Week 2 Discussion Questions.
1) Lisa’s strategy to borrow money to purchase shar
Week 2 Discussion Questions.
1) Lisa’s strategy to borrow money to purchase shares outright, leveraging the shares as collateral and expecting dividends to cover a portion of the loan payments, can indeed be a prudent approach, especially if the dividends are expected to be stable and provide a reliable source of income. To calculate the annual payments for the loan amortized over five, ten, or twenty years, we can use the formula for the periodic payment of an amortizing loan. The formula for the periodic payment (PMT) of an amortizing loan with an annual interest rate (r), a principal amount (P), and a loan term in years (n) is:
PMT = P * [r (1 + r) ^n] / [(1 + r) ^n – 1]
Where:
PMT is the periodic payment.
P is the principal amount of the loan
r is the annual interest rate divided by the number of payment periods per year
n is the total number of payments (loan term in years times the number of payments per year) Let’s say Lisa borrowed $100,000 at an annual interest rate of 5% to be paid annually. Here’s how we would calculate the annual payments for different loan terms:
1) For a 5-year term:
r = 0.05 (since payments are made annually)
n = 5
PMT = 100,000 * [0.05(1 + 0.05) ^5] / [(1 + 0.05) ^5 – 1] = $23,097.59 2)
2) For a 10-year term:
n = 10
PMT = 100,000 * [0.05(1 + 0.05) ^10] / [(1 + 0.05) ^10 – 1] = $13,020.78 3)
3) For a 20-year term:
n = 20
PMT = 100,000 * [0.05(1 + 0.05) ^20] / [(1 + 0.05) ^20 – 1] = $8,290.96
These are the annual payments Lisa would need to make to fully repay the loan in 5, 10, and 20 years, respectively.
2) Repeat Question 1 but assume that Lisa makes payments at the beginning of each year.
The formula for the periodic payment (PMT) of an amortizing loan with payments made at the beginning of each period is:
PMT = P * r(1+r) n / (1+r) n – 1 * 1 / (1+r)
Where:
• PMT is the periodic payment.
• P is the principal amount of the loan.
• r is the annual interest rate divided by the number of payment periods per year.
• n is the total number of payments (loan term in years times the number of payments per year).
1. For a 5-year term:
r = 0.05 (since the payments are made annually)
n = 5
PMT = 100,000 * 0.05(1+0.05)5 / (1+0.05)5 -1 * 1 / (1+0.05) = $24,252.44
2. For a 10-year term:
n = 10
PMT = 100,000 * 0.05(1+0.05)10 / (1+0.05)10 -1 * 1 / (1+0.05) = $14,083.85
3. For a 20-year term:
n = 20
PMT = 100,000 * 0.05(1+0.05)20 / (1+0.05)20 -1 * 1 / (1+0.05) = $9,228.36
These are the adjusted annual payments Lisa would need to make at the beginning of each year to fully repay the loan in 5, 10, and 20 years.
3) Build and analyze the amortization schedule below for a $10,000,000 loan at 7% with five equal end-of-year payments.
PMT = P*r (1 + r) n / (1+r) n – 1
Where:
• P is the principal amount
• r is the annual interest rate
• n is the total number of payments
PMT = 10,000,000 – 0.07 * (1 + 0.07)5 / (1 + 0.07)5 – 1
PMT = 10,000,000 * 0.07 * (1.07)5 / (1.07)5 – 1
PMT= 10,000,000 * 0.07 * 1.402551566 / 1402551566 – 1
PMT = 10,000,000 * 0.09817761562 / 0.402551566
PMT = 981,776.1562 / 0.402551566
PMT = $2,438,679.41
Year Beginning Balance Payment Interest Principal Ending Balance
1 $10,000,000 $2,438,678.41 $700,000,00 $1,738,679.41 $8,261,320.59`
2 $8,261,320.59 $2,438,678.41 $578,193.44 $1,860,485.97 $6,400,834.62
3 $6,400,834.62 $2,438,678.41 $448,058.42 $1,990,621.99 $4,410,212.63
4 $4,410,212.63 $2,438,678.41 $307,714.88 $2,130,964.53 $2,279,248.10
5 $2,279,248.10 $2,438,678.41 $158,747.37 $2,279,932.04 $0.00
In each year:
• The interest is calculated as the beginning balance multiplied by the annual interest rate.
• The principal payment is the difference between the annual payment and the interest.
• The ending balance is the beginning balance minus the principal payment.
4) Richard has offered to finance the purchase with a ten- year, 7%, interest-only loan. Explain how much is Lisa’s annual payment. Describe the pattern of payments over the ten years.
Lisa’s annual payment for the interest-only loan offered by Richard, we can use the formula:
Annual Interest Payment =P * r
Where:
P is the principal amount (the amount borrowed)
r is the annual interest rate (expressed as a decimal)
Given:
• Principal amount (P) is the amount borrowed, which we’ll assume is the same as the purchase price of the shares.
• Annual interest rate (r) is 7% or 0.07.
Annual Interest Payment=$10,000,000 * 0.07
Annual Interest Payment=$700,000
Lisa’s annual payment for the interest-only loan would be $700,000.
Now, let’s describe the pattern of payments over the ten years:
• Each year, Lisa would make an interest payment of $700,000 to Richard.
• Since it’s an interest-only loan, the principal amount remains unchanged throughout the term of the loan.
• At the end of the ten-year term, Lisa will still owe the original principal amount of $10,000,000 to Richard, as no principal payments are made during the loan term.
• Therefore, the pattern of payments over the ten years is consistent: Lisa pays $700,000 in interest each year, and the principal amount remains the same until the end of the loan term, at which point the full principal amount becomes due.
5) Assume that Lisa accepts Richard’s offer to finance the purchase with a ten-year, 7%, interest-only loan. If Richard can reinvest the interest payments at a rate of 7% per year, explain how much money will he have at the end of the tenth year.
If Richard can reinvest the interest payments at a rate of 7% per year, he will have accumulated a certain amount of money by the end of the tenth year due to the compounding effect of reinvesting the interest.
In an interest-only loan scenario, Richard receives $700,000 in interest payments annually from Lisa. If he reinvests each interest payment at a rate of 7% per year, it will compound over the ten-year period.
Calculate the future value of each interest payment using the formula for compound interest:
FV = PV * (1 + r)n
Where:
• FV is the future value of the investment.
• PV is the present value (initial investment or principal)
• r is the interest rate per period (expressed as a decimal)
• n is the number of periods
Given:
• PV=$700,000 (annual interest payment)
• r=0.07 (7% interest rate)
• n=10 (number of years)
Future value of each interest payment and then sum them up to find out how much money Richard will have at the end of the tenth year:
FV=700,000 * (1+0.07)10
FV=700,000 * (1.07)10
FV≈700,000 * 1.967151
FV≈$1,376,005.70
So, each $700,000 interest payment reinvested annually will grow to approximately $1,376,005.70 by the end of the tenth year.
Now, since Richard receives this interest payment every year for ten years, we need to calculate the total future value by summing up the future values of each interest payment:
Total Future Value=1,376,005.70 * 10
Total Future Value≈$13,760,057
Therefore, at the end of the tenth year, Richard will have approximately $13,760,057 if he reinvests each interest payment at a rate of 7% per year.