Calculation for interest in hire purchase

Under this method, the interest is apportioned evenly over the number of instalments agreed upon.

$Hire\ purchase\ interest\ per\ instalment = \frac{Total\ hire\ purchase\ interest}{Total\ number\ of\ instalments}$

$Hire\ purchase\ interest = Hire\ purchase\ price - Cash\ price$

Alternatively, when the interest rate is given:

$Hire\ purchase\ interest\ per\ instalment = (Hire\ purchase\ price - Cash\ price) \times rate$

Example

On 1st January, 2015 JOY Ltd. acquired a motor vehicle from PEC Ltd. on hire purchase. The terms of agreement were as follows:

Details	Amounts (TZS)
Cash Price	18,000,000
Down Payment	3,000,000
Hire Purchase Price	19,500,000
Nominal Rate of Interest	20%
Number of Annual Instalments	5
Total Interest	1,500,000
Annual Instalment	3,300,000

Required:

Calculate the amount of interest to be included in each instalment using straight line method.

Hire Purchase Interest Payable Per Instalment	Amount (TZS)
Hire purchase price	19,500,000
Less: Cash price	18,000,000
Total hire purchase interest	1,500,000
Number of instalments	5
Hire purchase interest payable per instalment	300,000

Date	Cash Price Outstanding (TZS)	Principal (TZS)	Interest (TZS)	Total Instalment (TZS)	Closing Balance (TZS)
01st Jan., 2015	18,000,000	3,000,000	-	-	15,000,000
31st Dec., 2015	15,000,000	3,000,000	300,000	3,300,000	12,000,000
31st Dec., 2016	12,000,000	3,000,000	300,000	3,300,000	9,000,000
31st Dec., 2017	9,000,000	3,000,000	300,000	3,300,000	6,000,000
31st Dec., 2018	6,000,000	3,000,000	300,000	3,300,000	3,000,000
31st Dec., 2019	3,000,000	3,000,000	300,000	3,300,000	-
TOTAL	18,000,000	15,000,000	1,500,000	16,500,000	-

Under this method, the hire purchase interest is apportioned according to the digits assigned. Usually, the digits are assigned in a descending order. The highest digit is given to the first instalment and the digit 1 to the last instalment. This method results in charging more interest in earlier periods compared to later periods. The logic behind this is that the acquired asset is assumed to be more productive in early years. The interest proportion to each accounting period is then calculated as follows:

$\text{Interest Proportion} = \frac{\text{Digit Given to Instalment} \times \text{Total Interest}}{\text{Sum of Digits}}$

$\text{Sum of Digits} = \frac{n(n + 1)}{2}$

Where:

• n = Number of years
• The "digit given to instalment" refers to the reverse order of the year (e.g., for a 3-year lease, Year 1 = 3, Year 2 = 2, Year 3 = 1)

Example

On 1st January, 2017 MKWAJUNI Ltd. acquired computers from KOA Ltd. on hire purchase agreement which provided the following terms:

Details	Amounts (TZS)
Cash price	12,400,000
Down payment	6,000,000
Hire price	13,020,000
Number of annual instalments	4

Required:

Calculate the hire purchase interest per instalment under each of the following situations:

1. If the first instalment is due at the end of 2017; and
2. If the first instalment is due at the date of sale.

Solution

a.

Details	Amounts (TZS)
Hire purchase price	13,020,000
Less: Cash price	12,400,000
Hire purchase interest	620,000

Number of instalments to be apportioned with interest is 4:

$Sum\ of\ digits = \frac{n(n + 1)}{2}$

Example with $n = 4$:

$\frac{4(4 + 1)}{2} = \frac{4 \times 5}{2} = \frac{20}{2} = 10$

Date	Digit Assigned	Digit Assigned / Sum of Digits	Hire Purchase Interest per Instalment (TZS)	Interest Payable (TZS)
31st December, 2017	4	4/10	620,000	248,000
31st December, 2018	3	3/10	620,000	186,000
31st December, 2019	2	2/10	620,000	124,000
31st December, 2020	1	1/10	620,000	62,000
Total	-	-	620,000	620,000

b.

If the first instalment is paid together with the down payment, then the remaining number of instalments to be charged interest becomes three (3) instead of four (4). Hence, the sum of digits will be:

$Sum\ of\ digits = \frac{n(n + 1)}{2}$

Example with $n = 3$:

$\frac{3(3 + 1)}{2} = \frac{3 \times 4}{2} = \frac{12}{2} = 6$

Hire purchase interest payable per instalment is calculated as follows:

Date	Digit Assigned	Digit Assigned / Sum of Digits	Hire Purchase Interest per Instalment (TZS)	Interest Payable (TZS)
31st December, 2017	3	3/6	620,000	310,000
31st December, 2018	2	2/6	620,000	206,667
31st December, 2019	1	1/6	620,000	103,333
Total	-	-	620,000	620,000

Sometimes, figures for cash price may not be given or made available directly. In such a situation, it is not possible to record the accounting transactions in the books of the buyer because the purchased asset is recorded using the cash price. Therefore, cash price should be ascertained using the following alternative ways:

1. With the help of annuity table
2. Without the help of annuity table

a. Calculation of cash price with the help of annuity table

The use of annuity table simplifies the calculation of interest amount to be included in each instalment. In the annuity table, the rate of interest is given in the rows and the years in the columns. When using the annuity table, the first step is to calculate the present value of each instalment. Then, the down payment is added to the summation of present values to obtain cash price. The procedure is as follows:

1. Observe the given rate of interest in the row and the year in the column to find out the corresponding figure to the given interest and year.
2. The obtained figure represents a present value of each 1 unit of currency.
3. Multiply the present value with the amount of the instalment to obtain the present value of the instalment. This is the amount of cash price included in the instalment.
4. Calculate the present values of all other instalments in the same manner.
5. Add the present values of all instalments to the down payment if any; the resulting figure will be the total cash price.

Example

MABARA Ltd. is a business firm dealing with manufacturing of home furniture using steel pipes and bars. On 1st July, 2021 the firm purchased a welding machine on hire purchase system, to assist in the manufacturing process. The hire purchase price was TZS 3,750,000, payable as follows:

Details	Amount (TZS)
Down Payment	697,500
1st Instalment on 30th June, 2022	1,069,500
2nd Instalment on 30th June, 2023	1,353,000
3rd Instalment on 30th June, 2024	630,000

The rate of interest is 5% per annum, the annuity table shows the present value of TZS 1 in different periods as:

Period	Interest Rate	Value
0	5%	1.000
1	5%	0.9524
2	5%	0.9070
3	5%	0.8639

Required:

Calculate the cash price of the welding machine.

Solution

Details	Instalments	Present Value of a Single Sum at 5%	Amount (TZS)
Down Payment	-	1	697,500
30th June, 2022	1,069,500	0.9524	1,018,592
30th June, 2023	1,353,000	0.9070	1,227,171
30th June, 2024	630,000	0.8639	544,257
Total Cash Price	-	-	3,487,520

b. Calculation of cash price without the help of annuity table

Under this method, interest is calculated starting with the last instalment. Suppose there are three instalments. The interest will be calculated first on the third instalment, then on the second and lastly on the first instalment. Notice that no interest is calculated on down payment as it does not involve any element of interest. Under this method, cash price will have to be calculated with the help of total amount due on hire purchase price. Interest involved in each instalment will have to be calculated first, then subtract the interest amount from the total amount due, to obtain the outstanding amount of cash price. This is done as:

$Interest = \frac{Total\ amount\ due\ at\ time\ of\ instalment \times interest\ rate}{100 + interest\ rate}$

Steps to be followed for the calculation of cash price due at the time of each instalment assuming there are three yearly instalments are as follows:

1. Calculate the interest on the instalment of the third year, deduct interest from this instalment. The resulting figure is the outstanding cash price at the time of third (last) instalment.
2. Add the cash price calculated under (a) above to the instalment amount of the second year, calculate the interest on the sum so obtained and subtract it from the total amount due at the end of the second year to get the outstanding cash price at the time of second instalment.
3. Add the cash price calculated under (b) above to the instalment amount of the first year and calculate the interest on the sum so obtained. Deduct the amount of interest from the total amount due at the end of the first year; the resulting figure is the cash price due at the time of the first instalment.
4. Add the cash price calculated under (c) above to the down payment, if any. Then, the sum obtained will be the cash price.

Example

LUTH GARMENTS Ltd. purchased a sewing machine on 1st January, 2021 on hire purchase basis for an interest of 5%. The hire purchase price was TZS 5,000,000 payable as follows:

Details	Amount (TZS)
Down Payment	930,000
1st Instalment	1,426,000
2nd Instalment	1,804,000
3rd Instalment	840,000
Total amount due	5,000,000

Required:

Calculate the cash price of the sewing machine and interest paid with each instalment.

Solution

Detail	Amount	Interest
Total amount due on 3rd instalment (last)	840,000	40,000
Less: Interest	-	-
Outstanding cash price on 3rd instalment	800,000	-
Add: 2nd instalment	1,804,000	-
Total amount due on 2nd instalment	2,604,000	124,000
Less: Interest	-	-
Outstanding cash price on 2nd instalment	2,480,000	-
Add: 1st instalment	1,426,000	-
Total amount due on 1st instalment	3,906,000	186,000
Less: Interest	-	-
Outstanding cash price on 1st instalment	3,720,000	-
Add: Down payment	930,000	-
Cash price	4,650,000	-
Hire purchase interest	-	350,000

$Cash\ price = Hire\ purchase\ price - Hire\ purchase\ interest$ $= TZS\ 5,000,000 - TZS\ 350,000 = TZS\ 4,650,000$