.04 - 0.02 - 0.02

- 0.0

I 0.04 0.03 - 0. I3 0.07 0.07

- 0.02

0.03 0.06 O.Oh 0.07 0.01 o.cn O.Oll

0.14

-

- 0.07

IJ.OX 0.07 0.0x 0. I I 0.02 0.05 0.00

“/(@I is the monthly rate of relurn on lhe portfolio of lirms in IIW IIIIINYSE size quinlile. The snxdlcs~ tirms are in quinlile I. Firms are rect;~ssiticd al the heginning of e;~ch year. I’11 is lhe monlhty I,c)llliI,g-periotl return on lhr tl,l,olsoi,-Sinquelirlll C’orpor~~t~ Ibnd tmtex. WY is the one-monlh ‘I‘-bill rale. RMV is the relurn on lhe value-wrighled NYSE porlt’olio. “It’ relurns are norm;dly distributed, eslimaled skewness has mean 0 and sland;~rd error O.tH. ‘II’ relurns are normdly dislrihulcd, eslimalcd kurlosis haa mean 0 ;mcl sl;lnctard error 0.37. “Markrl-atjusled returns arc raw returns minus RMV, the return on lhc NYSE porltiolio. ‘For each month and each assel, market-model par;imelrrs ti)r lhr forecast are compulcd using the returns t’rom the lirsl 00 of the talesl 72 monlhs. ‘The maximllm-likclihtx,tl numrraire proxy is lhe lixed-weight purlfolio that best approximates the maximum-likelihood elimule ot’ Ihe tJl~obserVed lir~leseries of returns on lhe ‘Irue’ numernire porltdio li)r this marker. 'l‘hc weights on HQI-/(C)S, c’8, ;ird WY arc, respeclivety, O.t9, 0.14, 0.12. 0.1 I, O.I4, 0. IO, ilnd 0.14. See lablc 2 lor dc~;lils.

x.77 7.3 I 7.20

6.31

4.x1

- I.11 - 1.51

- I.‘)3 0.46 0.40

20.27 5.46

3.3 I - 1.27

l’~~twl E: Nlrttr~rrrire-tk~trot,rbrcr~rtlwr~rh~s(trlotlercrir~,-I)orrlblio pmty = r,,rr.r.-li~rlil~twll prosy)

0.0532 0.02 I6 0.0 I8 I 0.0 I72

12 lesl tier zero me;ms p-value = 0.12Y

0.0066 wo2 I 0.w I6 O.MKl

R@lN RL)ZN RQ3N RQJN

-

J. B. Long. Jr.. The numerurre portfolio

‘szesm7==c,

I=

---=-_5=

z

-’

1

4

-’

i

_-4-e_--

II

I

x%x=rJFrc,P t--lnrc.m--P,-

== i__ i -II_d___

===rr=_--

r

r

i

x

z

2

O.M)40 0.0020 O.(H)13 - o.ow5 - 0.0087 - O.otlHI

R@ZN RQ3N RQ4N RC)SN CllN 7’UYN

Oh5 0.39 0.00

- o.su Oh7 0.5s

0.01 IO

o.wix 0.0363 0.0343

0.0045

O.oll25

RQZN

RQ3N RQ4N RQSN CBN TBYN

I .30 0. IO - 0.03 0.77 0.62 0.42

1.38

0.90

0.91

0.85

0.0132 0.0 I23 0.0140 0.0286 0.0274

0.0157

- 0.20 - 0.48 - 0.3 I 0.54 0.2h

0.07

0.45 0.77

0.56 0.82

0.79

0. I h 0.56

- 0.3 1 0.00 0.4 I (1.73

- 0.75 - O.XH - 0.88

- 0.69 -0.2x 0.x0

- 0.72 - 0.90 - 0.0 I

- 0.74 - 0.33

)

0. I I

0.18

0.13

0.13

0.17 O.O’J 0. I’)

0. I4 0. IX 0. I I

0. I2 - 0.01 0. I5 0. 10

.-lrXc~lil~otx/prory )’

-0.10 0. IO 0.93

- 0. I2

- 0. I I

- 0.02

0.17 0. IO 0.09 - 0.07 - 0. I4 0. I I 0.0x

(1.0’) OSN) - 0.01 - 0.01

0.08

u.00

O.Uh

- 0.03 0.04 0.04 - 0.03 -0.0x 0.04 0.0X

1).03 0.0’)

- 0. I3 - 0.07 - 0.05 0.05 0. I2

“RQrl is the monthly rate of return on IIIC ptrrrfolio of lirms in IIIC ~rih NYSll aizr quintilc. The ~m;dlus~ Iirms are in quinlilr I. Firms arc re&ssified aI the beginning of each yr;~r. (‘II is Ihe mollthly-llolJillg period relurn on the Ihboraon-Sillquelirld C’orporale Uod Index. 7BY is lhe one-month T-hill rate. RMV is the return on the value-weighled NYSE portfolio. ‘If returns iire normally dislrihulrd, rslimalrd skrwness has mean 0 and s!;lntl;trd error 0.18. ‘If return> arc’ normally tlistril)ulctI, cs~im;~tctl kurrosis IXIS I~C;~II 0 ;~nd ~t;rntlartl error 0.37. dMarkcl-;djusled rclurns arc raw rclurnh mimes RMV, lhc IC’IU~II OII IIIC NYSE porlfolit). “For each month ;mtl each ;ISSCI, market-model pmamelers for the lorcc;tsl are con~pu~ed using IIW relurus l’roni 111~ lirsl 60 IO the hle!d 72 months. ;Ipproxim;llus IIIC niaxirntrm-likelihood estimate 01 lhc ‘The In;lxinllllrl-likelil)ootl numcrairc prcjxy i\ lhu lixed-wuighl porlfolio lll;rl hl unobserved time series of rclurns on 111~.‘true numcrairc porllolio for Ihi.\ markcl. The wcighls on RL)/-KL)S, C/J, UI~ ‘/TIY are, rcspcclivcly, 0.10, 0.14, 0.12, 0.1 I, 0.14, 0.16, mid 0.14. See lablc 2 for delails.

I.30

0.56 0.50 I .32 0.82

I.24 I.11

,~tlrr

-0.10

- 0.07 0.15

0.3X

0.47

O.XH

-MB

0.02

0.2 I

0.14

0.82

0.70

I’crtttl I<: N~ctrrrrctir~,-~l~~~~~~tzi~~~~~~~~l ~CIW,~c ( n~cnrc,r~~i~c,-i~~~rf~~l;~~ pnay =

0.0203

T2 lest for zero me;ms ~~-v;duc = O.oOI

o.ooo2 - 0.0084 - 0.0077

0.0019

0.0064

RQIN

0.82

0.0209 0.01hl

0.0285

I’utrrl I): Ntrnfurcrirc,-c(~frw,rBlcrlrrlr&crtt.s t trru~~errrrrc,-/~l~r~~~f;(~ pmty = NYU: ,,~lttc-w,~*i~lrtt,llportjih~

K! test hr zero mc;ms p-value = 0.002

0.0060

R@lN

J.B. Long. Jr., The ntrmrratre portfolio

o.olm o.w32 0.0023 O.00I Y - 0.0004 - 0.olH)4 0.0007

0.0s 12 0.0346 0.0260 0.0184 0.0074 0.043X 0.0470

1.X’) 0.7’) 0.34 0.30 0.53 0.4 I 0.26

- U.0022 -0.0010

L’IIN

0.0367 0.022x 0.0 I flY 0.0157 O.OIOI 0.0407 0.0453

-

I .40 0.22 0.78 0.2s 0. I I 0.04 0.08

0.05

0.X9 0.90

0.76 0.x5

I .3Y I .h7

3.xX

x.14 1.X2( 1.61 I .3Y

0.84

Oh I 0.84

0.13 0.45 0.72

prcq - 0.50 - 0.24 I). 10 0.03

0.24 0.25 0.25 0.24 0.04

-0.2x - 0.32 - 0.3 I -0.3x - 0.20 0.7h

0.03

-0.x0 - 0.87 -0.x5 - 0.55

-

0. I4 0. IO

- 0.02 - 0.IY

(1.01 0.0 I

0.73

0.03 0. Ih 0.17

- 0.06 0.01 0.0’) 0.0’~ - 0.04 0.06 0.07

0.76 0.87 O.H2

= m~.t .-likc~li/mxl pro,ty )’

-

= NYSE fnl~rc-,vei~/~tr(l port&liv)

0.17 0.24 0.33 0.56

-0.13 - 0.04 (l.03 - 0.03 - 0.23

- 0.02 - 0.02

0.0 I - 0.07

- 0. I? 0.02 O.US

- 0.10

-0.21 - 0.N - 0.1’)

0.00 0.01 - lJ.lJS 0.07 0. I2

0.0s 0.07 0.04 0.03 - 0.0 I 0.04 - 0.02

“RQrr is the monthly rate of return on the portfolio of firms in the ,rth NYSE size quintile. The smallest lirms are in quintile I. Firm:, arc reclassified at the beginning of each year. C’fI is the monthly holding-prriotl return on the Ihbolsoll-Sinquefield Corporate I%ond Index. WY is the one-month T-bill rule. RMV is the return on the Calue-weighted NYSE portfolio. hlf returns are normally distributed, estimated skewness has mean 0 and standard error 0.1X. ‘If returns are normally distributed, estimated kurtosis has mean 0 and standard error 0.37. “Market-adjustrd returns are raw returns minus RMV, the rrlurn on the NYSE portfolio. ‘For each month and each ;ISS~I. mnrket-model p;lrameters for the forec;tst ;ire computed using the returns from the first 60 of the laxest 72 months. ‘The maximum-likelihood numerairr proxy is the lixed-weight portfolio that best approximates the maximum-likelihood estimate of 111~ unobserved time series of returns on the ‘true’ numeraire portfolio for this market. The weights on RL)I-RQS, CB. and TUY are, respectively, 0.10, 0.14, 0.12, 0.11, 0.14, 0.16, and 0.14. See table 2 for details.

72 lesl for zero means p-value = 0.3Y7

TUYN

HQSN

0.002x 0.00 I2 0.0004 0.otn~2 - O.ol)20

RUIN RQZN RQ.?N RQJN

0.94

1 I’tr~wl E: N~otrc,rcrin,-ck1,,l(~tttit~~i:~~~l rwuw ( rtrrtt,rrcrirc,port/i~iio

2.1x I .72 0.xX 0.66

I .93

4.25

‘).X3

I’mel I>: Nfortrrrrir~-tlrtto,,rbror~,~l r~t~~rt~.s (nrrnrc,r[rir~-jMtl~~~i~Jproq

72 test for zero means p-v;dur = 0.314

RQIN RL)2N RL).?N RQ4N RQSN C’UN TUYN

63

J. B. Long, Jr.. The nmnera~re porrfolro

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1. B. Long, Jr., The numeraire portfolio

69

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