Question

Equivalent Value Equations

3.- You. ask for a loan today to speculate in the quinoa market, $5,000,000 to be paid within 4 semesters at a simple interest rate of 4% quarterly. After failing in this purpose, you want to pay off the debt in four payments equal to cancel months 2,5,7 and 9 keeping the original interest rate and having as a focal date the end of year 2 . If your creditor accepts these conditions, you are asked to determine the value of each payment.

Answer 1

First Payment (Month 2): $649,248.57

Second Payment (Month 5): $697,317.94

Third Payment (Month 7): $721,359.06

Fourth Payment (Month 9): $823,333.07

Explanation: To determine the value of each payment, we can use the concept of equivalent value equations. We need to calculate the equal periodic payments that will pay off the debt within the specified time frame while considering the original interest rate.

Let's break down the problem and calculate the value of each payment:

Loan amount: $5,000,000

Interest rate: 4% quarterly

Time frame: 4 semesters (equivalent to 8 quarters)

To calculate the equal periodic payments, we'll use the formula for the present value of an annuity:

PV = PMT * (1 - (1 + r)^(-n)) / r

Where:

PV = Present value or loan amount

PMT = Equal periodic payment

r = Interest rate per period

n = Total number of periods

In this case, we have 8 quarters (n), and the interest rate is 4% quarterly (r). Let's calculate the equal periodic payment (PMT):

PV = PMT * (1 - (1 + 0.04)^(-8)) / 0.04

$5,000,000 = PMT * (1 - (1 + 0.04)^(-8)) / 0.04

Now, we need to calculate the value of each payment considering the focal date at the end of year 2 and the specified payment months (2, 5, 7, and 9).

To calculate the value of each payment, we need to discount the future payments back to the focal date using the interest rate of 4% quarterly.

Let's calculate the value of each payment:

First Payment (Month 2):

Discounting 6 quarters back to the focal date:

PV = Payment / (1 + r)^n

PV = Payment / (1 + 0.04)^6

Second Payment (Month 5):

Discounting 3 quarters back to the focal date:

PV = Payment / (1 + r)^n

PV = Payment / (1 + 0.04)^3

Third Payment (Month 7):

Discounting 1 quarter back to the focal date:

PV = Payment / (1 + r)^n

PV = Payment / (1 + 0.04)^1

Fourth Payment (Month 9):

Discounting 3 quarters forward from the focal date:

PV = Payment * (1 + r)^n

PV = Payment * (1 + 0.04)^3

Since the loan amount and the equal periodic payment are already known, we can solve for the value of each payment by rearranging the equations.

Here's the complete calculation:

Loan amount (PV) = $5,000,000

Interest rate (r) = 4% quarterly

Total periods (n) = 8 quarters

PMT = PV * (r * (1 + r)^n) / ((1 + r)^n - 1)

PMT = $5,000,000 * (0.04 * (1 + 0.04)^8) / ((1 + 0.04)^8 - 1)

PMT ≈ $750,051.74 (rounded to the nearest cent)

Using this equal periodic payment, we can calculate the value of each payment:

First Payment (Month 2):

PV = $750,051.74 / (1 + 0.04)^6 ≈ $649,248.57 (rounded to the nearest cent)

Second Payment (Month 5):

PV = $750,051.74 / (1 + 0.04)^3 ≈ $697,317.94 (rounded to the nearest cent)

Third Payment (Month 7):

PV = $750,051.74 / (1 +

0.04)^1 ≈ $721,359.06 (rounded to the nearest cent)

Fourth Payment (Month 9):

PV = $750,051.74 * (1 + 0.04)^3 ≈ $823,333.07 (rounded to the nearest cent)

Therefore, the value of each payment should be approximately as follows:

First Payment (Month 2): $649,248.57

Second Payment (Month 5): $697,317.94

Third Payment (Month 7): $721,359.06

Fourth Payment (Month 9): $823,333.07