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Valuation and term structure - Yield curves and bootstrappin...

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Learning Outcomes

After reading this article, you will be able to explain the structure and uses of yield curves, calculate spot rates and forward rates, and understand how to construct a zero-coupon yield curve using the bootstrapping method. You will also identify the relationship between bond prices, discount factors, and the term structure, and apply these concepts to valuation and exam-style problems.

CFA Level 1 Syllabus

For CFA Level 1, you are required to understand the construction and interpretation of yield curves, bootstrapping, and their role in bond pricing and term structure analysis. The following points highlight the exam’s focus for your revision:

  • Describe yield curve shapes, uses, and implications for interest rates and bond markets
  • Explain the calculation and application of spot rates and forward rates
  • Construct a zero-coupon yield curve using observed bond prices via bootstrapping
  • Relate the yield curve to valuation of fixed income instruments
  • Use bootstrapped spot rates for discounting and valuation

Test Your Knowledge

Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision.

  1. What is the difference between a par yield curve, a spot rate curve, and a forward rate curve?
  2. If a 1-year zero-coupon bond yields 3%, and a 2-year zero-coupon bond yields 3.5%, what is the implied 1-year forward rate for year two?
  3. Describe the bootstrapping process for deriving spot rates from coupon bond prices.
  4. Why are spot rates necessary for accurate bond valuation?

Introduction

The term structure of interest rates, commonly visualized as a yield curve, is a foundational concept in fixed-income valuation. Yield curves summarize how yields vary across maturities and help analyze and value bonds. For pricing and risk management, we often need spot rates—the yields on theoretical zero-coupon securities—for different maturities. Because actual zero-coupon bonds are not always available at every maturity, financial analysts use “bootstrapping” to infer spot rates from observed coupon bond prices. This article explains the construction and interpretation of yield curves, the mechanics of bootstrapping, and the application of spot and forward rates in bond valuation.

Key Term: yield curve
A graphical representation that plots annual yields (interest rates) of bonds with identical credit quality but different maturities, showing the term structure of interest rates.

Key Term: spot rate
The yield or discount rate for a single cash flow (zero-coupon bond) occurring at a specific future date. Also called the zero-coupon rate.

Key Term: forward rate
The implied future interest rate for a period starting at a future date, derived from current spot rates.

Key Term: bootstrapping
A step-by-step method for calculating spot rates from the prices of coupon-bearing and zero-coupon bonds by sequentially solving for each unknown discount rate.

Yield Curves and Their Interpretation

A yield curve plots the yields of bonds with equal risk but different maturities. The most common types are:

  • Par Yield Curve: Plots yields-to-maturity for coupon-paying bonds trading at par.
  • Spot (Zero-Coupon) Yield Curve: Plots spot rates, which are the yields on zero-coupon bonds for each maturity.
  • Forward Rate Curve: Shows implied future rates between periods, derived mathematically from spot rates.

Upward-sloping yield curves typically signal higher future rates or inflation expectations. Downward-sloping (inverted) curves may signal market expectations of lower rates or economic slowdown. Flat curves indicate little expectation of change in rates.

Bond valuation often requires spot rates because each cash flow may be discounted at a rate corresponding to its unique maturity.

Bootstrapping the Zero-Coupon Curve

Zero-coupon bonds are not always available for all maturities, and most coupon bonds' yields combine returns for a series of future dates. Bootstrapping is used to derive the spot rates for all periods.

The main principle: Use prices of bonds with increasing maturities, and sequentially solve for each new spot rate.

Key Term: discount factor
The present value today of $1 to be received at a specific future time, found by discounting by the relevant spot rate.

Worked Example 1.1

Suppose you have the following bonds (risk-free, annual coupons, price per $100 face value):

  • 1-year zero-coupon: Price = $97.09
  • 2-year 5% coupon: Price = $98.40
  • 3-year 6% coupon: Price = $99.05

Find the 1-, 2-, and 3-year spot rates.

Answer:

  • 1-year: $97.09 = $100 / (1 + s₁); s₁ = 2.999%
  • 2-year: $98.40 = $5/(1 + s₁) + $105/(1 + s₂)²; with s₁ known, solve for s₂: $98.40 = $5/1.02999 + $105/(1 + s₂)² \98.40 - $4.85 = $105/(1 + s₂)² ⇒ \93.55 = $105/(1 + s₂)² (1 + s₂)² = $105/$93.55 = 1.12289,so, so s₂ = (1.12289)^0.5 - 1 = 5.938%
  • 3-year: Use known s₁ and s₂ and solve for s₃ using the 3-year bond price, discounting all cash flows separately.

Exam Warning

A common exam error is to use the yield-to-maturity for all cash flows. Remember that for a true present value calculation, each cash flow should be discounted at its corresponding spot rate, not a single average yield.

Spot Rates and Bond Valuation

Plain-vanilla coupon bonds are valued by discounting each cash flow at the appropriate spot rate. The relation between bond prices and spot rates is:

P=∑i=1NCFi(1+si)iP = \sum_{i=1}^{N} \frac{CF_i}{(1 + s_i)^i}

where CFiCF_i is the cash flow at time ii, discounted by the spot rate for that period.

Worked Example 1.2

A 2-year bond pays $4 per year and $100 at maturity. Spot rates: s₁ = 3%, s₂ = 4%. What is the price?

Answer:
P = 4 / (1.03)^1 + 104 / (1.04)^2 = 3.88 + 96.29 = \100.17

Forward Rates

Forward rates represent the market's implied future yields and are calculated from spot rates as follows:

(1+sn)n=(1+sn−1)n−1×(1+fn−1,n)(1 + s_n)^n = (1 + s_{n-1})^{n-1} \times (1 + f_{n-1,n})

The 1-year forward rate beginning in year 2 (f1,2f_{1,2}):

(1+s2)2=(1+s1)×(1+f1,2)(1 + s_2)^2 = (1 + s_1) \times (1 + f_{1,2})

Worked Example 1.3

If s₁ = 2%, s₂ = 3%, what is the implied 1-year forward rate for year 2?

Answer:
(1.03)2=1.02×(1+f)(1.03)^2 = 1.02 \times (1 + f) ⇒ $1.0609 / 1.02 = 1.0401⇒⇒f = 4.01%$

Bootstrapping and the Exam

On the exam, you may be given a sequence of bond prices and asked to derive spot rates by bootstrapping, or asked to use bootstrapped spot rates to value another bond. Focus on the mechanics of solving for each unknown term, step by step.

Revision Tip

Practice setting up the bootstrapping equations for each maturity, writing out each cash flow’s present value equation before solving numerically.

Summary

Spot rates are the building blocks for bond valuation and yield curve construction. Bootstrapping derives the zero-coupon yield curve from market bond prices and ensures accurate pricing and consistency with no-arbitrage. Forward rates, derived from spot rates, reveal the implied path of future interest rates and are testable for CFA Level 1.

Key Point Checklist

This article has covered the following key knowledge points:

  • Distinguish par, spot, and forward yield curves and interpret their shapes
  • Apply spot rates for accurate bond cash flow discounting
  • Calculate missing spot rates from coupon bond prices by bootstrapping
  • Use bootstrapped spot rates to construct the zero-coupon yield curve
  • Compute forward rates from given spot rates and explain their meaning

Key Terms and Concepts

  • yield curve
  • spot rate
  • forward rate
  • bootstrapping
  • discount factor

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