Confused with IRR and YTM of a bond
Hi guys,
I have a very simple question that has been confusing me a bit recently. The best way to describe it is to give a simple example. Let's imagine a very simple 5 year %10 Annual Coupon Bond trading at 100. The current interest rate on similar securities on the market is also %10, that is YTM is %10.
We know that YTM of this bond is %10. However, assuming no reinvestment of coupons, we will be ending up with 1500$ at the end of 5th year. When I am trying to calculate IRR of PV=1000 and FV= 1500 and n=5 , I get an IRR of approximately %8,447 which is not equivalent to %10 YTM. However if we reinvest coupons we end up with approximately 1610,50 which gives us the %10 IRR which is equal %10 YTM based on the same step above(FV=1610,50, PV=1000 n=5) . So if we dont reinvest coupons, what yield will we get? Is it %10 YTM or %8,447 IRR?
Thank you very much
You're clearly not alone in your confusion. Have you tried to google this? If you haven't, you really should...
At any rate, search for "Yield-to-Maturity and the Reinvestment of Coupon Payments". It's a short and sweet paper by Forbes et al that should hopefully clarify things for you. I would post a link, but I am not allowed.
If you don't reinvest your coupons, your IRR will naturally be less, since you are not getting interest on the coupons that have been paid. IRR implicitly assumes that you reinvest any cash flows you get.
The YTM of a bond is its IRR when set equal to its price (ie, not an IRR in the traditional sense of the word, setting the cashflow equal to zero). The YTM is simply the interest rate that discounts all the incoming cashflows from owning the bond (coupon and face value) to the initial price.
Buddy, you are confused. Your numbers are off. In your example, if principal is $1000, 10% coupon p.a. for 5 years cash flows should be
yr 0 1 2 3 4 5 -1000, 100, 100, 100, 100, 100+1000
IRR = 10% (IRR is the rate where if you discount the cashflows above with it, you get 0)
YTM is essentially the discount rate that will discount the coupons and principal back to a present value equivalent to the bond principal of $1000.
i.e 100/(1+r)^1 + 100/ (1+r)^2 + ....+ 1100/(1+r)^5 = 1000 (Solve for r) which r should be = 10%
The yield to maturity equals the internal rate of return only if (1) the probability of default is zero and (2) the bond cannot be called. For bonds where there is some default risk, or where the bond may be called, there is some probability that the promised payments to maturity will not be received, in which case, the promised yield to maturity will be more than the internal return.
Also, don't forget that YTM assumes reinvestment of the coupon cash flows. If you assume no reinvestment, your IRR will not equal your YTM.
There are papers, this is true, and the one you posted earlier is a fantastic one. The debate comes in on whether or not the formula for YTM implicitly assumes reinvestment. I, personally, agree with you. However, in nearly all academic textbooks this assumption holds true. This includes Fabozzi books and the CFA. As I said earlier, I am assuming the OP was reagarding academics/ homework etc. and not a real world application.
No, that is not at all what I am saying. I am saying that since there is default risk, the IRR will not equal the YTM. I never said either rate couldn't be calculated.
What is wrong with you guys. Assuming you reinvest coupons, IRR = YTM BY DEFINITION. Why are you unnecessarily complicating things by talking about default rates and callability. They're not even relevant when talking about yield TO MATURITY.
I feel like this happens on every second thread; everyone here wants to show off how much they think they know by giving an unnnecessary amount of confusing, and often wrong, detail.
It isnt unnecessary. If you are talking about YTM on its OWN, then yes there is no need to involve callability or default risk. However, when you are relating YTM to IRR, as was the question in the OP, they become relevant factors. The OP was questioning why, under different circumstances the YTM and IRR may be different. I simply answered that question. I am assuming the OP is asking for academic purposes, in which case, these are things he/she should know.
You're just confusing him though if he's asking a question this basic. If someone asks me to add 2+3=5, I don't start by defining the set of natural numbers and the axioms for additions.
You're wrong anyways; if you want to invoke callability we would use YTC or YTW. YTM is a strictly mathematical formula that literally calculates to maturity. Lastly, the question was IRR and YTM; you can't assume no reinvestment for the latter.
Secondly, I think it is more beneficial that you explain why, so that the OP can determine the answer on their own. Rather than just throwing out that assuming no reinvestment the IRR is the correct return assumption. Just having the answer without knowing why provides ZERO long term benefit.
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