OK that was my main sticky point I think.
I don’t understand that.
A 5 years, 2,5% annual coupon which trade at 95 has a YTM of 4.08%
A 5 years, 3.75% annual coupon which trade at 95 has a YTM of 5.59%.
The “YTM” that you describe, the yield of the par curve, doesn’t seem like the same concept as the YTM which depends on the market price. Is this correct to call that concept (the yield of the par curve) a “YTM” when it doesn’t change with the price?
The magician is explaining that holding the time value components constant, they will give an answer in line with the required rate of return equal to the market (assuming no issues with credit, liquidity since we’re talking riskless). This is called the yield to maturity. It works well for explaining interest rates to beginners, but it’s full of so many assumptions that it’s pretty useless for putting money on the line. Also, to make this easy to visualize (at least for me), let’s assume we’re talking about annual interest rates, and annual coupons, and full years on the yield curve. With different coupons, or FVs, that means their prices will be different, the PV. Some market participants who are bearish will want premium bonds, the bulls will want discounts (or zeros if you’re Warren Buffet at the turn of the century). But the price issue will figure itself out. That’s all we as market participants can control (unless you’re the Fed and you can set the coupon rate, and even that is done with an auction). Why the par rate? The end goal, I’d say most typically, is to find the implied forward rates, because they show the extra little bit of return you get for lending your money longer. (They also have more complicated uses like bounding the stochastic processes behind a Monte Carlo interest rate forecast, or maybe helping to evaluate the key rate duration of a PAC tranche of an MBS based on forecasted prepayments). Example: If, on a five-year zero-coupon (that’s key) bond, most of the bang for my buck happens in years 1-4, should I limit my loan? The forward curve can help answer this. We do this by using spot rates and finding the pieces between them. The spot rate is just an annual average of all the year-over-year (it could be broken into second-by-second) forward rates that are behind it (remember pricing an FRA? Similar idea of building blocks). Instead of finding the value of my loan going from year 4 to 5, or buying a longer bond, now we’re talking in more basic terms. What’s the value of our 5-year zero in general? Well, it’s the price. And that price shows the average risk of a loan that goes out to the maturity of the zero we’re looking evaluating. Even if a bond has coupons, we can price it if option-free, since we just pretend they’re little zeros. This will be an arbitrage-free price. But it has to be option-free because the optionality could mean that on average the price is correct, but there could be times when the option gets in the way, and all bets are off. Here’s a practical issue, since you seem to like to separate it from theory: we can’t just go into the T-bill or STRIPS market, and pull those YTMs, because these bonds are only either for bulls or people with specific liabilities they’re matching, generally speaking. But we’re not like them typically… Instead, we need to back out the YTMs of coupon bonds, because they’re more representative of the ‘normal’ bond and have better liquidity, etc. As bonds’ prices move to trade at premium or discounts, they become problematic for the same reason zeros are, including other factors now added into the mix, like convexity. To get to said spot curve, bootstrapping is the closed-form (basic) way this is done, and I described this earlier. It would be done with the bid yield of the on-the-run Treasury yield curve, or the ‘par curve.’ (This has some issues in practice, which are way bigger than finding something selling close to par. Problems include the choppiness and noisiness of the bootstrapped implied forward curve, and also the overfitting from so many different maturities. In practice, since we’d wouldn’t be valuing Treasuries this way in all likelihood, we could use Eurodollar futures to construct the short end of our spot curve, and par swap rates for the longer ends, and after inverting the yield curve, interpolate using an instantaneous interest rate. This involves lots of, computers, coffee, patience, and smart people like the magician).
If they’re both Treasuries, they won’t both trade at 95. But they will trade at the same YTM.
Why would anyone pay the same price for a bond paying 2.5% as they would for a bond paying 3.75%? That’s silly.
You don’t seem to grasp the idea behind YTM.
If the maturity and face value are the same, then the prices can never be equal for different coupon rates!
This is arbitrage, short the 2.5 and long the 3.75!
Got it. Thank you.
So basically the par curve is just a YTM curve for treasuries, right?
Then why are people presenting it as “the coupon rate that is needed for a Treasury to trade at par” when they can say “the par curve represent the returns you get for buying a Treasury if you hold it until maturity (and reinvest the coupon at the same rate…)”? Are these two formulations identical?
Thanks for your post. I need an explanation of this point though:
"Some market participants who are bearish will want premium bonds, the bulls will want discounts (or zeros if you’re Warren Buffet at the turn of the century).’
Why the bear would buy premium bonds and the bulls discount bonds? I don’t get it.
My pleasure.
Like the stock market, buy low, sell high.
If you are bullish on the bond market, meaning you expect prices to go up, or yields and interest rates to go down, then you’d buy bonds at a discount because they exhibit a higher duration, and more potential for profit on interest rate changes.
Bears want premium bonds to get higher coupons.
Bulls want discount bonds for price appreciation.
Exactly, and another way of looking at it, BldSwtTrs, is the bulls want to ‘lock in’ their YTM as much as possible. They will look for well-seasoned disounts and maybe even zeros. Same reason somebody long a vanilla swap is short the bond market.
You might benefit from picking up a used copy of the first edition of “Inside the Yield Book,” which really forces you to get back to basics.
Hi. I know this is an old thread, but I couldn’t replicate this statement:
“The par curve simply has the YTM, which is the same no matter what the price of the bond is; i.e., if one Treasury with 5 years to maturity has a 2.5% coupon and another with 5 years to maturity has a 3.75% coupon, they’ll trade at the same YTM.”
I created a par curve (5%, 5,97% and 6,91%) and found the spot (5%, 6% and 7%). Then, I simulated 10 bonds, paying from 1% to 10% in yearly coupons, and found their prices using the spot.
Then I calculated their YTMs and they’re all different, they don’t trade at the same YTM. Did I make some error or do bonds of the same maturity not trade at the same YTM?
| y | par | spot | df |
|---|---|---|---|
| 1 | 5,00% | 5,00% | 0,952380952 |
| 2 | 5,97% | 6,00% | 0,88999644 |
| 3 | 6,91% | 7,00% | 0,816297877 |
| coupon | pmt | price | ytm | |
|---|---|---|---|---|
| 1,00% | $1,00 | $84,29 | 6,985% | |
| 2,00% | $2,00 | $86,95 | 6,971% | |
| 3,00% | $3,00 | $89,61 | 6,958% | |
| 4,00% | $4,00 | $92,26 | 6,945% | |
| 5,00% | $5,00 | $94,92 | 6,932% | |
| 6,00% | $6,00 | $97,58 | 6,920% | |
| 6,91% | $6,91 | $100,00 | 6,910% | the par bond, ytm calculation is therefore ok |
| 7,00% | $7,00 | $100,24 | 6,909% | |
| 8,00% | $8,00 | $102,90 | 6,897% | |
| 9,00% | $9,00 | $105,56 | 6,887% | |
| 10,00% | $10,00 | $108,22 | 6,876% |
Thanks in advance
You’re correct; that was an overstatement on my part. I should have said that they’ll all trade at similar YTMs (and at the same YTM when the yield curve is flat).
Because the bonds with higher coupons have a higher percentage of their present value at shorter terms with lower spot rates, they’ll trade at slightly lower YTMs; the opposite is true for bonds with lower coupons.
Maybe I am being a little frivolous. But I frankly did not understand much less appreciate the 6 years spanning thread on a rather simple concept 
Let’s take 5 years spot rate ( 5 maturity zeroes). Let’s say there is a SFR to be deduced because two parties want to enter a swap contract with the underlying being the treasury spot rates. So evidently I have to know the SFR. Some might equate to the coupon as well on the notional principal exchanged. How different is the Par rate for this arrangement than the SFR.
Now if I want to replicate the same exercise involving risky securities appropriate spread will be added to the SFR. Some might equate this to the coupon as well. But this will give rise to a new par rate pertaining to those risky securities ( bonds)
Is there anything else I am missing in my understanding of the par rate and thus par curve ?
The application are definitely myriad and any issuer before an NFO / issue would wanna know what the par rate could be … because that is the coupon should the issue be floated at par.
How difficult is this 