Part A (15 points) Included in Twombly-Cola\'s long-term debt as of December 31,
ID: 2496813 • Letter: P
Question
Part A (15 points)
Included in Twombly-Cola's long-term debt as of December 31, 2003, was $150 million of U. S.
dollar notes, which carry interest at 6%. Interest is paid semiannually. The notes are due December
31, 2013.
Twombly-Cola's management is considering establishing a sinking fund (an investment whose
purpose is to retire the notes at their maturity) in one of three different manners. In each case the
sinking fund would be used to retire only the principal amount of the notes. The sinking fund would
not be used to pay interest.
1. Suppose that Twombly-Cola were to establish a sinking fund by making a single lump-sum
payment on January 1, 2006. If the fund would earn 10 percent interest compounded
annually, what single amount should Twombly-Cola deposit on January 1, 2006, in order to
retire the notes on December 31, 2013?
2. Suppose, instead, that equal payments into the fund are to begin on January 1, 2009, and to
end on January 1, 2012. What is the amount of each annual payment required to retire the
notes on December 31, 2013? Interest on the fund will be 10 percent compounded annually.
3. How many consecutive payments of $20 million on January 1 of each year would have to be
made into the fund beginning on January 1, 2006, and ending on January 1, 2013, in order to
retire the debt on December 31, 2013? Interest on the fund will be 10 percent compounded
annually.
Part B (5 points)
Laura Ray bought a house for $150,000. She made a down payment amounting to $30,000 and
took out a 12 percent mortgage for the balance. The bank required that she pay off the mortgage
in 25 equal annual installments, beginning one year from the closing of the sale.
Required:
1. Determine the amount of the annual installments required by the lender.
2. What would be the total amount paid, including the down payment, by Ray (over the 25
years) for the $150,000 house (assuming that she did not retire the mortgage prematurely)?
3. How much of the total calculated in part 2 represents interest charges?
Explanation / Answer
Answer
PART A
The Company will require paying maturity value i.e. $150 Million at the time of maturity December 31, 2013.
So the value of Sinking Fund must become $150 Million as on Dec 31, 2013 in order to retire and repay the amount of principal of long term debt.
No of year to maturity = 10 years
Principal Maturity Amount = $150 Million
Case 1: Single Lump-sum payment on January 1, 2006
Rate of Interest (R) = 10% p.a. compounded annually
It means we need to find out what amount we will deposit on January 1, 2006 that will become $150 Million in 7 years i.e. on Dec 31, 2013
Amount Deposited on January 1, 2006 x (1+R)n = $150 Million
Amount Deposited on January 1, 2006 x (1+0.10)7 = $150 Million
Amount Deposited on January 1, 2006 x 1.9487171 = $150 Million
Amount Deposited on January 1, 2006 = $150 Million / 1.9487171 = $76,973,717.73
If Twombly-Cola deposit $$76,973,717.73 single amount on January 1, 2006 at 10% interest rate. This amount in 7 years i.e. on Dec 31, 2013 will become $150 Million
Case 2: Equal Payment into the fund beginning on January 1, 2009 and end on January 1, 2012
Rate of Interest = 10% p.a.
Equal Payment from Jan 1, 2009 to Jan 1, 2012 (4 equal payment) and maturity date is Dec 31, 2013
To understand, we will make this equation.
Let One equal payment amount is X, then 4 payment amount = 4X
4X amount further earn interest for 2 years (Jan 1, 2012 to 31 Dec, 2013) and will become $150,000,000.
Now we need to find out Value as on Jan 1, 2012 which will become the value of 4X.
Value of Money as on Jan 1, 2012 = $150,000,000 / (1+0.10)2 = $123,966,942
Equal payment beginning from Jan 1, 2009 to Jan 1, 2012 will become total value $123,966,942 on Jan 1, 2012.
This is Annuity Series and it is Annuity Due. Annuity Due means when cash flow (payment or receipt) of equal amount occur at the beginning of each period for a specified period of time.
Equal Payment x FVIFA (10%, 4) = $123,966,942 ---------equation 1
Value of FVIFA (10%, 4) can be calculated by using the below formula
FVIFA (10%, 4) = {[1 – (1+R)4] / R} x (1+R) = {[1 – (1.10)4]/0.10} x (1+0.10) = (0.4641 / 0.10) x 1.1 = 5.1051
Putting the value of FVIFA (10%, 4) in equation 1, we get
Equal Payment x 5.1051 = $123,966,942
Equal Payment Amount = $123,966,942 / 5.1051 = $24,282,960.57
Case 3: No of Consecutive payments of $20 million on January 1 of each year would have to be
made into the fund beginning on January 1, 2006 and ending on January 1, 2013 n order to
retire the debt on December 31, 2013
Jan 1, 2006 to Jan 1, 2013
$20,000,000 x (1+0.10)n= $150,000,000
(1+0.10)n = $150,000,000 / $20,000,000 = 7.5
From the Future Value Annuity table at 10% rate of interest in 6 years the value is 7.716, we need only 7.5 value. Hence only 6 payments are required of $20 Million amount to retire $150 million debt on Dec 31, 2013.
PART B
Amount to be paid in 25 Equal Annual Installments = House Value – Down Payment = $150,000 - $30,000 = $120,000
Rate of Interest = 12% p.a.
The present value of all 25 annual installments must be equal to $120,000
1) Amount of Annual Installment
Annual Installment x PVIFA (12%, 25) = $120,000
Value of PVIFA (12%, 25) can be taken from Present Value annuity table and also can be calculated by using following formula: PVIFA (12%, 25) = {[1 – (1+R)-n ] / R} = [1 – (1+0.12)-25] / 0.12 = (1 – 0.058823)/0.12 = 0.9411767 / 0.12 = 7.8431
Annual Installment x 7.8431 = $120,000
Annual Installment = $120,000 / 7.8431 = $15,300
2) Total Amount paid over 25 years including down payment = (Annual Installment x 25) + Down Payment = ($15,300 x 25) + $30,000 = $382,500 + $30,000 = $412,500
3) Interest Charges = $382,500 - $120,000 = $262,500