Fixed-Income Bond Valuation: Prices and Yields

Introduction

This module covers the core principles of bond valuation. We will focus on how to calculate a bond's price using discounted cash flow (DCF) analysis, understand the concept of Yield-to-Maturity (YTM), and explore the key relationships between a bond's price and its features. We will also learn about matrix pricing, a technique used to value bonds that do not trade frequently.

Bond Pricing and the Time Value of Money

The fundamental principle of bond valuation is that its price is the present value of all its expected future cash flows.

Bond Pricing with a Market Discount Rate

We use DCF analysis to find the present value (PV) of a bond's future cash flows. The "market discount rate" (or required yield) is the interest rate used to discount these cash flows. This rate reflects the return required by investors for a bond of similar risk and maturity.

Full Bond Price Formula:
PV = (PMT / (1+r)¹) + (PMT / (1+r)²) + ... + (PMT + FV) / (1+r)ᴺ

Where:
PV = Present Value (the bond's price)
PMT = Coupon Payment per period
FV = Future Value (the par or principal value)
r = Market discount rate per period
N = Number of periods until maturity

Example: Calculating a Bond's Price
Consider a 3-year bond with a par value (FV) of $1,000 and a 5% annual coupon rate. The market discount rate (r) for similar bonds is 6%.
1. Identify Cash Flows: The bond will pay $50 (5% of $1,000) each year for 3 years, plus a final principal payment of $1,000 at the end of year 3.
2. Discount Each Cash Flow:
Year 1: $50 / (1.06)¹ = $47.17
Year 2: $50 / (1.06)² = $44.50
Year 3: ($50 + $1,000) / (1.06)³ = $881.50
3. Sum the Present Values: The price of the bond is $47.17 + $44.50 + $881.50 = $973.17.

Price, Par, and Coupon Relationships

A bond's price relative to its par value depends on the relationship between its coupon rate and the market discount rate.

Bond Type	Relationship	Price vs. Par Value
Par Bond	Coupon Rate = Market Discount Rate	Price (PV) = Par Value (FV)
Discount Bond	Coupon Rate < Market Discount Rate	Price (PV) < Par Value (FV)
Premium Bond	Coupon Rate > Market Discount Rate	Price (PV) > Par Value (FV)

Yield-to-Maturity (YTM)

Yield-to-Maturity (YTM) is the total anticipated return on a bond if it is held until it matures. It is the single interest rate (internal rate of return) that equates the present value of a bond's future cash flows to its current market price. For valuation, YTM is often used interchangeably with the market discount rate or required yield.

Three Conditions for Achieving YTM: An investor's actual realized return will equal the calculated YTM only if all three of these conditions are met:

The bond is held to maturity.
The issuer makes all scheduled coupon and principal payments on time and in full.
All coupon payments are reinvested at the original YTM rate.

Flat Price, Accrued Interest, and Full Price

When a bond is traded between its coupon payment dates, the buyer must compensate the seller for the interest that has been earned but not yet paid.

Flat Price (Clean Price): The price of the bond excluding any accrued interest. This is the price that is typically quoted.
Accrued Interest (AI): The proportional share of the next coupon payment that belongs to the seller.
Accrued Interest Formula

AI = (t / T) × PMT

Where:
AI = Accrued Interest
t = Days from last coupon to settlement
T = Total days in coupon period
PMT = Coupon payment
Where: t = number of days from the last coupon date to the settlement date, and T = total number of days in the coupon period.
Full Price (Dirty Price): The total price the buyer pays, which is the sum of the flat price and the accrued interest. This represents the true economic cost.

Relationships Between Bond Prices and Bond Features

Understanding how a bond's price reacts to changes in market conditions and its own characteristics is fundamental to managing fixed-income risk.

Inverse Relationship with YTM: Bond prices and yields have an inverse relationship. When market yields rise, the price of existing bonds falls. When market yields fall, the price of existing bonds rises.
The Coupon Effect: For a given change in yield, bonds with lower coupons are more sensitive to price changes (have higher price volatility) than bonds with higher coupons. Zero-coupon bonds are the most sensitive.
The Maturity Effect: For a given change in yield, longer-term bonds are more sensitive to price changes than shorter-term bonds.
Constant-Yield Price Trajectory (Pull to Par): If a bond's yield remains constant, its price will gradually converge to its par value as the maturity date approaches. The price of a discount bond will rise, and the price of a premium bond will fall.
The Convexity Effect: The relationship between bond price and yield is not a straight line; it is curved (convex). For the same change in yield, the price increase for a decrease in yield is greater than the price decrease for an increase in yield. This is a beneficial characteristic for bondholders.

Matrix Pricing

Matrix pricing is a method used to estimate the market price and YTM of bonds that are not actively traded. It involves using the prices and yields of comparable, liquid bonds to infer the value of the illiquid one.

The Matrix Pricing Process

Identify Comparable Bonds: Find several actively traded bonds with similar characteristics to the bond being valued (e.g., same credit quality, similar maturity, similar coupon).
Determine Yields of Comparables: Find the YTM for each of the comparable bonds.
Interpolate to Estimate YTM: Calculate the average yield of the comparable bonds. If the target bond's maturity falls between the maturities of the comparable bonds, use linear interpolation to estimate its YTM.
Calculate the Price: Use the estimated YTM in the standard DCF bond pricing formula to calculate the price of the illiquid bond.

Use Case for New Issuance: Matrix pricing is also used by investment bankers to estimate the required yield spread over a benchmark rate (like a government bond) for a new bond issuance. This helps determine the appropriate coupon rate to offer investors.