A bond with face value of $500 redeemable at par in 13 years time, is paying interest at j2 = 9.0%. Find the purchase price of the bond to give a yield to maturity of j2 = 8.4%.
To find the purchase price of the bond, we need to use the present value formula. The present value formula is:
P = C / (1 + r)^n
Where:
P = Purchase price of the bond
C = Coupon payment (interest payment)
r = Yield to maturity (YTM)
n = Number of periods (years)
Given information:
Face value (FV) = $500
Redemption period = 13 years
Coupon rate (j2) = 9.0%
Yield to maturity (YTM) = 8.4%
Step 1: Calculate the coupon payment (C)
C = FV * (j2/100)
C = $500 * (9.0/100)
C = $45
Step 2: Calculate the number of periods (n)
n = Redemption period = 13 years
Step 3: Calculate the purchase price (P)
P = C / (1 + r)^n
P = $45 / (1 + 8.4/100)^13
Now let's calculate the value inside the parentheses:
(1 + 8.4/100)^13 ≈ 1.199695
P ≈ $45 / 1.199695
P ≈ $37.51 (rounded to the nearest cent)
Therefore, the purchase price of the bond to give a yield to maturity of 8.4% is approximately $37.51.
To find the purchase price of the bond, we need to calculate the present value of the bond's future cash flows.
The cash flows of the bond include the face value (the redemption value at maturity) and the interest payments. In this case, the face value of the bond is $500, and it will be redeemed at par in 13 years.
First, let's calculate the present value of the face value at maturity. We can use the formula for present value (PV) of a future cash flow:
PV = FV / (1 + r)^n
Where PV is the present value, FV is the future value, r is the discount rate, and n is the number of periods.
In this case, the future value (FV) is $500, the discount rate (r) is the yield to maturity of 8.4% or 0.084, and the number of periods (n) is 13.
PV = $500 / (1 + 0.084)^13
Calculating this equation will give us the present value of the face value at maturity:
PV = $500 / (1.084)^13
Next, let's calculate the present value of the interest payments. The bond pays interest at a rate of 9.0% or 0.09 annually. The interest payments are received for 13 years.
The present value of a stream of future cash flows, such as the interest payments, can be calculated using the present value of an ordinary annuity formula:
PV = PMT * [1 - (1 + r)^(-n)] / r
Where PV is the present value, PMT is the periodic payment (interest payment), r is the discount rate, and n is the number of periods.
In this case, the periodic payment (PMT) is calculated as a percentage of the face value:
PMT = $500 * 0.09
Now, let's substitute the values into the formula to calculate the present value of the interest payments:
PV = ($500 * 0.09) * [1 - (1 + 0.084)^(-13)] / 0.084
Finally, to find the purchase price of the bond, we sum up the present value of the face value at maturity and the present value of the interest payments:
Purchase Price = PV of Face Value + PV of Interest Payments
Purchase Price = [$500 / (1.084)^13] + [($500 * 0.09) * [1 - (1 + 0.084)^(-13)] / 0.084]
By calculating this expression, you'll find the purchase price of the bond that yields a yield to maturity of 8.4%.