Chapter 06: Interest Rates and Bond Valuation

Bonds and Bond Valuation

Key Concepts and Skills

• Identify bond features and types.
• Describe bond values and fluctuations.
• Discuss bond ratings.
• Evaluate inflation's impact.
• Explain the term structure and determinants of bond yields.

Chapter Outline

• Bonds and Bond Valuation
• Bond Features
• Bond Ratings
• Types of Bonds
• Bond Markets
• Inflation and Interest Rates
• Determinants of Bond Yields

Bond Definitions

• Bond: Debt contract, interest-only loan.
• Par Value (Face Value): Approximately $1,000.
• Coupon Rate: Stated interest rate.
• Coupon Payment: Interest payment.
• Maturity Date: Date when the principal is repaid.
• Yield to Maturity (YTM): Market required rate of return.

Key Features of a Bond

• Par value:
  • Face amount, repaid at maturity.
  • Assume $1,000 for corporate bonds.
• Coupon interest rate:
  • Stated interest rate.
  • Usually equals YTM at issue.
  • Multiply by par value to get coupon payment.
• Maturity:
  • Years until bond must be repaid.
• Yield to maturity (YTM):
  • Market required rate of return for bonds of similar risk and maturity.
  • The discount rate used to value a bond.
  • Return if bond held to maturity.
  • Usually equals the coupon rate at issue.
  • Quoted as an APR.

Bond Valuation

• Bond Value = PV(coupons) + PV(par).
• Bond Value = PV(annuity) + PV(lump sum).
• Remember:
  • As interest rates increase, present values decrease. $(r \uparrow \rightarrow PV \downarrow)$
  • As interest rates increase, bond prices decrease and vice versa.

The Bond-Pricing Equation

• Bond Value = PV(Annuity) + PV(lump sum)
• $Bond Value = C \times \frac{1 - (1 + YTM)^{-t}}{YTM} + \frac{F}{(1 + YTM)^t}$
  • Where:
    • $C$ = Coupon payment
    • $F$ = Face value

Texas Instruments BA-II Plus

• N = Number of periods to maturity.
• I/Y = Period interest rate = YTM.
• PV = Present value = Bond value.
• PMT = Coupon payment.
• FV = Future value = Face value = Par value.

Spreadsheet Formulas

• =FV(Rate,Nper,Pmt,PV,0/1)
• =PV(Rate,Nper,Pmt,FV,0/1)
• =RATE(Nper,Pmt,PV,FV,0/1)
• =NPER(Rate,Pmt,PV,FV,0/1)
• =PMT(Rate,Nper,PV,FV,0/1)
• Inside parentheses: (RATE,NPER,PMT,PV,FV,0/1)
• “0/1” Ordinary annuity = 0 (default), Annuity Due = 1 (must be entered)

Pricing Specific Bonds on the TI BAII+

• Bond Worksheet: 2nd BOND (above “9”)
• SDT CPN RDT RV ACT YLD PRI
  • SDT = Actual Settlement date (enter MM.DDYY)
  • CPN = Annual rate in percent
  • RDT = Actual Redemption (maturity) date
  • RV = Redemption value as a percent of par
  • ACT = ACT/360 day count setting
  • 2/Y = coupons per year, 2/Y – 1/Y
  • YLD = Yield to redemption
  • PRI = Dollar price per $100 of par value

Pricing Specific Bonds in Excel

• =PRICE(Settlement,Maturity,Rate,Yld,Redemption, Frequency,Basis)
• =YIELD(Settlement,Maturity,Rate,Pr,Redemption, Frequency,Basis)
  • Settlement = Actual date as a serial number
  • Maturity = Actual date as a serial number
  • Redemption and Pr(ice) = percent of par value
  • Rate (coupon) and Yld = Annual rates as decimals
  • Frequency = # of coupons per year
  • Basis = Day count convention (enter “2” for ACT/360)

Valuing a Premium Bond with Annual Coupons

• Coupon rate = 10% Annual coupons
• Par = $1,000
• Maturity = 5 years
• YTM = 11%
• Using the calculator:
  • 5 N
  • 11 I/Y
  • 100 PMT
  • 1000 FV
  • CPT PV = $-963.04$
• Using the formula:
  • $B = PV(annuity) + PV(lump sum)$
  • B = $100 \times \frac{1 - (1 + 0.11)^{-5}}{0.11} + \frac{1000}{(1.11)^5}
  • B = $369.59 + $593.45 = $963.04
• Using Excel: =PV(.11,5,100,1000,0)
• Note: When YTM > Coupon rate → Price < Par = “Discount Bond”

Valuing a Discount Bond with Annual Coupons

• Coupon rate = 10% Annual coupons
• Par = $1,000
• Maturity = 20 years
• YTM = 8%
• Using the calculator:
  • 20 N
  • 8 I/Y
  • 100 PMT
  • 1000 FV
  • CPT PV = $-1196.36$
• Using the formula:
  • $B = PV(annuity) + PV(lump sum)$
  • B = $100 \times \frac{1 - (1 + 0.08)^{-20}}{0.08} + \frac{1000}{(1.08)^{20}}
  • B = $981.81 + $214.55 = $1196.36
• Using Excel: =PV(.11,5,100,1000,0)
• Note: When YTM < Coupon rate → Price > Par = “Premium Bond”

Bond Prices: Relationship Between Coupon and Yield

• Coupon rate = YTM → Price = Par.
• Coupon rate < YTM → Price < Par.
  • “Discount bond” … Why?
• Coupon rate > YTM → Price > Par.
  • “Premium bond” … Why?

The Bond-Pricing Equation Adjusted for Semiannual Coupons

• $C$ = Annual coupon payment → $C ÷ 2$ = Semiannual coupon
• $r$ = Annual yield → $r ÷ 2$ = Semiannual yield
• $t$ = Years to maturity → $2t$ = Number of 6-month periods to maturity
• $Bond Value = \sum_{t=1}^{2t} \frac{C/2}{(1 + YTM/2)^t} + \frac{F}{(1 + YTM/2)^{2t}}$

Semiannual Bonds Example 6.1

• Coupon rate = 14 percent semiannually.
• $r$ = 16 percent.
• Maturity = 7 years
• Number of coupon payments? (2t or N) = 14 = 2 × 7 years
• Semiannual coupon payment? ($C / 2$ or PMT) = $70 = 14\% \times Face value / 2$
• Semiannual yield? ($r / 2$ or I/Y) = $8\% = 16\% / 2$

Example 6.1

• Semiannual coupon = $70.
• Semiannual yield = 8%.
• Periods to maturity = 14.
• Bond value =
• Using Excel: =PV(.08,14,70,1000,0)
• Using the calculator:
  • 14 N
  • 8 I/Y
  • 70 PMT
  • 1000 FV
  • CPT PV = $-917.56$
• Bond Value = $70 \times \frac{1 - (1 + 0.08)^{-14}}{0.08} + \frac{1000}{(1.08)^{14}} = $917.56

Interest Rate Risk

• Price Risk:
  • Change in price due to changes in interest rates.
  • Long-term bonds have more price risk than short-term bonds.
  • Low coupon rate bonds have more price risk than high coupon rate bonds.
• Reinvestment Rate Risk:
  • Uncertainty concerning rates at which cash flows can be reinvested.
  • Short-term bonds have more reinvestment rate risk than long-term bonds.
  • High coupon rate bonds have more reinvestment rate risk than low coupon rate bonds.

Computing Yield-to-Maturity (YTM)

• Yield-to-maturity (YTM): Market required rate of return implied by the current bond price.
• With a financial calculator:
  • Enter N, PV, PMT, and FV. Remember the sign convention.
    • PMT and FV need to have the same sign (+).
    • PV the opposite sign (−).
  • CPT I/Y for the yield.

YTM with Annual Coupons

• Consider a bond with a 10 percent annual coupon rate, 15 years to maturity, and a par value of $1,000. The current price is $928.09.
  • Will the yield be more or less than 10 percent?
  • 15 N
  • 928.09 PV (enter as a negative)
  • 1000 FV
  • 100 PMT
  • CPT PV = 11% ← Result = YTM
• Using Excel: =RATE(15,100,-928.09,1000,0).

YTM with Semiannual Coupons

• Suppose a bond with a 10 percent coupon rate and semiannual coupons, has a face value of $1000, 20 years to maturity, and is selling for $1,197.93.
  • Is the YTM more or less than 10 percent?
  • What is the semiannual coupon payment?
  • How many periods are there?
• 40 N
• 1197.93 PV (negative)
• 1000 FV
• 50 PMT
• CPT PV 4% (= ½ YTM)
• YTM = 4%×2 = 8%
• NOTE: Solving a semiannual payer for YTM results in a 6-month yield. The calculator and Excel solve what you enter.
• Using Excel: =RATE(40,50,-1197.93,1000,0) = 4%.

Summary of Bond Valuation

• Finding the value of a bond
• $Bond value = \sum_{t=1}^{T} \frac{C}{(1+r)^t} + \frac{F}{(1+r)^T}$
  • Where:
    • $C$ = Coupon paid each period
    • $r$ = Rate per period
    • $t$ = Number of periods
    • $F$ = Bond’s face value

Finding the yield on a bond

• Given a bond value, coupon, time to maturity, and face value, it is possible to find the implicit discount rate, or yield to maturity, by trial and error only.
• To do this, try different discount rates in the preceding formula until the calculated bond value equals the given bond value.
• Remember that increasing the rate decreases the bond value.

Debt or Equity

• Debt
  • Not an ownership interest.
  • No voting rights.
  • Interest is tax deductible.
  • Creditors have legal recourse if interest or principal payments are missed.
  • Excess debt can lead to financial distress and bankruptcy.
• Equity
  • Ownership interest.
  • Common stockholders vote to elect the board of directors and on other issues.
  • Dividends are not tax deductible.
  • Dividends are not a liability of the firm until declared. Stockholders have no legal recourse if dividends are not declared.
  • An all-equity firm cannot go bankrupt.

The Bond Indenture

• “Deed of Trust” Contract between issuing company and bondholders includes:
  • Basic terms of the bonds
  • Total amount of bonds issued
  • Secured versus Unsecured
  • Sinking fund provisions
  • Call provisions
    • Deferred call
    • Call premium
  • Details of protective covenants.

Bond Classifications

• Registered versus Bearer Bonds
• Security
  • Collateral – secured by financial securities
  • Mortgage – secured by real property, normally land or buildings
  • Debentures – unsecured
  • Notes – unsecured debt with original maturity less than 10 years
• Seniority
  • Senior versus Junior, Subordinated

Bond Characteristics and Required Returns

• Coupon rate
  • A function of the risk characteristics of the bond when issued
  • Usually ≈ yield at issue
• Which bonds will have the higher coupon, all else equal?
  • Secured debt versus a debenture
  • Subordinated debenture versus senior debt
  • A bond with a sinking fund versus one without
  • A callable bond versus a non-callable bond

Bond Ratings – Investment Quality

• High Grade
  • Moody’s Aaa and S&P AAA – capacity to pay is extremely strong
  • Moody’s Aa and S&P AA – capacity to pay is very strong
• Medium Grade
  • Moody’s A and S&P A – capacity to pay is strong, but more susceptible to changes in circumstances
  • Moody’s Baa and S&P BBB – capacity to pay is adequate, adverse conditions will have more impact on the firm’s ability to pay

Bond Ratings – Speculative

• Low Grade
  • Moody’s Ba, B, Caa and Ca
  • S&P BB, B, CCC, CC
  • Considered speculative with respect to capacity to pay. The “B” ratings are the lowest degree of speculation.
• Very Low Grade
  • Moody’s C and S&P C – income bonds with no interest being paid
  • Moody’s D and S&P D – in default with principal and interest in arrears

Government Bonds

• Municipal Securities
  • Debt of state and local governments
  • Varying degrees of default risk, rated similar to corporate debt
  • Interest received is tax-exempt at the federal level
  • Interest usually exempt from state tax in issuing state

Government Bonds

• Treasury Securities = Federal government debt
  • Treasury Bills (T-bills)
    • Pure discount bonds
    • Original maturity of one year or less
  • Treasury notes
    • Coupon debt
    • Original maturity between one and ten years
  • Treasury bonds
    • Coupon debt
    • Original maturity greater than ten years

Example 6.4

• A taxable bond has a yield of 8 percent and a municipal bond has a yield of 6 percent. If you are in a 40 percent tax bracket, which bond do you prefer?
  • 8%(1 − .4) = 4.8%
  • The aftertax return on the corporate bond is 4.8 percent, compared to a 6 percent return on the municipal
• At what tax rate would you be indifferent between the two bonds?
  • 8%(1 − t*) = 6%
  • t* = 25%

Zero Coupon Bonds

• Make no periodic interest payments (coupon rate = 0 percent)
• Entire yield-to-maturity comes from the difference between the purchase price and the par value (capital gains)
• Cannot sell for more than par value
• Sometimes called zeroes, or deep discount bonds
• Treasury Bills and U.S. Savings bonds are good examples of zeroes

Floating Rate Bonds

• Coupon rate floats depending on some index value
• Examples – adjustable rate mortgages and inflation-linked Treasuries
• Less price risk with floating rate bonds
  • Coupon floats, so is less likely to differ substantially from the yield-to-maturity
• Coupons may have a “collar” – the rate cannot go above a specified “ceiling” or below a specified “floor.”

Other Bond Types

• Structured notes
• Convertible bonds
• Put bonds
• Many types of provisions can be added to a bond
  • Important to recognize how these provisions affect required returns
  • Who does the provision benefit?

Bond Markets

• Primarily over-the-counter transactions with dealers connected electronically
• Extremely large number of bond issues, but generally low daily volume in single issues
• Getting up-to-date prices difficult, particularly on small company or municipal issues
• Treasury securities are an exception

Corporate Bond Quotations

• ABC 8.375 Jul 15, 2033 100.641 8.316 362 30 763,528
  • What company are we looking at?
  • What is the coupon rate?
  • If the bond has a $1,000 face value, what is the coupon payment each year?
  • When does the bond mature?
  • What was the trading volume on that day?
  • What is the quoted price? (Ask price)
  • What is the bond’s yield?

Treasury Quotations

• 5/15/2038 4.500 146.230 146.250 0.098 1.437
  • When does the bond mature?
  • What is the coupon rate on the bond?
  • What is the bid price? What does this mean?
  • What is the ask price? What does this mean?
  • How much did the price change from the previous day?
  • What is the YTM based on Ask price?
• 5/15/2038 4.500 146.230 146.250 0.098 1.437
  • Maturity = May 15, 2038
  • Coupon rate = 4.500 percent per year
  • Bid price = 146.230 percent of par.
    • Price at which dealer is willing to buy from you
  • Ask price = 146.250 percent of par
    • Price at which dealer is willing to sell to you
  • Bid-Ask spread = Dealer’s profit
  • Change = Ask price is up .098 percent since the previous day
  • Asked yield = 1.437 percent

Quoted Price versus Invoice Price

• Quoted bond prices = “clean” price
  • Net of accrued interest
• Invoice Price = “dirty” or “full” price
  • Price actually paid
  • Includes accrued interest
• Accrued Interest
  • Interest earned since last coupon payment is owed to bond seller at time of sale

Inflation and Interest Rates

• Real rate of interest = Change in purchasing power
• Nominal rate of interest = Quoted rate of interest = Change in purchasing power and inflation
• The ex ante nominal rate of interest includes our desired real rate of return plus an adjustment for expected inflation

The Fisher Effect

• The Fisher effect defines the relationship between real rates, nominal rates and Inflation.
• $R$ = Nominal rate (Quoted rate)
• $r$ = Real rate
• $h$ = Expected inflation rate
• Approximation: $R = r + h$
• $(1 + R) = (1 + r)(1 + h)$

Example 6.6

• If we require a 10 percent real return and we expect inflation to be 8 percent, what is the nominal rate?
  • $R = (1.1)(1.08) − 1 = .1880$, or 18.80%
  • Approximation: $R = 10\% + 8\% = 18\%$
• Because the real return and expected inflation are relatively high, there is significant difference between the actual Fisher effect and the approximation

Term Structure of Interest Rates

• Term structure: The relationship between time to maturity and yields, all else equal
  • The effect of default risk, different coupons, etc., has been removed
• Yield curve: Graphical representation of the term structure
  • Normal = upward-sloping → L/T > S/T
  • Inverted = downward-sloping → L/T < S/T

Factors Effecting Required Return

• Default risk premium – bond ratings
• Taxability premium – municipal versus taxable
• Liquidity premium – bonds that have more frequent trading will generally have lower required returns
• Maturity premium – longer term bonds will tend to have higher required returns
• Anything else that affects the risk of the cash flows to the bondholders will affect the required returns