A) A company has issued 10-year bonds, with a face value of $1,000,000 in $1000 units. Interest at 8% is paid quarterly. If an investor desires to earn 12% nominal interest (compounded quarterly) on $10000 worth of these bonds, what would the purchase price have to be? (5.3)
B) If the company plans to redeem these bonds in total at the end of 10 years and establishes a sinking fund that earn 8%, compounded semi-annually, for this purpose, what is the annual cost of interest and redemption? (5.5)
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A) To calculate the purchase price of the bonds to earn a 12% nominal interest, compounded quarterly, on a $10,000 investment, we need to find the present value of the expected future cash flows.
1. Determine the number of quarters in 10 years:
Number of quarters = 10 years x 4 quarters/year = 40 quarters
2. Calculate the quarterly interest rate:
Quarterly interest rate = 12% nominal interest rate / 4 quarters = 0.12 / 4 = 0.03
3. Calculate the future value of the investment after 10 years:
Future value = $10,000 x (1 + Quarterly interest rate)^Number of quarters
= $10,000 x (1 + 0.03)^40
= $10,000 x (1.03)^40
= $10,000 x 1.488643
= $14,886.43
4. Calculate the present value of the future cash flows:
Present value = Future value / (1 + Quarterly interest rate)^Number of quarters
= $14,886.43 / (1 + 0.03)^40
= $14,886.43 / 1.488643
= $10,000
Therefore, the purchase price of the bonds must be $10,000 to earn a 12% nominal interest rate on a $10,000 investment.
B) To calculate the annual cost of interest and redemption, we need to find the total annual cash outflows from the sinking fund.
1. Determine the number of semi-annual periods in 10 years:
Number of semi-annual periods = 10 years x 2 semi-annual periods/year = 20 semi-annual periods
2. Calculate the semi-annual interest rate:
Semi-annual interest rate = 8% / 2 semi-annual periods = 4%
3. Calculate the redemption amount at the end of 10 years:
Redemption amount = Face value of the bonds = $1,000,000
4. Calculate the semi-annual cash outflow for interest and redemption:
Semi-annual cash outflow = Redemption amount / (1 + Semi-annual interest rate)^Number of semi-annual periods
= $1,000,000 / (1 + 0.04)^20
= $1,000,000 / 1.819745
= $549,333.33
5. Calculate the annual cash outflow by multiplying the semi-annual cash outflow by 2:
Annual cash outflow = Semi-annual cash outflow x 2
= $549,333.33 x 2
= $1,098,666.67
Therefore, the annual cost of interest and redemption is $1,098,666.67.
To find the purchase price of the bonds in question A, we need to calculate the present value of the future cash flows from the bonds.
First, let's break down the information given:
- Face value of the bonds: $1,000,000
- Coupon rate (interest rate): 8%, paid quarterly
- Desired nominal interest rate: 12%, compounded quarterly
- Investment amount: $10,000
Step 1: Calculate the present value of the future coupon payments.
The coupon payments are paid quarterly and have a 8% interest rate. We can use the present value of an annuity formula to calculate the present value of these cash flows.
PV = C * (1 - (1 + r)^(-n)) / r
Where:
PV = Present value of the annuity
C = Cash flow per period (coupon payment)
r = Interest rate per period
n = Number of periods
In this case, the cash flow per period (C) is $1,000 (since we have $1,000,000 / $1,000 units) and the interest rate per period (r) is 8% / 4 = 2% (since we have quarterly payments). The number of periods (n) is 10 * 4 = 40 (since the bonds have a 10-year maturity and quarterly payments).
PV_coupon = $1,000 * (1 - (1 + 2%)^(-40)) / 2%
Step 2: Calculate the present value of the face value payment.
At the end of 10 years, the bondholder will also receive the face value of $1,000,000. We can calculate the present value of this future cash flow using the present value formula.
PV_face_value = $1,000,000 / (1 + 12%/4)^(10*4)
Step 3: Calculate the purchase price.
The purchase price of the bonds is the sum of the present values of the coupon payments and the face value payment.
Purchase price = PV_coupon + PV_face_value
Now, let's calculate the purchase price.
In question B, we are asked to calculate the annual cost of interest and redemption.
The annual cost of interest is the sum of the annual coupon payments (which we calculated in question A) and the annual cost of redemption.
To calculate the annual cost of redemption, we can use the sinking fund method. A sinking fund is set up to save money over time to be able to pay off the principal amount (face value of the bonds) at maturity.
The sinking fund will earn interest of 8% compounded semi-annually. We will need to calculate the amount that needs to be set aside each year to accumulate enough funds to redeem the bonds at maturity.
Using the formula for the future value of an annuity:
FV = P * (1 + r)^n
Where:
FV = Future value of the annuity (the face value of the bonds)
P = Annual deposit into the sinking fund
r = Interest rate per period (semi-annual in this case)
n = Number of periods (10 years, compounded semi-annually)
We know FV is $1,000,000, r is 8%/2 = 4% (since it's semi-annual), and n is 10 * 2 = 20 (since it's compounded semi-annually for 10 years).
We can rearrange the formula to solve for P:
P = FV / ((1 + r)^n - 1) / (1 + r)
Now, we can calculate the annual cost of redemption.
Annual cost of redemption = P + (PV_coupon * 10) - FV
In this calculation, the first term represents the annual deposit into the sinking fund, the second term represents the total coupon payments over 10 years, and the third term represents the face value of the bonds.