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.
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1 Answer
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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
$$r_c>y$$
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.