How Coupon Rates and Yield Curves Affect Bond Yield to Maturity
Summary
The document asks whether two bonds with the same maturity and spot curve must have the same internal rate of return when their coupon rates differ. The accepted response says the spot curve alone is not enough to determine a definite comparison without knowing its shape. Under a rising term structure, the higher-coupon bond has a lower yield to maturity; under a falling term structure, it has a higher yield. With a flat curve, the two yields match.
A second response argues that bonds with the same risk should offer the same return and illustrates prices consistent with a common yield. That reasoning does not resolve the stated limitation: a common spot curve discounts each cash flow according to its maturity, while yield to maturity compresses those cash flows into one rate. The document’s examples and competing answers show why the curve shape matters. The comparison concerns yield to maturity, not a guarantee that differently timed cash flows have identical returns under every reinvestment path.
Key ideas
- Equal spot curves do not by themselves determine which of two different-coupon bonds has the higher yield to maturity.
- A rising term structure implies a lower yield to maturity for the higher-coupon bond.
- A falling term structure implies a higher yield to maturity for the higher-coupon bond.
- A flat spot curve gives the two bonds the same yield to maturity under the stated comparison.
- Yield to maturity summarizes cash flows using one rate, whereas spot pricing discounts each cash flow at its own maturity rate.
Tags
Full text
# Compare the IRRs of two bonds
# Compare the IRRs of two bonds
Say i have two 3 year bonds, which pay an annual coupon of 8% (1st bond) and 10% (2nd bond) respectively. Also, let's assume, that the spot curve is the same for both bonds. Other things equal, how can i compare the IRRs of these 2 bonds? (Only using the fact, that the spot rates are the same)?
This was a question from my exam today, and i was really confused. I was given the IRR of the second bond, it was 8.87, i had to select one of 3 possible answers for the first bond's IRR: a)8.9 b)8.87 c)8.7
When i got home, i did a couple of simple simulations, and managed to get 3 sets of spot rates, for which the IRR of the second bond was 8.9 and 8.87 and 8.7
So is there some logic that i should have used in order to get to the answer or no? Is there a right answer? I mean, although i got the spots for each of the answers, they looked quite awkward (e.g. the spots for 8.7 were 15.43, 23.32, 7.88), so is the answer assuming that it's logic should be right for MOST of the spot rates (or at least those near to reality)?
## Answer by p.vitzliputzli (score 1, accepted)
https://quant.stackexchange.com/a/21409
Given no specific information about the term structure, no definite answer can be given. As you found out yourself, different term structures lead to different yield-to-maturities for the second bond. However, the following can be said:
- Rising term structures will give you a lower yield for higher coupon rates and
- Falling term structures will give you a higher yield for higher coupon rates.
The only time that the yield to maturity of both bonds will be the same is when the term structure is flat.
A good read on this topic is: Weingartner, H. Martin. "The generalized rate of return." Journal of Financial and Quantitative Analysis 1.03 (1966): 1-29.
## Answer by Neeraj (score -1)
https://quant.stackexchange.com/a/21393
For this, you donot need to perform any calculation. Examiner simply want to test the understanding. You just need to clarify would IRR of the First bond would be lower or higher or same as 2nd bond. Here, You can understand IRR as Yield to maturity. Assuming cashflow {+10,+10,+110} at time t=1,2,3 price of 2nd bond must be 102.867 to consistent with IRR of 8.87. Assuming both bond have same risk characteristic, market would ensure that both bond provide same return ie IRR(YTM). It means IRR for the first bond must be 8.87 otherwise it would lead to arbitrage opportunities. Difference would be reflected in the price of bond, hence price of 1st bond would be 97.79 at IRR of 8.87 assuming cashflow {+8,+8,+108}. I also uploaded excel spreadsheet assuming spot rate .0877.
```
Time Bond A Bond B Discount A Discount B
1 8 10 7.3482134656 9.185266832
2 8 10 6.749530142 8.4369126775
3 108 110 83.6949177157 85.2448235993
97.7926613233 102.8670031088
```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.