## **Learning Outcomes**

This article explains how spot rates, forward rates, and the term structure of interest rates are used to value fixed‑income securities in an arbitrage‑free framework. It clarifies the mathematical relationship between spot and forward curves, shows how to derive forward rates from a given set of spot rates, and applies these rates to price zero‑coupon and coupon‑paying bonds. The article explains how to detect and quantify arbitrage opportunities arising from inconsistencies between market prices and spot‑rate‑implied values, and how to construct trades that exploit such mispricing. It develops the logic of arbitrage‑free valuation using discount factors and extends it to the construction, calibration, and interpretation of binomial interest rate trees. It then uses a calibrated tree to compute bond values by backward induction, determine state‑contingent cash flows, and incorporate risk‑neutral probabilities into pricing. Finally, the article links term structure concepts, tree construction, and arbitrage‑free valuation into a coherent toolkit for tackling CFA Level 2 exam questions on fixed‑income pricing and interest rate modelling.

## **CFA Level 2 Syllabus**

For the CFA Level 2 exam, you are expected to understand how the term structure of interest rates affects fixed-income pricing and risk analysis, with a focus on the following syllabus points:

- Describing the relationship among spot rates, forward rates, and yield to maturity
- Calculating arbitrage-free values of option-free fixed-rate bonds
- Explaining and constructing binomial (interest rate) trees and using them for valuation
- Recognising and exploiting arbitrage opportunities in fixed-income markets
- Comparing present value calculations using spot rate curves and interest rate trees

## **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 spot rate and a forward rate for a given maturity?
2. How do you use spot rates to price a coupon-paying bond with multiple cash flows?
3. True or false? Arbitrage-free valuation means bond prices are constructed to preclude any riskless profit from mispricing.
4. Briefly describe how a binomial interest rate tree is used to value a bond.

## **Introduction**

Understanding how interest rates of different maturities relate is fundamental to fixed income analysis. The term structure—often presented as the yield, spot, or forward curve—determines pricing, risk, and return in bond markets. In this article, you will learn how to use spot and forward rates to value bonds, identify arbitrage opportunities, and apply binomial interest rate trees to ensure arbitrage-free pricing.

> **Key Term: term structure of interest rates**
> The pattern of yields or interest rates across different maturities, reflecting the market's expectations for short-term and long-term rates.

## THE SPOT AND FORWARD RATE RELATIONSHIP

Spot rates are yields on zero-coupon bonds of various maturities. Forward rates are the annualized returns for a loan beginning at a future date, derived from current spot rates. Both rates form the mathematical backbone of fixed-income valuation and arbitrage analysis.

> **Key Term: spot rate**
> The yield-to-maturity on a zero-coupon bond for a specific maturity, used to discount a single future cash flow.
>
> **Key Term: forward rate**
> The implied future interest rate for a given time period, starting at a future date, inferred from current spot rates.

### Calculating Spot and Forward Rates

The price of a zero-coupon bond is determined by discounting its face value using the appropriate spot rate for its maturity.

For a coupon bond, each cash flow is discounted using the corresponding spot rate. Forward rates can be derived algebraically from spot rates by ensuring there is no arbitrage between different investment strategies to a given horizon.

### **Worked Example 1.1**

Consider a one-year spot rate of 2% and a two-year spot rate of 2.8%. What is the implied one-year forward rate one year from today?

> **Answer:**
> Set up equivalence between investing at the two-year spot rate for two years and rolling over a one-year investment to a forward contract for the second year:
> (1 + 0.028)^2 = (1 + 0.02) × (1 + f)
> Solve for the forward rate:
> f = [(1.028^2) / 1.02] – 1 = (1.0578 / 1.02) – 1 ≈ 0.0371, or 3.71%

## ARBITRAGE-FREE VALUATION AND FORWARD RATE ARBITRAGE

Arbitrage-free valuation uses available spot rates to prevent the possibility of riskless profit through inconsistent pricing. A bond is priced by discounting each cash flow using the proper maturity spot rate. If market prices deviate from this model, arbitrage opportunities arise.

> **Key Term: arbitrage-free valuation**
> A pricing approach that ensures no combination of trades can provide a riskless profit due to mispriced securities.

### **Worked Example 1.2**

Suppose a two-year, 5% annual coupon bond has a price of \$97.20. The one-year spot rate is 3%, and the two-year spot rate is 4%. Is there an arbitrage opportunity?

> **Answer:**
> Price using spot rates:
> Year 1 cash flow: \$5 / 1.03 = \$4.854
> Year 2 cash flow: \$105 / (1.04^2) = \$97.061
> Total value = \$4.854 + \$97.061 = \$101.915
> Market price is \$97.20; theoretical price is higher. You could buy at \$97.20 and lock in a profit by selling the replicating set of cash flows (or by shorting the underpriced bond, buying correct synthetic cash flows). This is an arbitrage opportunity.

### **Exam Warning**

> Many exam questions require recognizing mispricing between theoretical and actual bond prices. Always use the correct spot rate for each cash flow—do not use yield to maturity when the spot curve is not flat.

## INTEREST RATE TREES AND BINOMIAL VALUATION

The binomial interest rate tree models various interest rate paths over time, capturing potential future rate changes. Each node represents a possible short-term rate; from any node, the rate can move up or down, leading to a recombining tree structure. This framework allows valuation of bonds—option-free or with embedded options—under risk-neutral assumptions.

> **Key Term: binomial interest rate tree**
> A discrete model showing possible short-term interest rate paths over multiple periods, used for risk-neutral bond valuation.

### Calibrating the Tree

The initial rates in the tree are calculated to align with the discount factors implied by the current spot curve (calibration). The spread between up and down rates at each step reflects interest rate volatility.

### Valuing a Bond by Backward Induction

Starting at the final nodes (bond maturity), work backwards to the present. Each node's bond value is the average of the discounted values at the two subsequent nodes (using the forward rates at that node). For bonds with coupons, add the coupon to the future values before discounting.

### **Worked Example 1.3**

A one-year spot rate is 3%. In a two-period binomial tree, the one-year forward rate one year from now can be either 4% or 2%. You have a two-year, 6% annual coupon, \$100 face value bond. Find its arbitrage-free value.

> **Answer:**
> At maturity (year 2), both upper and lower nodes: value = \$100 + \$6 = \$106
> At year 1 nodes:
>
> - Upper node: Value = \$106 / (1+0.04) = \$101.92
> - Lower node: Value = \$106 / (1+0.02) = \$103.92
>   At present (year 0): Value = [\$6 + (0.5)×\$101.92 + 0.5×\$103.92] / (1+0.03)
>   = [\$6 + 102.92] / 1.03 = \$108.92 / 1.03 ≈ \$105.83
>   The arbitrage-free price is \$105.83.

## PATHWISE AND RISK-NEUTRAL VALUATION

The pathwise approach considers all possible paths rates can take in the tree. For a risk-neutral measure, calculate expected payoffs along each path and discount at the risk-free rate. This method yields the same result as backward induction and ensures arbitrage-free valuation.

> **Key Term: risk-neutral valuation**
> A pricing method where cash flows are discounted using probabilities and risk-free rates, so all investors are indifferent to risk.

## SUMMARY

Spot and forward rates determine the pricing of bonds when applying arbitrage-free valuation. The absence of arbitrage arises from consistent application of spot rates to each cash flow. Interest rate trees, particularly the binomial model, are essential tools for modeling rate uncertainty and valuing bonds when cash flows may vary with rate movements.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Explaining the difference between spot rates and forward rates
- Pricing coupon bonds using individual spot rates for each cash flow
- Recognising and exploiting arbitrage opportunities in bond markets
- Understanding arbitrage-free valuation principles
- Constructing and interpreting binomial interest rate trees
- Applying backward induction and risk-neutral valuation for bond pricing

## **Key Terms and Concepts**

- term structure of interest rates
- spot rate
- forward rate
- arbitrage-free valuation
- binomial interest rate tree
- risk-neutral valuation

Term structure and interest rate trees - Spot and forward rates arbitrage-free valuation