Bond X has a coupon of 52 percent Bond Z has a coupon of 92 percent. Both bonds have 15 years to maturity and have a YTM of 74 percent a. If interest rates suddenly rise by 1.6 percent, what is the percentage price change of these bonds? (A negative value should be indicated by a minus sign. Do not rod inte calculations. Enter your answers as a percent rounded to 2 decimal places) b. If interest rates suddenly fall by 1.6 percent, what is the percentage price change of these bonds? (Do not round intermediate calculations. Enter your answers as a percent rounded to 2 decimal places.) - What is your conclusion?
a. Using the bond price formula:
Bond X price = (52/0.74)*(1 - 1/(1+0.74)^15) = $38.13
Bond Z price = (92/0.74)*(1 - 1/(1+0.74)^15) = $67.84
When interest rates rise by 1.6%, the new YTM is 0.74+0.016=0.756.
Bond X new price = (52/0.756)*(1 - 1/(1+0.756)^15) = $33.63
Bond Z new price = (92/0.756)*(1 - 1/(1+0.756)^15) = $59.91
The percentage price change for Bond X is (33.63-38.13)/38.13 = -11.79%
The percentage price change for Bond Z is (59.91-67.84)/67.84 = -11.71%
b. When interest rates fall by 1.6%, the new YTM is 0.74-0.016=0.724.
Bond X new price = (52/0.724)*(1 - 1/(1+0.724)^15) = $43.04
Bond Z new price = (92/0.724)*(1 - 1/(1+0.724)^15) = $77.72
The percentage price change for Bond X is (43.04-38.13)/38.13 = 12.88%
The percentage price change for Bond Z is (77.72-67.84)/67.84 = 14.55%
Conclusion: Bond Z has a higher percentage price change in both scenarios, indicating that it is more sensitive to changes in interest rates. This is because it has a higher coupon rate and longer maturity compared to Bond X.
To calculate the percentage price change of a bond, we can use the formula:
Percentage Price Change = - (Duration) × (Change in Yield)
Where the Duration represents a measure of the bond's sensitivity to changes in interest rates.
a. First, we need to find the Duration of each bond. The Duration is calculated as follows:
Duration = Coupon / Yield
For Bond X:
Coupon = 52% of the face value
Yield = 74%
Duration of Bond X = 52% / 74% = 0.7027
For Bond Z:
Coupon = 92% of the face value
Yield = 74%
Duration of Bond Z = 92% / 74% = 1.2432
b. Now, let's calculate the percentage price change when interest rates rise by 1.6%.
Change in Yield = 1.6%
Percentage Price Change for Bond X = - (Duration of Bond X) × (Change in Yield)
= - (0.7027) × (1.6%) = -1.1243%
Percentage Price Change for Bond Z = - (Duration of Bond Z) × (Change in Yield)
= - (1.2432) × (1.6%) = -1.9891%
c. Next, let's calculate the percentage price change when interest rates fall by 1.6%.
Change in Yield = -1.6%
Percentage Price Change for Bond X = - (Duration of Bond X) × (Change in Yield)
= - (0.7027) × (-1.6%) = 1.1243%
Percentage Price Change for Bond Z = - (Duration of Bond Z) × (Change in Yield)
= - (1.2432) × (-1.6%) = 1.9891%
Conclusion:
Based on the calculations, we can see that both Bond X and Bond Z have a negative percentage price change when interest rates rise by 1.6%, indicating a decrease in their prices. Similarly, both bonds have a positive percentage price change when interest rates fall by 1.6%, indicating an increase in their prices. This demonstrates the inverse relationship between bond prices and interest rates.