Introduction

In this chapter, we delve into the crucial relationship between key rates, partial 01s (often denoted as PV01), and forward-bucket 01s. Understanding how these elements interact is fundamental in risk management and fixed income analysis. We will explore how shifts in interest rates impact these metrics and learn to calculate forward-bucket 01 for rate shifts across one or more buckets. This knowledge is instrumental in managing interest rate risk in a portfolio effectively.

Key Rates and Their Significance

Key rates represent specific points on the yield curve that are commonly used as reference rates. They are critical in assessing how interest rate changes affect the value of fixed income securities. Key rates often correspond to the maturities of government bonds, such as 1-year, 2-year, 5-year, and 10-year rates.

Partial 01 (PV01)

Partial 01, also known as DV01 (Dollar Value of 01), measures the change in the price of a fixed income security for a 1 basis point $(0.01 \%)$ change in yield. It represents the sensitivity of a security’s price to small interest rate movements. The formula for PV01 is:

$$PV01=-\frac{\Delta P}{\Delta Y}$$

Where:

• $PV01$ = Partial 01
• $\Delta P$ = Change in the price of the security
• $\Delta Y$ = Change in yield (expressed in decimal form)

Key Rate Exposure and PV01

Key rate exposure is a concept that ties key rates and PV01 together. It measures how much a security’s price will change for a 1 basis point change in a specific key rate. Key rate exposures are calculated for each key rate, allowing us to assess a security’s sensitivity to changes in different segments of the yield curve.

$$\text{Key Rate Exposure} = -PV01_i$$

Where:

• Key Rate Exposure = Sensitivity of the security’s price to a specific key rate
• $PV01_i$ = Partial 01 for the corresponding key rate

Forward-Bucket 01 (FB01)

Forward-bucket 01, also known as bucket-shift PV01, extends the concept of PV01 to measure the sensitivity of a security’s price to a shift in rates within a specific future time period or bucket. It helps us understand how the security’s value will respond to rate shifts that are anticipated to occur in the future. The formula for FB01 is:

$$FB01=\sum_i\left( \text{Key Rate Exposure}_i \times \text{Shift in Rates}_i\right)$$

Where:

• FB01 = Forward-Bucket 01
• Key Rate Exposure$_i$ = Key Rate Exposure for each key rate within the bucket
• Shift in Rates$_i$ = Expected shift in rates for each key rate within the bucket

Example: Let’s consider a 5-year bond with the following key rate exposures for a rate shift in the 1year, 2-year, and 5-year key rates:

• Key Rate Exposure for 1-year rate (Key Rate Exposure$_1$) = -0.025
• Key Rate Exposure for 2-year rate (Key Rate Exposure$_2$) = -0.045
• Key Rate Exposure for 5-year rate (Key Rate Exposure$_5$) = -0.060

If we expect a 1 basis point shift in these rates within the next year:

• Expected Shift in 1-year rate (Shift in Rates$_1$) = 0.01
• Expected Shift in 2-year rate (Shift in Rates$_2) = 0.01
• Expected Shift in 5-year rate (Shift in Rates$_5$) = 0.01

Using the FB01 formula, we can calculate the bond’s sensitivity to rate shifts within the next year.

$$FB01=(-0.025 \times 0.01)+(-0.045 \times 0.01)+(-0.060 \times 0.01) = -0.0013$$

This means that for a 1 basis point shift in these rates, the bond’s value is expected to decrease by USD 0.0013.