Download Total interest on part payment credit (based on mean payment time)

Part payment and Annual percentage rate
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Part payment
Concepts:
Consumption credit is a credit given to a consumer so that he/she can get a
consumer product.
Single credit is a credit given for a single purchase, such as instalment credit.
Cash price is the price a consumer would pay if he/she bought it with cash.
Credit costs are all the costs, interest, expenses and other payments, that a
consumer needs to pay for his/her consumption credit.
Credit price is the total amount of cash price and credit costs.
Credit amount is the part of the cash price for which the consumer gets payment
time.
Down payment is the part of the payment paid at the conclusion of the sale.
Nominal interest is the annual interest of the credit mentioned in the contract.
Annual percentage rate is the rate when credit costs are calculated as annual
interest to the amount of credit and considering the instalments.
Mean payment time
Mean payment time is the time after which all credits can be paid in one time so
that debtor or creditor does not lose any interest loss.
If the instalments of a debt are of equal amount and take place even intervals,
mean payment time (t) is the mean of the first (t1) and the last (tn) payment time:
t
t1  t n
.
2
It is customary when part payments are applied that the time between signing the
contract and the first part payment is the same as the interval between
Part payment and Annual percentage rate
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payments. If this time in months is l and the payment time in months is t, the
mean payment time is then
t l
2
Total interest on part payment credit (based on mean payment time)
I
P  R  (t  l1 )
200  12
R = nominal annual interest rate as a percentage
P = credit amount
l1 = interval between purchase and first payment in months
t = credit time in months
Annual percentage rate (APR)
R% 
200  12  C
P  (t  l1 )
R = annual percentage rate
C = loan expenses (interest and all other charges)
P = credit amount
l1 = interval between purchase and first payment in months
t = credit time in months
N.B. Loan expenses are also discounts and compensations which are usually given to
those paying with cash without separately negotiated.
Part payment and Annual percentage rate
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Example. Cash price of a product is 490 €. When using part payment, the down
payment is 20 % of the cash price. After that there are 12 equal instalments
monthly. For the loan 16 % nominal rate is applied, and each monthly payment
also include 3 € billing charge.
a) Calculate the instalment and the credit price.
b) What is the annual percentage rate of the credit?
a)
Cash price: 490 €
Down payment: 20 % of 490 €: 98 €
Credit amount (P): 392 €
Credit time (t): 12 (months)
Payment interval (l): 1 (month)
Nominal annual interest rate (R): 16 (%)
Total interest of the part payment credit:
I
392  16  (12  1)
= 33,97 €.
200  12
From this we get the instalment by calculating:
Financing: 392 € / 12 = 32,67 €
Interest: 33,97 € / 12 = 2,83 €
Billing charge: 3 €
Instalment: 38,50 €
Credit price: (98 € + 12 × 38,50) = 560 €.
b) Total loan expenses: C = 33,97 € + 12 × 3 € = 69,97 €.
Annual percentage rate:
R
200  12  69,97
%  32,95 % .
392  (12  1)
Part payment and Annual percentage rate
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Example. Suppose snow tyres cost 600 €. We buy them using part payment.
Down payment is then 100 €. On top of that there will be nine monthly payments
each of equal amount. Nominal interest is 12 %. Each monthly payment include
3 € billing charge. Also, in the tyre business it is customary that cash discount is
five per cent. What is the annual percentage rate of the credit?
Price of the tyres
Down payment
600 €
100 €
Credit amount
500 €
Credit time
Payment interval
Nominal interest
9 months
1 months
12 %
Total interest of the part payment credit:
I
500  12  (9  1)
 25 €
200  12
Credit expenses:
Interest
Billing charges 9∙3 €
Cash discount 5% / 600 €
25 €
27 €
30 €
Total
82 €
Annual percentage rate:
R% 
200  12  82
 39,36%
500  (9  1)
We see that annual percentage rate is over triple compared to the nominal rate…
In car business instalments are often calculated using annuity technique as equal
monthly payments.
Part payment and Annual percentage rate
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Example. A car shop offers a 10 000 € part payment credit for customers to buy
a new car. Loan time is 36 months, nominal rate is 12 % and billing charge is 3 €
for each payment.
a) Calculate the amount of the instalment paid monthly.
b) What is the annual percentage rate of the credit?
a) The total number of payments is 36 and monthly interest rate is 1 %, and
therefore equal monthly payment is
payment =
1,0136  0,01
 10 000 = 332,14 €
1,0136  1
Instalment is then 332,14 € + 3 € = 335,14 €.
b) Amount of credit P = 10 000 €,
Credit charges: C = 36 × 335,14 € - 10 000 € = 2 065,04 €
Payment time: t = 36 months
Payment interval: l = 1 month
R% =
200  12  2 065,04
= 13,4 %
10 000  (36  1)