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Answer :
Final answer:
The price of a 10 year, 7.5% municipal bond quoted on a 6.00 basis would be above 101 and less than 105 since its coupon rate is between those of bonds with known prices and is above the market interest rate.
Explanation:
When considering the price of a bond, the prevailing interest rates in the market are a crucial factor. If the market interest rate is higher than the bond's coupon rate, the bond will sell for less than its face value. Conversely, if the market interest rate is lower, the bond will sell at a premium.
In the scenario where a 10 year 7.5% municipal bond is quoted on a 6.00 basis, we can infer its price based on the given information about other bonds. A bond with an 8% rate is priced at 105 (above par) and a bond with a 7% rate is at 101 (above par but below the 8% bond), as they offer a higher return than the market rate of 6%. Since 7.5% is in between these two rates, we can estimate that the bond in question will also be priced above par but less than the 8% bond since its coupon rate is closer to the market rate. Thus, the fair price would likely be slightly above 101 but less than 105.
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The price of a 10-year, 7.5% municipal bond, quoted on a 6.00 basis, would be approximately $102.99.
To solve this problem, we need to use the concept of bond pricing and yield. Bond pricing is the process of calculating the fair value of a bond, which is the present value of all future cash flows, including interest payments and principal repayment. Yield, on the other hand, is the rate of return earned by the investor on the bond.
In this case, we are given two bonds with different coupon rates and prices, but the same maturity and yield basis. We can use these bonds to calculate the yield, which we can then use to price the third bond.
To calculate the yield of the first bond, we can use the following formula:
[tex]\mathrm{Price} = \frac{\mathrm{Coupon\ Payment}}{\mathrm{Yield}} \times \left(1 - \frac{1}{(1 + \mathrm{Yield})^N}\right) + \frac{\mathrm{Face\ Value}}{(1 + \mathrm{Yield})^N}[/tex]
Where Price is the current market price of the bond, Coupon Payment is the annual interest payment, Yield is the annual yield, N is the number of years to maturity, and Face Value is the principal amount.
Using the values given in the problem, we can solve for the yield of the first bond as follows:
[tex]105 = \frac{0.08 \times \mathrm{Face\ Value}}{\mathrm{Yield}} \times \left(1 - \frac{1}{(1 + \mathrm{Yield})^{10}}\right) + \frac{\mathrm{Face\ Value}}{(1 + \mathrm{Yield})^{10}}[/tex]
Solving for Yield using a financial calculator or iterative process, we get Yield = 6.5908%
Similarly, we can calculate the yield of the second bond as follows:
[tex]\begin{equation}101 = \frac{0.07 \times \mathrm{Face\ Value}}{\mathrm{Yield}} \times \left(1 - \frac{1}{(1 + \mathrm{Yield})^{10}}\right) + \frac{\mathrm{Face\ Value}}{(1 + \mathrm{Yield})^{10}}\end{equation}[/tex]
Solving for Yield, we get Yield = 6.7328%
Now, we can use the yield of the first two bonds to price the third bond. Since the third bond has a coupon rate of 7.5%, we can assume that its price will be somewhere between the prices of the first two bonds. Using linear interpolation, we can estimate the price of the third bond as follows:
Price of 7.5% bond = Price of 7% bond + (Coupon Difference / Yield of 7% bond) * (Price of 8% bond - Price of 7% bond)
Where Coupon Difference is the difference in coupon rates between the second and third bonds.
Plugging in the values, we get:
Price of 7.5% bond = 101 + (0.5 / 6.7328%) * (105 - 101) = 102.99
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