Final answer:
To find the present value of the bond, the present values of the annuity (annual dividends) and the lump sum (face value at maturity) are calculated using their respective formulas and then added together. The total will approximate the market value of the bond.
Step-by-step explanation:
To determine the present value of a bond that pays $270 in dividends each year and has a face value of $2,700 at the end of 20 years, we need to calculate the present value of an annuity and the present value of a lump sum, both discounted at the interest rate of 4% per year, compounded annually.
First, we address the annual dividend payments, which represent an annuity. The formula for the present value of an annuity is given by:
PV(Annuity) = C × ((1 - (1 + r)^{-n}) / r)
where C is the annual payment, r is the annual interest rate, and n is the number of years.
Using the formula above, we get:
PV(Annuity) = $270 × ((1 - (1 + 0.04)^{-20}) / 0.04)
Next, we calculate the present value of the $2,700 which will be received at the end of 20 years. The formula for the present value of a lump sum is:
PV(Lump Sum) = F / (1 + r)^n
where F is the future value of the lump sum, r is the annual interest rate, and n is the number of years.
Applying the formula:
PV(Lump Sum) = $2,700 / (1 + 0.04)^20
After calculating both present values, we sum them to find the total present value of the bond; this will approximate the bond's market value or the price an investor should be willing to pay for the bond.