The Term Structure of Interest Rates: Spot, Par, and Forward Curves

How default-risk-free spot rates describe the term structure of interest rates, how to price bonds with them, how par rates and implied forward rates are derived from them (and spot rates back from forwards), and how the shapes of the spot, par and forward curves are linked.

1. 1. Spot rates and the spot curveA spot rate is the yield on a default-risk-free zero-coupon bond, and the spot curve plots those yields against maturity so that time-to-maturity is the only thing that differs.Free12 min
2. 2. Pricing a bond with spot ratesDiscount each cash flow at the spot rate for its own date: the sum is the bond's no-arbitrage price, and the YTM is just the single rate that gives the same total.Video · 6 min13 min
3. 3. Par rates and the par curveA par rate is the coupon rate that would make a bond priced off the spot curve worth exactly par, so it is that bond's coupon rate and its YTM at once.Video · 6 min13 min
4. 4. Implied forward rates from spot ratesAn implied forward rate is the rate for a future period that makes investing short and rolling over earn exactly the same as investing long today, so it is the breakeven reinvestment rate.Video · 6 min14 min
5. 5. Spot rates from forward rates, and pricing with forwardsChaining forward rates rebuilds the spot curve, because a spot rate is the geometric average of the forward rates that cover its period, so bonds can be priced with either set of rates.Video · 6 min12 min
6. 6. Comparing the spot, par and forward curvesPar and forward curves are both derived from the spot curve, so the spot curve's slope tells you where the other two lie: for an upward slope, par slightly below spot and forwards above.13 min