Prove that if a bond trades at a discount, its yield to maturity will exceed its coupon rate Can someone give me a mathematical proof of this statement? 


  If a bond trades at a discount, its yield to maturity will exceed its
  coupon rate.
If a bond trades at a discount, $P&lt;FV$.

$\displaystyle\implies CPN\times\frac{1}{YTM_n}\left(1-\frac{1}{(1+YTM_n)^n}\right)+\frac{FV}{(1+YTM_n)^n}&lt;FV$

$\displaystyle\implies CPN\times\frac{1}{YTM_n}\left(1-\frac{1}{(1+YTM_n)^n}\right)&lt;FV-\frac{FV}{(1+YTM_n)^n}$

$\displaystyle\implies CPN\times\frac{1}{YTM_n}\left(1-\frac{1}{(1+YTM_n)^n}\right)&lt;FV\left(1-\frac{1}{(1+YTM_n)^n}\right)$ --------------(1)

$\displaystyle YTM_n&gt;0\implies 1+YTM_n&gt;1\implies(1+YTM_n)^n&gt;1\implies\frac{1}{(1+YTM_n)^n}&lt;1\implies 1-\frac{1}{(1+YTM_n)^n}&gt;0$

So, we divide both sides of (1) by $\displaystyle 1-\frac{1}{(1+YTM_n)^n}$ to get

$\displaystyle CPN\times\frac{1}{YTM_n}&lt;FV$

$\displaystyle\implies\frac{CPN}{FV}&lt;YTM_n$

$\displaystyle\implies r&lt;YTM_n$

$\displaystyle\implies YTM_n&gt;r$

QED