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Bond and Mathematics

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A Simple and Student-Friendly Approach to the Mathematics of Bond Prices
Edward R. Lawrence* Florida International University Siddharth Shankar* Florida International University
In his seminal paper Malkiel (1962) rigorously developed and proved five theorems showing the relationship between changes in yield to maturity and bond price movements. These theorems are important in the eyes of the academicians and practitioners and find a place in virtually all finance textbooks dealing with the pricing of bonds. The proofs of the theorems are based entirely on calculus and seem to be challenging for finance professionals and students. Hence, finance textbooks usually describe these five theorems using figures and numerical examples to illustrate the results instead of providing any logical proof. In this paper we prove the Malkiel (1962) theorems using simple algebra and provide an alternate way of understanding the results.

Introduction
In his seminal paper Malkiel (1962) rigorously examined the relationship between the yield to maturity of bonds and their market prices. He developed and proved five theorems and showed how the relationship between the changes in yield to maturity and bond price movements depend on the coupon, the time to maturity, and the existing yield to maturity. Malkiel’s (1962) five bond theorems have become the cornerstone in understanding the relationship between bond prices and their various determinants. There is no book that we are aware of dealing with bond prices and related topics that does not refer to these five theorems. The original presentation by Malkiel is very elegant and uses calculus to prove these theorems. The proofs to the theorems seem so challenging that almost all of the professional and academic textbooks state the theorems and provide only numerical examples and figures to convey the points. For example, Sharpe

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