Final answer:
To find the coupon rate of a bond, an equation relating the bond's current price, par value, YTM, and number of periods is used. By solving this equation, the annual coupon payment is found and divided by the bond's face value to obtain the annualized coupon rate, which is then converted to a decimal with four places.
Step-by-step explanation:
The student has asked about calculating the coupon rate for a bond given its current price, yield to maturity (YTM), par value, and the number of years remaining until maturity. The bond also pays semi-annual coupons, and we are interested in the annualized coupon rate. The task involves a bit of financial mathematics where we use the concepts of present value and the price of a bond to figure out the coupon payments.
To find the coupon rate, we need to set up the equation for the present value of a bond, which equates the current bond price to the sum of the present value of all future cash flows, including semi-annual coupon payments (C/2) and the face value (FV) which is received at maturity. This is mathematically shown as:
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- The current bond price (P): $1,080
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- Par value of the bond (FV): $1,000
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- Years to maturity: 9
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- Yield to maturity (YTM): 6.5%
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- Number of periods (semi-annually): 9 years * 2 = 18 periods
Using the above information, the coupon rate can be found by solving the formula for the current price of the bond
P = (C/2) * [1 - (1 + (r/2))^(-n)] / (r/2) + (FV) / (1 + (r/2))^n
Where:
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- P is the current price of the bond
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- C is the annual coupon payment
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- r is the YTM
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- n is the total number of coupon payments left
By substituting the given values and solving for C (through iterative methods or using financial calculators), we can find the coupon payment and hence the coupon rate by dividing the coupon payment by the bond's face value.
Since we are interested in the annualized coupon rate and the bond pays semi-annually, we need to double the semi-annual coupon rate obtained from the equation to get the annual rate as a decimal, which can then be reported to four decimal places.