Final answer:
The price of a semi-annual 8 percent bond will increase by approximately 4.429% due to a 50 basis points decline in interest rates. The final price can be estimated if the initial price is known. For a bond with a $1,000 face value, the new price would be around $1,044.29.
Step-by-step explanation:
The question deals with the valuation of a semi-annual 8 percent bond with a 12-year period, a Macaulay duration of 9.5 years, and a yield to maturity (YTM) of 12 percent. When interest rates decline by 50 basis points, the price of the bond is expected to increase because the bond's fixed coupon payments and terminal value are more valuable when discounted at the new lower rates.
To determine the exact percentage change in the bond's price, we can make use of the modified duration, which is closely related to Macaulay duration. Modified duration is a measure of a bond's price sensitivity to changes in interest rates and can be calculated as the Macaulay duration divided by (1 + YTM/number of coupon periods per year).
Modified Duration = Macaulay Duration / (1 + YTM / 2)
Modified Duration = 9.5 years / (1 + 0.12 / 2)
= 8.858 years (approx.)
Percentage change in price = - Modified Duration * Change in Yield
Change in Yield = 50 basis points
= 0.005
Percentage change in price = - 8.858 * (-0.005)
= 0.04429 or 4.429%
The bond will therefore increase in price by approximately 4.429% due to a 50 basis points decline in interest rates. If at the time just before the decline, the bond price was equal to its face value of $1,000 (common baseline for bond price discussions), the new price would be $1,000 * (1 + 0.04429) = $1,044.29.
Given that the price increase is based on a percentage change, we cannot definitively say if you would pay more or less than $10,000 without knowing the bond's current market price; however, we can infer that the bond price will increase.