We shall look at two forms of loan that are calculated using simple interest, but involve higher interest payments than you might expect.

The first example is the add-on loan. In an add-on loan you add the whole amount of simple interest to the principal at the beginning of the loan period. If the principal is $P, the interest is R%, and the period is n years, then the total to be paid back is $P(1+ nR/100 ).

Very often these loans are for short periods, and in those cases the interest is high.

Sample Problem 1.1 What is your monthly payment on an add-on loan if you borrow $12,000 over 5 years at 8% per year?

Solution. Simple interest is $960 per year, so the total (simple) interest for 5 years is $4,800. Therefore the total to be paid is $16,800. There are 60 monthly payments, so your monthly payment will be $16,800/60, which is $280.

The second example is a discounted loan. In a discounted loan you subtract the interest from the amount borrowed. Suppose your loan says you will borrow $P at an interest rate of R%, and the period is n years. Instead of $P you receive $P(1 − Rn/100). For example, if your loan has principal $12,000 over 5 years at 8%, you only receive

                                   $12,000×(1−.4) = $7,200.

At the end of the period you repay the original principal, $12,000 in the example.

These loans are sometimes used by auto sales companies, for lease agreements with an option to buy.

Sample Problem 1.2 You need to pay $12,000. What will be your payments for a 5-year discounted loan at 8% per year?

Solution. If you need $12,000 then your principal will be $A where

                             A×(1−.4) = 12,000

so $A = $(12,000/0.6) = 20,000. Your monthly payments total $20,000 over 60 months, so they equal $333.33 per month, to the nearest cent. (In actual fact, you would probably pay $333.34 per month, with a last payment of $332.94, or maybe $334 per month, with the last payment adjusted down.)

Observe the difference between the payments in the two Sample Problems. This is not an isolated example. An add-on loan is always better than a discounted loan at the same (non-zero) interest rate.

To see this, suppose you need $100. If you borrow $100 for n years at R% interest, using an add-on loan, you eventually pay $100(1+ nR/100 ) = $(100+nR). In order to obtain $100 using a discounted loan at R%, your “principal” is $P, where    P(1−nR/100) = 100.

Suppose the discounted loan were as good a deal as the add-on. Then

                     P ≤ 100+nR. Then

100 = P(1−nR/100) ≤ (100+nR)(1−nR/100)=(100+nR)(100−nR)/100

    from which

               10,000 ≤ (100+nR)(100−nR) = 10,000−n²R².

This would mean n²R² ≤ 0. This is never true.

In any case, for a discounted loan or an add-on loan to be worthwhile, the interest rate must be low. They are better for short-term loans.

One can compare compound interest loans with these other sorts of loans. For example, an add-on loan of $1,000 at 5% interest for a 4-year period requires monthly payments of $25. If you took out a loan for 4 years at 6% compound interest, and found your monthly payment to be $25, your principal was $P,

 Where