Compute the price of a $1,000 par value, 8 percent (semi-annual payment) coupon bond with 21 years remaining until maturity assuming that the bond's yield to maturity is 14 percent? (Round your answer to 2 decimal places and record your answer without dollar sign or commas).
The price of the bond can be calculated using the following formula:
Price = (Coupon payment / (1 + Yield)^Time) + (Coupon payment / (1 + Yield)^(Time + 1)) + ... + (Coupon payment + Par value / (1 + Yield)^(Time + n))
Where:
Coupon payment = Par value x Coupon rate / 2
Yield = Yield to maturity / 2
Time = Years remaining until maturity x 2
n = Number of coupon payments remaining until maturity (42 in this case)
Plugging in the given values:
Coupon payment = 1,000 x 8% / 2 = 40
Yield = 14% / 2 = 7
Time = 21 x 2 = 42
Price = (40 / (1 + 7%)^2) + (40 / (1 + 7%)^3) + ... + (40 + 1,000 / (1 + 7%)^42)
Price = $482.80
Therefore, the price of the bond is $482.80.
To compute the price of the bond, we need to use the present value formula for a bond. The formula is:
\[ Price = \left(\frac{C}{(1 + r)^{n}}\right) \times \left(1 - \frac{1}{(1 + r)^{n}}\right) \times \left(\frac{1}{r}\right) + \left(\frac{M}{(1 + r)^{n}}\right) \]
Where:
- C = Coupon payment received every period (semi-annual in this case) = $1,000 * 8% / 2 = $40
- r = Yield to maturity = 14% / 2 = 0.07 (converted to a decimal)
- n = Number of periods remaining until maturity = 21 years * 2 = 42 (converted to semi-annual periods)
- M = Face value or par value of the bond = $1,000
Let's substitute these values into the formula:
\[ Price = \left(\frac{40}{(1 + 0.07)^{42}}\right) \times \left(1 - \frac{1}{(1 + 0.07)^{42}}\right) \times \left(\frac{1}{0.07}\right) + \left(\frac{1,000}{(1 + 0.07)^{42}}\right) \]
Now let's calculate:
\[ Price = \left(\frac{40}{1.07^{42}}\right) \times \left(1 - \frac{1}{1.07^{42}}\right) \times \left(\frac{1}{0.07}\right) + \left(\frac{1,000}{1.07^{42}}\right) \]
Using a calculator, we find that the price of the bond is approximately $483.78.