To find the price you are willing to pay for the bond, we need to calculate the present value of the bond's future cash flows, which consist of the annual coupon payments and the face value at maturity. The formula for the price of the bond (P) can be expressed as:
\[ P = \sum_{t=1}^{n} \frac{C}{(1+r)^t} + \frac{F}{(1+r)^n} \]
Where:
- \(C\) = annual coupon payment
- \(F\) = face value of the bond
- \(n\) = number of years to maturity
- \(r\) = required rate of return (discount rate)
Given Data:
- Coupon Rate = 7.25%
- Face Value \(F\) = $1,000.00
- Required Rate of Return \(r\) = 11.50%
- Number of Years to Maturity \(n\) = 4 years
Step 1: Calculate the Annual Coupon Payment
\[ C = \text{Face Value} \times \text{Coupon Rate} \] \[ C = 1,000.00 \times 0.0725 = 72.50 \]
Step 2: Calculate Present Value of the Coupon Payments
The present value of the coupon payments can be calculated using the formula:
\[ PV(Coupons) = \sum_{t=1}^{n} \frac{C}{(1+r)^t} \]
Calculating each term:
- For \(t = 1\): \[ \frac{72.50}{(1 + 0.115)^1} = \frac{72.50}{1.115} \approx 65.04 \]
- For \(t = 2\): \[ \frac{72.50}{(1 + 0.115)^2} = \frac{72.50}{1.247225} \approx 58.15 \]
- For \(t = 3\): \[ \frac{72.50}{(1 + 0.115)^3} = \frac{72.50}{1.392894} \approx 52.00 \]
- For \(t = 4\): \[ \frac{72.50}{(1 + 0.115)^4} = \frac{72.50}{1.561833} \approx 46.40 \]
Now, sum these present values of the coupon payments:
\[ PV(Coupons) \approx 65.04 + 58.15 + 52.00 + 46.40 = 221.59 \]
Step 3: Calculate Present Value of the Face Value
Now, calculate the present value of the face value of the bond at maturity:
\[ PV(Face Value) = \frac{F}{(1+r)^n} = \frac{1,000.00}{(1 + 0.115)^4} = \frac{1,000.00}{1.561833} \approx 640.66 \]
Step 4: Calculate the Price of the Bond
Finally, the price of the bond is the sum of the present value of the coupon payments and the present value of the face value:
\[ P = PV(Coupons) + PV(Face Value) \] \[ P \approx 221.59 + 640.66 = 862.25 \]
Conclusion
The price you are willing to pay for the bond is approximately $862.25.
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