Title LM09 The Term Structure of Interest Rates- Spot, Par, and Forward Curves IFT Notes.pdf ✅

LM09 The Term Structure of Interest Rates 2025 Level I Notes © IFT. All rights reserved 1 LM09 The Term Structure of Interest Rates Spot, Par, and Forward Curves 1. Introduction ........................................................................................................................................................... 2 2. Maturity Structure of Interest Rates and Spot Rates ............................................................................. 2 3. Par and Forward Rates ...................................................................................................................................... 4 4. Spot, Par, And Forward Yield Curves and Interpreting Their Relationship ................................. 7 Summary ...................................................................................................................................................................... 9 Required disclaimer: IFT is a CFA Institute Prep Provider. Only CFA Institute Prep Providers are permitted to make use of CFA Institute copyrighted materials which are the building blocks of the exam. We are also required to create / use updated materials every year and this is validated by CFA Institute. Our products and services substantially cover the relevant curriculum and exam and this is validated by CFA Institute. In our advertising, any statement about the numbers of questions in our products and services relates to unique, original, proprietary questions. CFA Institute Prep Providers are forbidden from including CFA Institute official mock exam questions or any questions other than the end of reading questions within their products and services. CFA Institute does not endorse, promote, review or warrant the accuracy or quality of the product and services offered by IFT. CFA Institute®, CFA® and “Chartered Financial Analyst®” are trademarks owned by CFA Institute. © Copyright CFA Institute Version 1.0
LM09 The Term Structure of Interest Rates 2025 Level I Notes © IFT. All rights reserved 2 1. Introduction In the previous learning modules, we priced fixed income securities by discounting future cash flows using a single interest rate (such as the yield-to-maturity or MRR plus a discount margin). In this learning module, we will cover how to price fixed income securities using a sequence of interest rates. We will introduce the concept of term structure of interest rates, and the three common ways of representing the term structure – spot rate curve, par curve, and forward curve. 2. Maturity Structure of Interest Rates and Spot Rates Maturity Structure of Interest Rates The term structure of interest rates is the relationship between interest rates and bonds with different times to maturity. The basic premise is that interest rates change when inflation rates are expected to change over a period of time. For example, a one-year zero- coupon bond may have an interest rate of 7.50%, while a two-year zero-coupon bond may have an interest rate of 9.75%, assuming all the other factors are the same (currency, credit rating, periodicity, etc.). The three common ways of representing the term structure – spot rate curve, par curve, and forward curve. Spot rate curve: In our example above, the interest rates of zero-coupon bonds, 7.50% and 9.75% are called spot rates. Spot rates are yields to maturity (or return earned) on zero- coupon bonds maturing at the date of each cash flow, if the bond is held to maturity. So, the spot rate curve is also called the zero or strip curve. The spot rate curve plots different maturities on the x-axis and corresponding spot rates on the y-axis. Exhibit 1 from the curriculum illustrates a spot curve.
LM09 The Term Structure of Interest Rates 2025 Level I Notes © IFT. All rights reserved 3 This spot curve is upward sloping, which means that longer-term government bonds have higher yields than the shorter-term bonds. This is the pattern typically observed under normal market conditions. However, unique circumstances can also result in a spot curve that is flat or downward sloping. While a zero-coupon government bond spot curve is ideal for analysis, there are several practical issues to consider.  Most actively traded government bonds make coupon payments (i.e. they are not zero- coupon).  Older bonds tend to be less liquid than newly issued debt. In practice, only the most recently issued actively traded government bonds are used to build a yield curve. Exhibit 2 illustrates a yield curve for a government that issues 2, 3, 5, 7, 10, and 30 year bonds. Straight line interpolation is used to complete the curve. Bond Pricing Using Spot Rates Spot rates are yields to maturity on zero-coupon bonds maturing at the date of each cash flow. Since zero-coupon bonds have no intermediate cash flows, the actual yield on a zero- coupon bond is used as the discount rate for a cash flow occurring at the same maturity date. Bond price (or value) determined using spot rates is sometimes referred to as the bond’s ‘no- arbitrage’ value. If a bond’s price differs from its no-arbitrage value, an arbitrage opportunity exists in the absence of transaction costs. PV = PMT (1+Z1 )1 + PMT (1+Z2 )2 + ⋯ + PMT+FV (1+ZN)N where: PMT = coupon payment
LM09 The Term Structure of Interest Rates 2025 Level I Notes © IFT. All rights reserved 4 FV = par value of the bond Z1 = spot rate or yield of zero-coupon bond for period 1 Z2 = spot rate or yield of zero-coupon bond for period 2 ZN = spot rate or yield of zero-coupon bond for period N Example The one-year spot rate is 2%, the two-year spot rate is 3%, and the three-year spot rate is 4%. What is the price of a three-year bond that makes a 5% annual coupon payment? Solution: Think of the bond as a portfolio of three zero-coupon bonds with one, two and three-year maturities with yields of 2%, 3%, and 4% respectively. Draw a timeline for the cash flows. Bond’s no-arbitrage value = 5 1.02 + 5 1.032 + 105 1.043 = 102.96 Is there a single rate (yield) for the bond that equals the present value of cash flows to its purchase price? The YTM can be calculated as: 102.96 = 5 (1 + r) 1 + 5 (1 + r) 2 + 105 (1 + r) 3 Computing for r, we get 3.93%, which is the bond’s yield to maturity. 3. Par and Forward Rates Par Rates from Spot Rates Spot rates are often used to determine 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 maturity dates are commonly used for term structure analysis because they incorporate tax, trading, and other potential distortions associated with actual bonds priced at a discount or premium. Given a sequence of spot rates Z1, Z2,....,Zn, we can use the following equation to calculate a par rate by solving for PMT. 100 = PMT (1+Z1 )1 + PMT (1+Z2 )2 + ⋯ + PMT+100 (1+ZN)N Notice that this equation is similar to the ‘bond pricing using spot rates’ equation, expect PV