Knowledge that Transforms

This note corrects recent results concerning qualitative portfolio composition in Roll's ‘Critique of Asset Pricing Theory Tests’.

This paper examines the relationship between forward exchange rates and subsequently observed spot rates. No evidence is found for a liquidity premium on forward exchange, indicating that the forward rate can be used as a proxy of the market's expectations and that open exchange positions involve little systematic risk. It is also shown that forward exhange is priced as if the exchange rate could be characterized by a diffusion process with a trend, although there is some evidence such a process does not adequately characterize the exchange rate in all cases.

This paper examines the problem of deriving Black's (1972) minimum-variance zero-beta portfolio. Long's (1971) methods, used by Morgan (1975), are briefly mentioned. Then the complementary pivot algorithm of Lemke (1965), which has been shown to be capable of deriving the optimal solution to certain quadratic programming problems that are subject to a non-negativity constraint, is described. Finally, Lemke's algorithm is shown to be capable of deriving the minimum-variance zero-beta portfolio efficiently from samples of risky assets where both long and short positions are allowed by reformulating the problem so as to avoid the difficulties encountered by having a non-negatively constraint.

This paper is concerned with the valuation of multiperiod cash flows in a world where prices are determined according to the Sharpe-Lintner-Black model of capital market equilibrium. We find that the current market value of any future net cash flow is the current expected value of the flow discounted at risk-adjusted discount rates for each of the periods until the flow is realized. The discount rates are known and non-stochastic, but the rates for the different periods preceding the realization of a cash flow need not to be the same, and the rates relevant for a given period can differ across cash flows. The risk adjustments in the discount rates arise because of uncertainties about reassessments through time of the expected value of a flow and the relationships between these reassessments and the corresponding reassessments of the expected cash flows of all firms.

A simple, necessary and sufficient condition is derived under which portfolios that are mean-variance efficient on a before-tax basis are also efficient on an after-tax basis and vice-versa. Under this condition and the hypothesis that investors demand after-tax efficient portfolios, the ‘no-tax’ form of the Capital Asset Pricing Model provides an accurate description of equilibrium asset returns even though investors in the economy may be subject to a wide variety of tax rates on dividends and capital gains. Evidence reported by Black and Scholes (1974), however, makes the condition for equivalence of before and after-tax efficiency empirically implausible. The paper thus concludes with a characterization of some essential differences between before and after-tax efficient portfolios and of the after-tax efficiency losses associated with before-tax efficient portfolios. The relation of these results to the issue of corporate dividend policy choices is also discussed.

Dual fund shares allow individuals to satisfy their divergent preferences for the ordinary income and the capital gains components of return. Institutional restrictions on short selling are shown to permit discounts on dual funds to fluctuate within wide bounds. However, these fluctuations are shown to be consistent with informational efficiency. The average discount is shown to have a contemporaneous association with net redemptions of diversified open-end funds. It is shown that the high turnover of dual fund portfolios is not warranted by their portfolio performance and causes redistributions of wealth between income and capital shareholders.

Assuming continuous trading in continuous time with Brownian motion processes, the basic capital asset pricing model of Sharpe, Lintner, and Mossin is developed under arbitrary distributions of investors' beliefs consistent with available information. Results on the processing of information are reported, and properties of investors' portfolios are derived.

A general formula is derived for the price of a security whose value under specified conditions is a known function of the value of another security. Although the formula can be derived using the arbitrage technique of Black and Scholes, the alternative approach of continuous-time portfolio strategies is used instead. This alternative derivation allows the resolution of some controversies surrounding the Black and Scholes methodology. Specifically, it is demonstrated that the derived pricing formula must be continuous with continuous first derivatives, and that there is not a ‘pre-selection bias’ in the choice of independent variables used in the formula. Finally, the alternative derivation provides a direct proof of the Modigliani-Miller theorem even when there is a positive probability of bankruptcy.

The relative performance of several portfolio selection strategies is assessed empirically. These strategies vary in sophistication from a ‘naive’ strategy of maintaining equal dollar investments in each stock available to a strategy that periodically uses updated parameter estimates to calculate new optimal proportions of portfolio value to be invested in the stocks available. Although it is to be expected a priori that relatively sophisticated strategies will perform at least as well as the more naive strategies, implementation costs will clearly differ across strategies and across investor-specific parameters such as total portfolio value. Thus estimation of the various strategies' performance gross of these costs is a necessary consideration in rational strategy selection by any given investor.

The option pricing model developed by Black and Scholes and extended by Merton gives rise to partial differential equations governing the value of an option. When the underlying stock pays no dividends – and in some very restrictive cases when it does – a closed form solution to the differential equation subject to the appropriate boundary conditions, has been obtained. But, in some relevant cases such as the one in which the stock pays discrete dividends, no closed form solution has been found. This paper shows how to solve these equations by numerical methods. In addition, the optimal strategy for exercising American options is derived. A numerical illustration of the procedure is also presented.