What must be the price of a $10,000 bond with a 6.5% coupon rate, semi-annual coupons, and two years to maturity if it has a yield to maturity of 8% APR?

9819.74

9727.76

To calculate the price of a bond, you can use the formula:

P = (C / (1 + r/n)) + (C / (1 + r/n)^2) + ... + (C / (1 + r/n)^N) + (M / (1 + r/n)^N)

Where:
P = price of the bond
C = coupon payment
r = yield to maturity (expressed as a decimal)
n = number of coupon payments per year
N = total number of coupon payments until maturity
M = maturity value of the bond

In this case, the bond has a coupon rate of 6.5% or 0.065, a yield to maturity of 8% or 0.08, semi-annual coupons, and a maturity of two years.

First, calculate the coupon payment:
Coupon payment = Coupon rate * Face value
= 0.065 * $10,000
= $650

Next, calculate the number of coupon payments until maturity:
Number of coupon payments until maturity = Number of years * Number of coupon payments per year
= 2 * 2
= 4

Now, plug these values into the formula and calculate the price of the bond:

P = (650 / (1 + 0.08/2)) + (650 / (1 + 0.08/2)^2) + (650 / (1 + 0.08/2)^3) + (650 / (1 + 0.08/2)^4) + (10,000 / (1 + 0.08/2)^4)

Using a financial calculator or spreadsheet software, evaluate this expression:

P = 650 / (1 + 0.04) + 650 / (1 + 0.04)^2 + 650 / (1 + 0.04)^3 + 650 / (1 + 0.04)^4 + 10,000 / (1 + 0.04)^4

The calculated value of P will give you the price of the bond.

To find the price of a bond, you need to calculate the present value of its future cash flows, which include the periodic coupon payments and the final principal payment. Here's how you can calculate the price of the bond:

Step 1: Determine the periodic coupon payment:
The coupon rate is 6.5%, which means the bond pays 6.5% of its face value as a coupon payment annually. Since the coupons are paid semi-annually, the periodic coupon payment is calculated as follows:
Coupon payment = (Coupon rate * Face value) / 2

Given that the face value of the bond is $10,000, the periodic coupon payment would be:
Coupon payment = (6.5% * $10,000) / 2 = $325

Step 2: Calculate the number of coupon payments:
The bond has a two-year maturity period, and since coupons are paid semi-annually, the number of coupon payments would be:
Number of coupon payments = 2 * 2 = 4

Step 3: Determine the discount rate:
The yield to maturity (YTM) of the bond is given as 8% APR (Annual Percentage Rate). However, since the bond pays semi-annual coupons, the semi-annual yield to maturity needs to be calculated:
Semi-annual YTM = YTM / 2 = 8% / 2 = 4%

Step 4: Calculate the present value of the coupon payments:
To calculate the present value of each coupon payment, we need to discount it to the present using the semi-annual yield to maturity. The formula to calculate the present value of an annuity is as follows:
Present value of an annuity = (Coupon payment * (1 - (1 + r)^(-n))) / r

Where:
- Coupon payment = periodic coupon payment
- r = semi-annual yield to maturity
- n = number of coupon payments

In this case, the formula becomes:
Present value of the coupon payments = (Coupon payment * (1 - (1 + r)^(-n))) / r
= ($325 * (1 - (1 + 0.04)^(-4))) / 0.04

Using a financial calculator or spreadsheet, the present value of the coupon payments is approximately $1,158.47.

Step 5: Calculate the present value of the principal payment:
Since the bond will mature in 2 years, the principal payment is equal to the face value of the bond. To calculate the present value of the principal payment, we use the formula for the present value of a single future cash flow:
Present value of the principal payment = Face value / (1 + r)^n

In this case, the present value of the principal payment would be:
Present value of the principal payment = $10,000 / (1 + 0.04)^4

Using a financial calculator or spreadsheet, the present value of the principal payment is approximately $8,848.98.

Step 6: Calculate the bond price:
The bond price is determined by adding the present values of the coupon payments and the principal payment. Therefore, the bond price is:
Bond price = Present value of the coupon payments + Present value of the principal payment
= $1,158.47 + $8,848.98

The calculated bond price is approximately $10,007.45.

Therefore, the price of the bond would be approximately $10,007.45.