Question
given interest 20% ,cash payment 4100 per year, calculate maturity on this $1000 ten year coupon bond, and explain the result
Answers
GPT-4o mini
To calculate the maturity of a coupon bond, we first need to determine the annual coupon payment, the interest rate, and then use this information to find out the bond's present value in relation to the cash payment.
### Bond Details:
1. **Face Value (F)**: $1,000
2. **Interest Rate (i)**: 20% (0.20 as a decimal)
3. **Annual Cash Payment (C)**: $4,100
4. **Number of Years (n)**: 10
### Understanding a Coupon Bond:
A coupon bond typically pays a fixed amount (interest) to the bondholder annually until maturity, when the face value is repaid. However, in this case, the annual cash payment ($4,100) is significantly higher than what would normally be expected from a face value of $1,000 with a 20% interest rate.
### Calculation of the Annual Coupon Payment:
The expected annual coupon payment for a bond with a face value of $1,000 and a 20% interest rate would be:
\[ C = F \times i \]
\[ C = 1000 \times 0.20 = 200 \]
### Result Comparison:
Given that the bond is paying $4,100 annually, which is vastly greater than the expected coupon payment of $200, this bond does not behave like a traditional coupon bond. Instead, it seems more like a situation where the bond is being sold at a high premium, possibly because it has some additional benefits or features (like a convertible option, etc.).
### Present Value Calculation:
To better understand the value of this bond, you could calculate the present value of receiving $4,100 a year for 10 years discounted at 20%.
The formula for the present value of an annuity is:
\[
PV = C \times \frac{(1 - (1 + i)^{-n})}{i}
\]
Where:
- \( PV \) = Present Value
- \( C \) = Cash payment per period ($4,100)
- \( i \) = Interest rate (0.20)
- \( n \) = Number of periods (10)
Substituting the values into the formula:
\[
PV = 4100 \times \frac{(1 - (1 + 0.20)^{-10})}{0.20}
\]
\[
PV = 4100 \times \frac{(1 - (1 + 0.20)^{-10})}{0.20}
\]
Calculating \( (1 + 0.20)^{-10} \):
\[
= (1.20)^{-10} \approx 0.1615
\]
Now plug this back into the PV formula:
\[
PV = 4100 \times \frac{(1 - 0.1615)}{0.20}
\]
\[
= 4100 \times \frac{0.8385}{0.20}
\]
\[
= 4100 \times 4.1925 \approx 17189.25
\]
This means the present value of the cash flows ($4,100 per year for 10 years) at a discount rate of 20% is approximately $17,189.25.
### Conclusion:
The bond appears to yield a very high cash payment, translating into a present value far exceeding its face value, indicating that it offers a much greater return than that of a typical bond when values are compared. Therefore, investors would be inclined to take such a bond for its higher cash payments.
Additionally, if it matures after 10 years, the bondholder also returns the principal of $1,000 at maturity. The bond’s cash flow and the discount rate imply it could be highly valuable under specific contexts or might be very risky, justifying the high return.
### Bond Details:
1. **Face Value (F)**: $1,000
2. **Interest Rate (i)**: 20% (0.20 as a decimal)
3. **Annual Cash Payment (C)**: $4,100
4. **Number of Years (n)**: 10
### Understanding a Coupon Bond:
A coupon bond typically pays a fixed amount (interest) to the bondholder annually until maturity, when the face value is repaid. However, in this case, the annual cash payment ($4,100) is significantly higher than what would normally be expected from a face value of $1,000 with a 20% interest rate.
### Calculation of the Annual Coupon Payment:
The expected annual coupon payment for a bond with a face value of $1,000 and a 20% interest rate would be:
\[ C = F \times i \]
\[ C = 1000 \times 0.20 = 200 \]
### Result Comparison:
Given that the bond is paying $4,100 annually, which is vastly greater than the expected coupon payment of $200, this bond does not behave like a traditional coupon bond. Instead, it seems more like a situation where the bond is being sold at a high premium, possibly because it has some additional benefits or features (like a convertible option, etc.).
### Present Value Calculation:
To better understand the value of this bond, you could calculate the present value of receiving $4,100 a year for 10 years discounted at 20%.
The formula for the present value of an annuity is:
\[
PV = C \times \frac{(1 - (1 + i)^{-n})}{i}
\]
Where:
- \( PV \) = Present Value
- \( C \) = Cash payment per period ($4,100)
- \( i \) = Interest rate (0.20)
- \( n \) = Number of periods (10)
Substituting the values into the formula:
\[
PV = 4100 \times \frac{(1 - (1 + 0.20)^{-10})}{0.20}
\]
\[
PV = 4100 \times \frac{(1 - (1 + 0.20)^{-10})}{0.20}
\]
Calculating \( (1 + 0.20)^{-10} \):
\[
= (1.20)^{-10} \approx 0.1615
\]
Now plug this back into the PV formula:
\[
PV = 4100 \times \frac{(1 - 0.1615)}{0.20}
\]
\[
= 4100 \times \frac{0.8385}{0.20}
\]
\[
= 4100 \times 4.1925 \approx 17189.25
\]
This means the present value of the cash flows ($4,100 per year for 10 years) at a discount rate of 20% is approximately $17,189.25.
### Conclusion:
The bond appears to yield a very high cash payment, translating into a present value far exceeding its face value, indicating that it offers a much greater return than that of a typical bond when values are compared. Therefore, investors would be inclined to take such a bond for its higher cash payments.
Additionally, if it matures after 10 years, the bondholder also returns the principal of $1,000 at maturity. The bond’s cash flow and the discount rate imply it could be highly valuable under specific contexts or might be very risky, justifying the high return.