In Section 7.5 of Hull 9/e, he states,

Note that 5-year swap rates are less than 5-year AA [LIBOR] borrowing rates

and gives a creditworthiness argument as to why this is so. However, on the next page, he works out the following example in which the swap rate is above the corresponding LIBOR rate:

Suppose that the 6-month, 12-month, and 18-month LIBOR/swap zero rates have been determined as 4%, 4.5%, and 4.8% with continuous compounding and that the 2-year swap rate (for a swap where payments are made semiannually) is 5%. This 5% swap rate means that a bond with a principal of \$100 and a semiannnual coupon of 5% per annum sells for par. It follows that, if $R$ is the 2-year zero rate, then $$ 2.5e^{-0.04\times 0.5} + 2.5e^{-0.045\times 1.0} + 2.5e^{-0.048\times 1.5} + 102.5e^{-R\times 2.0} = 100 $$ Solving this, we obtain $R = 4.953$%.

Note the 2-year borrowing rate is less than the 2-year swap rate: $4.953\% < 5\%$. In other words, there does not seem to be a mathematical reason as to why the $n$-year swap rate should be less than the $n$-year LIBOR rate.

Could someone explain this apparent contradiction?

  • 1
    $\begingroup$ Can you provide the full context ? a swap rate can be above or below its projection curve same maturity zero coupon rate, it only depends on the shape of the full zero coupon projection curve. What you are referring to seems to have to do with spreads between projection curves of different tenors. $\endgroup$ – Antoine Conze Jun 26 '17 at 14:20

This example you give from Hull is, unfortunately, poorly presented for a number of reasons.

The key point you initially state that Hull is trying to explain is: "5Y swap rates are less than the 5Y LIBOR borrowing rate."

There are a number of inherent problems with this statement;

  1. There are lots of different 5Y swap rates, i.e. IRSs whose floating leg is 1M LIBOR, or 3M LIBOR, or 6M LIBOR etc.. All of them have different rates, as do IRSs versus the same LIBOR tenor but with different fixed frequencies such as annual, semi, quarterly etc..
  2. It conveys the sense that LIBOR is some rate which permits borrowing money for any term and is an active market. It isn't liquid or active at all. All LIBOR is, is a set of fixed tenors (1M, 2M, 3M, 6M, 9M, 12M) per currency that gives an indication of borrowing rates based on banks' opinions only - no actual trading data. There certainly is no 5Y LIBOR rate.

The specific problem with the example, though, and its probably cause for contradiction is that by using exponentials in the bond calculation it is assuming the quoted tenor LIBOR rates are continually compounded. Continually compounded rates are never used in practice and only, really, in theory for more complicated problems using derivatives to simplify the mathematics. It would be better here to defined the discount factors as: $\frac{1}{1+d_ir_i}$ instead of $e^{-r_id_i}$. That changes your bond coupon calculation and instead of getting 4.953% you would end up with 5.212%. This is more directly comparable with swaps that are certainly not continually compounded.

From a more theoretical standpoint if instead of using 5Y, we arbitrarily chose a better example such as 6M we can claim: "The 6M swap rate (versus 3M floating LIBOR) is less than the 6M LIBOR rate". The reason this is the case is symmetric from both the point of view of the borrower and of the lender:

  1. A borrower who seeks to borrow for a tenor of 6M can choose to borrow for the entire tenor of 6M ensuring a fixed funding profile. Or he can first borrow for 3M and then roll the loan into a new 3M loan after the first has expired. If this works he ends up with a lower rate (the 6M swap rate versus 3M LIBOR) but if liquidity disappears then he is faced with a scarcity premium and securing the second loan can be very costly. Northern Rock of UK is an example of a bank that collapsed due to this precise feature.
  2. A lender who lends money can lend for a fixed tenor of 6M or for 3M and then choose at the 3M point to re-lend again. The choice to re-lend can be considered an option so if he synthetically sells it by issuing a loan for the full 6M tenor then logically he will charge a higher rate for that than two 3M loans.

The mathematics and a further description of these effects can be found in the section of term structure of ultra-short dated rates in this book.


Interestingly there is no such thing as 5 year LIBOR. As you know LIBOR is interbank borrowing rates up to 1 year. I know that Hull 9/e is very respected and popular book but that statement is simply wrong.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.