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, $P<FV$.

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

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

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

$\displaystyle YTM_n>0\implies 1+YTM_n>1\implies(1+YTM_n)^n>1\implies\frac{1}{(1+YTM_n)^n}<1\implies 1-\frac{1}{(1+YTM_n)^n}>0$

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

$\displaystyle CPN\times\frac{1}{YTM_n}<FV$

$\displaystyle\implies\frac{CPN}{FV}<YTM_n$

$\displaystyle\implies r<YTM_n$

$\displaystyle\implies YTM_n>r$

QED

• It’s not clear that you can assume that YTM_n is positive; it would be, but that probably needs to be validated. – Brian Romanchuk May 7 '18 at 19:56
• @BrianRomanchuk for many bonds YTM_n is negative, e.g. Schatz and Bobls. But the derivation would still result in the same since the inequality is flipped twice, the second time when multiplying by YTM – Attack68 May 9 '18 at 21:01

The tricky part of this question is that it does not specify that we are on a coupon date. If we are away from a coupon date, we normally assume that if we say that a bond “is trading at a discount”, we are talking about the clean price (which excludes accrued interest).

On a coupon date, we can use a simplified pricing formula and derive the relationship (as seen in another answer here).

The exact formula for the clean price will depend on the market convention. However, we could use the following structure for all market conventions.

1. Show that the clean price falls as yield increases, and vice-versa (by inspection of the clean price definition).
2. Show that the relationship between price and yield is one-to-one, i,e., for any yield, there is a single price. (This is not obvious, but I think this is a standard result for discounted cash flows where all future cash flows have the same sign.)
3. Show that if the yield on the bond equals the coupon rate, the clean price is par (for any date). Yield/price conventions were chosen precisely to get this behaviour, but one would have to validate depending on the convention. We should see that a lot of terms cancel out to get the desired result. This step requires the most algebra.
4. We turn to your original statement. We assume that the clean price is below par. (We use (2) to show that this price is associated with a unique yield. Not sure whether this step would be necessary.) This is less than the price when it is trading at par; we apply (1) to show that this implies that the yield is greater than the coupon rate.