As mentioned in a comment here, I am not qualified enough to answer this. However, I recently got to spend quite some time with the DCM, treasury and trading team at work and discussed this question with them during a break, out of personal interest.
Demand and supply in secondary markets will determine the price. Strictly speaking, computing yield from a given market price in secondary markets does not require discount curves either, unless you want to compute spreads (above a benchmark curve) as well. The standard yield to maturity, as shown in this answer, is the discount rate applied to all cashflows so that the NPV equals the quoted price of the bond. Insofar, everyone will get the same yield (discount rate), provided daycount and cashflows are mapped properly.
In general, there also different "types" of yield you compute as can be seen here. As always, demand (and supply) impacts pricing. Therefore, it is tempting to assume that it will also be the reason why most new issuances do not get issued at par.
The way it works with new bond issuances is different though. The price (and coupon) will be determined by the lead bank(s) and simply communicated to the subscribers.
Insofar, the discount rate does not matter much here, because the banks know exactly what curves they use to compute this price and as a potential buyer, you either decide to participate, or do not. In fact, you don’t even know the exact price when you bid for the bond.
The entire process, in a nutshell looks like this: If a firm wants to issue a bond, they use lead banks. If the firm already has bonds issued, former and potential lead banks will usually constantly quote the spread over the benchmark swap curve if you were to issue a new bond. If a firm decides to do so, they will usually have a so-called beauty contest where potential lead banks will show their ideas and strengths (this is especially important if it is the first issue for a firm that never issued bonds). Once it is decided what bank(s) help with the issuance, there will be road shows, general marketing and anything that creates interest in the bond. In this phase, you also get an idea about demand for the bond and look at comparative issuances to get an idea about the offered coupon.
Once everything is sufficiently set up (details about bond - senior preferred, covered ..., volume, prospectus etc), the books will open for subscription. Ideally, there will be oversubscription. In the rare cases it is not, the lead banks will often guarantee that the bond is sold. However, that will generate negative publicity that will usually be unfavourable for issuances in the future. During this phase, if it is quite oversubscribed, you often get an extra push and even more oversubscription for this issuance. Once subscription is over, the lead banks will suggest how to allocate the volume to the subscribers (Note, that is different for government bonds, where you frequently have auctions, that work differently). Ultimately, it will be the decision of the issuer who gets how much.
The final step is the pricing call with all lead banks, where the actual spread to the swap rate will be determined (the actual coupon so to say). That is the crucial point here, because coupons will not be written in strange numbers like the 2.639 or 2.597 you see in the Bloomberg screenshot of the question I linked above. This screenshot computes the yield in a secondary market but the point to take home is that the smallest increments for coupons will usually be in steps of 1/8th. While the actual price the banks compute based on the demand and market condition will be at par, the rounding of the coupon rate will result in a price that is almost never at par. This price will be communicated to the subscribers, alongside the allocation they got. Insofar, the buyer never knows up front what the price (or even coupon / yield) will be exactly before they get the "bill" for the bond. Almost always, there will be a new issue premium, which means that the new issue offers a more attractive yield compared to current yields in the secondary market.
In reality, there are a lot of factors that will have an impact
- Bonds can be fixed coupon, variable, step up coupons, callable, puttable, convertible, sinkable, pay in kind ....
- underwriters can agree on a firm commitment or best effort (in the former, the underwriter assumes all inventory risk, in the latter, there is no guarantee that the entire volume will be sold - used mainly in high risk or less than ideal market conditions) ...
Long story short, bonds usually do not price at par at the beginning, mainly because interest rates are set in discrete steps. Only once it trades in secondary market will market participants compute their own yield based on traded / quoted prices. Before that, all they can do is estimate where the new issue will be trading based on other new issues and existing bond curves. If you have access to Bloomberg, you can have a look at some functions like (
NI NEWBON and the like).