Financial Models

Question

7) A zero-coupon bond is a bond that is sold now at a discount and will pay its face value at the time when it matures no interest payments are made A zero-coupon bond can be redeemed in 20 years for $10,000. How much should you be willing to pay for it now if you want the following returns? 

(a) 12% compounded daily 

(b) 12% compounded continuously

Answer 1

(a) To find the present value of the zero-coupon bond with a 12% annual interest rate compounded daily, you can use the formula:

\[ PV = \frac{FV}{(1 + \frac{r}{n})^{nt}} \]

where:

- \( PV \) is the present value,

- \( FV \) is the future value (face value of the bond),

- \( r \) is the annual interest rate (in decimal form),

- \( n \) is the number of times interest is compounded per year, and

- \( t \) is the number of years.

For this scenario:

\[ PV = \frac{10,000}{(1 + \frac{0.12}{365})^{365 \times 20}} \]

(b) If the interest is compounded continuously, you can use the formula:

\[ PV = \frac{FV}{e^{rt}} \]

where:

- \( e \) is the base of the natural logarithm,

- \( r \) is the annual interest rate (in decimal form), and

- \( t \) is the number of years.

For this scenario:

\[ PV = \frac{10,000}{e^{0.12 \times 20}} \]

Calculate each expression to find the present value for both scenarios.