# Par Curve Derivation Question

Hello,

I was hoping someone could help me better grasp the process for deriving the par curve. From my understanding, the forward curve is derived from the spot curve, which is derived from the par curve through bootstrapping. A par rate is the coupon rate the US Treasury would set on a bond hypothetically issued at par today.

So, my question is:

If the par curve is obtained before the spot curve is, and a bond trading at par has its coupon = its YTM, then how does the Treasury actually determine what that coupon should be? If it needs to set a coupon = the YTM of that par bond, where does that YTM come from and how is it determined?

More broadly, I guess I’m a little cloudy on the order of events that determine the bond prices we actually see in the markets. As I understand it, bond prices are arrived at through discounting using treasury spot rates plus a spread for risk (except bonds with options, which use binomial interest rate tree models to arrive at prices). Those spot rates are bootstrapped from the par curve, and the par curve is composed of the coupon rates the US Treasury would hypothetically set on newly issued par bonds. But, by definition a bond trading at par has coupon = YTM, so how does the Treasury determine what that YTM is that it needs to set the coupon equal to to make it = par?

Let me know if there are any issues with my logic. Any help is appreciated. Thanks.

The coupon rate is set before the bonds are issued. I imagine that the Treasury typically tries to set the rate so that the bonds will sell at a price close to par, but,ultimately, the market will determine that. When the bonds are issued, they sell for whatever price the market will pay, and the YTM will be determined based on the selling price.

Thanks, S2000magician. So just to make sure I conceptually understand the process, could you let me know if the below line of reasoning sounds correct?

1. The US Treasury looks at market supply / demand and comes up with a coupon rate it believes will allow newly issued coupon-bearing Treasury securities to be priced at, or close to par. This would be kind of like the Treasury’s best guess as to what the market discount rate would be at the time of issuance. (If anyone has any clarity surrounding how exactly the Treasury comes up with this initial prediction, I would be interested to learn.)

2. By the time the bond is issued, market forces may have shifted to cause the actual market rates to deviate from the coupon rate initially set by the Treasury, causing the on the run bond to trade close to, but not exactly at par.

3. The combination of observable par rates set by the Treasury, as well as interpolated par rates for bonds that cannot be directly observed are used to form the par curve.

4. The Treasury spot curve is obtained from this Treasury par curve through bootstrapping. (Then the spot curve can be used to mathematically derive the forward curve.)

5. Prices for corporate bonds get determined by adding a spread representing the investment’s risk to those spot rates, and discounting that bond’s cash flows with those adjusted rates to arrive at the price. This is the price we see in the market.

6. YTM’s for these bonds can then be determined based on the observable market prices obtained through the process in steps 1 - 5.

If my understanding is correct, and please let me know if it’s not, it sounds like ultimately corporate bond prices are impacted by an initial estimate made by the US Treasury. That’s a bit interesting if the prices of foreign bonds are also influenced by an initial rate chosen by the US Treasury.

That all sounds good to me.

As a practical matter, what we see is the price of a corporate bond in the marketplace, and we infer from that the spread over the spot curve. I’d be amazed if the buyers in the market actively tried to estimate an appropriate spread, then determined from that the price that they offer for corporate bonds.

I see, so realistically the prices we see in the market are not really determined by a mathematical function of the bond’s risk, but rather the prices are observable in the market and we can determine the bond’s risk profile by inferring the spread between the bond’s YTM and the Treasury spot rate for a bond of the same maturity.

If that’s the case,

1. What is the actual basis buyers and broker dealers would use to determine bond prices and quotes on a practical level? Is it more of a general estimate of those risk factors combined with the current market supply / demand levels?

2. If market bond prices are not really determined by mathematical risk calculations on a practical level, would it be correct to say that in the inverse relationship between bond prices and rates, it is the prices that move first and cause the rates to change in response, as opposed to the rates moving and the prices changing in response?

Thank you very much for your help in demystifying this.

I have been struggling with understanding the precise mechanisms that determine actual prices and yields for awhile; throwing in the par and spot curves have made it difficult to understand which forces trigger which effects.

So, from my understanding, the Treasury engineers the par curve by setting a coupon based on what it expects the market discount rate to be at the time of issuance depending on projected supply / demand. From this we can we derive Treasury spot rates. In order for it to be true that the price of a Treasury can be mathematically found by discounting using Treasury spot rates, it must be the case that Treasury prices are a reflection of these rates. This would lead me to believe it is the rates that move first and the prices that change in response. However, this feels wrong because prices seem to be more directly tied to supply and demand, and if the original par rates are based on supply and demand, that would lead to a kind of chicken and egg situation in which the Treasury uses observable market prices to set rates which determine market prices.

So, if anyone could help me understand, is it a change in the prices or the rates that triggers the other to move? I have the same doubt with respect to whether coupon rates on OTR Corp bonds determine, or get determined by Corp bond prices. Perhaps I am oversimplifying or overcomplicating. Thank you.

You are both simultaneously right and wrong… and so could I be.

1. The FPO of an issue is definitely a fn of demand and supply and the prevailing rates.

2. Affixing rate is decidedly a policy function. The on the run issues adjust accordingly till a new supply demand imbalance disrupts and invite one more round of policy decisions

Thanks for your reply. I think one reason this seems so elusive is trying to reconcile policy decisions with supply and demand dynamics.

Here is my attempt to put it all together:

1. Supply / Demand drives the market prices of existing bonds (Treasuries, Corporates, etc.) up and down relative to their par values. The market prices of the individual bonds reflects the prevailing investor appetite for bonds with those characteristics, the liquidity of the market for that bond, as well as the opportunity cost of not realizing the return that could be achieved through other bonds in the market. The vendors who post bond quotes consolidate this information along with the prices of observable transactions, and come up with the final prices they post. Due to the number of inputs that go into a bond’s price, and the different methodologies of measuring those inputs, the quotes we see in the market are not entirely consistent across vendors (Especially for less liquid bonds that have a lack of trading activity that can be observed.)

2. These prices imply a market rate (I.E. the YTM) for the bond based on the PV formula. “The rate in the market must be X% to justify this bond’s price.”

3. Coupons get set on newly issued bonds prior to issuance. The issuers (I.E. Corporations for Corp bonds and the Treasury for Treasuries) choose a coupon rate based on these price-derived market rates, and try to set it at a level that will make the bond trade close to par at issuance. The coupon rates on these newly issued OTR bonds represent an opportunity cost to existing bond holders, which in turn influence the existing prices in the market (among the other previously listed price inputs) So, the coupons on OTR bonds get set by issuers based on the existing market rates, where the market rates come from the observable bond prices, which come from a number of inputs, one of which is the coupon rates set on the most recent OTR bonds.

In short: S/D and other factors drive bond prices of existing bonds → prices drive market rates → market rates drive coupons on OTR bonds → OTR bond coupon rates and other factors drive bond prices, and the cycle continues.

One such issuer is the Treasury, who sets coupons on par bonds by looking at current market rates and using a spline to interpolate data points for any bond maturities for which there is a lack of observable market data to form a par curve. The spot curve is derived from this par curve, and the forward curve from this spot curve. Spreads implied by the difference between the observable price and Treasury spot rate of the same maturity represents risk. So, while in theory we can calculate a bond’s price from spot rates, and while vendors may reference these curves when coming up with their bond quotes, there isn’t this hard mathematical relationship whereby the bond prices we see in the market perfectly reflect the price we would arrive at by discounting using the spot curve.

You are spot on. Just that any mathematical construct is only supposed to give you the "theoretical arbitrage free " VALUE of an asset. Any asset.

The PRICE of any Asset = The above VALUE
( also known as FAIR VALUE) +/- Premium/ Discount

Understood. Thank you for your help.