Suppose that 90-day LIBOR is 1.00% and 180-day LIBOR is 2.00%. What rate would a true believer in the expectations theory of the yield curve anticipate for 90-day LIBOR, 90 days into the future? That is, what's the 90 x 180 day forward LIBOR – 3.00%, calculated as a simple average? Unfortunately, you cannot use the approximation formula or even the accurate formula with money market instruments because their rates have different periodicities; 90-day LIBOR has a periodicity of 4 and 180-day LIBOR a periodicity of 2, assuming a 360-day year. Equations 5.2 and 5.4 simply do not apply.

Our true believer could delve back into Chapters 1 and 2 and retrieve the correct procedures to deal with the periodicity problem. First, the rates are converted to a 365-day year, and then the periodicities are converted to a common basis. For instance, if they are restated for a semiannual bond basis, the IFR can be obtained using equation 5.4. If they are restated to continuous compounding, equation 5.2 will suffice. But, for good reasons discussed below, that is not typically done in practice. Instead it is more useful to calculate the IFR given the manner in which the rates are quoted and traded in the money market.

For rates quoted on an add-on basis (e.g., bank CDs, LIBOR, repos), equation 5.5 parallels equation 5.3.

(5.5)

We are in the money market so the relevant time periods are ADays and BDays; Year is 360 in the U.S. but 365 in many other countries. The idea is that same total return is obtained from investing in the shorter-term security at AOR0×A and rolling over the proceeds at the implied forward rate for the A x B time period as is obtained when investing directly in the longer-term instrument earning AOR0×B.

The next step is to rearrange equation 5.5 algebraically to isolate the AORA×B term, which is the implied forward add-on money market rate between days A and B.

(5.6)

Notice that the first term in brackets is essentially the approximation formula in equation 5.2. The second term “adjusts” that approximation downward, the more so the higher the rate and the greater the number of days.

Now we can solve for the true believer's expectation. Let Year = 360, ADays = 90, and BDays = 180. The 90 × 180 day (or 3 × 6 month) implied forward LIBOR turns out to be 2.9925%.

One of the annoying realities of very low market interest rates is that some interesting bond math calculations turn out to be numerically insignificant (i.e., 3.00% versus 2.9925%). Imagine a world of much higher inflation. If 6-month LIBOR is 10% and 12-month LIBOR 20%, then the 6 x 12 implied forward LIBOR would be 28.57% – a more impressive departure from the simple approximation of 30%.

Money market rates quoted on a discount rate basis (e.g., commercial paper, bankers acceptances, and Treasury bills in the U.S.) are even more problematic for yield curve analysis than add-on rates. We saw in Chapter 1 that discount rates understate the investor's rate of return and are not even APRs in the traditional sense. It is very tempting to prescribe converting the discount rates to add-on rates so that equation 5.6 can be used. However, calculating the IFR on a discount rate basis can have its advantages.

Equation 5.7 is similar in structure to equation 5.3 for bond yields and to equation 5.5 for add-on rates but differs because it applies to money market discount rates.

(5.7)

The right side of the equation is the day-0 price per one unit of face value received BDays later given a discount rate of DRQ×B. This is based on the discount rate pricing formula 1.6 in Chapter 1. The left side obtains the same day-0 price per one unit of face value. It is first discounted back from day B to day A using the rate DRA×B. That amount is then discounted back from day A to day 0 at rate DRQ×A.

Rearranging equation 5.7 gives us a formula for the implied forward discount rate between days A and B.

(5.8)

Once again, the first term is the simple average and the second term “adjusts” the approximation. Now the presence of the minus sign in the denominator of the adjustment factor raises the IFR above the simple average. Suppose that 90-day and 180-day bankers acceptance (BA) rates are 1.00% and 2.00%. The 90 x 180 day implied forward BA discount rate turns out to be 3.0075%, above the simple average of 3.00%, albeit by a very small amount.

If 6-month and 12-month BA discount rates are 10% and 20%, the IFR is 31.58%. That's substantially more than the 30% approximation.

Why calculate implied forward money market rates on an add-on or discount rate basis? Why not just convert them to continuously compounded bond yields, as we would do in academia? The answer is in how the IFRs are used in practice. Typically, an IFR is compared to quoted market rates or to one's own expectation for future market rates. It simply is easier to keep all the rates on the same basis than to convert them. For instance, suppose you ask a money market trader for his or her view on the next T-bill auction. The response will be in terms of a T-bill discount rate – that is how the money market instrument trades and how the trader thinks about that market.

We soon get to some applications of IFRs, but first we have to deal with the reality that other than in the money market and with Treasury STRIPS, we just do not see very many zero-coupon bond yields.

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