Yield and Yield Spread Measures for Fixed-Rate Bonds

Introduction

This module delves into the nuances of bond yields. We will explore how compounding frequency affects a bond's yield, how to properly evaluate yields for bonds with embedded options (like callable bonds), and how to use yield spreads to compare bonds against benchmarks.

Periodicity and Annualized Yields

To compare bonds with different coupon payment frequencies, we must standardize their yields. This is done by converting them to a common periodicity.

The Importance of Annualizing

Common Periodicity: Most government and corporate bonds pay coupons semiannually (periodicity of 2). The annualized yield for these bonds is called the semiannual bond basis yield or semiannual bond equivalent yield.
Comparability: Annualized yields allow for a direct, apples-to-apples comparison between bonds with different payment schedules (e.g., comparing a semiannual-pay bond to a quarterly-pay bond).

The Relationship Between Compounding and Yield

General Rule: More frequent compounding means a lower annual percentage rate (APR) is needed to achieve the same effective annual return. For example, a 10% annual rate compounded annually is equivalent to a 9.76% annual rate compounded semiannually.

Converting Between Periodicities

We can convert an annual percentage rate (APR) from one compounding frequency to another using the following formula:

Periodicity Conversion Formula

(1 + APRₘ/m)ᵐ = (1 + APRₙ/n)ⁿ

Where:
APRₘ = APR with m compounding periods per year
APRₙ = APR with n compounding periods per year
m = Number of compounding periods for first rate
n = Number of compounding periods for second rate

Example: Converting Yields
A bond has a semiannual bond basis yield (APR with m=2) of 6%. What is its equivalent yield if it were compounded quarterly (n=4)?
1. Set up the equation: (1 + 0.06/2)² = (1 + APR₄/4)⁴
2. Solve for the quarterly rate: (1.03)² = (1 + APR₄/4)⁴ → 1.0609 = (1 + APR₄/4)⁴ → (1.0609)¹/⁴ = 1 + APR₄/4 → 1.014889 = 1 + APR₄/4.
3. Solve for APR₄: APR₄/4 = 0.014889 → APR₄ = 0.05956, or 5.956%.

Other Yield Measures and Conventions

While YTM is the standard measure, other conventions exist, and adjustments are needed for bonds with embedded options.

Yield Measures and Conventions

Measure/Convention	Description
Current Yield	A simple measure calculated as (Annual Coupon Payment / Bond Price). It ignores capital gains/losses and the reinvestment of coupons.
Street Convention Yield	The most common YTM calculation method. It assumes payments are made on their scheduled dates, ignoring weekends and holidays, which simplifies the calculation.
True Yield	A more precise YTM that accounts for the actual payment dates, considering weekends and holidays. It is rarely used in practice.
Government Equivalent Yield	Restates the yield of a corporate bond (which is typically quoted on a 30/360 day-count basis) to be comparable to a government bond (which uses an actual/actual day-count basis).

Evaluating Bonds with Embedded Options

Traditional YTM is not sufficient for bonds with embedded options because it doesn't account for the possibility of early redemption.

Concept for Callable Bonds	Description
Yield-to-Call (YTC)	Calculates the yield assuming the bond will be redeemed by the issuer on a specific call date. An investor should calculate the YTC for all possible call dates.
Yield-to-Worst (YTW)	This is the lowest possible yield an investor can receive from a callable bond. It is the minimum of the Yield-to-Maturity and all of the Yields-to-Call. This is the most conservative and appropriate yield measure for a callable bond.
Option-Adjusted Price & Yield	The price of a callable bond is lower than an identical option-free bond because the call option benefits the issuer. The Option-Adjusted Yield is the yield on the bond after accounting for the effect of the call option. It will be lower than the yield of a non-callable bond.
Value of the Call Option	The value of the embedded call option can be calculated as: Call Option Value Formula Value of Call Option = Price of Option-Free Bond − Price of Callable Bond Note: The call option has value to the issuer, reducing the price investors pay for callable bonds

Yield Spread Measures

A yield spread measures the additional return an investor receives for taking on risks beyond those of a risk-free benchmark rate (like a government bond).

Yield Spread Analysis

YTM Breakdown: YTM = Benchmark Rate + Yield Spread.
The yield spread compensates investors for credit risk, liquidity risk, and other issuer-specific factors. Analyzing spreads helps in assessing a bond's relative value independent of broad macroeconomic interest rate movements.

Types of Yield Spreads

Spread Type	Description	Benchmark Used
G-Spread (Benchmark Spread)	The difference between a bond's YTM and the YTM of a government bond with the same maturity.	Government Bond Yield
I-Spread (Interpolated Spread)	The difference between a bond's YTM and a swap rate of the same currency and tenure.	Swap Rate Curve
Z-Spread (Zero-Volatility Spread)	The constant spread that must be added to each spot rate on the government yield curve to make the present value of a bond's cash flows equal to its market price. It is a more precise measure than the G-spread.	Government Spot Rate Curve
Option-Adjusted Spread (OAS)	The most advanced spread measure, used for bonds with embedded options. It is the Z-spread adjusted to remove the value of the embedded option. Option-Adjusted Spread (OAS) Formula OAS = Z-Spread − Option Value Where: OAS = Option-Adjusted Spread Z-Spread = Zero-Volatility Spread Option Value = Value of embedded options The OAS represents the pure credit and liquidity spread of a bond.	Government Spot Rate Curve (adjusted for interest rate volatility)