A $10,000 bond with a coupon rate of 4.55% was redeemable on July 01, 2031. It was purchased on February 10, 2025 when the yield rate was 4.80% compounded semi- annually. Please include a well-labelle
d timeline.
To calculate the price of the bond, we need to discount all future cash flows to present value using the given yield rate of 4.80% compounded semi-annually. The bond has a coupon rate of 4.55% and a face value of $10,000.
Timeline:
February 10, 2025 - Purchase Date
July 01, 2031 - Redemption Date
To calculate the bond price, we need to determine the cash flows from coupons and the principal at redemption.
Coupon Cash Flows:
The bond pays a coupon of 4.55% of the face value semi-annually.
Number of coupon payments = (2031 - 2025) * 2 = 12
Coupon Payment = Coupon Rate * Face Value / 2 = 0.0455 * $10,000 / 2 = $227.50
Principal Cash Flow:
The bond redeems at the face value of $10,000 on July 01, 2031.
Now we can calculate the present value of each cash flow and sum them up to find the bond price.
Step 1: Calculate the discount rate per period
Yield Rate per Period = Yield Rate / 2 = 4.80% / 2 = 2.40%
Step 2: Calculate the present value of each cash flow
Present Value of Coupon Payments = Σ(Coupon Payment / (1 + Yield Rate per Period)^n) for n = 1 to 12
Present Value of Coupon Payments = Σ($227.50 / (1 + 2.40%)^n) for n = 1 to 12
Present Value of Principal = $10,000 / (1 + 2.40%)^14
Step 3: Sum up the present values of cash flows
Bond Price = Present Value of Coupon Payments + Present Value of Principal
The present value calculations need to be done manually, but you can use a financial calculator or spreadsheet software to make it easier.
Once you find the bond price, you can compare it to the market price to determine if it is a good investment.
To calculate the price of the bond, we need to determine the present value of the future cash flows.
Step 1: Determine the number of periods until redemption
The bond was purchased on February 10, 2025, and is redeemable on July 01, 2031. Therefore, the bond will be held for 6.37 years (from 2025 to 2031).
Step 2: Determine the semi-annual yield rate
The annual yield rate is given as 4.80%. We need to convert it to a semi-annual rate by dividing it by 2, giving 2.40%.
Step 3: Determine the number of semi-annual periods
Since the bond pays coupons semi-annually, the total number of periods will be twice the number of years. In this case, since the bond is held for 6.37 years, there will be 12.74 semi-annual periods.
Step 4: Calculate the semi-annual coupon payment
The coupon rate is given as 4.55% on a $10,000 bond, so the annual coupon payment will be $455. Since the bond pays coupons semi-annually, the semi-annual coupon payment will be $227.50 ($455 divided by 2).
Step 5: Calculate the present value of the future cash flows
To determine the price of the bond, we need to calculate the present value of both the coupon payments and the redemption value on maturity.
- Present value of coupon payments:
Using the formula for the present value of an annuity, we can calculate the present value of the semi-annual coupon payments using the semi-annual yield rate and the number of semi-annual periods. The formula is:
PV = C * [1 - (1 + r)^(-n)] / r
Where:
PV = Present Value
C = Semi-annual coupon payment
r = Semi-annual yield rate
n = Number of semi-annual periods
PV_coupon = $227.50 * [1 - (1 + 0.024)^(-12.74)] / 0.024
- Present value of the redemption value:
To calculate the present value of the redemption value, we use the formula for the present value of a single future sum. The formula is:
PV = F / (1 + r)^n
Where:
PV = Present Value
F = Future value (redemption value in this case)
r = Semi-annual yield rate
n = Number of semi-annual periods
Since the bond has a face value of $10,000 and is redeemable on July 01, 2031, the future value is $10,000.
PV_redemption = $10,000 / (1 + 0.024)^12.74
Step 6: Calculate the total present value
The total present value of the bond is the sum of the present value of the coupon payments and the present value of the redemption value.
Total PV = PV_coupon + PV_redemption
Step 7: Calculate the bond price
The bond price is equal to the total present value of the bond.
Bond Price = Total PV
Please note that the calculations shown here are based on the information provided.