Bonds and Fixed Income Analysis Notes

Introduction to Bonds

Bonds, classified under debt securities, are commonly known as fixed-income securities. They represent a loan from the bondholder to the issuer, wherein the issuer agrees to pay the bondholder specified coupon payments at predetermined intervals and to repay the principal (face value) upon maturity. The coupon rate dictates the interest payment owed to bondholders. Historically, bonds were issued with physical coupons that needed to be clipped and presented by the investor to claim their interest payments.

Example of Bonds

Consider a bond with a face value of $1,000$, a coupon rate of $8 ext{ extperthousand}$, and a term to maturity of 30 years. This bond is sold for $1,000$ and would generate payments of $80$ annually for 30 years:

• Year 0: -$1,000$

• Year 1: +$80$

- Year 2: +$80$

• …

• Year 30: +$1,080$

In a different structure, semiannual payments might be structured as:

• Year 0: -$1,000$

• Every 0.5 years: +$40$

• Year 30: +$1,040$

Zero-Coupon Bonds

Zero-coupon bonds, unlike regular bonds, do not provide periodic coupon payments; rather, they pay the face value upon maturity. An example would be a zero-coupon bond with a price $P$ being purchased at a discount to face value $F$:

• Year 0: -$P$

• Year 30: +$F$

These types of bonds are sold at a price that is lower than the face value to account for the absence of regular cash flows.

Market Overview

Bonds are primarily traded in the over-the-counter (OTC) market, offering less liquidity compared to equities. As of July 2024, the global bond market held a valuation of approximately $140.7$ trillion, surpassing the public equity market's valuation of around $115$ trillion. Bond issuance is typically undertaken by governments and corporations seeking to finance their operations and investments.

Yield and Pricing of Bonds

Key Terminology

• Nominal Yield: The coupon rate of the bond.

• Current Yield: Calculated as the coupon payment divided by the bond's price.

• Yield to Maturity (YTM): The overall return expected if the bond is held until maturity.

• Yield to Call (YTC): The yield if the bond is called prior to maturity.

• Realized Yield: Equal to YTM if all coupon payments are reinvested at the bond's yield rate.

Interest Rates

Interest rates directly correlate with the coupon rates and discount rates of bonds. The nominal interest rate reflects the growth rate of money, while the real interest rate accounts for inflation:
$1 + rr = (1 + r)(1 + i)$
Where $rr$ is the real interest rate, $r$ is the nominal rate, and $i$ is the inflation rate. For low inflation scenarios, we can approximate:
$rr ext{ extapprox} r - i$

Bond Pricing Example

Consider a one-year zero-coupon bond with a face value of $100,000$ selling at $96,618.36$. The YTM can be computed as:

$r = rac{F}{P} - 1 = rac{100,000}{96,618.36} - 1 ext{ resulting in a yield of } r = 3.5 ext{ extpercent}$

For an n-year zero-coupon bond, the price is given by:
$P = rac{F}{(1 + YTM)^n}$

To calculate YTM:
$YTM = rac{F}{P}^{ rac{1}{n}} - 1$

Coupon Bonds

Unlike zero-coupon bonds, coupon bonds provide periodic interest payments and principal at maturity. The price of a coupon bond can be evaluated through:
$P = C imes rac{1 - (1 + y)^{-N}}{y} + rac{F}{(1 + y)^N}$
Where $P$ is the bond price, $C$ is the coupon payment, $F$ is the face value, $y$ is the yield to maturity, and $N$ represents the number of periods until maturity.

Example Calculation

For a three-year, $1,000$ coupon bond with $10 ext{ extpercent}$ annual coupons:

• Cash Flow at Year 0: -$1,000$

• Year 1: +$100$

• Year 2: +$100$

• Year 3: +$1,100$ [Final payment includes face value]

Using the present value calculation of these cash flows will provide a price for the bond, which could be replicated with zero-coupon bonds as earlier demonstrated.

Relationship Between Bond Prices and Yields

Bond prices generally exhibit an inverse relationship with yields. Bond pricing strategies must be aware of current yield trends—whether approximate for coupon frequency or adjusted for increasing interest rates as observed in current economic climates.