Actual and warranted relations between asset prices.

1. Introduction

A VARIETY of efficient markets models can be represented in the form P[sub it] = E[sub t]P[sup *, sub it] where P[sub it], is the price of asset i at time t and P[sup *, sub it] ex post value, i.e., fundamental value.(1) in this paper, we inquire what such models imply for the covariance and for the correlation between the prices of the assets in terms of the covariance matrices of the ex post values. We show that while knowledge of the covariance matrix between ex post values does not permit us to predict, using efficient markets models, the covariance between prices, it does allow us to put bounds on this covariance.

Certainly, there is a common-sense presumption that these price covariances, correlations, as well as betas and factor loadings, depend on the covariances or correlations of ex post values. If an observed covariance between prices is to be justified in terms of these models, then there must be enough covariance between ex post values to warrant thc covariance between prices. But apparently the limits on the actual covariances between prices that may be warranted by the covariance matrix of ex post values have never been set forth in a general form before. These are important limits to set forth, since empirical finance is widely concerned with observed correlations among asset prices, and much work is based on the general notion that these have something to do with fundamentals. We will apply our theory to a study of the covariance and correlation of log price-dividend ratios between the United Kingdom and the United States. 2 In so doing, we will be able to offer some evidence on the claims of some who, observing the UK and US stock markets often rise and crash together, have doubted that information about fundamentals in the two countries could justify the extent of the co-movements.

Knowing the relations among these covariances and correlations is important for a number of purposes. :Uhey would help us to understand whether intemational transmission of asset price movements must be understood in terms of something other than the simple present value models; that is our immediate objective here. Beyond that, they may help us to understand how fundamentals interact with investor information to determine betas or factor loadin s of asset prices.

Many empirical studies of relations among financial prices refer either explicitly or implicitly to covariances among fundamentals to motivate the construction of the study or the interpretation of results. For example, Fama and French (1989, pp. 3-4) examine whether forecasting variables related to business conditions track common variation of expected returns on bonds and stocks for the US 1927-87. They ask: `Are the relations [between returns! consistent with intuition, theory and existing evidence on the exposure of different assets to changes in business conditions? and they conclude that the results are comforting.' Chen et al. (1986, p. 402), in their study of economic forces in the US stock market 1953-84 and in their selection of macroeconomic factors to use for stock market returns appear to base their selection on the likely correlation of fundamental with these. They write: 'Our conclusion is that stock returns are exposed to systematic economic news, that they are priced in accordance with their exposures, and that the news can be measured as innovations in state variables whose identification can be accomplished through simple and intuitive financial theory.' Pindyck and Rotemberg 1988, 1990, p. i) estimate multiple indicator multiple cause (MIMIC) models of asset prices where the indicators are returns and the causes are economic variables related to fundamentals. They conclude from thir study of US stock prices 1969-87 that we show that comovements of individual stock prices cannot be justified by economic fundamentals'. King et al. (1990, p. i) estimate a factor model for 16 national stock markets 1970-88 using ten macroeconomic variables. They conclude that 'the main empirical finding is that only a small proportion of the time-variation in the covariances between national stock markets can be accounted for by observable economic variables'. The conclusions in these different studies seem rather varied; it is worth examining whether these claimi can be evaluated in terms of a consistent and rigorous theoretical framework.

The key problem in carrying out the objective of reconciling correlations of prices with correlations of fundamentals is that we do not observe the full information set available to market participants to forecast present values, and in the framework of efficient markets theory, we must assume that market participants might have superior information. This means we cannot observe the optimal forecast at time t of P[sup *, sub it] cannot observe directly its covariance with anything, and thuszannot calculatejust what the covariance of prices should be.

We can only put bounds on the warranted covariances of prices from the knowledge of variance matrices of ex post values. In Section 2 we derive covariance bounds for the case when no forecasting information is available for statistical analysis, while in Section 3 we show that using more information is helpful both in deriving more efficient covariance bounds and in deriving bounds for the warranted correlation between the two assets. Section 4 contains a description of the pricing theories that we use for stocks, Section 5 describes the data. In contrast to most of the aforementioned studies, we use very long time series data, extending from 1919 to the present, to enable us to see more of the low frequency variation in fundamentals that might explain covariances between prices, and in contrast to all of these studies we use direct measures

of ex post or fundamental value. Section 6 describes the econometric methodology, and Section 7 gives the results.

2. The case of no forecasting information available for statistical analysis Suppose first that we wish to base our statistical analysis only on the covariance matrix of the vector P[sup *, sub t] = [P[sup *, sub it,] P[sup *, sub 2t,!!', whose ith element is the present value of the dividends accruing to asset i. The corresponding vector of prices P[sub t] has as its ith element the price of asset i. By basing our analysis only on these covariance matrices, we are attempting to see in very simple and basic terms whether we can find evidence of excessive co-movement of prices. We will suppose that the present values and corresponding prices have been suitably transformed so that they are stationary, and so that variance matrices var(P[sup *!) and var(P) exist.

How large can the covariance between P[sub 1t], and P[sub 2t] be, given var(P[sup *])? is, knowing how much the fundamental variables vary and co-move, how much can P[sub 1t] and P[sub 2t], co-move? To answer this, we must solve a nonlinear programming problem: maximize cov(P[sub 1t,] P[sub 2t]) with respect to [theta](P[sub 1] and [theta](P[sub 2]) subject to the inequality …