Final answer:
The market value of a bond is calculated by finding the present value of all future cash flows, including both the semi-annual interest payments and the par value at maturity. It takes into account the current yield of similar bonds and the time value of money. The given formulas help in determining the bond's price in the market.
Step-by-step explanation:
The question asks about the market value of a 20-year bond with a par value of $1000 that pays 9% annually, with interest rates for similar bonds currently at 8%. The bond pays interest semi-annually, which means that every six months, the bond pays 4.5% of its par value ($1000), resulting in a $45 payment.
To find the market value of the bond when similar bonds are yielding 8% annually (or 4% semi-annually), we need to calculate the present value of all future cash flows from the bond discounted at the new market interest rate of 4% per period.
The present value of the bond (PV) can be calculated using the formula for an annuity for the interest payments and the present value of a lump sum for the par value repaid at the end of the bond's term. The formula for the present value of an annuity is PV = Pmt [1 - 1/(1 + r)^n] / r, where Pmt is the payment each period, r is the interest rate per period, and n is the number of periods. The formula for the present value of a lump sum is PV = FV / (1 + r)^n.
The bond's interest payments are $45 every six months for 40 periods (20 years), and the lump sum repayment is the $1000 par value at the end of the term. Using the given formulas, we calculate the present value of the semi-annual payments and the repayment of the par value, then add them together to find the total current market value of the bond.