Bonds

Bond Basics

Definition of a Bond

A bond is an agreement between a borrower and a lender that specifies:

  • Issuer: can be a government or corporate entity

  • Face (Par) Value (F): the nominal value of the bond

  • Redemption Value: typically the face value

  • Coupon Rate (C): the interest rate, generally paid annually or semi-annually

  • Coupon Frequency: how often coupon payments occur

  • Maturity Date (T): the date on which the bond will be redeemed

Bond Valuation

Valuing Bonds

The value of financial securities is determined by the present value (PV) of expected future cash flows. The bond value consists of:

  1. Present value of coupon payments

  2. Present value of face value

  3. Yield to Maturity (YTM): the required market interest rate on the bond (y)

Inverse Relationship

Interest rates are inversely related to bond values.

Bond Pricing Equation

Formula for bond valuation:

PV = [ C \frac{1 - (1 + y)^{-T}}{y} + \frac{F}{(1 + y)^{T}} ]

Where,

  • PV = Present Value of the bond

  • C = Annual coupon payment

  • F = Face value of the bond

  • y = Yield to maturity

  • T = Number of years to maturity

Types of Bonds

Pure Discount Bonds

Also known as Zero Coupon Bonds.Features:

  • Do not make periodic interest payments.

  • YTM arises from the difference between purchase price and par value.

  • Cannot sell for more than par value.

  • Example: Treasury Bills.

Valuation of Pure Discount Bonds

Required information for valuing:

  • Time to Maturity (T) = Maturity date - current date

  • Face Value (FV)

  • Discount Rate (y)

Valuation Formula:PV = [ \frac{FV}{(1 + y)^{T}} ]Interpretation: This formula calculates the present value of a bond that pays no interest but is sold at a discount to its face value, which will be paid at maturity.

Coupon Bonds

Make periodic coupon payments alongside the face value at maturity.Payments remain equal each period.Viewed as a combination of an annuity (the periodic payments) and a terminal value (the par value).Coupon payments are generally annual (Europe) or semiannual (US).

Example of Coupon Bonds:Review a bond with a coupon of 6.375% that expires in 10 years:

  • Par Value = 100

  • Annual Coupon Payments = [ 100 \times 0.06375 = 6.375 ]

  • Total value calculation:

    • If required yield (y) is 5%:

    • PV = [ 6.375 \frac{1 - (1 + 0.05)^{-10}}{0.05} + \frac{100}{(1 + 0.05)^{10}} ]

    • This equation computes the total present value of all cash flows (the annual coupons and the face value) to determine the bond's price.

Bond Pricing and Market Behavior

Bond Pricing Examples

Calculate bond value:Given a required yield of 5%, apply the bond pricing formula to get the present value.

Concepts of Bond Pricing

Bond prices and market interest rates move inversely:

  • When coupon rate = y, price = par value.

  • If coupon rate > y, price > par value (premium bond).

  • If coupon rate < y, price < par value (discount bond).

Yield to Maturity (YTM) and Bond Values

Computing Yield to Maturity

YTM is determined using trial and error or financial calculators. It is derived from the current bond price and expected periodic payments.

YTM with Annual Coupons Example

Given a bond with a 10% annual coupon rate, 15 years to maturity, and current price at 92.809, hypothesize:

  • Use the bond pricing formula to compute YTM.

  • Assumed YTM can be adjusted based on calculated present value compared to current price.

YTM with Semiannual Coupons

Review a bond with a 10% coupon rate, selling at 119.793 with 20 years to maturity to calculate YTM:

  • Present Value = [ C \frac{1 - (1 + y/2)^{-2T}}{y/2} + \frac{F}{(1 + y/2)^{2T}} ]Where:

  • C = [ \frac{10}{2} = 5 ]

  • Adjust y accordingly for semiannual periods.

Bond Market Insights

Bond Market Characteristics

  • Transactions mainly occur over-the-counter with electronically connected dealers.

  • There's a large variety of bond issues but typically low daily volume in single issues.

  • Treasury securities are an exception, with more accessibility to up-to-date prices.

Term Structure of Interest Rates

Understanding Yield Curve

The yield curve shows how bond (zero-coupon) yields change with maturity.Commonly assumed as flat, but real-world scenarios show variations.

Spot Rates

Defined as rates determined from YTMs on discount bonds and compared using annualized rates (EAR).Knowledge of today’s yield rates helps in pricing bonds.

Forward Rates

Forward rates are set today for future loans, reflecting investor expectations and risk premiums.

Spot Rates and Bond Value

Calculate the bond value incorporating annual coupon payments, face values, and spot rates across years.For example, calculate a bond value with a 4.5% annual coupon and a maturity of 3 years at varying spot rates.

  • Use the formulae shown to compute values based on different spot rates.