What is the current value of a $1,000 bond with a 7% annual coupon rate (paid semi-annually) that matures in 7 years if the appropriate stated annual discount rate is 11%?
The Correct Answer and Explanation is:
To calculate the current value of a bond, we can use the present value formula for bonds. The current value (price) of the bond is the sum of the present value of the future cash flows, which include the annual coupon payments and the face value of the bond at maturity. Here’s how to compute it step by step:
Given Data:
- Face Value (FV): $1,000
- Coupon Rate: 7%
- Annual Coupon Payment: ( 0.07 \times 1000 = 70 )
- Semi-Annual Coupon Payment: ( \frac{70}{2} = 35 )
- Years to Maturity: 7
- Total Coupon Payments: ( 7 \times 2 = 14 )
- Stated Annual Discount Rate: 11%
- Semi-Annual Discount Rate: ( \frac{11\%}{2} = 5.5\% )
Present Value Calculation:
- Calculate Present Value of Coupon Payments:
The formula for the present value of an annuity (coupon payments) is:
[
PV_{\text{coupons}} = C \times \left(1 – (1 + r)^{-n}\right) / r
]
Where:
- ( C = 35 ) (semi-annual coupon payment)
- ( r = 0.055 ) (semi-annual discount rate)
- ( n = 14 ) (total number of payments) Plugging in the values:
[
PV_{\text{coupons}} = 35 \times \left(1 – (1 + 0.055)^{-14}\right) / 0.055
] Calculating this gives approximately:
[
PV_{\text{coupons}} \approx 35 \times 9.0996 \approx 318.49
]
- Calculate Present Value of Face Value:
The present value of the face value is calculated using:
[
PV_{\text{face}} = \frac{FV}{(1 + r)^n}
]
[
PV_{\text{face}} = \frac{1000}{(1 + 0.055)^{14}} \approx \frac{1000}{2.1131} \approx 472.12
] - Total Present Value (Current Value of the Bond):
Now, add the present values of the coupon payments and the face value:
[
PV_{\text{total}} = PV_{\text{coupons}} + PV_{\text{face}} \approx 318.49 + 472.12 \approx 790.61
]
Conclusion:
The current value of the bond is approximately $790.61.
Explanation:
The bond price reflects the present value of future cash flows discounted at the market interest rate. Because the coupon rate (7%) is lower than the market rate (11%), the bond sells for less than its face value, resulting in a discount. This calculation highlights the importance of discounting future cash flows at the correct rate to determine the bond’s market value, demonstrating how interest rates affect bond pricing.