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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 answer is :

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 ?
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  • $\begingroup$ (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). $\endgroup$ – Alex C Jul 4 at 16:08
  • $\begingroup$ (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. $\endgroup$ – Alex C Jul 4 at 16:10
  • $\begingroup$ 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 ? $\endgroup$ – TmSmth Jul 4 at 16:45
  • $\begingroup$ 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%. $\endgroup$ – Alex C Jul 4 at 17:21
  • $\begingroup$ 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." $\endgroup$ – TmSmth Jul 4 at 17:35
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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:

enter image description here

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.

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    $\begingroup$ Thanks, couldn't be clearer !! $\endgroup$ – TmSmth Jul 6 at 23:50
  • $\begingroup$ Could you please check the answer below and see if you have to change something in your answer ? thanks $\endgroup$ – TmSmth Sep 29 at 16:06
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    $\begingroup$ thanks for the notification! will add comment there $\endgroup$ – Magic is in the chain Sep 30 at 16:51
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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: $$c^*=ln(1+c)=ln\frac{1+e^{-R_1}}{e^{-R_1}+e^{-2R_2}}<ln\frac{1+e^{-R_2}}{e^{-R_2}+e^{-2R_2}}=ln(e^{R_2})=R_2$$

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  • $\begingroup$ So do we really need R12 ? If we just say c is a kind of average of R1 and R2 it's enough ? $\endgroup$ – TmSmth Sep 29 at 16:03
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    $\begingroup$ 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? $\endgroup$ – Magic is in the chain Sep 30 at 16:53
  • $\begingroup$ 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. $\endgroup$ – Magic is in the chain Sep 30 at 17:02
  • $\begingroup$ We don't need $R_{12}$ as I showed. $\endgroup$ – user3700927 Oct 1 at 18:50
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    $\begingroup$ @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 $R_2$. $\endgroup$ – Helin Oct 1 at 19:21
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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:

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

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

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