Why a Discount Bond’s Yield to Maturity Exceeds Its Current Yield

Summary

The document gives a proof that a discount bond’s yield to maturity exceeds its current yield. It defines current yield as annual coupon divided by price, and frames yield as a function of the purchase price, coupon cashflow, and redemption amount. A bond bought at par has a yield equal to its coupon, while scaling all cashflows by the same factor leaves yield unchanged.

The argument first compares the target bond with one whose purchase price and redemption amount are both equal to the discount bond’s price. That bond has yield equal to the coupon divided by price. Replacing its redemption amount with the higher face value increases the maturity cashflow, so its yield must be higher. The document provides a conceptual cashflow comparison rather than a full derivation of the yield equation. It assumes a bond that redeems above its purchase price and does not discuss variations in coupon timing or yield conventions.

Key ideas

• Current yield is the coupon amount divided by the bond’s price.
• A bond bought at par has a yield equal to its coupon rate under the stated setup.
• Scaling every bond cashflow by the same factor does not change its yield.
• A greater redemption payment raises yield when price and coupon are held fixed.
• For a discount bond, the yield to maturity exceeds its current yield under the argument’s assumptions.

Tags

# Proving that YTM > Current Yield on Discount Bond


# 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?

## Answer by dm63 (score 1)

https://quant.stackexchange.com/a/46010

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 $(100>P)$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.