Libor, Treasury Bill Yields, and Discounting in the Libor Market Model
Summary
The discussion questions whether the Libor Market Model’s link between a Libor rate and a zero-coupon bond price is realistic, given perceived differences between Libor and Treasury bill movements. The response distinguishes short-term co-movement from daily changes: three-month Libor and Treasury bill yields may track over longer horizons even when their day-to-day changes differ. It attributes some of that difference to Libor’s poll-based setting versus market-traded bill yields.
The answer also notes a change in discounting practice. Older work commonly used Libor for discounting, while modern practice uses the federal funds rate as a closer proxy for risk-free discounting. This is a brief qualitative answer rather than a detailed empirical test; it does not present data, define the relevant instruments precisely, or address all markets and model assumptions. The point is to distinguish a modeling convention from the choice of discount curve in practice.
Key ideas
- Libor and Treasury bill yields can move together over longer periods while differing in daily changes.
- Libor’s poll-based construction differs from yields observed in a traded market.
- The answer describes Libor discounting as common in older literature.
- It identifies federal funds as the more representative discount rate in modern practice.
- The exchange offers qualitative context but no data analysis or comprehensive model assessment.
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# T-Forward Measure, LMM & the Zero T-bond
# T-Forward Measure, LMM & the Zero T-bond
the zero-coupon T-bond is widely used in the industry as a tool to derive pricing formulas: for example it is used in the derivation of the Libor Market Model. The way in which it is often used implies that we assume a no-arbitrage relationship between the zero-coupon bonds and the Libor rates. As an example: if we assume discrete rates (as in the case of the Libor Market Model), we could write:
$P(t_0,T):=\frac{1}{1+\delta L(t_0,t_0,T)}$
Where $P(t_0,T)$ is today's price of the zero-coupon bond that pays 1 unit of currency at T, $\delta$ is the annual fraction corresponding to the Libor rate and $L(t_0,t_0,T)$ is the Libor rate between $t_0$ and $T$, observed at time $t_0$ (of course, from the above it follows that $\delta=T-t_0$).
My question: In reality, there is a very weak relationship between bonds and Libor rates. Even looking at the short-dated treasury bills in the US against the US Libors, the two instruments live their own, independent life. The above relationship certainly doesn't hold on most days, if on any day. The Libors are very stable, whilst the T-bills are quite volatile. Even the forward Libors implied by Eurodollar Futures or FRAs are quite independent of the T-bills. Doesn't this invalidate the usage of relationships such as the one stated above (and hence the entire LMM derivation)?
PS: I haven't even even mentioned non-US markets, where the relationship would be even weaker (because some governments don't even issue short-dated discount securities, and the ones that do certainly don't do it as often as the US). I would say that in most G10 markets, the assumption that we can price Libor rates from bonds is simply plain wrong.
## Answer by dm63 (score 1)
https://quant.stackexchange.com/a/50669
Look at the data again. 3month libor rates and 3month TBill yields follow each other very closely. It’s true that the daily changes may not be highly correlated , because Libor is the result of a poll , whereas TBill yields are true market rates. However they do move very much together over longer periods.
you are right , older literature assumes the value of a zero coupon bond is obtained by discounting using Libor rates. Nowadays we use Fed Funds as the discount rate, which is more representative of risk free rates.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.