Question

While shopping for a car loan, you get the following offers: Solid Savings & Loan is willing to loan you $10,000 at 5% interest for 4 years. Fifth Federal Bank & Trust will loan you the $10,000 at 3% interest for 3 years. Both require monthly payments. You can afford to pay $250 per month. Which loan, if either, can you take?

Solid Savings & Loan

Fifth Federal Bank & Trust   

neither loan

Answer 1

We can afford to pay $250 per month. So we have to compute the monthly payments for both loans. Then we can decide which loan we can take.

When we take a loan and then pay it back with interest by making fixed periodic payments over some numbers of years, the loan is an annuity from lender's point of view.

An annuity is an account earning compound interest from which periodic withdrawals are made. The balance that the account starts with is known as the present value of the annuity.

First consider the loan offered by Solid Savings & Loan.

Solid Savings & Loan will loan you the $10,000 at 5% interest for 4 years. So in this case, we have to find the monthly payments (withdrawal) for an annuity that starts with a balance of $10,000 and earns interest at the annual rate of 5% for 4 years.

Consider an annuity that earns interest at an annual rate of r compounded m times per year. Suppose the account starts with a balance of PV. If you receive a payment of PMT at the end of each compounding period, and the account is down to $0 after t years, then the value of PMT is given by the formula:

Where

And

n = mt

Here,

PV= $10,000

t=4

r=5%

=5/100

=0.05

And

m=12

Now,  

=0.0041

And

n=mt

=(12)(4)

=48

Therefore,

=$229.93

So if we take the loan from Solid Savings & Loan we will have to make monthly payments of $229.93

Now we consider the loan offered by Fifth Federal Bank & Trust

They will loan you $10,000 at 3% interest for 3 years. So in this case, we have to find the monthly payments for an annuity that starts with a balance of $10,000 and earns interest at the annual rate of 3% for 3 years.

So if we will have to make monthly payments of $290.81

We should take loan from Solid Savings & Loan.