November 2011
Beginner
576 pages
13h 31m
English
book
Fundamentals of Financial Instruments: An Introduction to Stocks, Bonds, Foreign Exchange, and Derivatives
by Sunil Parameswaran
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APPENDIX 1
Partial Derivative of the Price with Respect to the Yield
From the bond-pricing equation, we know that
The derivative of the second term with respect to the yield is
which is obviously negative.
We will now demonstrate that the derivative of the first term is also negative by considering the present value of an annuity that pays $A per period.
We need to show that the term in the bracket is positive or, in other words, that
From the expression for a Maclaurin series, we know that
Thus, it is indeed the case that
Hence, the partial derivative of the present value of the annuity with respect to the discount rate is negative. Because the bond price consists of an annuity and a terminal cash flow, the derivative of whose present value has already been shown to be negative, we conclude that the partial derivative of the price of a plain vanilla bond with respect to the YTM is negative. ...
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