The characteristics of the mean-variance, efficient portfolio frontier have been discussed at length in the literature. However, for more than three assets, the general approach has been to display qualitative results in terms of graphs. In this paper, the efficient portfolio frontiers are derived explicitly, and the characteristics claimed for these frontiers are verified. The most important implication derived from these characteristics, the separation theorem, is stated and proved in the context of a mutual fund theorem. It is shown that under certain conditions, the classic graphical technique for deriving the efficient portfolio frontier is incorrect.

Despite the enormous attention received by the single-period mean-variance model in the literature, its structural relationship to the empirical world is still largely unexplored. The purpose of this note is to show that when certain consistency requirements and equilibrium conditions in the financial markets are taken into account, the collective judgment of the present literature concerning the mean-variance approach is in some respects too lenient and in other respects too harsh. In addition, it will be noted that the mean-variance model can only achieve consistency with the von Neumann-Morgenstern postulates and absolute preference (also known as first-order stochastic dominance) at the price of a severe upper bound on the risk aversion that can be possessed by the decision maker.

Portfolio selection models based on expected value-semivariance (E-S) criteria have been suggested as offering certain advantages over the expected value-variance (E-V) approach. Although variance is more tractable mathematically, it has not always been satisfying to financial theorists ([3, pp. 278–284], [5], [6], [7, pp. 193–194], and [10, pp. 72–73]). In the pioneering work in portfolio analysis, Markowitz [7, p. 194] observed that semivariance concentrates on reducing losses as opposed to variance which considers extreme gains, as well as extreme losses, as undesirable. In the presence of nonsymmetrical probability distributions, this equal weighting of gains and losses may not adequately describe the alternative portfolios available to the decision maker.

This paper examines the random-walk hypothesis in the forward exchange market by applying spectral analysis to the three-month forward rates for dollars against sterling in the period 1961–1967.

This paper will demonstrate that different margin levels are associated with the price behavior differences of certain commodity futures. In 1959, Harry Roberts suggested the methodology of rational subgrouping as a means of testing random versus systematic price changes. His methodological suggestion has had only limited testing in security markets and no direct application to domestic futures markets. This paper uses margin levels as a basis for rational subgrouping of selected commodity futures. Evidence supports the argument that, when a series of price changes is grouped according to margin levels and these levels are analyzed separately, nonrandom characteristics that tend to be offsetting in the aggregate series become evident. The nonrandom behavior observed is consistent with the hypothesis that, in certain periods, margin levels have been set too high to attract a volume of speculative services necessary for the maintenance of market balance.

We have seen a recent surge in the interest in some revision of the consumer finance laws, the most comprehensive of which has been the Uniform Consumer Credit Code. In the past, there have been many attempts to determine independently the effect one aspect of consumer finance legislation had on the market for consumer loans. However, the different regulatory provisions are not enacted independently. Thus, it may be more appropriate to investigate jointly the effects of several of the features of the laws.

This study deals with the problem of debt — composition of the firm — specifically, the optimal mix of short- and long-term debt, or the optimal maturity composition of the total debt. It can be viewed as a subproblem of the more familiar determination of the optimal mix between total debt and owner's equity (Modigliani and Miller [15 and 16], Durand [3], Schwartz [18], Solomon [20], etc.).

There now exists a formidable documentation of the hypothesis that time series of common stock prices follow random walks. Taking these empirical demonstrations as convincing, additional work has largely fallen into two classes: (a) direct theorizing and testing of the exact nature of the random price-generating process; and (b) theorizing and testing of the implications of random walks in common stock price for the values of related securities such as warrants and convertibles.

The literature of Index numbers contains much discussion of the relative merits of geometric and arithmetic averages of prices and quantities. The controversy on this subject dates from the middle of the nineteenth century and is fully described by Crowe [2], In recent times both types of averages have been applied to security price relatives to measure the performance of groups of securities over time. The purpose of this paper is to demonstrate the properties of these security indexes and to show the relationships between them and an index based upon a more general type of average called the power mean. The concluding section of this paper contains the proof of an interesting and important limit property which provides the conceptual link between geometric and arithmetic security indexes.

In [2] Hakansson and Liu presented a multiperiod portfolio model in which there is an optimal myopic policy. In particular, at any decision point j and state m the optimal amount to invest in opportunity i, namely , may be found by maximizing

(42a)

subject to

(42b)

(42c) ,

where the expectation is taken with respect to the β's, and the p's and r are positive constants (r > 1). Assumptions are made in [2] which guarantee that (42) has a unique optimal solution and that the set of vijm which satisfies (42b and 42c) is a nonempty, compact, convex set for all j and m.

In a recent article in this Journal, Haugen and Pappas (H-P, hereafter) [1] attempted to prove, within the framework of the capital asset pricing model, the already proven proposition that the cost of capital is invariant with respect to leverage even with risky debt. The H-P proof contains a serious error which we would like to point out in this note.

Although Imai and Rubenstein are correct that our proof — that the Miller- Modiglianl (M-M) and Sharpe-Lintner-Mossin (S-L-M) capital asset pricing models are mutually consistent — is incomplete, their comments indicate some confusion about the relationships involved in the equilibrium pricing of assets in these models. Further, they seem to imply that Stiglitz's proof in terms of dollar returns is in some sense superior to a proof in terms of rates of return. This is erroneous. Accordingly, we shall further clarify the relationships inherent in the models and correct our presentation of the proof of the invariance of capital costs in the context of the S-L-M model.