# Standardize YTM's of coupon bonds with different coupon frequencies

In Financial Mathematics for Actuaries by Wai-Sum Chan and Yiu-Kuen Tse, the following formula is given for a $$n$$-year annual coupon bond with transaction price $$P$$ where the yield to maturity is $$i_Y$$.

$$P=F r \sum_{j=1}^{ n} \frac{1}{\left[1+i_{Y}\right]^{j}}+\frac{C}{\left[1+i_{Y}\right]^{n}}.$$

And then for a $$n$$ year semiannual coupon bond:

$$P=F r \sum_{j=1}^{2 n} \frac{1}{\left[1+\frac{i_{Y}}{2}\right]^{j}}+\frac{C}{\left[1+\frac{i_{Y}}{2}\right]^{2 n}}.$$

Question 1: Is this generalization correct? Where $$f$$ is frequency of coupon payments.

$$P=F r \sum_{j=1}^{f n} \frac{1}{\left[1+\frac{i_{Y}}{f}\right]^{j}}+\frac{C}{\left[1+\frac{i_{Y}}{f}\right]^{f n}}.$$

Question 2: That formula would give a nominal rate per annum. If I were to graph a yield curve of coupon bonds with different frequency, would converting the nominal $$i_Y$$'s with $$YTM = \left ( 1+\frac{i_Y}{f} \right )^{f}$$ first be correct?

Note: The bonds are not callable.

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