Compounding
Annual compounding means paying interest once a year, while semiannual compounding means paying interest once every six months.
Deriving Discount Factors from Swap Rates
Similar to a series of bond prices, discount factors can also be derived from a series of interest rate swap rates. To make this calculation, swap rates are treated as bond coupon payments and the swap notional amount represents the bond’s par value.
Spot Rates
A t-period spot rate is the yield to maturity on a zero-coupon bond that matures in t-years. The spot rate curve is the graph of the relationship between spot rates and maturity. The spot rate curve can be derived from either a series of STRIPS prices, or the comparable discount factors.
Forward Rates
Forward rates are interest rates corresponding to a future period implied by the spot curve. Bootstrapping is the process of computing forward rates from spot rates.
Par Rates
The par rate at maturity is the rate at which the present value of a bond equals its par value. Par rates are the same as swap rates and can be accessed via the swap rate curve.
Pricing a Bond
A spot rate is approximately equal to the average of the forward rates of equal or lower term. As spot rates increase over time, forward rates are greater than corresponding spot rates.
Given an upward-sloping spot rate curve, par rates are near, but slightly below, corresponding spot rates. This relationship occurs because the spot rate curve is not flat.
Effect of Maturity on Bond Prices and Returns
In general, bond prices will increase with maturity when coupon rates are above relevant forward rates. A bond’s return will depend on the duration of the investment and the relationship between spot and forward rates.
Yield Curve Shapes
When the yield curve undergoes a parallel shift, the yield on all maturities change in the same direction and by the same amount. The slope of the yield curve remains unchanged following a parallel shift.
When the yield curve undergoes a nonparallel shift, the yields for the various maturities do not necessarily change in the same direction or by the same amount. The slope of the yield curve after a nonparallel shift is not the same as it was prior to the shift.
- Twists refer to yield curve changes when the slope becomes either flatter or more steep. A flattening of the yield curve means that the spread between short- and long-term rates has narrowed.
- Butterfly shifts refer to changes in curvature of the yield curve. A positive butterfly means that the yield curve has become less curved. A negative butterfly means that there is more curvature to the yield curve.
