Bonds: Features and Prices

• Bonds are long-term debt securities issued by corporations and by federal, state, and local governments
• A bond is normally an interest-only loan, where the borrower pays interest periodically and repays the principal at the end of the loan term
• It is commonly called level-coupon bond.

Bond Features

• Coupon: The stated interest payment made on a bond is $120
• Coupon rate: This is the annual coupon divided by the face value of a bond, or 12%
• Face value (par value): The principal amount of a bond that is repaid at the end of the term is $1,000
• Maturity date: The date on which the principal amount of a bond is paid and the day of the last coupon payment
• Time to maturity: Number of years until the face value is paid; as a bond ages, the time to maturity decreases

Bond Prices

• Relationship between coupon rate and yield to maturity (YTM) exists
• If YTM equals the coupon rate, then the bond price equals the face value
• If YTM is greater than the coupon rate, then the bond price is less than the face value, and the bond is selling at a discount (discount bond)
• If YTM is less than the coupon rate, then the bond price is greater than the face value, and the bond is selling at a premium (premium bond)
• Par value bonds have a market price equal to the face value and a yield to maturity equal to the coupon rate
• Discount bonds sell below par value and have a yield to maturity exceeding the coupon rate
• Premium bonds sell above par value and have a coupon rate exceeding the yield to maturity

Interest Rate Risk

• Interest rate risk arises from fluctuating interest rates
• Sensitivity of a bond price to interest rate changes depends on time to maturity and the coupon rate
• The longer the time to maturity, the greater the interest rate risk, all else being equal
• The lower the coupon rate, the greater the interest rate risk, all else being equal

Duration Introduction

• Definition: measurement of the effective maturity of a bond
• Duration is shorter than maturity for all bonds except zero-coupon bonds and equal to maturity for zero coupon bonds
• More complete measure of interest rate sensitivity than maturity, since it considers the timing of all cash flows

Duration Explained

• Technically weighted-average time to maturity using the relative present values of cash flows
• Measures the time required to recover the initial investment
• Cash flows received before the loan's duration reflect the initial investment recovery
• Cash flows received after the loan's duration but before its maturity are profits

Duration Formula

• Key assumptions: yield curve is flat, yield curve shifts in a parallel fashion when rates change, borrower will pay all interest and principal as promised (no default risk)

Duration Definition

• Weighted-average time to maturity on a security
• The interest elasticity of a security's price to small interest rate changes

Duration Features

• Duration increases with the maturity of a fixed-income security, but at a decreasing rate
• Duration decreases as the yield on a security increases
• Duration decreases as the coupon or interest payment increases
• Holding maturity constant, a bond's duration is higher when the coupon rate is lower
• Holding the coupon rate constant, a bond's duration generally increases with its time to maturity

Duration Considerations

• Holding other factors constant, the duration of a coupon bond is higher when the bonds' yield to maturity is lower

Risk Management with Duration

• Duration equals the maturity of an immunized security
• Duration gap is used by FIs to measure and manage interest rate risk of a balance sheet

Duration of a Zero-Coupon Bond

• Zero-coupon bonds sell at a discount, pay face value at maturity, and do not have coupon payments
• The price an investor pays equals the present value of the future face value
• Duration of a zero-coupon bond equals its maturity
• Duration and maturity are only equal for zero-coupon bonds; for bonds with cash flows prior to maturity, duration is less than maturity

Duration of a Perpetuity

• Perpetuity: security with no maturity date and pays a fixed cash flow regularly, like preferred stock
• While the maturity is infinite, the duration of a perpetuity is finite

Features of Duration

• Positive relationship between duration and maturity, but duration increases at a decreasing rate
• Negative relationship between duration and yield; higher yields discount later cash flows more heavily
• Inverse relationship between duration and coupon rate; higher coupon payments lead to lower duration

Duration and Interest Rate Risk

• The duration is a measure of the average life and an asset or liability
• Duration is also a direct measure of the interest rate sensitivity, or elasticity, of an asset or liability
• The larger the numerical value of duration, the more sensitive the price of that asset or liability is to changes or shocks in interest rates.

Managing Interest Rate Risk with Duration

• Duration is a measure for managing interest rate risk exposure
• Duration helps to reduce or eliminate interest rate risk on a balance sheet