Study Notes on Valuing Bonds

6-1 Distinctions Between a Bond’s Coupon Rate and Yield to Maturity

  • Coupon Rate: The annual interest payment made by the bond issuer expressed as a percentage of the bond's face value. For example, a bond with a face value of $1,000 and annual coupon payment of $75 has a coupon rate of 7.5%.
  • Yield to Maturity (YTM): Calculated as the discount rate that equates the present value of a bond’s future cash flows (both coupon payments and face value upon maturity) to its current market price. YTM reflects the total return an investor can expect if the bond is held until maturity.

6-2 Calculation of Market Prices and Yields

  • The relationship between bond prices and yields is inversely correlated. When bond yields increase, bond prices decrease, and vice versa.
  • To Find Market Price of a Bond Given YTM: Use the present value formula:
    extPV=C(1+r)1+C(1+r)2+C+F(1+r)next{PV} = \frac{C}{(1 + r)^1} + \frac{C}{(1 + r)^2} + \frac{C + F}{(1 + r)^n}
    where C is the coupon payment, r is the YTM, F is the face value, and n is the number of years to maturity.
  • To Find YTM Given Price: Solve for r in the above equation given the known PV (the bond price).

6-3 Interest Rate Risk of Bonds

  • Interest Rate Risk: The risk that bond prices will fluctuate due to changes in interest rates. When interest rates rise, bond prices fall, and when they decrease, bond prices increase.
  • Bonds with longer maturities exhibit greater interest rate risk. For instance, a 30-year bond will react more significantly to interest rate changes than a 4-year bond.

6-4 Importance of the Yield Curve

  • Yield Curve: A graphical representation showing the relationship between bond yields and their maturities. Typically, longer-term bonds have higher yields to compensate for greater risk related to interest rates and other factors.
  • Investors analyze the yield curve to assess potential interest rate movements and to make informed investment decisions based on expected future yields.

6-5 Significance of Bond Ratings

  • Bond Ratings: Indicators of creditworthiness assigned by rating agencies (e.g., Moody’s, Standard & Poor’s). High ratings correspond to low default risk, while lower ratings indicate higher risk.
  • Investors demand a higher yield for bonds with lower ratings to compensate for the greater risk of default, referred to as the default premium.

6.1 Bond Pricing

  • A bond represents a loan made by an investor to a borrower (issuer). Bonds are characterized by their maturity and coupon payments.
  • Types of Bonds: Corporate bonds, Treasury bonds (government-issued), and municipal bonds (issued by local governments).

6.2 Cash Flows of a Bond

  • Example Bond: 7.5% coupon, maturing in 2024, face value $1,000.
  • Annual Coupon Payment: 7.5% of $1,000 = $75.
  • Cash Flows: Receive $75 annually until maturity and a repayment of $1,000 (face value) at maturity.

6.3 Valuing Bonds Using Present Value

  • The present value of cash flows can be computed using a discount rate equal to the expected yield. Example calculation using a 3% interest rate:
    • [ ext{PV} = 75 imes rac{1}{(1.03)} + 75 imes rac{1}{(1.03^2)} + 75 imes rac{1}{(1.03^3)} + 1075 imes rac{1}{(1.03^4)} = 1167.27 ]

6.4 Semiannual vs. Annual Payments

  • Many U.S. bonds (including Treasuries) make coupon payments semiannually. For a 7.5% coupon payment, each semiannual payment would be
    752=37.50\frac{75}{2} = 37.50
  • Quoted interest rates used in calculations reflect semiannual rates where necessary: a 3% annual rate translates to 1.5% semiannual.

6.5 Impact of Yield on Bond Prices

  • When interest rates rise, existing bonds pay relatively lower rates compared to new issues, leading to a drop in their price to reach competitive yield levels.
  • Conversely, if interest rates fall, the existing bonds become more valuable leading to a rise in their prices.
  • Example Market Changes: If YTM falls to 2% for the same bond, price increases to $1,209.43 indicates a loss for original investors if they sell at the market rate.

7 Interest Rate Risk Sensitivity

  • The magnitude of price change due to interest rate shifts is greater for longer-term bonds (i.e., those maturing in 30 years) than for shorter-term bonds (i.e., those maturing in 4 years).

8 Summary of Key Points

  • Coupon Rate vs. YTM: Coupon rates are fixed upon issuance and do not change; YTMs fluctuate with prevailing interest rates. Returns are derived from both coupon payments and price changes in the bond.
  • Understanding the yields and market dynamics is essential for informed bond investing and managing associated risks effectively.

9 Calculations and Formulas for Bonds

  • Key Formulae:
    1. Pricing a Bond: extBondprice=PV(extcoupons)+PV(extfacevalue)ext{Bond price} = PV( ext{coupons}) + PV( ext{face value})
    2. Investment Return: extRateofReturn=extCouponincome+extPricechangeextInvestmentext{Rate of Return} = \frac{ ext{Coupon income} + ext{Price change}}{ ext{Investment}}

10 Problems and Exercises

  • Review practice questions related to bond yields, pricing, and interest rate risk to solidify understanding of concepts covered in this chapter.