Question 468258: I need help with this question please..?
Emma decides to buy a home entertainment system costing $5000. She takes out a loan from her Credit Union for the full cost of the system. The loan is based on a reducing balance interest rate of 10.5% pa with equal monthly installments for three years.
a.) find the size of each repayment for the loan.
b.)Hence find the total paid for the Entertainment System and the amount of interest paid.
c.) after two years,Emma finds that she is in a position to pay off the loan in full.How much does she still owe the bank after two years?
d.)How much does she save by paying off the loan early?Would you advise Emma to pay off the loan early?
I think we need to use the formula of Annuities in arrears here..
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! interest rate is equal to .105 per year compounded annually.
to get the monthly rate, divide the yearly rate by 12.
you will get a monthly interest rate of .00875.
to get the annual interest rate, you divided the annual interest rate percent by 100.
10.5% became .105
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the number of time periods for the loan is 3 years times 12 months per year equals 36 months.
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the payments on the loan are calculated to be $162.51221756 per month.
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over the 36 months of the original loan of $5,000, you made 36 payments.
The total of those payments = 36 * $162.51221756 = $5850.439831
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The total payments made minus the amount of the original loan is the total interest paid on the loan.
That would be equal to $5850.439831 - $5000 = $850.439831
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After 24 months of paying off the loan, you will have 12 payments left.
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the present value of these payments is calculated to be $1843.616486.
that's the amount of money that is still owed on the loan after 24 months.
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She could pay off the remainder of the loan or she could continue to make payments.
If she continues to make payments, she will have paid 12 * $162.5122175 = $1950.14661.
Since she still owes $1843.616486, that means she has paid interest of $106.5301242.
If she pays off the loan right away, she will not have to pay this interest.
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Whether she should pay off the loan right away or continue to make payments is largely dependent on the interest rate she can earn on her money.
If she can earn 10.5% per year on her money, then it's a wash. It doesn't matter whether she pays off the loan or continues to make the payments.
If she can earn less than 10.5% per year on her money, then she should pay off the loan because it is costing her more than she can earn.
If she can earn more than 10.5% per year on her money, then she should keep paying off the loan, because she is paying off at 10.5% per year, but she is earning more than that, so she will have a surplus in her account.
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This should answer most of your questions.
They are:
a.) find the size of each repayment for the loan.
*** Amount paid each month is $162.5122175
b.)Hence find the total paid for the Entertainment System and the amount of interest paid.
*** Total paid was $5850.43983. Total Interest paid was $5850.439831.
c.) after two years,Emma finds that she is in a position to pay off the loan in full.How much does she still owe the bank after two years?
*** She still owes the bank $1843.616486.
d.)How much does she save by paying off the loan early?
*** She saves approximately $106 in interest payment for the remainder of the loan.
Would you advise Emma to pay off the loan early?
*** It depends on the interest rate she can earn on her money. If she can earn exactly 10.5% per year, then it's a wash. If she can earn less than 10.5% per year then she should pay off the loan. If she can earn more than 10.5% per year then she should continue to pay off the loan.
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You may wish to check out the following tutorial that discusses the formulas and assumptions used in the financial formulas.
I used a financial calculator and Excel to make my calculations.
It's so much easier.
If you use the formulas correctly, though, you will get the same answer plus or minus some very insignificant digits.
BASIC FORMULAS AND ASSUMPTIONS USED IN FINANCIAL FORMULAS
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The rationale for the decision whether to pay off the loan is shown below:
You make a loan of $100.
The interest rate on the loan is 10% per year.
you make one payment at the end of the year.
Your payments needs to be $100 + .10 * $100 = $110.
This covers the loan amount ($100) plus the interest amount ($10).
Right after you take the loan, somebody gives you $100.
You can use that $100 to pay off the loan or you can invest the $100.
Assuming the interest rate you can earn is 10% per year, it's a wash.
If you pay off the loan, you have 0 dollars in your account and you owe 0 dollars.
If you invest the $100 at 10%, you will have $110 in your account at the end of the year, but you will then need to pay off the loan, so you wind up with $0 in your account anyway.
If you can only earn 5% on the $100, then you should pay off the loan.
If you pay off the loan, you have 0 in your account at the end of the year.
If you don't pay off the loan, you have $105 in your account at the end of the year but you still owe $110, so your account is minus $5 which you will have to get from somewhere else (your pay check presumably).
If you can earn 20% on the $100, then you should NOT pay off the loan.
If you do not pay off the loan, you will have $120 at the end of the year.
Once you pay off the loan at $110, you will still have $10 in your account.
If you pay off the loan right away, you will have 0 in your account.
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The same principles apply with a loan of any duration.
If the interest rate you can earn is greater than the interest rate of the loan, than you should continue to make payments on the loan.
If the interest rate you can earn is less than the interest rate of the loan, then you should pay off the loan.
If the interest rate you can earn is the same as the interest rate of the loan, then it doesn't matter whether you continue to make payments on the loan or you pay off the loan.
A time point Analysis will show this up as well.
First we'll look at 10% interest rate you can earn.
We'll assume you are re-investing the proceeds of the loan and / or reinvesting the proceeds of the gift.
Case 1:
You get a loan of $100.
you invest the loan at 10%
at the end of the year you have $110 in your account, but you have to shell out $110 to pay off the loan, so you are left with 0.
Case 2A:
KEEPING THE GIFT AT 10% REINVESTMENT RATE
You get a loan of $100.
Somebody gives you a gift of $100 immediately after.
You invest the loan and the gift at 10%
at the end of the year you have $220 in your account, but you have to shell out $110 to pay off the loan, so you are left with $110 in your account.
Case 3A:
KEEPING THE GIFT AT 15% REINVESTMENT RATE
You get a loan of $100.
Somebody gives you a gift of $100 immediately after.
You invest the loan and the gift at 15%.
at the end of the year you have $230 in your account, but you have to shell out $110 to pay off the loan, so you are left with $120 in your account.
Case 4A:
KEEPING THE GIFT AT 5% REINVESTMENT RATE
You get a loan of $100.
Somebody gives you a gift of $100 immediately after.
You invest the loan and the gift at 5%
at the end of the year you have $210 in your account, but you have to shell out $110 to pay off the loan, so you are left with $100 in your account.
Case 2B:
PAYING OFF THE GIFT IMMEDIATELY 10% REINVESTMENT RATE
You get a loan of $100.
Somebody gives you a gift of $100 immediately after.
You use that $100 to pay off the loan immediately.
at the end of the year you have $110 in your account.
Case 3B:
PAYING OFF THE GIFT IMMEDIATELY AT 15% REINVESTMENT RATE
You get a loan of $100.
Somebody gives you a gift of $100 immediately after.
You use that $100 to pay off the loan immediately.
at the end of the year you have $115 in your account.
Case 4B:
PAYING OFF THE GIFT IMMEDIATELY AT 5% REINVESTMENT RATE
You get a loan of $100.
Somebody gives you a gift of $100 immediately after.
You use that money to pay off the loan immediately.
at the end of the year you have $105 in your account.
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Here's a summary of the money in your account at the end of the first year.
Case 2A: make payment on the loan at 10% reinvestment rate: $110
Case 2B: pay off the loan immediately at 10% reinvestment rate: $110
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Case 3A: make payment on the loan at 15% reinvestment rate: $120
Case 3B: pay off the loan immediately at 15% reinvestment rate: $115
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Case 4A: make payment on the loan at 5% reinvestment rate: $100
Case 4B: pay off the loan immediately at 5% reinvestment rate: $105
the analysis confirms the theory.
if the interest rate you can earn on your money is the same, then you can go either way and have the same money in your account at the end of it.
if the interest rate you can earn on your money is greater than the interest rate of the loan, then you should continue to make payments on the loan.
if the interest rate you can earn on your money is less than the interest rate of the loan, then you should pay off the loan.
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