When evaluating the present value of bonds, it is essential to understand the relationship between the bond's cash flows and the current market interest rates. In this example, a company issues $100,000 of 10% bonds due in 10 years, with interest payments made semiannually. The market interest rate is 8%, which is lower than the bond's stated interest rate of 10%. This discrepancy means the bonds will sell for more than their face value.

The cash interest payment for the bonds is calculated using the stated interest rate. Since the bonds pay interest semiannually, the cash interest payment every six months is determined as follows:

Cash Interest = Face Value × Stated Rate × (1/2) = $100,000 × 0.10 × (1/2) = $5,000.

This results in a total annual interest payment of $10,000.

To find the present value of the bond, we need to consider two components: the present value of the annuity (interest payments) and the present value of the lump sum (principal repayment). The timeline for cash flows consists of 20 semiannual periods (10 years × 2), with $5,000 paid every six months and a $100,000 principal payment at the end of the 20 periods.

For the annuity, we use the market interest rate of 8%, which translates to 4% per semiannual period. The present value factor for an annuity can be found in present value tables. For 20 periods at 4%, the factor is approximately 13.590. Thus, the present value of the annuity is calculated as:

Present Value of Annuity = Cash Payment × Present Value Factor = $5,000 × 13.590 = $67,950.

Next, we calculate the present value of the principal payment. The present value factor for a lump sum at 20 periods and 4% is approximately 0.456. Therefore, the present value of the principal is:

Present Value of Principal = Face Value × Present Value Factor = $100,000 × 0.456 = $45,600.

Finally, to find the total present value of the bond, we add the present value of the annuity and the present value of the principal:

Total Present Value = Present Value of Annuity + Present Value of Principal = $67,950 + $45,600 = $113,550.

This means that the current market value of the bonds is $113,550, reflecting the higher interest payments compared to the market rate. Investors are willing to pay a premium for these bonds because they offer a better return than what is currently available in the market.

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