Bond J Has A Coupon Rate Of 43 Percent Bond S Has A Coupon Rate Of 1 ✓ Solved

Bond J has a coupon rate of 4.3 percent. Bond S has a coupon rate of 14.3 percent. Both bonds have eleven years to maturity, make semiannual payments, a par value of ,000, and have a YTM of 9.6 percent. If interest rates suddenly rise by 3 percent, what is the percentage price change of these bonds? (A negative answer should be indicated by a minus sign. Do not round intermediate calculations and enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.) Percentage change in price Bond J % Bond S % If interest rates suddenly fall by 3 percent instead, what is the percentage price change of these bonds? (Do not round intermediate calculations and enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.) Percentage change in price Bond J % Bond S % -20..37

Paper for above instructions

Bond Price Sensitivity to Interest Rate Changes

Introduction

Bonds are fixed-income securities that provide investors with regular interest payments, known as coupon payments, until maturity when the principal is repaid. One crucial characteristic affecting a bond's price is its sensitivity to changes in market interest rates. When interest rates rise, bond prices typically fall, and when interest rates fall, bond prices typically rise. This concept of interest rate risk is particularly important for investors to understand, especially when analyzing bonds with different coupon rates and maturities.
In this paper, we will analyze two specific bonds: Bond J with a coupon rate of 4.3% and Bond S with a coupon rate of 14.3%, both having a face value of ,000, a maturity of 11 years, and initially a yield-to-maturity (YTM) of 9.6%. We will explore how a sudden increase or decrease in interest rates affects their prices and calculate the percentage price change for both scenarios.

Bond Pricing Calculation Formula

The price of a bond can be calculated using the present value of its future cash flows, which consist of its periodic coupon payments and its par value at maturity. The formula for calculating the price \( P \) of a bond is:
\[
P = \sum_{t=1}^{N} \frac{C}{(1 + r)^t} + \frac{F}{(1 + r)^N}
\]
Where:
- \( C \) = the semiannual coupon payment
- \( r \) = the semiannual yield to maturity (YTM)
- \( N \) = total number of periods (years to maturity times 2)
- \( F \) = par value of the bond

Bond Characteristics

1. Bond J:
- Coupon Rate: 4.3% of ,000 = (annual), hence semiannual = .50
- Years to Maturity: 11 years
- Par Value: ,000
- Initial YTM: 9.6% (semiannual = 4.8%)
- Total Periods: 11 * 2 = 22
2. Bond S:
- Coupon Rate: 14.3% of ,000 = 3 (annual), hence semiannual = .50
- Years to Maturity: 11 years
- Par Value: ,000
- Initial YTM: 9.6% (semiannual = 4.8%)
- Total Periods: 11 * 2 = 22

Price Calculation at Initial YTM (9.6%)

Bond J:

Using the bond pricing formula, we calculate Bond J’s price as follows:
\[
P_J = \sum_{t=1}^{22} \frac{21.50}{(1 + 0.048)^t} + \frac{1000}{(1 + 0.048)^{22}}
\]
Calculating this gives:
1. Present Value of Coupons:
- \( PV_{coupons} = 21.50 \sum_{t=1}^{22} \frac{1}{(1 + 0.048)^t} = 21.50 \cdot 15.234 \approx 327.22 \)
2. Present Value of Par Value:
- \( PV_{face} = \frac{1000}{(1 + 0.048)^{22}} \approx 1000 \cdot 0.523 = 523.00 \)
Thus, the price of Bond J:
\[
P_J \approx 327.22 + 523.00 \approx 850.22
\]

Bond S:

Similarly, for Bond S:
\[
P_S = \sum_{t=1}^{22} \frac{71.50}{(1 + 0.048)^t} + \frac{1000}{(1 + 0.048)^{22}}
\]
Calculating this gives:
1. Present Value of Coupons:
- \( PV_{coupons} = 71.50 \sum_{t=1}^{22} \frac{1}{(1 + 0.048)^t} = 71.50 \cdot 15.234 \approx 1,090.12 \)
2. Present Value of Par Value:
- \( PV_{face} = \frac{1000}{(1 + 0.048)^{22}} \approx 1000 \cdot 0.523 = 523.00 \)
Thus, the price of Bond S:
\[
P_S \approx 1,090.12 + 523.00 \approx 1,613.12
\]

Price Change Calculation with Interest Rate Changes

When Interest Rates Rise by 3%:

New YTM = 9.6% + 3% = 12.6% (semiannual = 6.3%)
1. Bond J Price Change:
\[
P'_J = \sum_{t=1}^{22} \frac{21.50}{(1 + 0.063)^t} + \frac{1000}{(1 + 0.063)^{22}}
\]
Calculating gives us approximately 7.83.
2. Bond S Price Change:
\[
P'_S = \sum_{t=1}^{22} \frac{71.50}{(1 + 0.063)^t} + \frac{1000}{(1 + 0.063)^{22}}
\]
Calculating gives us approximately ,179.12.
Percentage Change Calculation:
- Bond J:
\[
\frac{(677.83 - 850.22)}{850.22} \times 100 \approx -20.25\%
\]
- Bond S:
\[
\frac{(1,179.12 - 1,613.12)}{1,613.12} \times 100 \approx -27.06\%
\]

When Interest Rates Fall by 3%:

New YTM = 9.6% - 3% = 6.6% (semiannual = 3.3%)
1. Bond J Price Change:
\[
P''_J = \sum_{t=1}^{22} \frac{21.50}{(1 + 0.033)^t} + \frac{1000}{(1 + 0.033)^{22}}
\]
Calculating gives us approximately 5.97.
2. Bond S Price Change:
\[
P''_S = \sum_{t=1}^{22} \frac{71.50}{(1 + 0.033)^t} + \frac{1000}{(1 + 0.033)^{22}}
\]
Calculating gives us approximately ,582.80.
Percentage Change Calculation:
- Bond J:
\[
\frac{(985.97 - 850.22)}{850.22} \times 100 \approx 15.92\%
\]
- Bond S:
\[
\frac{(1,582.80 - 1,613.12)}{1,613.12} \times 100 \approx -1.88\%
\]

Results and Conclusion

The percentage price changes when interest rates rise by 3% and fall by 3% are as follows:
- When Interest Rates Rise by 3%:
- Bond J: -20.25%
- Bond S: -27.06%
- When Interest Rates Fall by 3%:
- Bond J: 15.92%
- Bond S: -1.88%

References

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