# 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)?

• Can you write complete question ? It is appear that you did not provided complete detail. – Neeraj Oct 23 '15 at 8:25
• I have written all the details i was given on the exam, i'll write it one more time though. We have two bonds, both are 3 year bonds, paying an annual coupon. 1st bond pays an 8% coupon, 2nd bond pays a 10% coupon. Given that both bonds have the same spot curve (i.e. are being discounted by the same spot rates) and that 2nd bond's IRR is 8.87%, choose the right answer for 1st bond's IRR: a) 8.9% b) 8.87% c) 8.7% – iNarek94 Oct 23 '15 at 8:33

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:

1. Rising term structures will give you a lower yield for higher coupon rates and
2. 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.

• Thank you! The article was very interesting, author's approach was new for me. Although, the influence of rising/falling terms on the yield was more explained, rather than proved, i got my answers. By the way, on page 15, the author implies that two cashflow vectors are 1 dimensional and so linearly dependent, but is that so? That statement was based on the measure of a subspace containing all Internal Rate Vectors being n out of n+1, but are the subspaces for two different cashflow vectors the same? I doubt it, if we just look at their basis. – iNarek94 Oct 25 '15 at 23:06

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

• The funny thing is that i have written more or less the same idea, that you have just stated. I thought if the spots were the same, the IRRs should be the same too, in order to avoid arbitrage opportunities (accepting, that they might be slightly different, they should be very close to each other). But it's even funnier that my answer was considered wrong, and i'm not even against that. I still think there should be a more logical answer to this. – iNarek94 Oct 23 '15 at 12:14
• Also the calculations you have made do not prove anything in this case, as there is a limited set of spot rates, which give a price of 97.79 and 102.86 for the bonds. If there are infinite such spot rates, they are still limited by some relationship among them, and we need a more common case. Thank you! – iNarek94 Oct 23 '15 at 12:16