Introduction

In this chapter, we will explore the concept of spot rates, a fundamental element in understanding interest rates. Spot rates represent the interest rates applicable to a single period in the future. We will delve into what spot rates are, their significance in the financial world, and how to compute discount factors based on spot rates. Understanding spot rates and discount factors is crucial for various financial calculations, including bond pricing and valuation.

Spot Rates

• Definition: Spot rates, also known as zero-coupon rates, are the interest rates applicable to a single period or a specific point in time. They represent the yield on an investment with a maturity of one period.

• Significance: Spot rates serve as building blocks for constructing the entire yield curve, which is essential for pricing various financial instruments such as bonds and derivatives.

• Calculation: The spot rate for a specific period can be calculated using the formula:

$$\text{Spot Rate} =\left(\frac{\text {Face Value}}{\text {Present Value}}\right)^{\frac{1}{n}}-1$$

Where:

• Face Value is the future value of an investment.
• Present Value is the current value of the investment.
• $n$ is the number of periods.

Computing Discount Factors

Discount factors are used to calculate the present value of future cash flows. They represent the factor by which future cash flows should be multiplied to determine their current value. The discount factors can be computed using below formula.

$$\text{Discount Factor} =\frac{1}{(1+\text {Spot Rate})^n}$$

Where:

• Spot Rate is the interest rate for the specific period
• $n$ is the number of periods.

Example: Let’s consider a scenario where the spot rate for a one-year investment period is 5%. We will calculate the discount factor for this period. Solution: Given that:

• Spot Rate $(r)=5 \%$
• Number of Periods $(n)=1$

Applying the formula:

$$\text{Discount Factor} =\frac{1}{(1+0.05)^1}=\frac{1}{1.05} \approx 0.9524$$

The discount factor for a one-year investment at a 5% spot rate is approximately 0.9524.