Footnotes 1. F. R. Macaulay, Some Theoretical Problems Suggested by the Movement of Interest Rates, Bond Yields, and Stock Prices in the United States Since 1856 (New York: National Bureau of Economic Research, 1938). 2. See M. L. Dunetz andj M Mahoney, Using Duration and Convexity in the Analysis of Callable Bonds, Financial Analysts Journal, May/June 1988. 3. F Black and M. Scholes, Pricing of Options and Corporate Liabilities, Journal of Political Economy 81 (1973), pp. 637-54. 4. For a more formal treatment, see J Hull, Options, Futures, and Other Derivative Securities (Englewood Cliffs, NJ: Prentice Hall, 1989), chapter 8. 5. See Hull, Options, Futures, op. cit. 6 The derivation is available from the authors on request. 7. See, for example, J Ingersoll, Examination of Corporate Call Policies on Securities, Journal of Finance 2 (1977), pp. 463-78; E. I. Altman, Debt Market: Are Returns Worth the Risk? Financial Analysts Journal July/August 1989; and R. King, Convertible Bond Valuation: An Empirical Test, Journal of Financial Research 1 (1986), pp. 53-69. 8. The derivation is available from the authors on request. 9. The derivation is available from the authors on request.