Par and Forward Rates

Par Rates from Spot Rates

An important use of spot rates is determining par rates. A par rate is a yield-to-maturity that makes the present value of a bond’s cash flows equal to par (100% of face value). Par rates derived for hypothetical government bonds with different times-to-maturity are commonly used for term structure analysis because they control for tax, trading, and other potential distortions associated with actual bonds priced at either a discount or premium.

On a coupon payment date, the following equation can be used to calculate a par rate by solving for PMT given a sequence of spot rates Z₁, Z₂, . . . , Z_n . Between coupon dates, we set the flat price, rather than the full price, equal to 100.

100=\frac{PMT}{1+z_1}+\frac{PMT}{(1+z_2)^2}+\dots+\frac{PMT+100}{(1+z_N)^N}

Forward Rates from Spot Rates

Spot rates can also be used to calculate implied forward rates (also known as forward yields), which are breakeven reinvestment rates. They link the return on an investment in a shorter-term zero-coupon bond to the return on an investment in a longer-term zero-coupon bond.

A general formula for the implied forward rate, IFR_A_,_B_–_A, for a security begins at t = A and matures at t = B (tenor B – A). To solve for it, we need the spot rate, z_A, and the longer-term spot rate, z_B .