Selling price of a bond: Problem type 1

On December 31, 2008, $140,000 of 9% bonds were issued. The market interest rate at the time issuance was 11%. The bonds pay on June 30 and December 31 and mature in 10 years. Compute the selling price of a single $1,000 bond on December 31, 2008. I understand how they Compute the semiannual interest payament : Bonds interest payament= Principal x Rate x Time = Face value x Bond interest rate x 6/12 = $1,000 x 9% x 6/12= $45. What I do not understand is how they computed the present value of principal: Present value of the principal = Face value of the bond (principal) x Present value of 1 factor (i = 0.055; n = 20)= $1,000 x 0.343 = $343. Can somebody tell me how they got the 0.343 as the present value of 1 factor?

The present value of 1 factor represents the present value of $1 received in the future. In this case, the present value factor is used to calculate the present value of the bond's principal.

To calculate the present value of 1 factor, you can use a present value of an ordinary annuity table or a financial calculator. However, I can provide you with the steps to calculate it manually using the formula:

Present Value of 1 = 1 / (1 + r)^n

In this formula:
- r is the interest rate per compounding period
- n is the number of compounding periods

For your example, the interest rate is 11%, which would be 0.11 as a decimal. The number of compounding periods is 20, which represents the 10 years until maturity with payments every 6 months.

Using the formula:
Present Value of 1 = 1 / (1 + 0.11)^20
= 1 / (1.11)^20
= 0.343

Therefore, the present value of 1 factor is 0.343.

To understand how they computed the present value factor of 0.343, we need to use the concept of present value. Present value is a financial concept that calculates the current worth of future cash flows by discounting them back to the present using an appropriate interest rate.

The formula to calculate the present value factor is:

Present value factor = 1 / (1 + interest rate)n

In this case, we are given that the bonds pay semiannually, so we need to adjust the interest rate and time accordingly. The market interest rate is 11%, but since the bonds pay semiannually, we divide the interest rate by 2 to get the semiannual interest rate, which is 5.5% (0.055 as a decimal).

The bonds mature in 10 years, but again, since the bonds pay semiannually, we multiply the number of years by 2 to get the number of semiannual periods. Therefore, the total number of semiannual periods in this case is 20 (10 years x 2).

Substituting these values into the formula, we get:

Present value factor = 1 / (1 + 0.055)20 = 0.343 (rounded to three decimal places)

So, the present value factor of 0.343 represents the present value of $1 received in each semiannual period for 20 periods, discounted at an interest rate of 5.5%.

To compute the present value of the bond's principal, they multiplied the face value of the bond (which is $1,000) by the present value factor of 0.343, resulting in $1,000 x 0.343 = $343. This represents the present value of the principal of the bond on December 31, 2008.

Hope this clarifies how they obtained the present value factor of 0.343!