FINC 335 - Investments: Bond Prices and Yields- Topic 10

FINC 335: Investments - Bond Prices and Yields

Course Information

• Course Title: FINC 335 Investments

• Instructor: An Qin

• Institution: Loyola University Chicago

• Term: Spring 2026

Review

• Behavioral Finance:

  • Critiques conventional finance approaches by introducing psychological factors that affect investor behavior.

• Technical Analysis:

  • Analyzes past price patterns and trading volumes to forecast future price movements.

Agenda

• Main Features of Bonds

• Yield to Maturity (YTM)

• Realized Return

• Bond Pricing

• Bond Sectors and Yield Spreads

• The Yield Curve (Term-Structure of Interest)

• The Slope of the Yield Curve

Main Features of Bonds

• Issuer Types:

  • US Treasury/Government Bonds: Issued by the federal government.

  • State and Municipal Bonds: Issued by state and local governments.

  • Corporate Bonds: Issued by private and public corporations.

  • Foreign Government Bonds: Sovereign bonds issued by other countries.

• Term (Maturity):

  • Short-term Bonds: < 1 year (e.g., T-bills, CDs, Commercial papers).

  • Long-term Bonds: > 1 year (e.g., T-bonds, corporate bonds, perpetuities or consols).

• Price vs. Par Value:

  • Par Bond: Bonds sold at face value.

  • Discount Bond: Bonds sold below face value.

  • Premium Bond: Bonds sold above face value.

Coupon Payment Characteristics

• Coupon Rate: Total annual interest payment per dollar of face value.

• Payment Period: Usually semi-annual.

• Types of Coupons:

  • Fixed Rate: Constant coupon rate.

  • Variable Rate: Floater and inverse floater structures.

  • Zero-Coupon Bonds: No periodic payments, pay face value at maturity.

• Currency and Trading Venue:

  • Yankee Bonds: U.S. dollar-denominated bonds sold in the U.S. by foreign entities.

  • Eurobonds: Issued in multiple currencies and traded internationally.

• Credit Risk:

  • Risk-Free Bonds: Generally considered free of default risk (e.g., Treasury bonds).

  • Defaultable Bonds: Risk of issuer defaulting on payments.

  • Seniority and Security: Some bonds have seniority in claims on assets or are secured.

• Covenants and Option Provisions: Include agreements between issuer and bondholders about restrictions and options.

Bond Pricing and Cash Flow

• Investor Action: Investors lend money to bond issuers in exchange for interest payments and repayment of face value at maturity.

Payment Structure Timeline

• Period Timeline:

  • 0, 1, 2, …, n

  • Price of bond over time, coupon payments at defined intervals, and maturity payment.

Yield to Maturity (YTM)

• Definition: The discount rate that makes the present value of all cash flows equal to the bond price. It acts as the internal rate of return (IRR) for bond investments.

• YTM Equation: P = rac{C}{(1+r)^1} + rac{C}{(1+r)^2} + … + rac{C}{(1+r)^n} + rac{F}{(1+r)^n} Where:

  • $P$ = Price of bond

  • $C$ = Annual coupon payment

  • $F$ = Face value of the bond

  • $r$ = YTM

  • $n$ = Number of periods to maturity

Example 1: Annual-Pay Coupon Bond

• Given Parameters:

  • 3-year bond, face value = $1000, annual coupon payment = $80.

Case Analysis

1. Case 1 (Par Bond):

   • Sell Price = $1000

   • YTM: Solve ext{YTM} = rac{C}{F} = rac{80}{1000} = 8 ext{%}

   • Current Yield: ext{Current Yield} = rac{80}{1000} = 8 ext{%}

2. Case 2 (Discount Bond):

   • Sell Price = $900

   • YTM Calculation:
     ext{YTM} = 12.17 ext{%} ext { (calculate from cash flows)}

   • Coupon Rate: rac{80}{1000} = 8 ext{%}

   • Current Yield: rac{80}{900} = 8.88 ext{%}

3. Case 3 (Premium Bond):

   • Sell Price = $1100

   • YTM Calculation:
     ext{YTM} = 4.37 ext{%} ext { (calculate from cash flows)}

   • Coupon Rate: rac{80}{1000} = 8 ext{%}

   • Current Yield: rac{80}{1100} = 7.27 ext{%}

• Key Insight: When YTM is higher than the coupon rate, the bond price must be below par value (discount bond).

Premium and Discount Bonds Definitions

• Premium Bonds ($P > F$):

  • Coupon rate > YTM.

  • Current yield > YTM.

  • Bond price declines to par over maturity.

• Discount Bonds ($P < F$):

  • Coupon rate < YTM.

  • Current yield < YTM.

  • Bond price increases to par over maturity.

Bond Pricing Over Time

• Price Trends:

  • Bond Price ($) vs Time (years) graph displays different price paths for:

  • Par bond.

  • Discount bond.

  • Premium bond.

• Indicates trajectory towards par upon maturity.

Yield to Maturity on Semi-Annual Pay Coupon Bonds

• Calculation Steps:

  1. Compute the semi-annual IRR (YTM).

  2. Correspond this semi-annual rate to an annual percentage rate (APR).

Example 2: Cash Flows of a Semi-Annual Coupon Bond

• Bond Parameters:

  • Face value = $1000, coupon = 4%, semi-annual payments.

  • Semi-annual payment = rac{1000 imes 4 ext{%}}{2} = 20

  • Cash Flow Timeline:

  • Period 1: $20

  • Period 2: $20

  • … (20 total semi-annual payments)

  • Final Payment: $20 + face value ($1000) at maturity.

Example 3: Semi-Annual Pay Coupon Bond Pricing

• Parameters:

  • 7-year bond, face value = $1000, coupon rate = 8%, YTM = 6.75%.

• Trading Premium: Since YTM < Coupon Rate, the bond is trading at a premium.

• Price Calculation if YTM rises to 7%:

  • Given n = 14, rate = 3.50%, payment = $40, FV = $1000.

  • Solve using bond price formula (calculate present value).

Pricing of Bonds Formula

• Pricing formula encompasses the present value of cash flows:
  P = rac{C}{(1+r)^1} + rac{C}{(1+r)^2} + … + rac{C}{(1+r)^n} + rac{F}{(1+r)^n}

• To generalize: P = C imes rac{(1-(1+r)^{-n})}{r} + rac{F}{(1+r)^n} Where:

  • $C$: Coupon payment, $r$: YTM, $n$: Years to maturity, $F$: Face value.

Realized Return vs. YTM

• Return Variables: The return on investment equals YTM when:

  • Coupons can be re-invested at the same YTM.

  • The bond is held until maturity or sold at YTM corresponding price.

• Practical Considerations: Market yields may change, affecting the actual realized return.

Example 4: Realized Return on a Zero-Coupon Bond

• Bond Characteristics:

  • Face value: $1000, YTM = 5%.

  • Current Price Calculation:
    ext{Price} = rac{1000}{(1 + 0.05)^3} = 863.84

YTM Change Scenario

1. If YTM changes to 7%, New Price:
   ext{New Price} = rac{1000}{(1 + 0.07)^2} = 873.44

   • Calculate realized holding period return (HPR) from previous price to new price.

2. If YTM remains at 5%: Calculate the price accordingly.

Realized Holding Period Return Calculation

• Generic Holding Period Calculation:
  Vt = V0 (1 + ext{aHPR})^t

• Annualized Holding Period Return Formula:
  ext{aHPR} = igg( rac{Vt}{V0}igg)^{1/t} - 1

Example 5: Multiple Year ZCB Holding Period Return

• Investment Parameters:

  • Buy at t=0, face value $1000, YTM = 5%.

  • YTM jumps to 6% after purchase, calculate returns.

1. Determine purchase price at t=0 and selling price at t=2.

   • Price at t=0: 863.84

   • Price at t=2 (selling price): 943.40

   • Calculate aHPR.

Example 6: Realized Return on a 4-Year Coupon Bond

• Bond Parameters:

  • Face value: $1000, coupon payment: $80, YTM: 8%.

  • Case 1: Reinvest at 8%: Current Price and Future Value calculations.

• Cash Flows and Accumulated Values:

  • Cash flow calculations across different yield scenarios. Reinvestment variabilities between different rates.

Major Bond Sectors

• Market Sectors Include:

  • U.S. Treasury Issues: Secure, low-risk bonds.

  • Municipal Bonds: Tax-exempt, issued by cities or states.

  • Corporate Bonds: Higher risk, potential higher yield.

• Yield Spreads: The differences in interest rates between varied bond sectors.

Factors Affecting Yield Spreads

• Municipal bond rates: Generally 20-30% lower than corporate bonds.

• Treasury bonds usually have lower rates due to no default risk.

• Credit ratings influence yields: Lower ratings = Higher risk = Higher rates.

• Revenue municipal bonds: Typically yield higher than general obligation bonds due to associated risks.

• Callable bonds yield more than non-callable bonds, reflecting greater risk to investors.

• Longer maturities typically yield more due to increased risk and time value of money.

Factors Influencing Bond Prices and Interest Rates

• Primary Influencer: Interest Rates

  • Inverse relationship: Rates up = Prices down and vice versa.

• Inflation's Role:

  • Inflation goes up = Interest rates go up generally via the Fisher equation:
    1 + ext{Nominal Rate} ext{ }= (1 + ext{Real Rate}) imes (1 + ext{Inflation Rate})

  • Approximation:
    ext{Nominal Rate} ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ }

The Yield Curve or Term-Structure of Interest

• Definition: The yield curve reflects the relationship between the interest rates and the maturity of bonds.

• Graphical Representation: A visual form mapping yields against maturities.

• Common Shapes:

  • Upward Sloping (Normal)

  • Flat

  • Downward Sloping (Inverted)

  • Hump-Shaped

Constructing the Yield Curve

• Data Points for Construction: Various Treasury bond yields captured on specific dates, considered for different maturities to construct the curve.

Interpreting the Shape of the Yield Curve

• Upward-Sloping Curves: Suggest higher inflation expectations, greater liquidity risk, and want for shorter-term loans.

• Flat or Downward Sloping Curves: Indicate lower inflation expectations and liquidity risks; suggest a higher supply of long-term loans.

Appendix: Derivation of Annuity Formula

• Basic Annuity Calculation:
  A = rac{PV}{(1+r)^1} + rac{PV}{(1+r)^2} +…
  

• This fundamental approach is applied to derive both bond pricing and stock pricing models:

  • Bond Pricing Formula:
    P = C imes rac{(1 - (1 + r )^{-n})}{r} + rac{F}{(1 + r )^n}

• Stock Pricing (DDM) Formula:
  P = rac{D1}{(1+r)} + rac{D2}{(1+r)^2} + … $$