# Par Yield theory

#### ericbmoreira

##### New Member
So I searched the forum and couldn't find an answer to this, which obviously does not mean I didn't miss it. I found one of your videos in a link that describes it as a Par Yield Coupon, which makes a whole lot more sense to me. You usually use discount factors of .5, 1, 1.5, and 2. I am confused when I see questions asking what the "per annum" and "two year par yield" are but the answers are derived in the same way. So here is my question;

If you had a zero rate spot curve of 15 years would the per annum, 2 year, 13 year, 7.2673 year rate be all the same? Your video would certainly imply that, which is why the Par Yield Coupon makes much more sense.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @ericbmoreira I don't understand your question. I think we need to clarify the terms. First, by the way, I have recorded a more recent video specifically on the par yield. It is based on Tuckman and shows the specifics of constructing the par yield curve; it is called "Par yields are swap rates, see below.

Second, the "per annum" is throwing me off because, in general, any of the interest rates (e.g., spot rate, par yield) will be expressed in per annum terms. In this way, the term "per annum" is not really adding additional information. We can have a 3-year spot (aka, zero) rate of 5.0% per annum, or we can have a 10-year par yield of 6.3% per annum. In general the "per annum" can be dropped because it is implicit.

Third, my label of "par yield coupon" has the potential to be confusing (which i did not anticipate). My intention was to distinguish from the illustrated 4% and 5% coupon bonds. But "par yield coupon" by itself, to me, is a confusing/nonsensical. The par yield is a coupon rate that prices a bond to par. In the above video, the 2-year par yield is 8.57% and we can interpret it (at least) two ways:
• [per the video demonstration] A 2-year bond with an annual-pay coupon rate of 8.57% (per annum, implicitly) has a par price (\$100.00) when its cash flows are discounted by the zero rate curve. Put another way: coupon rates above (below) the par yield will price the bond at a premium (discount), while a coupon rate set at the par yield will price the bond at par.
• [per my video below] As a 2-year par yield, 8.57% is the 2-year swap rate implied by the spot/zero (equivalently, discount function) rate curve. That is, 8.57% is a "fair" fixed rate to pay in a 2-year swap, which anticipates in exchange, by definition, floating rate payments informed by the forward rates embedded in the spot/zero rate curve. I hope that's helpful!
Par yields are swap rates (FRM T3-13)