Valuation of Financial Assets: Bonds

Bond Terminology

  • A bond is a debt instrument where an issuer promises to pay periodic interest payments and repay the principal at maturity.
  • Bond Certificate: States the contractual terms.
  • Face Value: The principal amount repaid at maturity.
  • Maturity Date: The date when the face value is repaid.
  • Term: The time remaining until the maturity date.
  • Coupon: The promised interest payments, paid periodically until maturity (usually semi-annually).
  • Coupon Rate: Determines the amount of each coupon payment; expressed as an APR and set by the issuer.
  • Coupon Payment: Calculated as CPN=Coupon Rate×Face ValueNumber of Coupon Payments per YearCPN = \frac{Coupon \ Rate \times Face \ Value}{Number \ of \ Coupon \ Payments \ per \ Year}

Zero-Coupon Bonds

  • Only two cash flows:
    • Bond's market price at purchase.
    • Bond's face value at maturity.
  • Compensation is the difference between the initial price and face value.

Yield to Maturity of a Zero-Coupon Bond

  • The yield to maturity (YTM) is the rate of return earned if the bond is held until maturity.
  • It's the discount rate that sets the present value of promised bond payments equal to the current market price.
  • The per-period yield to maturity of an n-period zero-coupon bond is calculated as: YTMn=(Face ValuePrice)1n−1YTM_n = (\frac{Face \ Value}{Price})^{\frac{1}{n}} - 1
Lecture Example 1
  • Maturity: 1 year; Price: $96.62; YTM1=(10096.62)1−1=3.50%YTM_1 = (\frac{100}{96.62})^1 - 1 = 3.50\%
  • Maturity: 2 years; Price: $92.45; YTM2=(10092.45)12−1=4.00%YTM_2 = (\frac{100}{92.45})^{\frac{1}{2}} - 1 = 4.00\%
  • Maturity: 3 years; Price: $87.63; YTM3=(10087.63)13−1=4.50%YTM_3 = (\frac{100}{87.63})^{\frac{1}{3}} - 1 = 4.50\%
  • Maturity: 4 years; Price: $83.06; YTM4=(10083.06)14−1=4.75%YTM_4 = (\frac{100}{83.06})^{\frac{1}{4}} - 1 = 4.75\%

Computing the Price of a Zero-Coupon Bond

  • The price PP of an n-period zero-coupon bond with a per-period yield to maturity of YTM<em>nYTM<em>n is calculated as: P=Face Value(1+YTM</em>n)nP = \frac{Face \ Value}{(1 + YTM</em>n)^n}
  • The price of a zero-coupon bond is the present value of a single cash flow.
Lecture Example 2
  • Maturity: 1 year; Yield to maturity: 3.50%; P = \frac{100}{1.035} = $96.62
  • Maturity: 2 years; Yield to maturity: 4.00%; P = \frac{100}{1.04^2} = $92.45
  • Maturity: 3 years; Yield to maturity: 4.50%; P = \frac{100}{1.045^3} = $87.63
  • Maturity: 4 years; Yield to maturity: 4.75%; P = \frac{100}{1.0475^4} = $83.06

Coupon Bonds

  • Pay face value at maturity.
  • Pay regular coupon interest payments.

The Cash Flows of a Coupon Bond

Lecture Example 3
  • Australian Government bonds issued in May 2025, maturing in May 2030, with a $1,000 face value and a 2.2% coupon rate with semi-annual coupons.
  • Face value = $1,000, paid at maturity in May 2030.
  • CPN = \frac{0.022 \times 1,000}{2} = $11, paid every six months.

Bond Price and Yield to Maturity

  • The price PP of a coupon bond is the present value of the bond’s cash flows:
    • Where the cash flows are discounted by the bond’s yield to maturity yy.
  • P=PV<em>coupons+PV</em>face value=PV<em>annuity+PV</em>single cash flowP = PV<em>{coupons} + PV</em>{face \ value} = PV<em>{annuity} + PV</em>{single \ cash \ flow}
  • P=CPN×1y(1−1(1+y)n)+FV(1+y)nP = CPN \times {\frac{1}{y}(1 - \frac{1}{(1 + y)^n})} + \frac{FV}{(1 + y)^n}
Lecture Example 4

  • Five-year, $1,000 bond with a 2.2% coupon rate and semi-annual coupons.

  • The bond is trading for a price of $963.11.

  • Cash flows: an annuity of 10 payments of $11, paid every six months, and a single cash flow of $1,000 in five years (10 six-month periods).

  • Compute the yield to maturity yy by solving the equation: P=11×1y(1−1(1+y)10)+1,000(1+y)10P = 11 \times {\frac{1}{y}(1 - \frac{1}{(1 + y)^{10}})} + \frac{1,000}{(1 + y)^{10}}

  • Can be solved using the RATE function in Excel:

    • Excel formula: = RATE(nper, pmt, pv, fv) = RATE(10, 11, -963.11, 1000) ≈ 1.50% per six months or 1.50% × 2 = 3.00% APR with semi-annual compounding.

  • If the bond’s yield to maturity decreases to 2% APR with semi-annual compounding (1% per six months):

    • P = 11 \times {\frac{1}{0.01}(1 - \frac{1}{1.01^{10}})} + \frac{1,000}{1.01^{10}} = $1,009.47
    • Excel formula: = PV(rate, nper, pmt, fv) = PV(0.01, 10, -11, -1000)

Why Bond Prices Change

  • Zero-coupon bonds always trade at a discount (to face value).
  • Coupon bonds may trade at a discount or at a premium (a price greater than their face value).
  • Most issuers of coupon bonds choose a coupon rate so that the bonds will initially trade at, or very close to, par (i.e. at the bond’s face value).
  • After the issue date, the market price of a bond generally changes over time for two reasons:
    1. As time passes, the bond gets closer to its maturity date. Holding fixed the bond’s yield to maturity, the present value of the bond’s remaining cash flows changes as the time to maturity decreases.
    2. At any point in time, changes in market interest rates affect the bond’s yield to maturity and its price (the present value of the remaining cash flows).

Interest Rate Changes and Bond Prices

  • If a bond sells at par (price = face value), the bond’s coupon rate will exactly equal its yield to maturity.
  • Changes in yields will affect the price investors are willing to pay to purchase a bond.
  • If market interest rates rise, the yield to maturity on the bond also rises. Because the bond pays less than the going rate, investors will only be willing to pay something less than the $1,000 face value payment. Because the bond sells for less than its face value, it is trading at a discount to its face value.
  • If market interest rates fall, and the yield to maturity on the bond also falls. Because the bond pays more than the going rate, investors will be willing to pay something more than the $1,000 face value payment. Because the bond sells for more than its face value, it is trading at a premium to its face value.
  • A higher yield to maturity means that investors demand a higher return for investing. They apply a higher discount rate to the bond’s remaining cash flows, reducing their present value and, hence, the bond’s price.
  • The reverse holds when interest rates fall. Investors then demand a lower yield to maturity, reducing the discount rate applied to the bond’s cash flows and raising the price.
  • As interest rates and bond yields rise, bond prices will fall, and vice versa, so that interest rates and bond prices always move in the opposite direction.
  • When coupon rate = yield to maturity, price = face value: bond trades at par
  • When coupon rate > yield to maturity, price > face value: bond trades at a premium
  • When coupon rate < yield to maturity, price < face value: bond trades at a discount
Lecture Example 5
  • Three 30-year bonds with annual coupon payments:
    • 10% coupon rate
    • 5% coupon rate
    • 3% coupon rate
  • The yield to maturity of each bond is 5% per year.
  • Cash flows for each bond:
    • 10% coupon bond: FV = $100, CPN = 0.10 \times 100 = $10
    • 5% coupon bond: FV = $100, CPN = 0.05 \times 100 = $5
    • 3% coupon bond: FV = $100, CPN = 0.03 \times 100 = $3
  • Bond prices:
    • P = 10 \times {\frac{1}{0.05}(1 - \frac{1}{1.05^{30}})} + \frac{100}{1.05^{30}} = $176.86 (trades at a premium)
    • P = 5 \times {\frac{1}{0.05}(1 - \frac{1}{1.05^{30}})} + \frac{100}{1.05^{30}} = $100.00 (trades at par)
    • P = 3 \times {\frac{1}{0.05}(1 - \frac{1}{1.05^{30}})} + \frac{100}{1.05^{30}} = $69.26 (trades at a discount)

Time and Bond Prices

  • No, a bond does not always have the same price if the yield to maturity does not change. Money has a time value and cash flows become more valuable as the time to receive them decreases.
  • Within a coupon period, the bond price will slowly rise as we get closer and closer to the coupon payment date.
  • The bond price will then fall after the coupon is paid because fewer coupons remain. This is because the bond price is the present value of the remaining cash flows.
  • Additionally, holding the yield to maturity constant, the bond price will move towards the face value over time.
  • The price of a zero-coupon bond rises smoothly, but the prices of the coupon bonds form zigzag lines (i.e. a sawtooth pattern).
Lecture Example 6
  • Purchase a 30-year, zero-coupon bond with a yield to maturity of 5% per year.
  • For a face value of $100, the bond will initially trade for: P = \frac{100}{1.05^{30}} = $23.14
  • If the yield to maturity remains at 5% per year, the price five years later is: P = \frac{100}{1.05^{25}} = $29.53
  • The rate of return on investment is: r=(29.5323.14)15−1=5%r = (\frac{29.53}{23.14})^{\frac{1}{5}} - 1 = 5\%
  • If a bond’s yield to maturity does not change, then the rate of return of an investment in the bond equals its yield to maturity even if you sell the bond early.

Interest Rate Risk and Bond Prices

  • Interest rate risk measures the change in a bond’s price due to a change in interest rates.
  • Long-term bonds have more interest rate risk than short-term bonds.
  • Low-coupon-rate bonds have more interest rate risk than high-coupon-rate bonds.
  • Consider a 10-year coupon bond and a 30-year coupon bond, both with 10% annual coupons and a face value of $100. Suppose the yield to maturity of each bond increases from 5% per year to 6% per year. The bond prices at these yields are:
    • YTM 5%.
      • 10-year 10% annual coupon bond: P = 10 \times {\frac{1}{0.05}(1 - \frac{1}{1.05^{10}})} + \frac{100}{1.05^{10}} = $138.61
      • 30-year 10% annual coupon bond: P = 10 \times {\frac{1}{0.05}(1 - \frac{1}{1.05^{30}})} + \frac{100}{1.05^{30}} = $176.86
    • YTM 6%.
      • 10-year 10% annual coupon bond: P = 10 \times {\frac{1}{0.06}(1 - \frac{1}{1.06^{10}})} + \frac{100}{1.06^{10}} = $129.44
      • 30-year 10% annual coupon bond: P = 10 \times {\frac{1}{0.06}(1 - \frac{1}{1.06^{30}})} + \frac{100}{1.06^{30}} = $155.06
  • The percentage change for each bond from an increase in the yield to maturity from 5% per year to 6% per year is:
    • 10-year 10% annual coupon bond: (129.44−138.61138.61)=−6.6%(\frac{129.44 - 138.61}{138.61}) = -6.6\%
    • 30-year 10% annual coupon bond: (155.06−176.86176.86)=−12.3%(\frac{155.06 - 176.86}{176.86}) = -12.3\%
  • The 30-year bond is almost twice as sensitive to a change in the yield than the 10-year bond.
Lecture Example 7
  • Two bonds, each paying semi-annual coupons and having five years until maturity.
    • One has a coupon rate of 5%.
    • The other has a coupon rate of 10%.
  • Both currently have a yield to maturity of 8% APR with semi-annual compounding.
  • Both bonds have a face value of $100.
  • Suppose the yield to maturity decreases from 8% (4% per half year) to 7% (3.5% per half year).
  • Note, 𝑛 = 5 × 2 = 10.
    • YTM 8%.
      • 5-year 5% coupon bond: P = 5 \times {\frac{1}{0.04}(1 - \frac{1}{1.04^{10}})} + \frac{100}{1.04^{10}} = $87.83
      • 5-year 10% coupon bond: P = 10 \times {\frac{1}{0.04}(1 - \frac{1}{1.04^{10}})} + \frac{100}{1.04^{10}} = $108.11
    • YTM 7%.
      • 5-year 5% coupon bond: P = 5 \times {\frac{1}{0.035}(1 - \frac{1}{1.035^{10}})} + \frac{100}{1.035^{10}} = $91.68
      • 5-year 10% coupon bond: P = 10 \times {\frac{1}{0.035}(1 - \frac{1}{1.035^{10}})} + \frac{100}{1.035^{10}} = $112.47
  • 5-year 5% coupon bond percentage change: (91.68−87.8387.83)=4.4%(\frac{91.68 - 87.83}{87.83}) = 4.4\%
  • 5-year 10% coupon bond percentage change: (112.47−108.11108.11)=4%(\frac{112.47 - 108.11}{108.11}) = 4\%
  • The bond with the smaller coupon payments is more sensitive to changes in interest rates. Because its coupons are smaller relative to its face value, a larger fraction of its cash flows is received later.

Corporate Bonds

Credit Risk

  • Australian Government (Treasury) bonds are widely regarded to be risk-free (default-free).
  • Credit risk is the risk of default, so the bond’s cash flows are not known with certainty.
  • Corporations with higher default risk will need to pay higher coupons to attract buyers to their bonds.
  • In November 2024, Woolworths issued a 10-year bond with a 5.91% coupon and face value of $800 million. The average yield to maturity on a 10-year Australian Government bond during November 2024 was 4.54%.

Corporate Bond Prices and Yields

  • Investors pay less for bonds with credit risk than they would for an otherwise identical default-free bond.
  • The yield of bonds with credit risk will be higher than that of otherwise identical default-free bonds.

Bond Ratings

  • Several companies rate the creditworthiness of bonds and make this information available to investors.
  • By consulting these ratings, investors can assess the creditworthiness of a particular bond issue.
  • The ratings therefore encourage widespread investor participation and relatively liquid markets.
  • The two best-known bond rating companies are:
    • Standard & Poor’s
    • Moody’s.
  • Australian Government bonds are rated AAA by Standard & Poor’s.
  • Woolworths bonds are rated BBB by Standard & Poor’s.

Credit Spreads

  • The credit spread (or default spread) is the difference between the yields of corporate bonds and Government bonds (Treasuries).
  • Credit spreads fluctuate as perceptions regarding the probability of default change.
  • 1 basis point = 0.01%