Comparing Coupon Bond Yields with Zero-Coupon Yields
Summary
The document considers whether a coupon bond maturing on the same date as a zero-coupon bond should have a higher or lower yield, given an upward-sloping set of zero-coupon prices. It explains that a bond’s yield to maturity is the single rate that discounts its cash flows to its quoted market price. Thus, the comparison cannot be determined from maturity alone: the coupon schedule and the price being compared matter.
One response says that if the coupon bond and zero trade at the same price, the coupon bond’s yield is higher. Another uses the supplied discount factors to value a par coupon bond and obtains a fair coupon below the 2010 zero’s yield; earlier coupon payments pull the coupon rate down when discounted along the upward-sloping curve. These statements concern different comparisons—yield to maturity at a shared price versus the coupon rate that makes a bond worth par. The discussion’s numerical illustration depends on the stated annual cash-flow and discounting assumptions.
Key ideas
- Yield to maturity is the rate that discounts a bond’s cash flows to its market price.
- Comparing bonds requires their prices and coupon cash flows, not just their maturity dates.
- At the same price, a coupon bond can have a higher yield to maturity than a zero-coupon bond.
- Discounting earlier coupon payments using an upward-sloping spot curve can imply a par coupon below the long zero’s yield.
- A coupon rate and a yield to maturity are distinct quantities.
Tags
Full text
# Estimate yield of coupon bond given yield of zero coupon bond
# Estimate yield of coupon bond given yield of zero coupon bond
Suppose that now is August 2006 and we have the following zero-coupon bonds:
`Maturity: August 2007, Price: 95,53`
`Maturity: August 2008, Price: 91,07`
`Maturity: August 2009, Price: 86,2`
`Maturity: August 2010, Price: 81,08`
Would you expect the yield on a non-zero coupon bond maturing in August 2010 to be higher or lower than the yield on the 2010 zero-coupon bond?
My attempt:
I think we should somehow use the fact that if we calculate the yields on these zero-coupon bonds then the term structure will be upward-sloping. By the expectations hypothesis, an upward sloping yield curve implies that the market is expecting higher spot rates in the future. But I don't know what conclusion we can draw about non-zero coupon bond yield from it.
## Answer by KT8 (score 1)
https://quant.stackexchange.com/a/68793
It depends on the value of the bonds that you are comparing. The yield of a bond it's related to its market value. Note that the value of a bond can be given in term of its yield as
$$V (y) = \sum^N_{i = 1} \dfrac{C_i}{\left(1 + y \right)^{t_i}},$$ where $C_i$ are the coupons. Note that I'm implicitly including the principal in the last coupon, $C_N$.
Then, determining the yield of a bond its just a matter of solving $$0 = V_{\rm quoted} - \sum^N_{i = 1} \dfrac{C_i}{\left(1 + y \right)^{t_i}}.$$
If both bonds, the one with zero coupons and the one with coupons trade at the same price $V$, then the one with coupons will have a higher yield. Note that the yield is a way of measuring cost-effectiveness. Therefore, the quote-price of the bond it's an important feature.
## Answer by Alper (score 0)
https://quant.stackexchange.com/a/68405
The yield on a discount (zero-coupon) bond maturing in 2010 should be higher than that of a coupon bond maturing in 2010 under the stated circumstances.
This is because some of the cash flow of the coupon bond will be realized earlier than that of the discount bond, and as shown in the table below, the yield curve, as far as these two bonds are concerned, is upward sloping.
| Maturity | Price | Annualized Yield (%) |
| Aug 2007 | 95.53 | 4.7 |
| Aug 2008 | 91.07 | 4.8 |
| Aug 2009 | 86.20 | 5.1 |
| Aug 2010 | 81.08 | 5.4 |
## Answer by dm63 (score 0)
https://quant.stackexchange.com/a/68413
Let’s say the fair coupon on the 2010 coupon paying bond is C. Then this bond is worth 100. Its cash flows in 2007,2008,2009,2010 are respectively C,C,C, (100+C) and we can use the discount factors in the OP to calculate the present value as follows: $$ PV = 0.9553C +0.9107 C+ 0.8620C+ 0.8108(100+C)$$. Equating this to 100 and solving, we find C to be 5.34, less than the yield of the 2010 zero coupon bond.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.