Interest Rates and Bond Valuation

Chapter 6 Interest Rates and Bond Valuation

Key Concepts and Skills

  • After studying this chapter, you should be able to:

    • Identify important bond features and types of bonds.

    • Describe bond values and why they fluctuate.

    • Discuss bond ratings and what they mean.

    • Evaluate the impact of inflation on interest rates.

    • Explain the term structure of interest rates and the determinants of bond yields.

Chapter Outline

  • 6.1: Bonds and Bond Valuation.

  • 6.2: More on Bond Features.

  • 6.3: Bond Ratings.

  • 6.4: Some Different Types of Bonds.

  • 6.5: Bond Markets.

  • 6.6: Inflation and Interest Rates.

  • 6.7: Determinants of Bond Yields.

Bond Definitions

  • Bond:

    • A debt contract.

    • Interest-only loan.

    • Par value (face value) approximately $1,000.

    • Coupon rate: The rate of interest the bond pays.

    • Coupon payment: The actual amount paid to bondholders typically on an annual or semi-annual basis.

    • Maturity date: The date on which the bond's principal is repaid.

    • Yield to maturity (YTM): The total return anticipated on a bond if it is held until it matures.

Key Features of a Bond

  1. Par value:

    • Also known as the face amount.

    • This amount is repaid at maturity, usually assumed to be $1,000 for corporate bonds.

  2. Coupon interest rate:

    • The stated interest rate of the bond.

    • Usually equals the yield to maturity (YTM) at the time of issuance.

    • To find the coupon payment, multiply the coupon rate by the par value.

  3. Maturity:

    • Refers to the number of years until the bond must be repaid.

  4. Yield to maturity (YTM):

    • Represented as “r”.

    • The market required rate of return for bonds of similar risk and maturity.

    • It is the discount rate used to value a bond and represents the return if the bond is held to maturity.

    • Typically equals the coupon rate at the time of issuance.

    • Quoted as an Annual Percentage Rate (APR).

  5. Current Yield:

    • Defined as the annual coupon divided by the current price of the bond.

Bond Value

  • The formula for bond value is:
    extBondValue=PV(extcoupons)+PV(extpar)ext{Bond Value} = PV( ext{coupons}) + PV( ext{par})

  • This expands to:
    extBondValue=PV(extannuity)+PV(extlumpsum)ext{Bond Value} = PV( ext{annuity}) + PV( ext{lump sum})

  • Important relationships:

    • As interest rates increase, present values decrease.

    • As interest rates increase, bond prices decrease, and vice versa.

The Bond-Pricing Equation

  • The bond-pricing equation for cash flows is formulated as follows:

    • $C$ = Coupon payment

    • $F$ = Face value

Practicing Bond Valuation with Texas Instruments BA II Plus

  • Utilize the following keys:

    • N = Number of periods to maturity.

    • I/Y = Period interest rate (YTM).

    • PV = Present value (The bond value).

    • PMT = Coupon payment.

    • FV = Future value (Face value or Par value).

Valuing a Discount Bond with Annual Coupons

  • Example:

    • Coupon rate = 10%

    • Par value = $1,000

    • Maturity = 5 years

    • YTM = 11%

    • Using a calculator:

    • Input:

      • 5 (N)

      • 11 (I/Y)

      • 100 (PMT)

      • 1000 (FV)

    • Calculate PV = −963.04

    • Using formula:

    • B=PV(extannuity)+PV(extlumpsum)B = PV( ext{annuity}) + PV( ext{lump sum})

    • B=369.59+593.45=963.04B = 369.59 + 593.45 = 963.04

    • Excel formula: = PV (0.11, 5, 100, 1000, 0).

    • Note: When YTM > Coupon rate → Price < Par = "Discount Bond".

Valuing a Premium Bond with Annual Coupons

  • Example:

    • Coupon rate = 10%

    • Par value = $1,000

    • Maturity = 20 years

    • YTM = 8%

    • Using a calculator:

    • Input:

      • 20 (N)

      • 8 (I/Y)

      • 100 (PMT)

      • 1000 (FV)

    • Calculate PV = −1196.36

    • Using formula:

    • B=PV(extannuity)+PV(extlumpsum)B = PV( ext{annuity}) + PV( ext{lump sum})

    • B=981.81+214.55=1196.36B = 981.81 + 214.55 = 1196.36

    • Excel formula: = PV (0.08, 20, 100, 1000, 0).

    • Note: When YTM < Coupon rate → Price > Par = "Premium Bond".

Bond Prices: Relationship Between Coupon and Yield

  • Understanding the relationships:

    • When Coupon rate = YTM, Price = Par.

    • When Coupon rate < YTM, Price < Par; known as a "Discount bond".

    • When Coupon rate > YTM, Price > Par; known as a "Premium bond".

The Bond-Pricing Equation Adjusted for Semiannual Coupons

  • Adjustments include:

    • C = Annual coupon payment → Cimesrac12C imes rac{1}{2} for semiannual

    • YTM = Annual yield → YTMimesrac12YTM imes rac{1}{2} for semiannual

    • t = Years to maturity → 2t2t = Number of six-month periods to maturity.

Semiannual Bonds Example

  • Example 6.1:

    • Coupon rate = 14% semiannually.

    • r = 16%

    • Maturity = 7 years

    • Number of coupon payments = 14 (2 x 7 years).

    • Semiannual coupon payment:

    • Calculating semiannual yield results in:

      • Bond value = Using the calculator:

      • Input:

      • 14 (N)

      • 8 (I/Y)

      • 70 (PMT)

      • 1000 (FV)

      • Calculate PV = −917.56

Computing Yield-to-Maturity (YTM)

  • Yield-to-maturity (YTM) is defined as follows:

    • The market required rate of return implied by the current bond price.

  • Calculator steps:

    • Enter N, PV, PMT, and FV.

    • Remember the sign convention:

    • PMT and FV must share the same sign.

    • PV is the opposite sign (negative).

    • Calculate using CPT I/Y for the yield.

YTM with Annual Coupons

  • Example:

    • Consider a bond with:

    • Coupon rate = 10%

    • 15 years to maturity

    • Par value = $1,000

    • Current price = $928.09

    • Market yield deduction:

    • Using calculator inputs for YTM computation:

      • 15 (N)

      • −928.09 (PV)

      • 1000 (FV)

      • 100 (PMT)

    • YTM = 11%

YTM with Semiannual Coupons

  • Suppose a bond has:

    • Coupon rate = 10% (semiannual)

    • Face value = $1,000

    • Maturity = 20 years

    • Current price = $1,197.93

  • Calculate YTM using:

    • Inputs:

    • 40 (N = 20 years x 2)

    • −1197.93 (PV)

    • 1000 (FV)

    • 50 (semiannual PMT = $100 \div 2)

    • Result: Semiannual yield = 4%, hence YTM = 8%.

Summary of Bond Valuation (Table 6.1)

  • The valuation of a bond is summarized:

    • extBondvalue=racCr+racF(1+r)text{Bond value} = rac{C}{r} + rac{F}{(1+r)^t}

    • Where:

    • C = Coupon paid each period.

    • r = Rate per period.

    • t = Number of periods.

    • F = Bond’s face value.

Is It 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 may lead to financial distress and bankruptcy.

  • Equity:

    • Ownership interest in the firm.

    • Common stockholders vote for board members.

    • Dividends paid are not tax-deductible.

    • Dividends are not liabilities until declared.

    • Stockholders do not have recourse if dividends are not declared.

    • An all-equity firm cannot go bankrupt.

The Bond Indenture

  • The bond indenture, or "Deed of Trust," is a contract containing:

    • Basic terms of the bonds.

    • Total amount of bonds issued.

    • Whether the bonds are secured or unsecured.

    • Sinking fund provisions.

    • Call provisions, including deferred call and call premium.

    • Protective covenants.

Bond Ratings – Investment Quality

  • High Grade:

    • Moody’s Aaa and S&P AAA – extremely strong capacity to pay.

    • Moody’s Aa and S&P AA – very strong capacity to pay.

  • Medium Grade:

    • Moody’s A and S&P A – strong capacity to pay, but more susceptible to changes.

    • Moody’s Baa and S&P BBB – adequate capacity to pay, with adverse conditions impacting ability to pay.

Bond Ratings – Speculative

  • Low Grade:

    • Moody’s Ba, B, Caa and Ca.

    • S&P BB, B, CCC, CC.

    • Considered speculative regarding capacity to pay; "B" ratings are the lowest degree of speculation.

  • Very Low Grade:

    • Moody’s C and S&P C – income bonds with no interest paid.

    • Moody’s D and S&P D – in default with principal and interest in arrears.

Government Bonds

  • Treasury Securities = Federal government debt:

    • Treasury Bills (T-bills): Pure discount bonds with a maturity of one year or less.

    • Treasury notes: Coupon debt with maturity between one and ten years.

    • Treasury bonds: Coupon debt with maturity greater than ten years.

Government Bonds - Municipal Securities

  • Municipal Securities are:

    • Debt of state and local governments.

    • Varying degrees of default risk and rated similarly to corporate debt.

    • Interest received is tax-exempt at the federal level and usually exempt from state tax in the issuing state.

Example 6.4: Tax Consideration

  • A taxable bond yields 8% and a municipal bond yields 6%.

  • In a 40% tax bracket:

    • After-tax return on corporate bond: 8%(1 − 0.4) = 4.8%.

    • Municipal bond return: 6%.

    • At what tax rate would be indifferent? Setting:

    • 8%(1 - t) = 6% → t = 25%.

Zero Coupon Bonds

  • Characteristics:

    • Make no periodic interest payments (coupon rate = 0%).

    • Yield to maturity comes from the difference between purchase price and par value (capital gains).

    • Cannot be sold for more than par value.

    • Also referred to as zeroes or deep discount bonds.

    • Examples include Treasury Bills and US Savings bonds.

Floating Rate Bonds

  • Definition:

    • Coupon rate floats based on an index value.

    • Examples: Adjustable rate mortgages and inflation-linked Treasuries.

  • Advantages:

    • Less price risk as coupon adjusts, maintaining closeness to yield to maturity.

    • Coupons may have a "collar" limiting the maximum and minimum rates.

Bond Markets

  • Market characteristics:

    • Primarily consist of over-the-counter (OTC) transactions with electronically connected dealers.

    • Although a large number of bond issues exist, daily volume for individual issues is often low.

    • Obtaining current prices can be challenging, especially for small company or municipal bonds.

    • Treasury securities are an exception with more consistent pricing.

Inflation and Interest Rates

  • Key definitions:

    • Real rate of interest: Represents a change in purchasing power.

    • Nominal rate of interest: Quoted rate of interest, reflecting changes in purchasing power and inflation.

    • The ex ante nominal rate includes desired real return plus an adjustment for expected inflation.

The Fisher Effect

  • Definition:

    • The Fisher effect illustrates the relationship among real rates, nominal rates, and inflation defined mathematically as:

    • $R$ = Nominal rate (Quoted rate).

    • $r$ = Real rate.

    • $h$ = Expected inflation rate.

    • Approximates as: R=r+hR = r + h

Example 6.6: Calculation of Nominal Rate

  • If a real return of 10% is required and expected inflation is 8%, calculate the nominal rate:

    • Actual Calculation:

    • R = (1.10)(1.08) - 1 = 0.1880, ext{ or } 18.80 ext{%}.

    • Using approximation:

    • R = 10 ext{%} + 8 ext{%} = 18 ext{%}.

  • Note: The difference between actual Fisher effect and approximation is significant when real return and inflation are relatively high.