To determine the maximum amount Sammy should pay for the bond, we need to calculate the present value of the bond's future cash flows. The bond pays semi-annual coupons, and we will use the current market interest rate to discount those cash flows.
Given Information:
- Face Value (FV) = $1,000
- Coupon Rate = 5.84% per year
- Maturity = 10.00 years
- Current Market Rate = 4.08% per year
- Coupons are paid semi-annually.
Step 1: Calculate the semi-annual coupon payment.
The annual coupon payment is calculated as follows: \[ \text{Annual Coupon Payment} = \text{Face Value} \times \text{Coupon Rate} \] \[ \text{Annual Coupon Payment} = 1000 \times 0.0584 = 58.40 \]
Since the coupons are paid semi-annually, the semi-annual coupon payment is: \[ \text{Semi-Annual Coupon Payment} = \frac{58.40}{2} = 29.20 \]
Step 2: Determine the number of periods and the semi-annual market rate.
- Number of Years = 10
- Since the coupons are paid semi-annually, total periods (N) = \(10 \times 2 = 20\)
- Semi-annual Market Rate = \( \frac{4.08%}{2} = 2.04% = 0.0204\)
Step 3: Calculate the present value of the coupon payments.
The present value of an annuity (the coupon payments) can be calculated using the formula: \[ PV(\text{Coupons}) = C \times \left(1 - (1 + r)^{-n}\right) / r \] Where:
- \(C\) = Coupon payment per period = $29.20
- \(r\) = Semi-annual market rate = 0.0204
- \(n\) = Total number of periods = 20
Calculating \(PV(\text{Coupons})\): \[ PV(\text{Coupons}) = 29.20 \times \left(1 - (1 + 0.0204)^{-20}\right) / 0.0204 \] \[ PV(\text{Coupons}) = 29.20 \times \left(1 - (1.0204)^{-20}\right) / 0.0204 \]
Calculating \( (1.0204)^{-20} \): \[ (1.0204)^{-20} \approx 0.6667 \]
Thus: \[ 1 - (1.0204)^{-20} \approx 1 - 0.6667 = 0.3333 \] Now substitute back: \[ PV(\text{Coupons}) = 29.20 \times \left(0.3333\right) / 0.0204 \] \[ PV(\text{Coupons}) \approx 29.20 \times 16.333 = 476.24 \]
Step 4: Calculate the present value of the face value.
The present value of the face value is calculated using the formula: \[ PV(\text{Face Value}) = \frac{FV}{(1 + r)^n} \] Where:
- \(FV\) = Face Value = $1,000
- \(r\) = 0.0204
- \(n\) = 20
Calculating: \[ PV(\text{Face Value}) = \frac{1000}{(1.0204)^{20}} \] Calculating \( (1.0204)^{20} \): \[ (1.0204)^{20} \approx 1.4859 \] So: \[ PV(\text{Face Value}) = \frac{1000}{1.4859} \approx 673.47 \]
Step 5: Total Present Value of the Bond
Now, add the present value of the coupon payments and the present value of the face value: \[ PV(\text{Bond}) = PV(\text{Coupons}) + PV(\text{Face Value}) \] \[ PV(\text{Bond}) = 476.24 + 673.47 \approx 1149.71 \]
Conclusion
The maximum amount Sammy should pay for the bond is approximately $1,149.71.
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