Consider Three Bonds With 6.6% Coupon Rates All Selling At F

Consider Three Bonds With 66 Coupon Rates All Selling At Face Value

Consider three bonds with 6.6% coupon rates, all selling at face value. The short-term bond has a maturity of 4 years, the intermediate-term bond has a maturity of 8 years, and the long-term bond has a maturity of 30 years.

a. What will be the price of each bond if their yields increase to 7.6%? (Do not round intermediate calculations. Round your answers to 2 decimal places.)

b. What will be the price of each bond if their yields decrease to 5.6%? (Do not round intermediate calculations. Round your answers to 2 decimal places.)

Paper For Above instruction

The valuation of bonds is fundamentally based on the present value of their future cash flows, which include periodic coupon payments and the face value repaid at maturity. When interest rates or yields change, bond prices adjust accordingly because the discount rate applied to future cash flows alters their present value. This paper examines the bond price calculations for three bonds with identical coupon rates but different maturities, under two different yield scenarios: an increase to 7.6% and a decrease to 5.6%. The analysis demonstrates how maturity length influences price sensitivity to interest rate changes, consistent with the principles outlined in bond valuation theory.

Bond price calculations are based on the present value formula, which discounts future cash flows using the current market yield. The formula for a bond's price (P) is:

\[

P = \sum_{t=1}^{n} \frac{C}{(1 + y)^t} + \frac{F}{(1 + y)^n}

\]

where:

  • C is the periodic coupon payment (Coupon Rate * Face Value)
  • y is the yield per period (expressed as a decimal)
  • F is the face value of the bond (assumed to be 100 for simplicity unless otherwise specified)
  • n is the number of periods until maturity

Given that all bonds are initially selling at face value, their coupon payments are calculated as 6.6% of face value. Assuming a face value of 100, the annual coupon payment (C) is:

\[ C = 0.066 \times 100 = 6.60 \]

### Calculations for Changing Yields

Case 1: Yields increase to 7.6% (0.076)

For each bond, we calculate as follows:

  • Short-term (4 years, n=4)
  • Intermediate-term (8 years, n=8)
  • Long-term (30 years, n=30)

4-Year Bond

Yield: 7.6% (0.076)

Price: \[ P = \sum_{t=1}^{4} \frac{6.60}{(1 + 0.076)^t} + \frac{100}{(1 + 0.076)^4} \]

Calculating each term:

\[

PV = 6.60 \times \left( \frac{1 - (1 + 0.076)^{-4}}{0.076} \right) + \frac{100}{(1 + 0.076)^4}

\]

Similarly, for the 8-year and 30-year bonds, we replace n accordingly and compute the present values.

8-Year Bond

Yield: 7.6% (0.076)

\[

P = 6.60 \times \left( \frac{1 - (1 + 0.076)^{-8}}{0.076} \right) + \frac{100}{(1 + 0.076)^8}

\]

30-Year Bond

Yield: 7.6% (0.076)

\[

P = 6.60 \times \left( \frac{1 - (1 + 0.076)^{-30}}{0.076} \right) + \frac{100}{(1 + 0.076)^{30}}

\]

Case 2: Yields decrease to 5.6% (0.056)

Repeat the same process with y=0.056 for each bond maturity.

4-Year Bond

\[

P = 6.60 \times \left( \frac{1 - (1 + 0.056)^{-4}}{0.056} \right) + \frac{100}{(1 + 0.056)^4}

\]

8-Year Bond

\[

P = 6.60 \times \left( \frac{1 - (1 + 0.056)^{-8}}{0.056} \right) + \frac{100}{(1 + 0.056)^8}

\]

30-Year Bond

\[

P = 6.60 \times \left( \frac{1 - (1 + 0.056)^{-30}}{0.056} \right) + \frac{100}{(1 + 0.056)^{30}}

\]

Calculations involve plugging in these values and computing the present values, demonstrating how the bond prices increase when yields decrease and decrease when yields increase, with longer maturities showing greater sensitivity.

Conclusion

This exercise illustrates the inverse relationship between bond prices and interest rates. Longer-term bonds are more sensitive to rate changes due to their extended exposure to interest rate risk, which is reflected in the larger fluctuations in their prices when yields change. The calculations reaffirm the principles of duration and convexity, emphasizing that bondholders and investors must consider how changes in interest rates can impact bond valuations, especially for bonds with longer maturities.

References

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