# On coupon rate versus yield

I have seen the following roughly spelled out (or else tacitly used) in a few of my finance textbooks and I was hoping for confirmation and, ideally, a proof:

Coupon rate equals APR of yield if and only if the coupon bond trades at par.

To be sure we're all working with the same definitions (mine are from Chapter 6 of Berk and Demarzo's Corporate Finance text), the "coupon rate" is the annual simple interest (rate) which accrues on the par value of the stock and the "APR of yield" is the APR of the yield to maturity on the bond, where the YTM is defined as the IRR for the period of the coupon cash flows (which may or may not be annual).

P.S. Is my definition of "yield" as generally used for coupon bonds correct? That is, is the yield the APR on the actual effective rate for the given period of the coupon, or is it the EAR of the actual effective rate for the given period?

This question came to mind while answering the following problem from Bodie, Kane, and Marcus's Investments:

A municipal bond carries a coupon of 6.75% and is trading at par. What is the equivalent taxable yield to a taxpayer in a combined federal plus state 34% tax bracket?

They give the solution simply as:

The equivalent taxable yield is 6.75/(1-0.34) = 10.23%.

That is, they plunge ahead using the (APR) coupon rate as the appropriate "yield" and I wanted to understand why.

• Both quotes say "trading at par". I'm not seeing a conflict? Commented Feb 1 at 16:33
• 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
– EE18
Commented Feb 1 at 16:35
• Isn't that the definition of "at par"? Commented Feb 1 at 20:54
• 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
– EE18
Commented Feb 1 at 22:53

APR is the periodic rate annualized by multiplying by the number of periods in a year, so a bond that pays semi-annual coupons and has a semi-annaul yield (IRR) of 3.5% will have a 7% "yield". There is also an "Effective Annual Yield" that does compound, but the common definition of yield is the periodic IRR multiplied by the period (e.g. semi-annual yield multiplied by 2).

To understand why a bond trading at par has a yield equal to the coupon, it's easier (to me) to think about what yield means. Yield (IRR) means "at what constant interest rate could I invest money at to have the same amount of cash at the end of the cash flow stream".

If a bond with a par amount of 100 trades at par and has a coupon payment of C, the cash flow stream of investing 100 at interest rate C and getting the coupons from the bond would be identical, so the "yield" of that bond is equal to C.

Take the first coupon as en example - if you invest 100 at annual rate C (semiannual rate C/2), in 6 months you get a cashflow of C/2, the same as you get with the bond. In one year you get another C/2, the same as you get with the bond (in both cases it's assumed that the first coupon is reinvested at the yield, which is C/2). Iterating that formula until the bond matures, you get the exact same cash flow stream, which by definition means that the (annual) yield is C.

The mathematics of discounting each coupon by C and adding to the present value of the redemption amount also holds, but it's not as easy for me to visualize.

So it's a common shortcut in academic finance to take a bond with a fixed coupon that trades at par to give the yield rather than giving the yield directly (or forcing you to calculate the yield based on a different market value than par).

• 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...
– EE18
Commented Feb 1 at 18:08
• assumptions about how we reinvest coupons etc. in order to round the argument out, right?
– EE18
Commented Feb 1 at 18:09
• 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.
– EE18
Commented Feb 1 at 18:11
• No this site does now support LaTeX. "there are some underlying assumptions about how we reinvest coupons etc. in order to round the argument out right?" no, the "coupon reinvetsment" assumption is largely misunderstood. You don't have to reinvest coupons, but the definition of PV assumes that you can invest them at the yield rate. In reality it doesn't matter if you can or not, that's just a way to help interpret the "yield" in a PV context. Commented Feb 1 at 20:37
• Mathematically your PV formula also works out if y = c/n but it's not as intuitive (to me) as thinking about it the other direction (investing at the yield ending up with the same FV as the bond coupons) Commented Feb 1 at 20:39