Bond Prices and Yields

Bond Prices and Yields

10.1 Bond Characteristics

  • Bond

    • A security that obligates the issuer to make payments to the holder over time.

  • Face Value, Par Value

    • The payment to the bondholder at maturity of the bond.

  • Coupon Rate

    • The bond’s annual interest payment per dollar of par value.

  • Zero-Coupon Bond

    • A type of bond that pays no coupons, sells at a discount, and provides only the payment of par value at maturity.

10.2 Bond Pricing

  • General Bond Pricing Formula

    • Bond value = Present value of coupons + Present value of par value.

  • Detailed Equation

    • Bond value=∑C=1E(Coupon(1+r)t+Par value(1+r)T)Bond\,value = \sum_{C=1}^{E} \left( \frac{Coupon}{(1+r)^t} + \frac{Par\,value}{(1+r)^T} \right)

    • Where:

    • TT = Maturity date

    • rr = Discount rate

  • Alternate Formulation

    • Bond price=Coupon×1−(1+r)−Tr+Par value×1(1+r)TBond\,price = Coupon \times \frac{1 - (1+r)^{-T}}{r} + Par\,value \times \frac{1}{(1+r)^{T}}

    • This can also be expressed as:

    • Bond price=Coupon×Annuity factor(r,T)+Par value×PV factor(r,T)Bond\,price = Coupon \times Annuity\,factor(r,T) + Par\,value \times PV\,factor(r,T)

  • Interest Rate Impact

    • Prices fall as market interest rates rise.

    • Interest rate fluctuations are a primary source of bond market risk.

    • Bonds with longer maturities are more sensitive to fluctuations in interest rates.

10.3 Inverse Relationship between Bond Prices and Yields

  • Graphical Representation

    • A figure showing the inverse relationship between bond prices and yields, with bond prices decreasing as interest rates rise.

  • Quantitative Data

    • Example: A bond price chart indicating values at various interest rates including points like $3,500 to $0$ at rates from $0\%$ to $20\%$.

10.4 Bond Prices at Different Interest Rates

  • Table of Bond Prices

    • Bond Prices at Given Market Interest Rate

    • For different maturities (1 year, 10 years, 20 years, 30 years) and various interest rates (2%, 4%, 6%, 8%, and 10%):

      • 1 year:

      • 2%: $1,695.22

      • 10%: $981.41

      • 10 years:

      • 2%: $1,541.37

      • 10%: $875.38

      • 20 years:

      • 2%: $1,985.04

      • 10%: $828.41

      • 30 years:

      • 10%: $810.71

10.5 Bond Yields

  • Yield Definitions

    • Yield to Maturity (YTM)

    • The discount rate that makes the present value of the bond's payments equal to its price.

    • Current Yield

    • Annual coupon payment divided by the bond price.

  • Bond Classifications

    • Premium Bonds

    • Bonds selling above par value; the coupon rate is greater than the yield to maturity.

    • Discount Bonds

    • Bonds selling below par value; the coupon rate is less than the yield to maturity.

10.6 Bond Price Example

  • Example Calculation

    • Consider a five-year, $1,000 par value bond with a 5% coupon rate and semiannual coupons. If the yield to maturity is 6.30% (as expressed as an APR with semiannual compounding), the bond price is calculated as:

    • Financial Calculator Inputs:

      • N = 3.15

      • I/Y = 25

      • PMT = 1,000

      • FV = (Future Value)

    • Result: $PV$ (Present Value) = -$944.9757.

10.7 Bond Prices Over Time

  • Comparison of Yields

    • Yield to Maturity vs. Holding Period Return (HPR)

    • Yield to maturity measures average return on investment if the investment is held until the bond matures.

    • HPR represents the return over a particular investment period; it depends on the market price at the end of that period.

10.8 Zero-Coupon Bonds

  • Definition and Valuation

    • Zero-Coupon Bond

    • A bond that carries no coupons but provides all returns in the form of price appreciation.

    • Valuation Formula

    • The value of a zero-coupon bond is formulated as:

      • Zero−Coupon Bond Value=1(1+r)TZero-Coupon\,Bond\,Value = \frac{1}{(1+r)^T}

10.9 Price Dynamics Over Time

  • Graphical Representation

    • Figure illustrating the price of a 30-Year Zero-Coupon Bond over time, assuming a yield to maturity of 10%. Prices range from $1,000 to $0$ spanning the bond's life.

10.10 Default Risk and Bond Pricing

  • Bond Ratings

    • Investment Grade Bond

    • Rated BBB and above by S&P or Baa and above by Moody’s, indicating lower default risk.

    • Speculative Grade or Junk Bond

    • Rated BB or lower by S&P, Ba or lower by Moody’s, or unrated indicating higher risk.

10.11 Bond Rating Classes

  • Classification Overview

    • Bond ratings vary from high quality (AAA by S&P, Aaa by Moody’s) to very speculative (CCC, D ratings).

    • Both agencies occasionally employ adjustments to classifications, where S&P uses plus and minus signs and Moody's incorporates numerals to designate strength within ratings.

10.12 Yield Spreads between Corporate and Treasury Bonds

  • Yield Spread Data

    • A chart illustrating yield spreads between Aaa-rated, Baa-rated, and B-rated corporate bonds as compared to the 10-Year Treasury bonds from 1997 to 2012, showing variations in spreads over the years.

10.13 The Yield Curve

  • Definition

    • The yield curve graphically represents the yield to maturity as a function of the term to maturity, illustrating the relationship between these financial metrics.