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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.

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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.

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  • $\begingroup$ Thank you. I am aware of the Fed Funds rate and the OIS discounting. Wouldn't it still be the case though that the implied risk-free yield from 3-moth T-bills isn't the same as the Fed Fund rate on most days, but rather it fluctuates around it? I am trying to say that we treat the relationship as an absolute equality in our models and derivations, but in fact they almost never equal. Maybe it's correct to conclude that it is a rough approximation? $\endgroup$ – Jan Stuller Jan 12 '20 at 9:16
  • $\begingroup$ It’s true that TBill yields and Fed funds are slightly different rates. For derivatives modeling we don’t care about this, because Fed funds is the ‘correct’ rate anyway. That’s because most derivatives are cleared, and the clearing house is using Fed funds discounting to calculate the valuation. $\endgroup$ – dm63 Jan 12 '20 at 11:04
  • $\begingroup$ Sorry to be a pest: so I understand that the Fed funds rate is the correct rate to use for discounting and that the T-bill yields are different to it: so doesn't this essentially mean that using the T-bills as a proxy for the zero coupon bonds to infer the Fed funds rate (or any other rate such as Libor) is not correct? $\endgroup$ – Jan Stuller Jan 12 '20 at 16:17
  • $\begingroup$ Yes. Fed funds is usually a more appropriate discount rate than TBills, for many applications. $\endgroup$ – dm63 Jan 12 '20 at 20:51

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