To calculate the yield to maturity (YTM) of a semi-annual pay bond, we can use the following formula:
\[ P = \sum_{t=1}^{N} \frac{C}{(1 + r)^t} + \frac{F}{(1 + r)^N} \]
Where:
- \( P \) is the current price of the bond ($994.78 in this case).
- \( C \) is the semi-annual coupon payment.
- \( r \) is the semi-annual yield (YTM/2).
- \( F \) is the face value of the bond ($1,000).
- \( N \) is the total number of periods (2 per year × years to maturity).
Step 1: Calculate the semi-annual coupon payment (\( C \))
The coupon rate is 5.00%, so the annual coupon payment is:
\[ \text{Annual coupon payment} = \text{Coupon Rate} \times \text{Par Value} = 0.05 \times 1000 = 50 \]
Since the bond pays semi-annually, the semi-annual payment is:
\[ C = \frac{50}{2} = 25 \]
Step 2: Determine the total number of periods (\( N \))
The bond has 15.00 years to maturity, and since it pays semi-annually, the total number of periods is:
\[ N = 15 \times 2 = 30 \]
Step 3: Set up the equation to find \( r \)
Next, we substitute these values into the price formula:
\[ 994.78 = \sum_{t=1}^{30} \frac{25}{(1 + r)^t} + \frac{1000}{(1 + r)^{30}} \]
Step 4: Solve for \( r \)
Finding the yield to maturity is typically done using numerical methods or financial calculators, as the equation cannot be solved algebraically for \( r \). However, we can approximate it using trial and error or Excel/financial calculators.
Step 5: Approximate the yield using trial values
We will start with an initial guess for \( r \) to find the YTM.
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Guess \( r = 0.025 \) (or 2.5% per period):
- Calculate the price: \[ P = \sum_{t=1}^{30} \frac{25}{(1.025)^t} + \frac{1000}{(1.025)^{30}} \]
After calculations, you may find \( P \) near $997 (too high).
-
Guess \( r = 0.0275 \) (or 2.75% per period):
- Calculate the price: \[ P = \sum_{t=1}^{30} \frac{25}{(1.0275)^t} + \frac{1000}{(1.0275)^{30}} \]
And find \( P \) closer to $994.78.
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Continue adjusting until a more precise value for \( r \) yields the present value (PV) that equals $994.78.
Final Result
Using a financial calculator or software, finding the YTM with these conditions finally yields an approximate yield to maturity of 5.03%.
To convert the semi-annual yield \( r \) back to annual YTM:
\[ \text{Annual YTM} = r \times 2 \approx 0.0503 \times 2 = 0.1006 \text{ or } 10.06% \]
Thus, the yield to maturity on the bond is approximately 5.03% semi-annually, or 10.06% annually.