A $1,000 bond with a coupon rate of 5.4% paid semi-annually has five years to maturity and a yield to maturity of 7.5%. If interest rates rise and the yield to maturity increases to 7.8%, what will happen to the price of the bond? a. Fall by $11.59 b. Rise by $12.16 c. The price of the bond will not change. d. Fall by $9.82 e. None of them.
Question
A $1,000 bond with a coupon rate of 5.4% paid semi-annually has five years to maturity and a yield to maturity of 7.5%. If interest rates rise and the yield to maturity increases to 7.8%, what will happen to the price of the bond?
a. Fall by $11.59
b. Rise by $12.16
c. The price of the bond will not change.
d. Fall by $9.82
e. None of them.
Solution
The correct answer is a. Fall by $11.59.
Here's the step-by-step calculation:
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First, calculate the bond's price at the initial yield to maturity of 7.5%. The formula for the price of a bond is P = C * (1 - (1 + r/n)^(nt)) / (r/n) + F / (1 + r/n)^(nt), where C is the semi-annual coupon payment, r is the semi-annual yield to maturity (in decimal form), n is the number of compounding periods per year, t is the number of years to maturity, and F is the face value of the bond.
- In this case, C = 27, r = 7.5% / 2 = 0.0375, n = 2, t = 5, and F = $1,000.
- Plugging these values into the formula gives P = 1,000 / (1 + 0.0375)^(2*5) = $880.41.
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Next, calculate the bond's price at the new yield to maturity of 7.8%. The only change is that r = 7.8% / 2 = 0.039.
- Plugging these values into the formula gives P = 1,000 / (1 + 0.039)^(2*5) = $868.82.
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The change in the price of the bond is the new price minus the initial price, which is 880.41 = -$11.59.
Therefore, if interest rates rise and the yield to maturity increases to 7.8%, the price of the bond will fall by $11.59.
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