Historically, preferred stock ranks well below common stock and bond issues as a source of financing in the private capital market. Relative to issuance of bonds, the issuance of preferred stock burdens the issuer with a fixed financing cost without benefitingthe issuer with the tax deductability associated with interest payments.

To sum up, Emery and Cogger [5] have raised several interesting questions concerning the derivation of the safety index (as well as the related risk of ruin) and the interpretation of that index which needed to be addressed. While the potential limitations discussed are theoretically possible, closer examination reveals that most of the concerns raised are unlikely to occur in practical applications, although certain of the procedures utilized were in need of further explanation. Several of these issues also provide extensions of the present work to make the estimation of the risk of ruin an even more robust measure of the potential for corporate failure.

The usual formulation of the portfolio selection problem through meanvariance analysis assumes that the variance-covariance matrix of the rate of returns on risky assets is non-singular. In view of the literature discussing the creation of riskless portfolio from carefully balanced quantities of risky securities (e.g., shares and warrants as in Black-Scholes [1]), the assumption of non-singularity may be challenged. Consequently, the proofs of the classical theorems of portfolio management may no longer be developed as originally presented. Buser [2] presents a means of using a singular variance-covariance matrix in the derivation of portfolio weights, which, although slightly flawed by a mathematical error, still provides some interesting insights about the efficient frontier. We present a corrected version of Buser's method in Section II; in Section III, we comment on some of the implications of his presentation and indicate some more precise results.

In “The Pricing of Premium Bonds,” Livingston [4] presents an erroneous analysis of the coupon effect on yield to maturity (YTM). This comment will present a correct analysis and briefly indicate Livingston's error. Following [4], we will assume no transaction costs, etc., and confine our analysis to N-period bonds (N = 2). The prices of premium bonds (P+) and discount bonds (P−) can be represented as:

where

TP = tax rate on income of marginal investors,

TG = capital gains tax rate of marginal investors,

A = market price of an untaxed N-period $1- annuity,

D = market price of an untaxed N-period $1 discounted note,

C = coupon,

F = “face” or principal amount of bond,

and PS, PA, PDS, and PD are the prices of the annuities and discounted notes implicitly in taxable bonds. Expressions (1) and (2) show equilibrium prices as functions of A, D, TP, and TG; cf. McCulloch [6], Caks [2], and Livingston [3, 4, 5].

In a recent paper [3], Livingston analyzes the relationship between bond prices and market discount rates in a market with taxes, but which is otherwise frictionless. Two main cases are addressed. In the first, all investors are assumed to be in the same tax bracket and a relationship [17] is derived between the price of a bond, the rates of tax on income and capital gains, and the aftertax term structure. In a subsequent section, it is claimed that, when the assumption of a common tax bracket is removed, but maintaining the frictionless market assumption, the analysis is more or less unchanged:

”..there must still be a unique price of an annuity of maturity j and a unique price of a discount note of maturity j. There will be implicit tax rates in these annuities and discount notes for each maturity.… The bond pricing equation is the same except that TP and TG are replaced by TPj. and TGj., implicit marginal tax rates for each maturity” [3, p. 20].

Professors Sorensen and Hawkins (hereafter SH) have utilized regression analysis to examine the pricing of preferred stocks both before and after a particular event. This event, the NAIC event, occurred in 1979 when the National Association of Insurance Commissioners (NAIC) adopted a rule permitting insurance companies to carry sinking fund preferred issues at book value rather than at the market value required before. SH results indicate nine to 12 variables have a significant effect on the pricing of preferred stock.

This page replaces the original in Richard Roll's article, “Orthogonal Portfolios,” in the December 1980 issue. Tear out and insert.

Since the orthogonal frontier extends indefinitely with nonzero slope, every nonefficient portfolio has an orthogonal portfolio with the same mean return.

Models of security markets invariably assume that market participants act intelligently since any other assumption generally leads to absurd results. Schaefer's Comment makes the incorrect assumption that one market participant, namely the government, acts unintelligently by allowing investors to reap tax arbitrage profits. Consequently, the conclusions in the Comment are erroneous. As I will show below, a government can (and the U.S. Government does) prevent tax arbitrage and any resulting disequilibrium in the bond market through two simple tax rules. My earlier paper [1] primarily showed that bond price must be a linear function of coupon for any maturity under the assumption of constant tax brackets for all investors, and then suggested that a linear relationship will also hold with investors in different tax brackets. I will prove this last result below.

This paper is concerned with two decisions facing firms with defined benefit, trusteed pension plans: the amount of funds to contribute to the pension fund in any given period; and the selection of a particular mix of assets for that fund. The pension decision area is of importance for many firms today due to the recent rapid growth in pension costs in absolute terms and as a percent of firm profits. Unfunded liabilities (pension benefits payable net of pension assets) have become a sizable component of the capital structure of many firms. Pension funds (assets) also have grown rapidly due to liberalizationof pension benefits, inflation, the higher funding requirements of the pension reform law (ERISA), and the overall aging of the labor force.

My paper, “The Pricing of Premium Bonds [2],” is the first to show deductively that the taxation of premium bonds makes it possible for yield to maturity to reach a maximum at par. My paper also supports this analytical result with empirical examples.

The paper by Frankfurter and Hill is an interesting contribution to our understanding of the interrelationship between funding and portfolio investment decisions in pension fund management. This research summary is written with grea clarity, and the first three sections will be must reading for anyone doing serious work on the theory of corporate pension fund management. The reader would be well advised to go ahead and read the fourth section (“a numerical demonstration”) as well, for there theauthors clearly demonstrate both the power and serious weaknesses of a linear programming approach to this multiperiod, fixed horizon problem.