Constructing Long-Dated Curves and Choosing a Forward-Measure Numeraire
Summary
The document addresses how a T-forward measure can be used when a directly observed zero-coupon bond for the required maturity is unavailable. It explains that market curves are built from instruments spanning different horizons: spot borrowing rates anchor the curve, while forward rate agreements, futures, and swaps provide information about borrowing costs in future periods. Bootstrapping these instruments can produce curves extending beyond the maturities of quoted spot rates.
The answer notes that each reference rate may have its own discount curve and that a zero-coupon bond is not the only possible numeraire. A suitable numeraire must remain valid over the horizon of interest; a hybrid involving a zero-coupon bond and a bank account is offered as an alternative. The explanation is conceptual and does not give curve-building equations or address modern benchmark and collateral conventions, so its LIBOR examples should be read in their stated context.
Key ideas
- Spot rates alone do not determine a long-dated curve; instruments covering future periods supply additional market information.
- Forward rate agreements, futures, and swaps can be combined to bootstrap curves beyond quoted spot maturities.
- Different reference rates may be associated with separate curves.
- A zero-coupon bond is a common numeraire, but another suitable numeraire can be used if it remains valid over the relevant horizon.
- A hybrid of a bond and a bank account is presented as one possible alternative numeraire.
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Full text
# T-Forward measure and tenors # T-Forward measure and tenors As far as I understand, a T-forward measure is associated with a situation when a zero-coupon bond with the same maturity, i.e. $P(t,t+T)$, is used as a numeraire. However, given that the yield curves, LIBOR, used to derive this bods have in the markets maturities only upto $12M$, what do we do when $T > 12M$? In such cases $P(t,t+T)$ is no longer a tradable (or even observable) asset and hence can't be used as a numerarie. ## Answer by Magic is in the chain (score 1) https://quant.stackexchange.com/a/49913 The current LIBORs (say O/N, 1wk, 1m, 2m, 3m, 6m, and 1y) refer to the borrowing cost for borrowing starting today (technically this is the spot date, which differ by currency,e.g. T+1, but we can call it today!). Each of these rates will have its own discounting curve. Let's focus on the 3 months rate. The current 3 month LIBOR refers to borrowing today for 3 months. To construct the discount curve, as you correctly pointed out, we need the borrowing costs for future periods, so we need additional contracts/intruments. Luckily you have the 3 months FRAs (forward rate agreements) which give you the cost of borrowing for 3 months, starting in the future. You have the future contracts, which are similar to FRAs, but there are some subtle differences between FRAs and futures as you would know. You then have the swaps referencing 3 months LIBORs, and these swaps can go to very long maturities, say 50 years. So you can construct the discount curve for 3 months LIBOR by combining these contracts (e.g.,bootstrapping). You can do the same for the other LIBORs, so each LIBOR (3m, 6m etc) can have its own discount curve. Normally you would use the longest maturity zero coupon as the nuemraire, but you don't have to. If the nuemraire asset is such that it stays alive over the horizon of your interest, then you are obviously fine. But if not then you can use a hybrid numeriare- e.g., hybrid of zero coupon and bank account.
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