Lesson 1.5

Loans and Investments

In life you will be faced with many financial situations, such as car loans, checking accounts, credit cards, long-term investments, life insurance, retirement accounts, and home mortgages.

You will need to make intelligent choices about your money. Fortunately, the mathematics involved in no more complicated than the recursive formula

0 

n n-1

u =principal(starting amount)

u   r u  d.

Life’s Big ExpendituresYou plan to borrow $22,000 from a bank to

purchase a new car. You will make a payment every month to the bank to repay the loan, and the loan must be paid off in 5 years (60 months). The bank charges interest at an annual rate of 7.9%, compounded monthly. Part of each monthly payment is applied to the interest, and the remainder reduces the starting balance, or principal.“Who gives whom money when there’s a loan or mortgage?”

Step 1:

What is the monthly interest rate?

What is the first month’s interest on the $22,000?

If you make a payment of $300 at the end of the first month, then what is the remaining balance?

7.9% 0.6583%

12

 0.658% 22,000 $144.83

$ 22,000 $144.83-$300= 0.658% 22,000 $21,844.83

A loan of $22,000 to be paid off in 5 years (60 months) at 7.9%, compounded monthly.

Step 2 Record the balances for the first 6 months

with monthly payments of $300.

Step 2 How

many months will it take to pay off the loan?

Do you pay the same amount of interest each month?Why does the amount of interest change each month?

Step 3 Experiment with other values for the monthly payment. What monthly payment allows you to pay off the loan in exactly 60 months?

Step 4 How much do you actually pay for the car using the monthly payment you found in Step 3?

445.03(59) 445.03 .11

$26,701.69

Use the techniques that you discovered in the previous part to find the monthly payment for a 30-year home mortgage of $146,000 with an annual interest rate of 7.25%, compounded monthly.

How much do you actually pay for the house?A payment of $995.98 will pay off the mortgage in 360 payments or 30 years.

359(995.98) 992.61

$358,549

Investments are mathematically similar

to loans. With an investment, deposits

are added on a regular basis so that your balance increases.

ExampleGwen’s employer offers an investment plan that

invests a portion of each paycheck before taxes are deducted. Gwen gets paid every week. The plan has an annual interest rate of 4.75%, compounded weekly, and she decides to contribute $10 each week. What will Gwen’s balance be after 5 years?

Gwen’s starting balance is $10. Each week, the previous balance is multiplied by and Gwen adds another $10. Use your calculator to generate the first two weeks of her investment.

0.04751

52

A recursive formula that generates the balance is

Enter this sequence into a list in the calculator to find the amount in the savings after 5 years (260 weeks)

0

1

10

.04751 10, 1

52n n

u

u u where n

After 5 years Gwen will have $2945.89 in her account.