The Term Structure of Interest Rates: Spot, Par, and Forward Curves
Overview
Prior lessons priced fixed-income instruments by discounting all future cash flows using a single interest rate, such as the yield-to-maturity or a market reference rate (MRR) plus a discount margin. The next three lessons relax this assumption by introducing the term structure of interest rates, or the fact that interest rates vary with time-to-maturity. The ideal data to use for term structure analysis are default-risk-free zero-coupon bonds, known as spot rates or the spot curve. Since these are generally not directly observable, various estimation techniques are used. The spot curve is used to derive two other important yield curves: the par curve and the forward curve. A par curve involves bond yields for hypothetical benchmark securities priced at par, while the forward curve involves rates for interest periods starting in the future. All three of these curves are fundamental to fixed-income analysis and other applications because they represent default-risk-free rates of return for time periods that start today and in the future. We show the pricing of bonds using these different rates and establish their relationships.
- Spot rates are market discount rates on default-risk-free zero-coupon bonds, sometimes referred to as zero rates. By using a sequence of spot rates in calculating bond prices, a no-arbitrage bond price is obtained.A par rate is the market discount rate for a specific maturity that would result in a bond priced at par. A par rate is derived from the spot rates up to and including the maturity date.
- Implied forward rates are calculated using spot rates and can be interpreted as an incremental, or marginal, return for extending the time-to-maturity for an additional time period. As such, they reflect a breakeven reinvestment rate.
- Since par and forward rates can be derived from spot rates, the shape of the spot curve is closely related to the shape of the par and forward curves.
- In upward-sloping term structures, par rates will be lower than their corresponding spot rates and forward rates will be greater than spot rates. In downward-sloping term structures, par rates will be greater than spot rates and forward rates will be lower than spot rates.
Learning outcome
The candidate should be able to:
- define spot rates and the spot curve, and calculate the price of a bond using spot rates;
- define par and forward rates, and calculate par rates, forward rates from spot rates, spot rates from forward rates, and the price of a bond using forward rates;
- compare the spot curve, par curve, and forward curve.
1 PL Credit
If you are a CFA Institute member don’t forget to record Professional Learning (PL) credit from reading this article.