Introduction

Day-count conventions play a crucial role in bond pricing. They determine how the interest accrues between coupon payments and are vital for calculating accrued interest accurately. In this chapter, we will delve into the common day-count conventions used in bond pricing. Additionally, we’ll explore how these conventions can be explained and applied using arbitrage arguments.

Common Day-Count Conventions

Day-count conventions dictate how the fraction of a year is calculated for interest accrual. There are several conventions, but we will focus on three common ones:

1) Actual/Actual (A/A): This convention is considered the most precise. It calculates the actual number of days between two coupon payments and divides it by the actual number of days in the year. The accrued interest under this convention is given as follows.

$$\text{Accrued Interest} = \left(\frac{\text{Actual Days}}{\text{Actual Days in Year}}\right) \times \text{Coupon Payment}$$

2) 30/360: In this convention, each month is assumed to have 30 days, and the year is assumed to have 360 days. It simplifies calculations. The accrued interest under this convention is given as follows.

$$\text{Accrued Interest} = \left(\frac{\text{Days between Payments}}{360}\right) \times \text{Coupon Payment}$$

3) Actual/360 (A/360): Similar to A/A, but it assumes a 360-day year for simplicity. The accrued interest under this convention is given as follows.

$$\text{Accrued Interest} = \left(\frac{\text{Actual Days}}{360}\right) \times \text{Coupon Payment}$$

Using Arbitrage Arguments

Now, let’s explore how day-count conventions can be explained using arbitrage arguments. Arbitrage ensures that the bond’s price remains consistent in the market. Suppose we have two bonds, Bond A and Bond B, with identical characteristics except for the day-count convention.

Bond A (A/A): This bond calculates accrued interest precisely based on actual days.

Bond B (30/360): This bond assumes a fixed 30-day month and 360-day year, simplifying calculations.

To demonstrate the arbitrage argument, let’s consider a scenario where Bond A and Bond B have different accrued interest amounts. An arbitrage opportunity arises for traders who can profit from this discrepancy.

Application to Bond Pricing

Day-count conventions are crucial in bond pricing because they affect the accrued interest portion. When calculating a bond’s clean price (the price without accrued interest), it is essential to use the correct day-count convention to determine the accrued interest.

Here’s how it works in practice:

1) Calculate the accrued interest using the appropriate day-count convention as shown in the formulas above.

2) Add the accrued interest to the quoted clean price of the bond to obtain the dirty price (the price with accrued interest).

3) The dirty price is the actual price at which the bond will be traded.