# Different maturities but same tenor to obtain the yield

My question is in regards to obtaining the yield of a specific tenor at any date (for example, when constructing the yield curve). For example, when calculating the yield for a specific zero-coupon (zc) bond with the tenor (i.e. time to maturity) of 3 months, one has multiple options: the 3-month on the run zc bonds, all the longer maturity zc bonds with the time-to-maturity of 3 months (i.e. bonds with 6M, 1y, ... maturity).

My question is whether there is some kind of averaging of the yields of all these bonds which have the same tenor but different maturities? In theory, with a fixed par value, the price of a 1y zc bond with 3 month to maturity should be the same as the price of a 3M zc bond in the secondary market.

There are two main approaches. For traded bonds, firms like Bloomberg tend to pick a representative benchmark bond for a given tenor, which is typically a recently issued liquid bond. They often show yield curves computed from the yields of these benchmark bonds.

One issue with this approach is that it gives yields for a discrete set of points but often one would instead like to have a continuous yield curve. The above curves tend to also be for standard coupon bonds but for standardisation and simplicity, zero-coupon or par curves are used, see e.g. https://www.federalreserve.gov/data/yield-curve-tables/feds200628_1.html. Then one needs to fit a yield function to the prices that minimises pricing error with respect to traded bonds.

Here one can include several bonds with similar tenors so effectively there can be some "averaging" described in your question. Usually the practice is to include all Treasury bonds that satisfy certain conditions. For example one might exclude bonds with anomalous yields.

One particular choice concerns whether to include on-the-run bonds, which often trade at liquidity premium relative to other bonds with similar tenors. One argument for excluding them is that their yields are not representative. However, if you are interested in trading these bonds due to their liquidity then focusing on them can be justified.

When you observe two or more bonds from the same issuer, and with about the same time left to maturity, trading at materially different yields, it's a good exercise to ponder, why are the yields different? To add to @fes's answer, in addition to bondholder's inexpicable preference for on-the-runs, here are a few more possible reasons to consider:

• are you looking at yield quotes at which someone said they're willing to transact now, or at quotes at which these bonds were last traded (e.g. from Finra Trace)? Many bonds don't trade often, so perhaps one was traded a day ago, another was traded a week ago, the markets have moved, and if they traded today, the yields would be something else yet. In this case, you may want to change from yields into spreads over some benchmark that you can observe more frequently.

• in some markets, taxation causes zero-coupon and coupon-bearing bonds to, basically, trade on different yield curves. E.g. in Brazil, the government treasury issues zero-coupon bonds with maturities >1 year, and bonds that pay coupons. Because of materially different after-tax returns, you shouldn't try to combine them.

• especially with corporate debt, there may be some subtle differences in the terms and conditions pertaining to what can go wrong, such as the issuer not paying as promised. E.g. maybe one prospectus says that if there is a dispute, the bondholder can sue in New York court, while the other says that all disputes will be addressed by arbitrage in some frontier market hellhole. Maybe one bond requires 100% of bondholders to agree to amend some terms, while the other bond requires >50%. Or maybe only one obligation is collateralized / secured / involves some liens, i.e. if the issuer doesn't pay, then the bondholder can easily seize something valuable instead. Or maybe they have different seniority, e.g. if the issuer can't afford to pay both, then only the senior debt gets paid first, but then the money runs out and the subordinated one doesn't get paid in full. Etc. When you see different yields, always check to make sure you're not averaging apples and oranges.