Question

suppose that a bond has a time-to-maturity of 10 years, a coupon rate of 7% paid semi-annually, and a face value of $1,000. how much would you pay for the bond at most? $800 you answered $1,000 correct answer $1,800 infinity

Answer 1

The maximum price for the bond is approximately $1,525, which is not among the provided options. The closest option is C. $1,800, but it exceeds the calculated value.

To determine the maximum price you would be willing to pay for the bond, you can use the present value of future cash flows formula. The present value of a bond is the present value of its future cash flows, which include both the coupon payments and the face value repayment at maturity.

The formula for the present value of a bond is:

`\[PV = \left( \frac{C} {2} \right) * \left(1 - \frac{1} {(1 + r/2)^(2n)}\right) * \frac{2} {r/2} + \frac{F} {(1 + r/2)^(2n)}\]`

Where:

- C is the semi-annual coupon payment,

- r is the semi-annual yield to maturity rate,

- n is the total number of periods (in this case, 2 periods per year for 10 years, so
`\(2 * 10 = 20\)` periods), and

- F is the face value of the bond.

In this case, the coupon rate is 7%, so the semi-annual coupon payment
`(\(C\))` is
`\(0.07 * 1000 / 2 = $35\)`. The face value F is $1,000, and the number of periods n is 20.

Assuming an interest rate of 0%, the present value PV would be the sum of the present values of the future cash flows. However, as interest rates increase, the present value decreases. Therefore, the maximum price you would be willing to pay for the bond is when the interest rate is 0%.

Let's calculate the present value with an interest rate of 0%:

`\[PV = \left( \frac{35} {2} \right) * \left(1 - \frac{1} {(1 + 0)^(2 * 10)}\right) * \frac{2} {0} + \frac{1000} {(1 + 0)^(2 * 10)}\]`

Simplifying this, we get:

`\[PV = \frac{35} {2} * 10 + 1000 = 525 + 1000 = $1,525\]`

Therefore, the correct answer is not among the provided options. However, the closest option is C. $1,800.

The complete question is:

Suppose that a bond has a time-to-maturity of 10 years, a coupon rate of 7% paid semi-annually, and a face value of $1,000. How much would you pay for the bond at most?

A. $800

B. $1,000

C. $1,800

D. Infinity