12 - Asset Pricing II - Bonds

Asset Pricing II: Bonds

Where we are coming from

• Completion of analysis on pricing formulas:

  • Uniform (finite) Annuity:

    • Formula: [ A = C \frac{1 - (1 + r)^{-n}}{r} ]

  • Uniform (infinite) Annuity:

    • Formula: [ A = \frac{C}{r} ]

Agenda

• Application of the uniform annuity framework to price a bond contract:

  • Discuss parameters of a standard bond contract.

  • Describe the bond's cash flow stream from payments.

  • Apply uniform pricing formula to bond's cash flow to price it.

  • Analyze relationship between yield and price.

Bond Contract Basics

• Fundamental elements of a bond contract:

  • Face (par) value: Denoted as ( F ) to be received at maturity.

  • Coupon Payments: Made at a constant rate ( c ) as a percentage of the face value.

    • Payments start in period 1 and end with the final coupon and face value at maturity in period ( n ).

  • Current Market Price: Denoted as ( P ).

  • Yield-to-maturity: Interest rate on the bond, denoted as ( Y ).

The bond’s cash flow stream

• Analysis of cash flow stream without initial investment:

  • Two components:

    • Uniform annuity flow of coupon payments for periods 1 to ( n ): ( C \cdot (1, 2, ..., n) )

    • Final payment of face value ( F ) in period ( n ): ( [0, 0, ..., 0, F] )

  • Full cash flow stream is the sum of both components:[ C \cdot (1, 2, ..., n) + [0, 0, ..., 0, F] = (C, C, ..., C + F) ]

Example

• For a bond with:

  • ( F = 100, c = 5%, n = 3 )

• The cash flow streams:

  • Coupon CFs: ( (5, 5, 5) )

  • Final Payment CFs: ( (0, 0, 100) )

  • Combined CFS: ( (5, 5, 105) )

Pricing the bond via classical theory

• Applying classical theory to bond's CFS for pricing:

  • Price equals present value of cash flows:

    • Formula: ( P = PV(coupons) + PV(face value) )

  • Present value of uniform annuity flow applied as:

    • Price Formula:[ P = C \cdot \frac{1 - (1 + Y)^{-n}}{Y} + \frac{F}{(1+Y)^{n}} ]

Relationship of Prices and Yields

• Explicit pricing formula demonstrates inverse relationship:

  • Higher YTM leads to lower prices due to being in the denominator.

Yield-to-maturity: Internal Rate of Return (IRR)

• The bond cash flow stream including initial negative investment:

  • ( CFS = (-P, C, ..., F) )

  • Computing IRR seeks the yield such that: [ 0 = -P + \sum_{t=1}^n \frac{C}{(1 + r)^t} + \frac{F}{(1+r)^n} ]

Selling the bond example

• For( F = 100, c = 5%, n = 3, Y = 2% ):

• Price when collecting first coupon: ( P_1 \approx 105.82 )

• Price when collecting second coupon: ( P_2 \approx 102.94 )

Effects on Bond Pricing

1. Collected payments reduce remaining payments.

2. Time passage brings remaining payments closer.

3. Changes in interest rates influence present value of future payments:

   • Increases yield decrease price.

   • Decreases yield increase price.

Strategies for Trading Bonds

• To profit (buy): anticipate falling interest rates.

• To profit (short selling): anticipate rising interest rates.

• Profits hinge on changes in interest rates.

Fixed Income Securities Overview

• Bonds have deterministic cash flow streams unlike equities.

• Precise knowledge of cash flows at purchase.

Understanding Prices and Yields

• Prices can be derived from yields and vice versa.

  • Yield determines price but based on prevailing market conditions.

  • Price reporting as % of face value clarifies relationship.

Conclusion and Key Takeaways

• Bond price expressed either as yield-to-maturity or investment cost ratio compared to par value.

• Trading conditions impact bond evaluation and price performance.