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Corporate Finance Unit 4: Bond Valuation and Interest Rates

A bond is a promise to pay fixed coupons and a face value, so its price is the present value of those payments at the market's required yield. This unit covers bond pricing, yields, interest rate risk and what drives interest rates.

Covers: Bond features and terminology, Bond pricing and yield to maturity, Premium, discount and par bonds, Interest rate risk and duration, Yield to call, current yield and zero-coupon bonds, Credit ratings, yield spreads and the term structure

Study guide

A bond is a loan to a government or company. The face (par) value, usually $1,000, is repaid at maturity. The coupon is the regular interest payment, and the coupon rate equals the annual coupon divided by face value. Most U.S. bonds pay coupons twice a year. The indenture spells out terms such as seniority, collateral, sinking funds and call provisions. A callable bond lets the issuer repay early, usually at a call price above par, which hurts investors when rates fall. Bond prices are quoted as a percentage of par.

A bond's price is the PV of its coupons (an annuity) plus the PV of its face value, discounted at the yield to maturity (YTM): P = C × [1 − (1 + r)⁻ⁿ] ÷ r + F ÷ (1 + r)ⁿ. For semiannual bonds, use half the coupon, half the yield and twice the periods. When the YTM equals the coupon rate, the bond sells at par; when YTM is higher, it sells at a discount; when YTM is lower, it sells at a premium. Solving for YTM given a price requires trial and error or a calculator. The current yield, coupon ÷ price, ignores capital gains or losses.

Bond prices move opposite to interest rates. Interest rate risk is larger for bonds with longer maturities and lower coupons, because more of their value comes from distant cash flows. Duration measures this sensitivity: Macaulay duration is the weighted-average time of a bond's cash flows, and modified duration (Macaulay ÷ (1 + y)) estimates the percentage price change for a change in yield: %ΔP ≈ −modified duration × Δy. A zero-coupon bond's duration equals its maturity. Because the price-yield curve is convex, duration overstates price drops and understates price gains for large rate changes.

Yields differ across bonds for several reasons. The real rate and expected inflation set the base level; the Fisher relation links them: (1 + nominal) = (1 + real)(1 + inflation). Default risk adds a premium measured by credit ratings: BBB− and above (S&P) are investment grade, while BB+ and below are speculative or junk bonds. Liquidity, call features and taxes also matter; municipal bond interest is usually exempt from federal tax, so munis offer lower yields, and the taxable-equivalent yield is the muni yield ÷ (1 − t).

The term structure of interest rates, shown by the Treasury yield curve, plots default-free yields against maturity. The expectations theory says long rates reflect expected future short rates, so an upward-sloping curve signals rising expected rates. The liquidity (maturity risk) premium theory adds that investors demand extra return to hold long bonds, which tilts the curve upward even if rates are not expected to change. An inverted curve, with short rates above long rates, has often preceded recessions. Investors also face reinvestment risk: YTM assumes coupons are reinvested at the YTM, so realized returns differ when rates change.

Key ideas

• A bond's price is the PV of its coupons plus the PV of its face value at the market yield.
• Price and yield move in opposite directions.
• YTM above the coupon rate means a discount bond; below means a premium bond.
• Longer maturity and lower coupons mean greater interest rate risk and longer duration.
• Duration approximates price sensitivity; convexity explains why the approximation errs for big moves.
• Yield spreads reflect default risk, liquidity, taxes and embedded options such as calls.
• The yield curve reflects expected future short rates plus a maturity risk premium.

Formulas and rules

Common mistakes

• Using the full annual coupon and yield with semiannual periods instead of halving both.
• Confusing the coupon rate (fixed, based on par) with the yield to maturity (market-determined).
• Thinking a premium bond's YTM exceeds its coupon rate; it is lower.
• Assuming a 1-point yield change moves price by 1%, ignoring maturity and duration.
• Forgetting that YTM assumes coupons are reinvested at the same yield.

Before the test, make sure you can

• Calculate the price of annual-coupon, semiannual-coupon and zero-coupon bonds.
• Calculate YTM, current yield and yield to call.
• Identify whether a bond sells at a premium, discount or par from its coupon and yield.
• Explain how maturity and coupon affect interest rate risk.
• Calculate Macaulay and modified duration and use them to estimate price changes.
• Explain yield spreads using default risk, liquidity, taxes and call features.
• Use the Fisher relation and the expectations theory to find real and forward rates.
• Calculate a bond's realized or horizon return given reinvestment rates.

Words to know

Face (par) value

The principal amount repaid at maturity, usually $1,000 per bond.

Coupon rate

Annual coupon payment divided by face value.

Yield to maturity (YTM)

The discount rate that makes the PV of a bond's payments equal its price.

Current yield

Annual coupon divided by current bond price.

Discount bond

A bond selling below par because its YTM exceeds its coupon rate.

Premium bond

A bond selling above par because its YTM is below its coupon rate.

Zero-coupon bond

A bond with no coupons, sold at a discount and paying face value at maturity.

Callable bond

A bond the issuer can repay before maturity at a set call price.

Yield to call

The yield earned if the bond is called at the first call date at the call price.

Interest rate risk

The risk that bond prices fall when market interest rates rise.

Reinvestment risk

The risk that coupons must be reinvested at lower rates than expected.

Macaulay duration

The weighted-average time until a bond's cash flows are received, weighted by PV.

Modified duration

Macaulay duration ÷ (1 + y); the approximate % price change per 1-point yield change.

Convexity

The curvature of the price-yield relationship; prices rise more than they fall for equal yield moves.

Investment grade

Bonds rated BBB− or higher by S&P (Baa3 or higher by Moody's).

Junk bond

A speculative-grade bond rated BB+ or lower, paying higher yields for higher default risk.

Yield curve

A plot of yields on default-free bonds against time to maturity.

Expectations theory

The view that long-term rates equal the average of expected future short-term rates.

Sample practice questions

30 more questions like these, from level 1 to 5, are in the practice set.