finance 200

Bond Pricing and Yield

• Cash Flow Structure:

  • Bonds have coupon payments made at regular intervals for a specified number of periods (t).

  • At maturity, the bond pays the final coupon plus the principal amount (face value).

• Investor Considerations:

  • When investing in a bond, understanding the yield is crucial.

  • Spot rates can vary and significantly impact the bond's yield.

• Yield Calculation:

  • Average spot rate:

    • Formula:[ \text{Average} = \frac{R_1 + R_2 + \ldots + R_t}{t} ]

    • This can be a simplistic approach, which may not account for differing cash flow magnitudes or timings.

Example Scenario

• Term Structure of Interest Rates:

  • Given rates for 4 years:

    • R1 = 2%

    • R2 = 4%

    • R3 = 4.3%

    • R4 = 6%

• Bond Cash Flows:

  • Annual coupon payment of $50 (5% of $1000).

  • Cash flows: Year 1: $50, Year 2: $50, Year 3: $50, Year 4: $1050.

• Discounting Cash Flows:

  • Highest present value cash flows affect bond yield.

  • To price the bond, each cash flow needs to be discounted:

    • Year 1: ( \frac{50}{(1 + 0.02)^1} )

    • Year 2: ( \frac{50}{(1 + 0.04)^2} )

    • Year 3: ( \frac{50}{(1 + 0.043)^3} )

    • Year 4: ( \frac{1050}{(1 + 0.06)^4} )

  • Total bond price: $970.38.

Yield to Maturity (YTM)

• Concept of YTM:

  • YTM represents the constant interest rate that equates the present value of cash flows to the bond price.

  • Can be computed using iterative methods or financial calculators.

• Example Calculation:

  • YTM was calculated to be 5.85%, which is closer to the highest cash flow spot rate (6%) than other lower rates.

  • This indicates larger cash flows are significantly more pertinent to yield calculations than smaller cash flows.

Duration

• Definition of Duration:

  • A measure of the sensitivity of a bond's price to changes in interest rates, reflecting the average time to receive the bond's cash flows.

• Importance of Duration:

  • Duration provides a better risk measure than maturity because it accounts for the timing of cash flows.

• Duration Calculation:

  • Formula:[ ext{Duration} = \frac{1\cdot CF_1 + 2\cdot CF_2 + \ldots + t \cdot CF_t}{\text{Price}} ]where CF is the cash flow at each period.

• Example Calculation:

  • For two bonds with cash flows in different periods, the duration was calculated to be:

    • Bond A: Duration = 1.988 years

    • Bond B: Duration = 1.008 years

  • Despite both bonds having a maturity of two years, their durations reflect their cash flow distributions, showing their time value differently than mere maturity.

Conclusion

• Key Takeaways:

  • When evaluating bonds, focus on cash flow distributions over the time period.

  • YTM and Duration provide critical insights into the bond's performance, potential return, and risk.