Final answer:
The present value of a $3,000 bond with a discount rate of 8% is equal to its face value, meaning no discount. However, if the discount rate is 11%, the present value is $2846.87, resulting in a discount of $153.13 at issuance.
Step-by-step explanation:
To calculate the present value of the bond when the discount rate is 8%, we need to discount the bond's future cash flows to their present values using the 8% rate. The bond pays $240 in interest at the end of the first year and $240 in interest plus the $3,000 principal at the end of the second year.
For the first year: Present Value of Interest = $240 / (1 + 0.08) = $240 / 1.08 = $222.22
For the second year: Present Value of Interest + Principal = ($240 + $3,000) / (1 + 0.08)^2 = $3240 / 1.1664 = $2777.78
Summing up these two present values gives us the bond's present value at the 8% discount rate:
Present Value = $222.22 + $2777.78 = $3000
There is no discount at this rate since the bond's present value is equal to its face value.
Now, if the discount rate rises to 11%, the calculations change accordingly:
For the first year: Present Value of Interest = $240 / (1 + 0.11) = $240 / 1.11 = $216.22
For the second year: Present Value of Interest + Principal = ($240 + $3,000) / (1 + 0.11)^2 = $3240 / 1.2321 = $2630.65
Therefore, the present value at an 11% discount rate is:
Present Value = $216.22 + $2630.65 = $2846.87
Since the face value of the bond is $3,000, the bond is issued at a discount of $3,000 - $2846.87 = $153.13.
The finally, considering the mentioned change in interest rates, one would expect to pay less than the face value for the bond due to increased market interest rates, hence a higher discount rate leading to a lower present value.