To calculate the effective annual yield of a semi-annual pay bond, we first need to determine the yield to maturity (YTM) based on the bond's price, coupon rate, and time to maturity. Here's how to do that:
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Identify the bond details:
- Market Price (P) = $988.75
- Par Value (FV) = $1,000
- Coupon Rate = 7.00% (this means a coupon payment of $70 per year, or $35 every six months)
- Years to Maturity (N) = 10.00 years (which means there are 20 semi-annual periods)
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Calculate the semi-annual coupon payment: \[ \text{Semi-Annual Coupon Payment} = \frac{7% \times 1000}{2} = 35 \]
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Set up the equation for the bond price: The price of the bond can be expressed as the present value of future cash flows, which include the semi-annual coupon payments and the par value at maturity: \[ P = \sum_{t=1}^{20} \frac{C}{(1 + r)^t} + \frac{FV}{(1 + r)^{20}} \] where \( C \) is the semi-annual coupon payment, \( FV \) is the par value, \( r \) is the semi-annual yield (YTM/2), and \( t \) is the period.
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Using trial and error (or a financial calculator), we can solve for \( r \). We know that: \[ 988.75 = \sum_{t=1}^{20} \frac{35}{(1 + r)^t} + \frac{1000}{(1 + r)^{20}} \]
For simplification, we can determine \( r \) approximately. Let's try a few values:
- For \( r = 0.035 \) (or 3.5% per period): \[ P \approx \frac{35 \times (1 - (1 + 0.035)^{-20})}{0.035} + \frac{1000}{(1 + 0.035)^{20}} \] Calculating:
- First term (coupons): \[ PV(\text{coupons}) \approx \frac{35 \times (1 - (1 + 0.035)^{-20})}{0.035} = \frac{35 \times (1 - (1.035)^{-20})}{0.035} \approx 35 \times 14.478 \approx 507.83 \]
- Second term (par value): \[ PV(\text{par}) \approx \frac{1000}{(1.035)^{20}} \approx \frac{1000}{1.979} \approx 505.02 \]
Combining these gives: \[ P \approx 507.83 + 505.02 \approx 1012.85 \quad (\text{too high}) \]
After testing several interest rates (using a financial calculator or spreadsheet for precision), you may converge toward approximately \( r \approx 0.0375 \) (or 3.75%).
- Calculate the Effective Annual Yield (EAY): The effective annual yield can be calculated using the formula: \[ EAY = (1 + r)^n - 1 \] where \( n \) is the number of compounding periods per year. In this case, there are 2 compounding periods per year (since it's semi-annual): \[ EAY = (1 + 0.0375)^2 - 1 \approx (1.0375)^2 - 1 \approx 0.0764 \text{ or } 7.64% \]
Thus, the effective annual yield is approximately:
\[ \boxed{7.64%} \]
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