Math Help - Can you guys help me with this business math problem?

1. Can you guys help me with this business math problem?

Hey guys, our professor gave this as our final problem set for the semester but I don't really know how to do it or how to start. Any help would be very much appreciated

Here it is:

Mr. Go borrowed from Ms. Sy a certain amount payable at 9% compounded monthly after three years. Ms. Sy required Mr. Go to make regular payments of 9,000 pesos at the end of each month to settle his obligation. Everything went as agreed upon for a year. After the first year, however, Mr. Go missed remitting his monthly payments for five months. In order to make up for his missed payments, Mr. Go paid 60,000 pesos a month after the fifth month of missed payments and made regular installments of 9,000 pesos at the end of each month only for six months. After which he decided to pay Ms. Sy the rest of his loan through three quarterly payments of 40,000 pesos each.

At the end of the three-year loan term, Ms. Sy claims that Mr. Go has not adequately paid the principal and all accumulated interests on the loan. Mr. Go, on the other hand, maintains that he has paid all his dues and has no obligation left unpaid. He adds that he, in fact, has paid more than he was supposed to.

Show how much Mr. Go either still owes or has overpaid Ms. Sy (if any).

3. Originally Posted by jenn17sanders
Hey guys, our professor gave this as our final problem set for the semester but I don't really know how to do it or how to start. Any help would be very much appreciated

Here it is:

Mr. Go borrowed from Ms. Sy a certain amount payable at 9% compounded monthly after three years. Ms. Sy required Mr. Go to make regular payments of 9,000 pesos at the end of each month to settle his obligation. Everything went as agreed upon for a year. After the first year, however, Mr. Go missed remitting his monthly payments for five months. In order to make up for his missed payments, Mr. Go paid 60,000 pesos a month after the fifth month of missed payments and made regular installments of 9,000 pesos at the end of each month only for six months. After which he decided to pay Ms. Sy the rest of his loan through three quarterly payments of 40,000 pesos each.

At the end of the three-year loan term, Ms. Sy claims that Mr. Go has not adequately paid the principal and all accumulated interests on the loan. Mr. Go, on the other hand, maintains that he has paid all his dues and has no obligation left unpaid. He adds that he, in fact, has paid more than he was supposed to.

Show how much Mr. Go either still owes or has overpaid Ms. Sy (if any).

First work out the principle of the loan p_0. Then set up a spreadsheet showing for the duration of the repayments, the final value will be the outstanding debt.

Repayment of a loan of initial value $p_0$ over 36 months at 9% compounded monthly with monthly repayments of 9000 gives:

$\rho=1+0.09/12 = 1.0075$

$p_0=9000\frac{1-\rho^{36}}{(1-\rho)\rho^{36}}\approx 283021.25$

Now set up a spreadsheet for the total duration of the actual repayment scheme showing the interest and payments.

Also this is not entirely clear, how many payments of 60000 were there? I assume just one.

He has overpaid by about 17000

RonL

4. Mr. Go borrowed from Ms. Sy a certain amount payable at 9% compounded monthly after three years. Ms. Sy required Mr. Go to make regular payments of 9,000 pesos at the end of each month to settle his obligation. Everything went as agreed upon for a year. After the first year, however, Mr. Go missed remitting his monthly payments for five months. In order to make up for his missed payments, Mr. Go paid 60,000 pesos a month after the fifth month of missed payments and made regular installments of 9,000 pesos at the end of each month only for six months. After which he decided to pay Ms. Sy the rest of his loan through three quarterly payments of 40,000 pesos each.

At the end of the three-year loan term, Ms. Sy claims that Mr. Go has not adequately paid the principal and all accumulated interests on the loan. Mr. Go, on the other hand, maintains that he has paid all his dues and has no obligation left unpaid. He adds that he, in fact, has paid more than he was supposed to.

Show how much Mr. Go either still owes or has overpaid Ms. Sy (if any).

To find the outstanding loan balance, if any, at the end of 3 years, write an equation of value with the end of 3 years as the comparison date. Thus,
Outstanding loan balance = (Future value of the loan amount) minus (Future value of the past payments)
Accordingly, if Ms. Sy required Mr. Go to make regular payments of 9,000 pesos at the end of each month for 3 years payable at 9% compounded monthly, it is conceivable that the loan amount is
$9,000 \cdot \frac{{1 - \left( {1 + \tfrac{{.09}}
{{12}}} \right)^{ - \left( {3 \times 12} \right)} }}
{{\tfrac{{.09}}
{{12}}}}$

The future value of this loan amount at the end of 3 years is
$\left[ {9,000 \cdot \frac{{1 - \left( {1 + \tfrac{{.09}}
{{12}}} \right)^{ - \left( {3 \times 12} \right)} }}
{{\tfrac{{.09}}
{{12}}}}} \right] \cdot \left( {1 + \tfrac{{.09}}
{{12}}} \right)^{\left( {3 \times 12} \right)}
$

On the other hand, the future value of the past payments (or installment payments) is
$\left[ {9,000 \cdot \frac{{\left( {1 + \tfrac{{.09}}
{{12}}} \right)^{\left( {1 \times 12} \right)} - 1}}
{{\tfrac{{.09}}
{{12}}}}} \right] \cdot \left( {1 + \tfrac{{.09}}
{{12}}} \right)^{\left( {2 \times 12} \right)} + 60,000\left( {1 + \tfrac{{.09}}
{{12}}} \right)^{\left( {1.5 \times 12} \right)}
$

$+ \left[ {9,000 \cdot \frac{{\left( {1 + \tfrac{{.09}}
{{12}}} \right)^{\left( {.5 \times 12} \right)} - 1}}
{{\tfrac{{.09}}
{{12}}}}} \right] \cdot \left( {1 + \tfrac{{.09}}
{{12}}} \right)^{\left( {1 \times 12} \right)}
$

$+ 40,000\left( {1 + \tfrac{{.09}}
{{12}}} \right)^{\left( {.75 \times 12} \right)} + 40,000\left( {1 + \tfrac{{.09}}
{{12}}} \right)^{\left( {.5 \times 12} \right)} + 40,000\left( {1 + \tfrac{{.09}}
{{12}}} \right)^{\left( {.25 \times 12} \right)}
$

I now leave the calculation in your hands to show for yourself if Mr. Go either still owes or has overpaid Ms. Sy (if any).
If your calculation results in positive amount, then Mr. Go does still owes Ms. Sy.
If your calculation results in negative amount, then Mr. Go has overpaid Ms. Sy and should be entitled to a refund.
Think of it this way:
If Ms. Sy decided to place the loan amount in a bank that pays 9% compounded monthly, she would be entitled to
$\left[ {9,000 \cdot \frac{{1 - \left( {1 + \tfrac{{.09}}
{{12}}} \right)^{ - \left( {3 \times 12} \right)} }}
{{\tfrac{{.09}}
{{12}}}}} \right] \cdot \left( {1 + \tfrac{{.09}}
{{12}}} \right)^{\left( {3 \times 12} \right)}
$
at the end of 3 years.
If Mr. Go decided instead to deposit his installment payments in bank that pays 9% compounded monthly rather than paying Ms. Sy directly, the value of his installment payments at the end of 3 years would be equal to
$\left[ {9,000 \cdot \frac{{\left( {1 + \tfrac{{.09}}
{{12}}} \right)^{\left( {1 \times 12} \right)} - 1}}
{{\tfrac{{.09}}
{{12}}}}} \right] \cdot \left( {1 + \tfrac{{.09}}
{{12}}} \right)^{\left( {2 \times 12} \right)} + 60,000\left( {1 + \tfrac{{.09}}
{{12}}} \right)^{\left( {1.5 \times 12} \right)}
$

$+ \left[ {9,000 \cdot \frac{{\left( {1 + \tfrac{{.09}}
{{12}}} \right)^{\left( {.5 \times 12} \right)} - 1}}
{{\tfrac{{.09}}
{{12}}}}} \right] \cdot \left( {1 + \tfrac{{.09}}
{{12}}} \right)^{\left( {1 \times 12} \right)}
$

$+ 40,000\left( {1 + \tfrac{{.09}}
{{12}}} \right)^{\left( {.75 \times 12} \right)} + 40,000\left( {1 + \tfrac{{.09}}
{{12}}} \right)^{\left( {.5 \times 12} \right)} + 40,000\left( {1 + \tfrac{{.09}}
{{12}}} \right)^{\left( {.25 \times 12} \right)}
$

Check your results with the spreadsheet amortization schedule that I’ve attached.

5. THANK YOU SO MUCH for everything. Especially you jonah. You have no idea how much I appreciate the time you guys took to help me out. It gives me hope in humankind to know that there are still people like you who help complete stangers with their problems. Again, thank you so much