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BFinancial Mathematics
Credit and Loans:
Simple Interest and Flat Rate Loans:
A flat rate loan is one where flat or simple interest is charged on an amount borrowed or principal for the term of the loan. Interest is always charged on the full amount of the loan.
I = Prn
P = principal
r = rate per period expressed as a decimal
n = number of periods
E.g. Phil borrowed $4000 for three years at 8%p.a. (per annum) (flat rate)
a) What is his interest?
b) What is the total repaid?
c) What are the monthly repayments?
Solution:
a) 4000x8/100x3
Interest=$960
b) Total Repaid: interest + principal
=960+4000
=$4960
c) Monthly repayments:
=496036 (36 is the number of months is 3yrs)
=137.777
=137.78
Buying on Terms:
Time payment agreement to pay for goods over a certain period of time
This is also called a hire purchase as the customer actually hires (borrows) the good until they are paid off.
Goods can be reposed if payments are failed to be paid
Deferred Payment Plan deposit, interestfree period,
E.g. Jem borrowed $200 at $120 per month for two years. He also paid $300 deposit
What was the cash price of the item
=$2300
What did he pay in total?
= 120x24+300
=$3180
What was the flat interest rate?
= Int= 31802300
=$880
=Rate: I=PRN
=880=2000xrx2 (2 means years of interest rate in % p.a.)
r=88020002
r=0.22 (x100 to find %)
=22% p.a.
Reducing Balance Loan:
Reducible interest
Interest is calculated on the balance still owing, not on the total principal borrowed with flat rate interest
Interest calculated one period at a time
Shortcut Calculator, operations between = signs get repeated on the calculator
e.g. 5000= x1.09800 = Ans=Ans=Ans etc. Push equals require amount of times.
E.g. 1) Teri borrows $5000 at 9% reducible interest. If she pays $800 per year, show the first four years balances.Solution: (Big R means Repayment)
Year(P) Starting balanceP+Int.(P+IR) End Payment 150005000x1.09=54505450800=4650246504650X1.09=5068.505068.50800=4268.534268.50=4652.6=3852.643852.6=4199.4=3399.404
Published Loan Repayment tables:
 What banks uses to calculate large loans
E.g. using table in textbook page 90.
Mr. and Mrs. Pitt obtain a premium home loan of $370,000 at 7%p.a. reducible interest for a term of 20yrs find:
I) the monthly repayment:
Monthly repayment for $1000=7.75
Monthly repayment for $370,000=370x$7.75 = $2867.50
II) The total amount repaid:
=$2867.50x20x12 =$688,200
III) The total interest paid
Interest Paid=total amount repaidamount borrowed
=4688,200$370,000 =$318,200
Credit Card Payments:
Two types of credit cards:
No interestfree period and no annual fee at a lower interest rate
An interestfree period and an annual fee account is paid in full before the period
Ends otherwise interest is charged from the date of purchase
The due date on a statement is when the interestfree period ends.
E.g. 1) Manual has a credit card with no interestfree period and interest rate of 14% p.a. He makes the following purchases for the period 1 August to 31 August:
2 August Dinner Set $65.5
16 August Pair Trousers $85.00
23 August Haircut $24.00
26 August Diner $36.80
29 August White Shirt $32.00
a) What is the total amount of his purchases?
Total purchases=$65.50, $85.00, $24.00, $36.80, $32.00 = $243.30
b) Manuel pays his account in full on 3 September. How much does he pay?
Interest is charged on each purchase from the date of purchase until the date payment is received. For example the dinner set is bought on 2 August and paid for on 3 September.
Number of days=29+3= 32
Interest rate per annum= 14%
Interest rate per day= 14/36500
Purchase AmountNo. of days interestInterest to 3 September ($)$65.003265.00x14/36500x32=0.8048$85.003214 =1885.00x14/36500x18=0.5875$24.0011=0.1013$36.808=0.1130$32.005=0.0614
Total interest= $1.6672=$1.67
Manuels total payment= $243.30+$1.67
= $24409
See text book for another example pg. 98
Compound Interest:
A = P(1 + r) or I = A P
A = final amount
I = compound interest
P = principal
r = interest rate per compounding period (decimal)
n = number of compounding periods
Effective Interest Rates
E = (1 + r)  1
r = stated rate per compounding period (decimal)
n = number of compounding periods
Annuities:
An annuity is an investment in which a series of periodic equal contributions made to an account for a specified term.
The Future Value of an Annuity:
The future value of an annuity is the total value at the end of the term. It is sometimes called the amount of the annuity and includes all payments deposited as well as the accrued interest.
A = M (1 + r)  1
r
M = contribution made at the end of each compounding period
r = rate of interest pre compounding period (decimal)
n = number of compounding periods
The Present Value of an Annuity:
The present value of an annuity is the single sum of money (principal) that you could invest today at the same compounded interest rate to produce the same amount as you would by investing a series of regular payments over the same term.
A = M (1 + r)  1 or N = A .
r(1 + r) (1 + r)
Loan Repayments:
Present value of annuity formula used to find the amount borrowed N or the loan repayment M.
The amount owing on a loan at any time is equal to the present value of the remaining payments.
Depreciation:
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&FgdKMgd"Kn asset is its purchase price
Salvage value of an asset is its value at a particular time: book value or current value
Total depreciation of an asset is the difference between the initial value and the salvage value
Straight Line Method of Depreciation:
When the item decreases by the same amount each year.
S = V  Dn
S = Salvage/current value at the end of n periods
(Salvage value = initial value total depreciation)
V = purchase/original price
D = depreciation per period
n = number of periods
Declining Balance Method of Depreciation:
Reduces the value of the asset each year by a constant percentage.
S = V(1 r)
S = Salvage/current value at the end of n periods
V = purchase/original price
r = depreciation per period
n = number of periods
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