The purposes of this article are two. Some extensions to Mayers’ [1] classical work on portfolio building in the presence of nonmarketable assets are presented. In addition the implications of certain simple forms of taxation and redistribution are investigated.

The assumption that investors can borrow and lend at a riskless interest rate reduces the Mean-Variance (M-V) efficient set to only one optimal unlevered portfolio. However, once we realize that the market is generally imperfect and that the borrowing rate is higher than the lending rate, we can no longer use the mean-variance Separation Theorem. Instead, a number of unlevered portfolios must be included in the efficient set, while the optimal unlevered portfolio is selected on the basis of the investor's preference. The size of the efficient set of unlevered portfolios is a function of the type of empirical data used and of the disparity between the borrowing and lending interest rates.

The existence of risk aversion in portfolio theory can be explained by positing a concave utility function of wealth. In some cases it is useful to construct some measure of risk aversion rather than merely accept its existence.

The solution to the problem of portfolio choice is relevant in a positive financial economics context because it provides models of individual maximizing behavior which when aggregated to the level of the market provide models of equilibrium asset pricing. These models generally assume that the parameters of the probability distribution of security returns are known to individual investors. In practice, however, the individual has to estimate these parameters. To the extent that there is parameter uncertainty or “estimation risk”, what are the observable implications of a market equilibrium derived on the assumption that the information set of all investors is equivalent to a given set of sample data?

Most research in modern portfolio theory and capital market theory is based on investor selection of portfolios that are efficient in the sense that they are not dominated by other portfolios in terms of their risk-expected return characteristics. The most widely used measure of portfolio risk is the variance about the mean of the exante distribution of portfolio returns. The theoretical framework from which this measure of risk is usually derived was initially suggested by Markowitz [12], and is by now well known. Although variance has the attention of most researchers, another measure, semivariance, had some early support from Markowitz himself, and from Quirk and Saposnik [17], Mao [10], and others. Semivariance as a measure of risk can be derived from the same theoretical framework as is variance; it requires only a slightly different utility function. The semivariance of returns of portfolio p below some point h is defined as

where fp (R) represents the probability density function of returns for portfolio p. Semivariance portfolio theory is enjoying something of a revival in the works of Porter [15, 16], Hogan and Warren [6] and Klemkosky [8], and semivariance capital market models have been developed by Hogan and Warren [7] and Greene [5].

The problem of the portfolio demand for money was first rigorously studied by Tobin [22]. It has been analyzed since then, by Hicks [8] and Arrow [1], among many others. Many interesting results and implications regarding liquidity preference and risk-taking are derived in these studies. However, the effect of purchasing power risk on liquidity preference has been overlooked in these studies.

There appears to be some confusion in the literature on asset-pricing models about the role of default-risk in short-selling and borrowing by consumers. For example, in the Sharpe-Lintner ([16] [9]) model it is usual to assume that all consumers are able to borrow or lend without restriction, at the riskless rate of interest. This assumption has been recognized to involve an inconsistency in the theory, because with personal borrowing there is always a positive probability of default with the S-L assumptions, so that the consumer's bond (as seen by any lender) is not a perfect substitute for the safe asset.

In a recent issue of this journal Barnea [1] presented an empirical study of the impact of a specialist (market-maker) on the variability of the price of a stock. He concludes with others that “the chief cost of dealing with a market maker is the difference between the theoretical but unobservable equilibrium price and the transaction price, rather than the bid-ask spread.” This paper presents a rigorous dynamic model in which the specialist, who is uncertain about the future arrival of tenders, determines transaction prices periodically over the trading day. The structure of the model permits a direct comparison of the specialist's prices to the “equilibrium price,” to be defined below, and also to what prices would be in the absence of a specialist.

The need for an empirical measure of the relative value of financial theories becomes more important as we increase their number and sophistication. Although there are several methods of determining value, here we emphasize relative predictive power. According to this single criterion, a theory is insignificant if it fails to increase our present ability to predict future events. We recognize, however, that a theory can be useful if it merely increases our understanding of real events, or if it is based upon realistic assumptions.

There is a considerable body of empirical research in accounting devoted to the analysis of relationships between accounting numbers and security prices. Very roughly, this body of research may be classified into three different categories: (i) share price valuation models and the determination of market equity values; (ii) the measurement of “unexpected earnings” and their contemporaneous association with security returns; (iii) the forecasting of future security returns. The selection and definition of accounting numbers in most of these types of studies have, by and large, been quite heuristic. The accounting variables are usually selected with little consideration given to their empirical time-series behavior; more important appears to be their intrinsic economic connotations. The approaches can thus be thought of as stipulating the existence of “real” economic variables, e.g., real income for a period, and then using numbers of published accounting statements as estimates of the real variables. The errors in estimates of the true variables are then often minimized by the use of aggregation procedures and the diversification effects of such procedures. The postulating of real economic variables has another methodological advantage: it permits the use of comparative statics analysis of corporate behavior and its effect on equity risk and return. For example, Hamada [7], among others, has shown that leverage affects risk in the usually hypothesized manner but this analytical result depends on the assumption that leverage and earnings are real and unambiguous economic variables without “measurement” errors.

In a past issue of the Journal of Financial and Quantitative Analysis, Norstrπm [7] has presented a very simple sufficient condition for detecting whether a given pattern of cash flows over time has a unique nonnegative internal rate of return. Nor strum's condition is now widely cited in the literature and included in stock computer routines for analyses using the internal rate of return. See, e.g., de Faro [5] and Newnan [6].

In the literature dealing with bond price volatility, there have been two divergent approaches. On the one hand, theoretical papers have looked at bond price volatility in the instantaneous framework of the calculus. Using the derivative of bond price (P) with respect to yield to maturity (y), it has been shown that volatility is linearly related to this derivative (dP/dy). (See [10].)

Since Markowitz [15, pp. 98–101] and Sharpe [19] developed the single-index market model (SIMM hereafter) it has received considerable research attention. Empirical tests have established the model's econometric significance [3, 12, 13] in partial equilibrim analysis. However, research into the relationship between the SIMM and its macroeconomic environment has been meager. It has been shown that the market factor changes intertemporally [13, 16, 18, 20]. However, whether these changes in the market factor and, more basically, changes in the macroeconomic situation affect the SIMM is unknown.

The composite measures of investment performance: the reward-to-variability index, by Sharpe ([29], [30]) and Lintner [23], and the reward-to-volatility index, by Treynor [33], were developed after Markowitz ([24], [25]) and Tobin [32] popularized the mean-variance framework of analyzing the problems of certain investments. Since these are ex ante measures they are not directly applicable to the evaluation of ex post performance. A theoretical basis for doing so has been provided by Jensen ([17], [18]) who also developed another composite performance measure, the predictability index. In practice, these composite measures have been found to have problems. Foremost, they have been observed to exhibit systematic biases. Various causes of the biases have been proposed. These are: the existence of unequal lending and borrowing rates, the failure to consider higher moments of return distributions, and the elusive “true” holding period.

This paper develops an exact theoretical test of the presence or absence of a filter effect for a portfolio of securities and a general number of different filter sizes. It is a natural development from Praetz [8], which obtained exact expressions for the mean and variance of rates of return of the investment strategies under filter tests assuming the underlying stochastic process is a random walk. These expressions showed that expected returns from filter strategies are, in fact, less than the return from a buy-andhold alternative with which filter returns are usually compared.

The concept of a relationship between assumed risk and realized return is intuitively pleasing and has become widely accepted in the field of finance. Until recently this acceptance was anchored largely in what Hirschleifer [11] has called the “notorious fact” that stocks yield more in the long run than bonds and Hickman's finding [10] (since challenged by Fraine [7]) that over the years 1900–1943 the average ex post yield on publicly issued corporate debt was higher the lower the initial quality rating. However, with the advent of the capital asset pricing model of Sharpe [21], Lintner [16], and Mossin [19] the risk-return tradeoff concept has grown in importance and scope. The capital asset pricing model itself has weathered the years well, but has been theoretically and empirically revised, extended, and otherwise altered. Little remains of the original formulation except the proposition that in equilibrium more risk leads to more return--where “risk” for common stocks now means the nondiversified component as measured by the “Beta” coefficient of return volatility vis-à-vis the general market. As Modigliani and Pogue [18] observe after a review of the “more important” empirical tests of the capital asset pricing model: “Obviously, we cannot claim that the CAPM is absolutely right. On the other hand, the empirical tests do support the view that beta is a useful risk measure and that high beta stocks tend to be priced so as to yield correspondingly high rates of return.”

According to the current state of knowledge in finance, the expected rate of return adjusted for risk is independent of the stock price. The basic proposition of the capital asset pricing model (CAPM) is that the expected rate of return for each security is a function of the “risk” of that security, and that this risk is measured by the contribution of the security to the variability of the market portfolio. The implication of the CAPM is that knowing the price of a security perse will add nothing to predicting its expected rate of return.

The widespread notion that dollar-cost averaging can help an investor minimize the risk of investing all of one's capital in the market at an inappropriate time is aptly stated by Malkiel [4, p. 242]:

Periodic investments of equal dollar amounts in common stocks can substantially reduce (but not avoid) the risks of equity investment by insuring that the entire portfolio of stocks will not be purchased at temporarily inflated prices. The investor who makes equal dollar investments will buy fewer shares when prices are high and more shares when prices are low.