ACCURATE IMPLIED FORWARD RATES

Equation 5.2 provides an excellent, and easy-to-calculate approximation for the implied forward rate (RateA×B) that connects a shorter-term rate (Rate0×A) to a longer-term rate (Rate0×B). For now, we'll stay with bond yields so the relevant time periods continue to be AYears and B Years. Money market rates will require some special attention because of the unique manner in which they are quoted – recall the add-on rates and discount rates and 360-day years in Chapter 1.

An accurate formula for an implied forward rate that includes compounding and the specific periodicity (PER) for the yields is based on the expression shown in equation 5.3.

(5.3)

The first term is the proceeds per unit invested for AYears assuming that the 0 × A rate is an APR quoted for PER periods per year. That amount is

reinvested for the remaining time period out to year B (i.e., for B Years minus AYears, at the A x B implied forward rate). The compounded total return equals the proceeds per unit invested for BYears at the 0 × B rate. It is important that all three rates are annualized for the same periodicity.

The accurate implied forward rate formula comes from rearranging equation 5.3 to isolate the Rate^ term.

(5.4)

Let's return to the example of 1-year, 2-year, and 3-year rates of 1.00%, 2.00%, and 2.50% quoted on a semiannual bond basis for zero-coupon government bonds. The 1×2 IFR, which we approximated to be 3.00%, more accurately is 3.0050%.

Similarly, the 2×3 and 1×3 IFRs, approximated earlier at 3.50% and 3.25%, are really 3.5037% and 3.2542%, respectively.

These examples indicate that the simple weighted averages are excellent approximations. In fact, the difference between IFRs calculated with equations 5.2 and 5.4 diminish with increasing periodicity. When the rates are quoted for continuous compounding, the approximation formula produces exact results. In general, approximations are fine for back-of-the-envelope calculations, but there is no reason not to use the accurate formula when building a spreadsheet program. Still, these formulas are for zero-coupon rates having the same periodicity. If you care only about U.S. Treasury STRIPS, you're fine. If you care about the rest of the debt market, you need more bond math.