To calculate the effective annual yield (EAY) of the bond, we first need to determine the bond's cash flows and then calculate the yield to maturity (YTM), which we can convert to EAY.
Step 1: Calculate Cash Flows
The bond has a coupon rate of 5.00%, and since it pays semi-annually, the coupon payment will be: \[ \text{Coupon payment} = \frac{\text{Coupon rate} \times \text{Par value}}{\text{Number of payments per year}} = \frac{0.05 \times 1000}{2} = 25 \]
The bond makes semi-annual payments for 10 years, which gives us a total of: \[ \text{Total payments} = 10 \text{ years} \times 2 = 20 \text{ payments} \]
Step 2: Determine the Yield to Maturity (YTM)
The bond's price ($P$) can be expressed as the present value of its future cash flows: \[ P = \sum_{t=1}^{n} \frac{C}{(1 + r)^t} + \frac{F}{(1 + r)^n} \] Where:
- \(C\) = Coupon payment = $25
- \(F\) = Face value = $1,000
- \(n\) = Total number of payments = 20
- \(P\) = Market price of the bond = $986.50
- \(r\) = Yield per period (semi-annual)
Our equation becomes: \[ 986.50 = \sum_{t=1}^{20} \frac{25}{(1 + r)^t} + \frac{1000}{(1 + r)^{20}} \]
Step 3: Solve for \(r\)
This equation does not lend itself to straightforward algebraic manipulation, so we typically need to use numerical methods or a financial calculator to find \(r\).
Using a financial calculator or software to solve for \(r\):
- Input \(N = 20\) (total semi-annual periods),
- \(PV = -986.50\) (present value is negative because it's an outflow),
- \(PMT = 25\) (semi-annual coupon),
- \(FV = 1000\).
The resulting \(r\) (YTM per semi-annual period) will be approximately around 0.025 or 2.5%.
Step 4: Convert to Effective Annual Yield (EAY)
The relationship between the nominal yield per period and the effective annual yield is given by: \[ \text{EAY} = (1 + r)^{m} - 1 \] Where \(m\) is the number of compounding periods per year. For semi-annual compounding, \(m = 2\): \[ \text{EAY} = (1 + 0.025)^{2} - 1 \] Calculating this: \[ \text{EAY} = (1.025)^{2} - 1 \approx 1.050625 - 1 \approx 0.050625 \]
Convert to Percentage
\[ \text{EAY} \approx 0.050625 \times 100 \approx 5.06% \]
Final Answer:
The effective annual yield (EAY) is approximately 5.06%.