Agreed! The interest rate expressed as an APR to confirm (I know we said this before but just want to mention it again for posterity and for my future reference). Thanks again for your help :)

Got it, thank you. I've accepted but, just to confirm, you agree with the mathematical formulation of the claim I made in that comment above?

I think they are equivalent (and that's what my question is about -- showing said equivalence) in which case I agree that it can serve as a definition, but the definition that I am working from is that "at par" means "at the same price as par value". @keshlam

Hmm, it seems like it's not letting me use Tex syntax here, I'm sorry about that. Will try to figure that out but hopefully my question is somewhat clear.

assumptions about how we reinvest coupons etc. in order to round the argument out, right?

Thanks very much for your answer! I don't completely follow I'm afraid. Let's assume we are at the point in time where the bond has just been issued and the bond is for \$M\$ years. I guess you are arguing that if $c$ is the coupon rate paid out \$n\$ times per year and $K$ is the par value, then if the price $P$ equals $K$ we have that the yield, defined as that compound rate \$y\$ (i.e. the YTM) which sets $$P = K = \frac{K}{(1+y)^{nM}}+ \sum_{i=1}{nM} \frac{cK/n}{(1+y)^i}$$ is always going to be $y = c/n$? The reason I don't think I follow your answer is I think there are some underlying...

I'm trying to understand why "trading at par" implies that yield (which is a priori unknown) is equal to coupon rate (which is given). Just to be clear, above the line is my question whereas below the line is a question from BKM which I've given just for context of where my question above the line came from. @keshlam

I am not sure I follow. You are saying that holders of Greek debt had the option to swap and, therefore, that their debt was worth at the very least the value of the new bond instruments being offered (if it was worth any less then they could get more value by doing the swap). This puts a lower bound on their instruments, but why does it place an upper bound too?

...where the "coverage" is in the sense of expected value (and hence this is not an arbitrage opportunity as it would be if the cash flows were risk-free)? But we nevertheless have $500 in our pocket right now?

Thank you for your answer. If it's possible, I'd like to reiterate the same question as I did to the other answerer:are you able to elaborate specifically on how we can make any positive NPV be "cash in our pocket" right now, as we can do in the risk-free case? Are you saying the following (for example)? Suppose we have a project with a certain risk profile and NPV of +500. Are you saying that (we assume) there is a market to sell this set of cash flows, and we can then buy an a set of cash flows with the same risk profile in order to cover this sale...

Thank you very much for your answer. If possible, are you able to elaborate specifically on how we can make that $500 be "cash in our pocket" right now, as we can do in the risk-free case?