# Par Yield, Bond Yield and Zero Rate

In the Hull's book, chapter 4.4, it says :

The par yield for a certain bond maturity is the coupon rate that causes the bond price to equal it's par value.

Then for this question (4.18) :

“When the zero curve is upward sloping, the zero rate for a particular maturity is greater than the par yield for that maturity. When the zero curve is downward sloping the reverse is true.” Explain why this is so.

The par yield is the yield on a coupon-bearing bond. The zero rate is the yield on a zero-coupon bond. When the yield curve is upward sloping, the yield on an N-year coupon-bearingbond is less than the yield on an N-year zero-coupon bond. This is because the coupons are discounted at a lower rate than the N-year rate and drag the yield down below this rate. Similarly, when the yield curve is downward sloping, the yield on an N-year coupon bearing bond is higher than the yield on an N-year zero-coupon bond.

• First, it's not very clear in my mind, is the par yield a coupon rate or a yield ?
• Secondly, i don't understand the answer, what does he mean by "drag the yield down below this rate" ? When he says "are discounted at a lower rate", of which rate does he talk about ? The bond's yield ?
• (1) When a bond is selling at par the Coupon Rate and the Yield to Maturity are the same, and this number is called the Par Yield or Par Rate. (nevertheless much of what he says later is applicable to any coupon bonds, whether at par or not). Commented Jul 4, 2019 at 16:08
• (2) When you discount individual cash flows to find their present value you discount at the ZCB rate for the corresponding maturity of the cash flow. Commented Jul 4, 2019 at 16:10
• Ok for (1). For the second one, why are we not discounting in this case at the par yield ? As i understood, the YTM (so here the par yield also) is the rate at which we can discount at every period (factor the length between present and this period) instead different ZCB rate. That's my problem here, how do i know this rate we use for each period (the ytm) is lower than the ZCB rate of the last period ? Commented Jul 4, 2019 at 16:45
• We don't know the par yield, all we know is the ZCB rates, say 1%, 2%, 3%. In this situation any three year nonzero coupon bond yields less than 3%: we make 3% on the final cash flow, but on the first and second coupon we make 1% and 2% respecitvely, which "drags down" the return to less than 3%. So all the coupon bonds, and in particular the par coupon bond, yield less than 3%. Commented Jul 4, 2019 at 17:21
• Ok so what you call "return" or "yield less than 3%" is the par yield in his question because in that case the coupon rate equal the return ? Because we don't konw the par yield but the question is "...the zero rate for a particular maturity is greater than the par yield for that maturity." Commented Jul 4, 2019 at 17:35

Let's assume we have yearly cash flows, and let's focus on just two years - year 1 and year 2. Let $$R_1$$ and $$R_2$$ represent the zero rates of year 1 and year 2. So if you want to borrow for one year, you pay $$R_1$$ percent, and if you want to borrow for 2 years, you pay $$R_2$$ percent per year. So in an upward sloping scenario, these will look like this:

Now, you can easily convert these into the forward rates, so the first rate remains unchanged and the second rate would then be the interest rate for borrowing between year 1 and year 2, let's call it $$R_{12}$$ so the graph of the implied forward would be similar though the second point will be taller.

A 2-year maturity coupon bond will need to pay coupon rate of $$R_1$$ in the first year and a coupon rate of $$R_{12}$$ in the second year to be valued at par. But standard coupon bonds have fixed coupon rate (not variable by year), so the par coupon($$c_2$$, which is the par yield in the question wording), as you can see, will be some average of $$R_1$$ and $$R_{12}$$, which in an upward sloping environment will be lower than $$R_2$$. This $$c_2$$ is your 2-year par rate.

And you can extend the same logic to the third year and so on, to get the term structure of par yields, and if you plot it against the term structure of zero coupon, you see the c's will be lower than the R's (except for year 1 when they are equal), and this is what is meant.

• Thanks, couldn't be clearer !! Commented Jul 6, 2019 at 23:50
• Could you please check the answer below and see if you have to change something in your answer ? thanks Commented Sep 29, 2019 at 16:06
• I often hear people talk about a "zero rate LIBOR curve". But what does it mean? As far as I can tell, a "LIBOR rate" assumes a certain frequency (tenor) of payment, while "zero rate" by definition does not have any intermediate payments? Say, we have 3M LIBOR quotes up to 1 year, then how can we even price a 1-year zero bond with that information since the former assumes a payment every 3 months, while the latter no payments until maturity? Commented Apr 22, 2020 at 9:23
• You should explain why c2 < R2. Saying that some average of R1 and R12 is lower than R2 is far from accurate, especially given that R12 > R2. Also, you could explain the "drag the yield down" from the question, which would lead to directly to the prove of c2 < R2.
– mfnx
Commented Nov 19, 2021 at 9:49

This question came to me as well, the reasoning in the solution is not convincing, neither is the solution given by Magic.

In Magic's answer, the reasoning "$$c_2$$ is some average of $$R_1$$ and $$R_{12}$$, then lower than $$R_2$$" is incorrect since $$R_{12}$$ is bigger than $$R_2$$ in upward sloping.

We can prove it rigorously in the 2-year setting as below.

In continuous compounding setting, discrete par yield is computed by solving: $$(100c)e^{-R_1}+(100c+100)e^{-2R_2}=100$$ which gives: $$c=\frac{1-e^{-2R_2}}{e^{-R_1}+e^{-2R_2}}$$ Next, convert c to continous par yield: $$c^*=ln(1+c)=ln\frac{1+e^{-R_1}}{e^{-R_1}+e^{-2R_2}}

• So do we really need R12 ? If we just say c is a kind of average of R1 and R2 it's enough ? Commented Sep 29, 2019 at 16:03
• So c is the par yield, which you showed is lower than R2 in an upward scenario, but is not that what the question says? Commented Sep 30, 2019 at 16:53
• As an aside, when you compare c (simple rate) to the continuously compounded zero rate, both will need to be converted to the same basis, as otherwise the hypothesis will get impacted by the differences in the compounding methods. Commented Sep 30, 2019 at 17:02
• We don't need $R_{12}$ as I showed.
– S55
Commented Oct 1, 2019 at 18:50
• @user3700927 TBH, I think you're comparing apples to oranges. Your zero rates are continuously compounded, but your par rate is compounded at discrete intervals. Assuming $R_1= 0.1$ and $R_2=0.2$ are annually compounded, you have annually compounded par yield of $c = 0.1905$, which is less than $R_2$. Commented Oct 1, 2019 at 19:21

Just to test understanding of the reasoning:

$$\frac{c}{1+R_{01}}+\frac{1+c}{\left(1+R_{01}\right)\left(1+R_{12}\right)}=1$$

Multiply through by 1+c:

$$c\frac{1+c}{1+R_{01}}+\frac{\left(1+c\right)^2}{\left(1+R_{01}\right)\left(1+R_{12}\right)}=1+c$$

Now $$\frac{1+c}{1+R_{01}}>1$$ as per the reasoning in the previous answer because the curve is upward sloping, which means:

$$\frac{\left(1+c\right)^2}{\left(1+R_{01}\right)\left(1+R_{12}\right)}<1$$

which by definition means:

$$\frac{\left(1+c\right)^2}{\left(1+R_{2}\right)^2}<1$$

In discrete and 2-year setting, the question can be formulated to: given $$R_1, show $$R_1 from $$\frac{c}{1+R_1}+\frac{1+c}{(1+R_2)^2}=1.$$

This can be proven by contradiction:

1) assume $$c<=R_1$$, then $$1=\frac{c}{1+R_1}+\frac{1+c}{(1+R_2)^2}<\frac{R_1}{1+R_1}+\frac{1+R_1}{(1+R_1)^2}=1$$ 2) assume $$c>=R_2$$, then $$1=\frac{c}{1+R_1}+\frac{1+c}{(1+R_2)^2}>\frac{c}{1+c}+\frac{1+c}{(1+c)^2}=1$$

• This helps prove that $c_2$ is between $R_1$ and $R_2$ from magic's answer. Commented Mar 11, 2021 at 15:43

Answering this question four years too late, but I stumbled upon this particular problem and racked my head with it for quite a while, so I'll post an answer here for the sake of completeness. I prefer to approach such questions in a mathematically rigorous method to leave as little room for hand-waving as possible.

Defining some notation and assumptions:

• We assume continuous compounding everywhere, as it is used in the book itself
• The coupons are paid once a year, thus $$m = 1$$ (following the notation from Section 4.6 of John C. Hull 11e)
• The zero rate for a bond with maturity $$i$$ years is given by $$z_i$$ (note that $$z_i$$ are not percentages, so if a 2-year zero coupon bond has a yield of 5%, then $$z_2 = 0.05$$)
• The time to maturity in question is $$M$$ years, so the zero rate for the maturity $$M$$ is $$z_M$$ as I understand in the question
• The par yield for maturity $$M$$ is given by $$c$$ Thus, $$c = \frac{(1 - d)}{A}$$ where, $$d = e^{-Mz_M}, \quad A = \sum_{i=1}^{M} e^{-iz_i}$$

Or equivalently, we need to prove that $$z_M < c$$ The downwards sloping nature of the zero curve implies that $$0 < z_M < \ldots < z_2 < z_1 \\ \therefore e^{-z_i} < e^{-z_M} \quad \forall i \in \{1, 2, \ldots, M-1\} \\ \therefore e^{-iz_i} < \left(e^{-z_M}\right)^i \quad \forall i \in \{1, 2, \ldots, M-1\} \\ \therefore \sum_{i=1}^{M} e^{-iz_i} < \sum_{i=1}^{M} \left(e^{-z_M}\right)^i$$ Since this is a geometric progression, we can evaluate it to $$A < \frac{e^{-z_M} \left(1 - e^{-Mz_M} \right)}{1 - e^{-z_M}} \\ \therefore c = \frac{\left( 1 - e^{-Mz_M} \right)}{A} > e^{z_M} - 1$$ Using basic calculus, it is not difficult to prove that $$\forall x > 0 \quad e^x - 1 > x$$ Thus, we have proved as asked $$c > e^{z_M} - 1 > z_M$$
Or equivalently, we need to prove that $$z_M > c$$ The upwards sloping nature of the zero curve implies that $$0 < z_1 < z_2 < \ldots < z_M$$ I'll make a really bold claim here and say that I believe this statement is incorrect or at least it not always true in the in the most general case. Consider the following zero rates for the maturity periods from 1 year to 6 years: 1.00%, 1.01%, 1.02%, 1.03%, 1.04% and 1.05%. $$d = e^{-6 \times 0.0105} = 0.93894 \\ A = e^{-0.01} + e^{-0.0202} + e^{-0.0306} + e^{-0.0412} + e^{-0.0520} + e^{-0.0630} = 5.78783 \\ \therefore c = 0.010550 = 1.055\% > 1.05\% = z_M$$ Thus, it is possible that either I am misunderstanding the question and making some unreasonable assumptions or there need to be some other conditions on the interest rates for the claim mentioned in the question to be true.