I know that if the coupon rate on a bond is larger than the yield-to-maturity, then the price must be higher than the par value. Yet I have the bond price equation $P=\frac{c}{y}\left(1-\frac{1}{(1+y)^T}\right)+\frac{Par}{(1+y)^T}$, and I wonder how I manipulate this equation to show that what I mentioned above is correct. Grateful for any tips to solve this.


Set the inequality $P>Par$, substitute the bond price equation, and then manipulate the inequality to find your result:

$$P>Par$$ $$\frac{c}{y}\left(1-\frac{1}{(1+y)^T}\right)+\left(\frac{Par}{(1+y)^T}\right)>Par$$ $$\frac{c}{y}\left(1-\frac{1}{(1+y)^T}\right)>Par\left(1-\frac{1}{(1+y)^T}\right)$$ $$\frac{c}{y}>Par$$

Let $c=r_cPar$ so that $r_c$ is the coupon rate (this is the fraction of the Par value that the coupon represents. Then the last equality is equivalent to


Therefore, the first inequality is true if and only if all of the following inequalities are. In particular, the last inequality gives the result you are looking for.


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