The Health Corp issued an 12%, 15 yr bond 5 yrs ago. At the time of issue the bond's par value was $1,000. Comparable bonds are yielding 10% today. What must Health Corp's bond sell for in today's market to yield 20% (YTM) to the buyer? Assume bond pays interest quarterly.
Formula: Pb=PMT[PVFAk,n]+FV[PVFk,n]
Pb=$30[PVF 5,40] + $1,000[PVF 5,40]
PMT= $30
k=5%
n=40
I don't know where to go from here... Please help!
To calculate the price of the bond in today's market to yield 20% to the buyer, you can use the formula:
Pb = PMT[PVFAk,n] + FV[PVFk,n]
PMT = $30
k = 5% (for each quarter, as interest is paid quarterly)
n = 40 (number of quarters in 10 years)
Firstly, let's calculate the present value factor for 5 years and 40 quarters using the yield of 20% (0.05 for each quarter):
[PVFk,n] = 1 / (1 + k)^n
[PVFk,n] = 1 / (1 + 0.05)^40
Now, substitute the values into the formula:
Pb = $30 x [PVFA0.05,40] + $1,000 x [PVF0.05,40]
You already calculated the present value factor for 5 years (0.78353). Now, let's calculate the present value factor for 40 quarters:
[PVF0.05,40] = 1 / (1 + 0.05)^40
To calculate this, raise (1 + 0.05) to the power of 40:
(1 + 0.05)^40 ≈ 3.69847
Now, divide 1 by 3.69847 to get the present value factor:
[PVF0.05,40] ≈ 1 / 3.69847 ≈ 0.27038
Substitute these values back into the formula:
Pb = $30 x 0.78353 + $1,000 x 0.27038
Calculate these values:
Pb ≈ $23.51 + $270.38 ≈ $293.89
Therefore, the bond must sell for approximately $293.89 in today's market to yield 20% to the buyer.
To determine the current market price of the Health Corp bond, you need to calculate the present value of the bond's future cash flows.
The formula you provided, Pb = PMT[PVFAk,n] + FV[PVFk,n], can be used to calculate the present value of the bond's interest payments and the future value of the bond's face value. Let's break down the formula step by step:
Pb = $30[PVF 5,40] + $1,000[PVF 5,40]
First, you need to find the present value factor (PVF) for the interest payments and the face value. The interest payments are $30, and the face value is $1,000.
To calculate the PVF, you'll need to use the present value of an ordinary annuity (PVFA) formula and the present value of a future value (PVF) formula.
PVFak,n = (1 - (1 + k)^(-n)) / k
PVFAk,n = (1 - (1 + k)^(-n)) / k * (1 + k)
In this case, k is the periodic interest rate (5% divided by 4 because interest is paid quarterly) and n is the remaining number of periods (15 years minus 5 years, multiplied by 4 because interest is paid quarterly).
Let's calculate the values step by step:
k = 5% / 4 = 0.05 / 4 = 0.0125
n = (15 - 5) * 4 = 40
PVFak,n = (1 - (1 + 0.0125)^(-40)) / 0.0125 ≈ 25.7028371
PVFAk,n = (1 - (1 + 0.0125)^(-40)) / 0.0125 * (1 + 0.0125) ≈ 29.196523
Now let's substitute the calculated values back into the original formula:
Pb = $30 * 29.196523 + $1,000 * 25.7028371
Pb ≈ $875.89 + $25,702.84
Pb ≈ $26,578.73
Therefore, the Health Corp bond must sell for approximately $26,578.73 in today's market to yield 20% yield to the buyer (YTM).