What is loan amount with payment of 326.50 for 36 months?

Answer 1

WITHOUT KNOWING the interest rate, and whether the interest is simple or compounded, and how often the latter is applied, it is IMPOSSIBLE to answer the question.

As the interest rate, the type of interest and how often it is applied is NOT given, the answer cannot be given.

All I can give you is a formula for you to work it out yourself:

A loan will usually be charged at compound interest compounded every month.

Let L = amount of Loan

Let r = monthly rate (see below)

Let p = payment per month

The rate r is not quite the specified APR.

The APR is given as a percentage: APR = apr% = apr/100

Assume that no repayments are made. Then interest is added onto the interest.

and the amount to pay will increase by L × apr/100

→ total to pay = L + L × apr/100 = L(1 + apr/100)

→ Annual rate = 1 + apr/100

Now assuming no payments are made for 12 periods with a period percentage rate of ppr%

→ amount to pay = L(1 + ppr/100)×(1 + ppr/100)×...×(1 + ppr/100) = L(1 + ppr/100)¹²

The rate for one of the periods is the 12th root of the rate to get the interest added after 12 periods.

If the period is a month, then 12 monthly periods = 1 year

→ the monthly rate is the 12th root of the annual rate

→ monthly rate (r) = (1 + apr/100)^(1/12)

Then:

After 1 month, the amount remaining to pay is Lr - p

After 2 months, the amount remaining to pay is (Lr - p)r - p = Lr² - p(r + 1)

After 3 months, the amount remaining to pay is (Lr² - p(r + 1))r - p = Lr³ - p(r² + r + 1)

After n months, the amount remaining to pay is Lrⁿ - p(rⁿ⁻¹ + rⁿ⁻² + ... + r + 1)

rⁿ⁻¹ + rⁿ⁻² + ... + r + 1 is a GP with sum = (rⁿ - 1)/(r - 1)

→ amount to pay after n months = Lrⁿ - p(rⁿ - 1)/(r - 1)

If the loan is paid off after n months, then the monthly payment can be calculated:

0 = Lrⁿ - p(rⁿ - 1)/(r - 1)

→ p(rⁿ - 1)/(r - 1) = Lrⁿ

→ p = Lrⁿ(r - 1)/(rⁿ - 1)

Similarly, The Loan amount can be calculated:

0 = Lrⁿ - p(rⁿ - 1)/(r - 1)

→ Lrⁿ = p(rⁿ - 1)/(r - 1)

→ L = p(rⁿ - 1)/(rⁿ(r - 1))

It is up to you to insert the relevant figures for p and r (calculated from APR).

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If it is simple interest applied for the full period, then:

r is calculated slightly differently:

As the interest is only dependent upon the number of periods:

Interest = L × apr/100 × number of years

The monthly rate is 1/12 the annual rate as simple interest assumes the same amount added per period and 1 year = 12 monthly periods

This gives:

amount to repay = L(1 + apr/100 × n/12) = pn

→ L = pn/(1 + n × apr/1200)

where apr = annual percentage rate

L = loan

p = monthly payment

n = number of months.

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Of note is that when I financed the purchase of my car, they calculated the monthly payment based on simple interest, ie they added simple interest for the period of the loan and then divided this by the number of months By doing this the compounded APR (the "representative APR" given to allow comparison with other loans available) is higher than the simple APR as the simple APR ignores the fact that part of the monthly payments are repaying part of the loan and so not the whole loan is borrowed for the whole period; the proportion of the monthly repayment that is the loan increases as the time progresses towards the end of the loan period.

An interest only mortgage works in a similar way to simple interest - your monthly payments are lower as you are only paying back the interest accrued each month. At the end of the mortgage period you have the find the capital to pay off the WHOLE of the value loaned from somewhere: this led to endowment mortgages where you paid interest only to the mortgage provider and also paid some money into a fund that you hoped would gain enough interest to have enough in the pot to cover the whole of the mortgage loaned (with some extra). If the fund did not perform as well as expected, you would end up with a deficit in your pot you would need to find from somewhere.

Answer 2

The answer will depend on the interest rate on the loan and the frequency at which interest is compounded.