 # What's the point of the par curve?

Hi,

Here’s my understanding of the par curve: it gives the coupon rate that bonds should be paying to trade at par (ie. in order to equalize their price and their face value).

But why is this information useful in the first place?

I mean, if we know the YTM of a bond, we already know what that coupon rate should be (ie. the same as the YTM), so… what information the par curve give that one couldn’t get by looking only at the YTM?

I don’t know.

The par curve gives the YTM on coupon-paying bonds.

It’s generally the only directly observable yield curve in the market: coupon-paying bonds are traded liberally, so we can observe their YTMs.

It’s also the starting point to construct the spot curve, then the forward curve.

Thanks for the answer S2000. So you are saying that par yields = YTMs is that correct or am I misunderstanding ? I am confused because the bonds rarely trade at par so I think the par yields should be different than the YTMs every time that price doesn’t equal face value, and therefore the par yields shouldn’t be usually observable. What am I missing?

YTM is the market interest rate for a specific maturity. Meaning that it’s the annualized coupon rate that would net you no capital gains or losses on your investment (priced at par).

The YTM changes with changes in the spot curve, and therefore the price of a bond bought at par would change as well, because it’s coupon was fixed on a previous YTM.

I am afraid I disagree. The YTM is the discount rate that equalize the price and the PV of future CF. In my understanding the face value is irrelevant to the YTM, only the market price is plug into the equation.

You can construct a zero coupon yield curve using the par curve ( boot strapping)

I believe, par rate = add on yield where as spot rate = YTM of that particular maturiy.

You need to first generate spot rate curve from the givem par rate curve to price a bond. If you have already given the spot rate curve the forget about par rate.

I never said face value.

Well if it’s price at par then it’s price at face value, isn’t it ?

I did and I am still confused. Sorry I must be an idiot. If you really want to help just try to expand on your answer, otherwise don’t bother.

I will try to explain my confusion another way. My understanding can be summarize with the two equations below (are they correct?)

Price = future cash flows / (1 + YTM)^time to maturity

Par value = future cash flows / (1 + PAR YIELD)^time to maturity

And given that the price is a random variable which rarely equals par, I don’t see why the YTM should be equal to PAR YIELD.

That’s it.

If the coupon equals the YTM, the bond trades at par. So it’s the YTM, and it’s the coupon rate for a bond to trade at par. Same thing.

I wrote an article on yield curves that may be of some help here: http://financialexamhelp123.com/par-curve-spot-curve-and-forward-curve/

I think MrSmart, who is usually extremely helpful, might have gotten a bit mixed up on this one–‘priced at par’ does indeed mean at 100% face value, no more, no less. The par curve is a theoretical concept, and our closest example of this is the US Treasury term structure of only on-the-run bills, notes, and bonds. Their superior credit quality, liquidity, and recent coupon print means they are by far our best indicator of what said theoretical concept would be. So, we can hold these factors constant to focus on the time value of money. I could imagine a practical use by a trader in the mid-1960s at one of the Fed’s primary dealer investment banks (there’re about 22 desks today). They buy directly from the government and make markets in Treasuries. On the buy side, as the magician noted here and on his incredibly insightful and thoughtfully constructed web page, the par curve is the starting point to derive the spot (zero) and forward (breakeven) term structure. All these curves are packed with the information we need to unwrap to get some arb-free analysis going. We can only (kind of) see the par curve (the US does have a solid zero market, but other countries often don’t). Practically, we want to move from the observed to the unobservable in a risk-reward tradeoff. Total reward = price, (compounded) average reward = spot, and marginal or incremental reward = forward rate. (In reality, as we already know, not all Treasuries always trade at par, even if they’re the new print, so fixed income analysts might use computers to fit a par curve, adjusting the current one a bit, but the USA curve is usually pretty close; I only mention this so you’re not too confused, that minutia is outside the scope of this exam). By plotting the par curve, you use price (the PV) and cash flows over time (the FVs @ t’s) to unwrap the discount rate, which is also known as the YTM. We’re now at a critical juncture; for a 3 or 6 month T-bill, this is the YTM and the spot rate and the f0,1 forward rate. Next, after plotting the the T-bills, which are basically freebies (although they can have liquidity problems–again, minutia), you remove the coupons, except for the final big one, which is the principal, for each maturity point, bootstrapping or creating the spot curve, or zero-coupon curve. With one variable for price, another for the face value, and all the data behind that maturity in the term structure, you finish it off with the last piece of the puzzle, which is also the the forward rate that gets you back home.

The compounded average of the previous maturity’s spot rate, along with this new building block/forward rate, creates the spot rate, which is a geometric link of your (implied) forward rates for a given maturity. It’s also the rate that should price a zero coupon bond at 100par, or face value. Once you go out as far as you want along the term structure, you can use that data and some geometric averaging to find the implied forward term structure, as you slice up the spot curve based on the start of the forward rate you are interested in and its maturity.

But since everyone robotically states that the three curves “all contain the same information,” assuming I’ve got this understood properly, the par curve, with a (not stale) print determined by the primary market and quickly evaluated by the secondary market, is the only one of these we can really see and trust, is really where it all begins. And I think that’s the point.

I don’t believe I’m mistaken. Maybe I should have made it more clear in my first post.

YTM is the market interest rate for a specific maturity _ on the par curve _.

What do you mean price is a random variable? Price is the summation of all future cash flows discounted at their matched opportunity costs.

In the case of a par curve (A fixed income security that retires with the same price at issue), the par yield is the same as the YTM. And the YTM in this case is the annualized coupon rate, because if you always discount the cash flows with the same cost as the yield it gives back, you’re FV will always equal the PV.

Consider this. If you are looking to save \$100 and earn 5% per year on your investment, and I am willing to take you’re \$100 and give you \$5 every year, what would I owe you when you feel like taking back your money just after the last coupon payment? Your initial investment, because I already gave you the return we agreed on before, no more no less.

In the bond market, it’s priced by no arbitrage when the coupon rate does not equal the required yield using spot rates on every single cash flow matching their term. Which would give you a yield-to-maturity as opposed to just ‘yield’. In the case of a par bond, the yield = YTM = Coupon. And the Price = PV = FV.

Hope this helps a little.

I have read most of your site which is extremely helpful (many thanks by the way), I understand very well that if market price = par value, then YTM = par yield. But I don’t understand why we are assuming that bonds trade at par , that’s my whole point.

In real life bonds don’t trade at par, so to me it looks like the observed YTM is not the same thing at all as the PAR YIELD. It seems that the par yields/the par curve are just a theoritical construction. We obersve the YTMs, but we have to make the fancy hypothesis of market price = par value to compute the par yields and construct the par curve.

I am going to read the anwsers of the others.

Thanks that’s clear things up.

So you are saying that the Treasury bonds trade close to par, therefore the observed YTMs are a close approximation of the par yields, and when it’s not we make an adjustment. Am I right?

But why bother with this approximation and calculus when instead one could just plot the actual YTMs of the Treasuries? Why use that abstraction that lies on a hypothesis (the par yield) instead of the raw empirical data (ie. the actual YTM)?

Can’t we derive the spot curve and the forward curve from the curve of the actual YTMs instead of the par curve?

We’re not assuming that bonds trade at par. Depending on their coupon rate vis-à-vis their YTM they can trade at par, above par, or below par.

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’m merely saying that the reason that the par curve is called the _ par curve _ is that bonds whose coupon rates equal the rate on the curve (the YTM) will trade at par. I’m definitely not saying that there are any such bonds.