Proving that YTM > Current Yield on Discount Bond

Question

I’m currently stuck in proving that for a discount bond: YTM > current yield, with:

$$\text{current yield} = c \frac{100}{P}$$ with $P=100-d$ the price of the discounted bond and $c$ the coupon rate.

With numerical simulations in Python, I’ve seen that indeed the relation is true, but I’m stuck in trying to prove it theoretically.

Here's what I've tried:

I’ve expressed the current yield as a function of YTM and considered the function f(YTM) = YTM – Current Yield.
To prove the relation, my approach was to derivate the function, see that the derivative is positive and then see that YTM – Current Yield > 0 for Current Yield > Coupon Rate. However, in that approach, the expression I get for the derivative is very complex and cannot be interpreted easily.

Do you have any advice on how to tackle this proof?

Answer 1

I think this proof works: Denote the annual yield of a bond as follows: $$ y(-Price, Annual Coupon Amount, Redemption Amount). $$ Then for example, $y(-100, C , 100) = C$ Which simply says that the yield of a bond purchased at par and with a coupon of $C$ also has a yield of $C$. Similarly, $y(-100, C/P, 100) = C/P$. Now scaling every cashflow by the factor $P/100$ cannot affect the yield, so $y(-P, C, P) = C/P$. Finally we must have $y(-P, C, 100) > C/P $ since the latter bond has a greater cashflow at maturity given that $(100>P)$.