# Proving that YTM > Current Yield on Discount Bond

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?

• The basic idea is that the current yield takes into account only the coupon income relative to the initial investment, but the YTM also includes the accretion of the discount d, due to the fact you pay P at the beginning and eventually get back 100. Hence the YTM is greater. – dm63 Jun 9 '19 at 13:10

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 $$(100>P)$$.