Why does the YTM equal the coupon rate at par?

Question

I know the YTM of a coupon bond is the interest rate $i$ which verifies

$ P =\frac{C}{(1+i)} + \frac{C}{(1+i)^2} + ...+ \frac{C}{(1+i)^n} + \frac{F}{(1+i)^n} $

where $P$ is price, $C$ is the coupon payment and $F$ is face value.

I don't understand why $i = C/F$ when $P=F$. In words: I can't grasp why the yield to maturity equals the coupon rate when the bond is priced at face value.

On the one hand I can't solve that equation above so that this fact is verified, but I might need some tools I don't have yet to do so. On the other hand it doesn't make intuitive sense to me on a conceptual level.

What am I missing?

Answer 1

Let $P$ denote the dirty price, $F$ the face value and $i$ the YTM. Using the geometric sum we get

\begin{align} P &= \sum_{j=1}^n \frac{C}{{(1+i)}^j} + \frac{F}{(1+i)^n}\\ &= C\frac{1-\frac{1}{{(1+i)}^n} }{i} + \frac{F}{(1+i)^n} \end{align}

and thus

\begin{align} P=F \Leftrightarrow & F= C\frac{1-\frac{1}{{(1+i)}^n} }{i} + \frac{F}{(1+i)^n} \\ \Leftrightarrow & C\left(1-\frac{1}{{(1+i)}^n}\right) =i \left( F- \frac{F}{(1+i)^n} \right)\\ \Leftrightarrow & \frac{C}{F} = i \end{align}

Answer 2

The price of a bond is determined by the sum of the discounted cashflows plus the discounted face value of the bond. Intuitively and academically, a bond cannot be worth more than the sum of the future cashflows plus future value. In the case of yield equaling coupon rate, the price is equal to par because the rate at which you are discounting makes it so that the sum of the discounted cashflows and discounted par equal present par. Understanding this, by looking at the equation you should be able to convince yourself that is the case. Here is a simple example using a 3 year note with a 3% coupon:

$$ 100 = \frac{3}{(1+0.03)^1} + \frac{3}{(1+0.03)^2} + \frac{100+3}{(1+0.03)^3}$$ $$ 100 = 2.912621 \ + \ 2.827788 \ + \ 94.25959 $$

Let me know if that helps.