Introduction

Understanding the relationship between spot rates and Yield-to-Maturity (YTM) is crucial in bond valuation and investment decision-making. Spot rates represent the yields on zero-coupon bonds with specific maturities, while YTM represents the total yield an investor can expect to earn on a bond if held until maturity. In this chapter, we will explore the concept of spot rates, their calculation, and how they are related to YTM. We will also delve into the implications of this relationship for bond pricing and portfolio management.

Spot Rates

Spot rates, also known as zero-coupon rates, are the yields on bonds that do not make periodic interest payments (coupon payments). These bonds are sold at a discount to their face value and pay the full face value at maturity. Spot rates are typically used as building blocks for valuing bonds with various maturities.

• Calculating Spot Rates: Spot rates can be derived from the prices of zero-coupon bonds with different maturities. The formula for calculating the spot rate for a specific maturity, denoted as “r”, is as follows:

$$\text {Price}=\frac{\text {Face Value}}{(1+r)^T}$$

Where:

• Price is the current price of the zero-coupon bond
• FaceValue is the face value of the bond
• $r$ is the spot rate for the given maturity
• $T$ is the time to maturity in years By rearranging this formula and solving for “r”, we can calculate the spot rate for a particular maturity

Relationship between Spot Rates and YTM

The spot rate curve, which plots spot rates against various maturities, plays a pivotal role in understanding the relationship with YTM:
1) YTM as a Weighted Average of Spot Rates: YTM represents the total yield an investor earns on a bond if held until maturity. It can be thought of as a weighted average of spot rates along the yield curve. The weights are determined by the bond’s cash flows (coupon payments and face value) and the time to each cash flow.

$$YTM=\sum\left(\frac{C}{\left(1+r_i\right)^2}\right)+\frac{F}{\left(1+r_n\right)^n}$$

Where:

• $YTM$ is the Yield-to-Maturity
• $C$ is the coupon payment
• $r_i$ is the spot rate for the $i$-th period
• $i$ is the time period
• $F$ is the face value of the bond
• $n$ is the total number of periods

2) Impact on Bond Prices: Changes in spot rates can significantly impact bond prices. When spot rates rise, bond prices generally fall, and vice versa. This inverse relationship between spot rates and bond prices is a fundamental concept in fixed-income markets.

For example, if the current YTM of a bond is 5%, and the spot rate for a specific maturity increases to 6%, the bond’s price will generally decrease because the fixed coupon payment is less attractive relative to the higher prevailing market rate.