Bond basics and valuation
Bonds are loans that investors make to governments and companies. Valuing them is a direct application of present value.
Bond features
A bond has a face (par) value, usually 1,000 dollars, repaid at maturity; a coupon rate, the annual interest as a percentage of face value; coupon payments, often semi-annual; and a maturity date. A 1,000-dollar bond with a 5% coupon pays 50 dollars a year, often as two 25-dollar payments.
Bond valuation
A bond's price is the present value of its coupons plus the present value of its face value, discounted at the market yield: \(P = C \times \frac{1 - (1+r)^{-n}}{r} + \frac{F}{(1+r)^n}\). A 10-year, 5% annual-coupon, 1,000-dollar bond priced to yield 6% is worth about 926 dollars.
P = C\frac{1-(1+r)^{-n}}{r} + \frac{F}{(1+r)^n}Price and yield move inversely
When market interest rates rise above a bond's coupon rate, its price falls below par (a discount bond); when rates fall below the coupon, its price rises above par (a premium bond). If the yield equals the coupon rate, the bond sells at par.
Yield to maturity
Yield to maturity (YTM) is the discount rate that makes the present value of a bond's cash flows equal its price, the total annual return an investor earns if they buy at today's price and hold to maturity, assuming coupons are reinvested at the same rate. Current yield = annual coupon / price is a simpler but incomplete measure.
Zero-coupon bonds
Zero-coupon bonds pay no coupons; they are sold at a deep discount and pay face value at maturity. Their price is \(\frac{F}{(1+r)^n}\). US Treasury bills are short-term zero-coupon securities. Zeros are especially sensitive to interest rate changes.
Interest rate risk and duration
Bond prices fall when rates rise; the sensitivity depends on maturity and coupon. Longer-maturity and lower-coupon bonds are more sensitive. Duration measures this: a bond with a duration of 7 years loses roughly 7% of its value if yields rise by 1 percentage point. In 2022, rapidly rising rates caused one of the worst years for bond prices in decades.
Reinvestment risk
Reinvestment risk is the risk that coupons or maturing principal must be reinvested at lower rates. Short-term bonds have more reinvestment risk but less price risk; long-term bonds have the reverse. Investors balance these risks according to their goals.
Worked example
A zero-coupon bond with a face value of 1,000 dollars matures in 8 years. If the market yield is 5%, its price is \(\frac{1{,}000}{1.05^8} \approx 677\) dollars. If yields rise to 6%, the price falls to \(\frac{1{,}000}{1.06^8} \approx 627\) dollars, a drop of about 7%.
Key takeaways
Bonds have face value, coupon rate, payment frequency, and maturity. Bond price = PV of coupons + PV of face value at the market yield. Prices and yields move inversely: discount bonds when yields exceed coupons, premium bonds when yields are lower. YTM is the total return held to maturity. Zero-coupon bonds pay only face value. Duration measures interest rate sensitivity; longer, lower-coupon bonds are more sensitive. Investors balance price risk and reinvestment risk.
Flashcard
Amount repaid at maturity
Check your understanding
A bond's coupon rate is 6% but market yields on similar bonds are 8%. The bond will trade: