Welcome Mathematics of Finance
This is a revision lesson. You should review your notes before going through these lessons.
Understanding the concepts will help you to use the formulae and apply in finance problems

Mathematics of Finance :

Mathematics of Finance If you save in a bank, you expect get to more than what you put in when you take out your money in the future.
$1 today is worth more than $1 tomorrow
How much more depends on
the interest rate,
interest rate plan and
time in years

Simple Interest :

Simple Interest You earn money only on the principal; the amount you put into the bank (save) at the beginning of the savings period.
P: Principal, r: nominal interest rate; per year, t: time in number of years
Interest earned : Prt
Total amount at the end of savings period: A= P + Prt = P(1+rt)

Simple Interest Example :

Simple Interest Example You deposit $100 into the savings bank account that pay 10% simple interest per year. How much will you have at the end of 1 year
A = P(1+rt) = 100(1+ 0.1x1) = $110
How much will you have in 2 years?
A = 100(1 + 0.1x2) = $120
How much will you have in 5 years?
A = 100(1 + 0.1x5) = $150

Simple Interest :

Simple Interest Alternatively if you loan a $100 at 10% simple interest, then you will be have to pay back the amounts mentioned before.
In reality, the loan interest rate is higher than the savings interest rate. A simple analogy is like a business making profit

Present Value on Simple Interest :

Present Value on Simple Interest Using the same example, how much will we need to save now in order to have $100 in 1 year from now.
A=P(1+rt). We have A but want to find P
P= A(1+rt)-1
A=100. The amount we need to have in the future. P = 100(1+0.1x1)-1= $90.91
How much do we need to save now to have $100 in 5 years time?
t=5. P=100(1+0.1x5)-1= $66.67

Simple Interest :

Simple Interest End of Lesson
Review
$1 today > $ 1 tomorrow
How much?
Interest rate
Time in years
A=P(1+rt) or P=A(1+rt)-1

Compound Interest :

Compound Interest In a simple interest plan, interest is only calculated on the principal (the initial amount)
In a compound interest plan, interest is calculated additionally on the interests already earned.
How much?
Interest rate
Time in years
Number of compounding per year

Compound Interest Example :

Compound Interest Example You save $100 in a bank that pay interests at 10% compounded once a year (or compounded yearly). How much will you have at the end of 5 years.
Compound interest formula
A=P(1+r/m)mt =100(1+0.1/1)1x5 =161.05
Compare with the amount you get in a simple interest plan from the previous example; $150. Compound interest is better if all other things are the same for savings.

Compound Interest Example :

Compound Interest Example You save $100 in a bank that pay interests at 10% compounded twice a year (or compounded half yearly). How much will you have at the end of 5 years?
Compound interest formula
A=P(1+r/m)mt =100(1 + 0.1/2)2x5 = $162.89
Compare with compounding once a year; $161.05. Compounding twice a year is better than once a year in a savings if all other things are the same.

Present Value on Compound Interest :

Present Value on Compound Interest How much will need to save now to receive $100 in 5 years at 10% compounded once a year?
A=P(1+r/m)mt, We have A but need to find P
P=A (1+r/m)-mt, P=100(1+0.1/1)-1x5= $62.09
Compare with the present value on Simple Interest; $66.67

Effective Interest Rate :

Effective Interest Rate We can compare between different interest rate plans by using the Effective Interest Rate. reff = (1+r/m)m -1
Consider the Principal is $1.The increase (change) at the end of 1 year is the effective interest rate.

Effective Interest Rate Example :

Effective Interest Rate Example You save $100 in a bank that pay interests at 10% compounded twice a year (or compounded half yearly). What is the effective interest rate?
reff = (1+r/m)m -1 = (1 + 0.1/2)2-1 = 1.1025-1 =0.1025 = 10.25%
reff of 10.25% is higher than the nominal rate of 10%

Continuous Compounding Interest :

Continuous Compounding Interest Our final type of interest rate plan is the Continuous Compounding Interest Rate plan
A=Pert
Example
You save $100 in a bank that pay interests at 10% compounded continuously. How much will you have at the end of 5 years?
A = 100 x e 0.1x5 = $164.87
Note that Continuous Compounding Interest Plan pays more than compounding twice a year; $162.89

Compound Interest :

Compound Interest End of Lesson - Review
All other things being the same
For savings compound interest plan is better simple interest plan
Conversely for a loan, we have to pay more in a compound interest plan
More compounding period is better for savings
A = P (1 + r/m)mt or P=A(1+r/m)-mt
Continuous Compounding - A=Pert

Annuities :

Annuities In the previous examples, we find single payment and then single receipt or vice versa
In annuities (singular: annuity), we could be
Make a single payment (-) at the beginning of the finance (savings or loan) period and then get identical periodic receipts (+)
Your parents deposit a large amount of money (-) and you receive identical amounts every semester (+) to pay your course fee and expenses.

Annuities :

Annuities Examples continued
Make identical payment (-) at the end of every installment period and then get single receipts (+) at the end of the finance (savings or loan) period
You deposit (pay) in equal installment at the end every installment period (e.g. every month) to receive a lump sum at the end of the savings period (withdrawal) to pay for a new computer
Get a single receipt (+) at the beginning of the finance (savings or loan) period and then make identical periodic payments (-)
You receive an amount at the beginning to buy a house. You pay back the loan in equal installment at the end every installment period (e.g. every month)
Note that there + then – or – then +

Annuity – Savings Example :

You started to save $100 every month to buy a computer in 1 year. The bank that pay interests at 10% compounded monthly. How much do you have at end of 1 year to spend?
Formula :
S: Future Value (Savings at the end), R: Periodic amount (Payment), r is nominal bank rate per year, m is number of compounding periods in a year, t is period of finance in years n=mt & i=r/m.
R = 100, r=0.10, t = 1, m=12 n=12, i=0.10/12
Savings S = $1,256.66 Annuity – Savings Example

Annuity – Loan Example :

Annuity – Loan Example You had taken a loan now to buy your computer. You are to pay back $100 every month for 1 year. The bank charges interests at 10% compounded monthly. How much is your loan
Formula:
P: Present Value (loan at the beginning), R: Periodic amount (Payment), r is nominal bank rate per year, m is number of compounding periods in a year, t is period of finance in years n=mt & i=r/m.
R = 100, r=0.10, t = 1, m=12 n=12, i=0.10/12
Loan P = $1137.45

Amortization & Sinking Fund :

Amortization & Sinking Fund In the previous annuity examples, we are looking for the single payment/receipt at the beginning/end of a finance period.
Using the same formulae, in amortization and sinking funds, we are finding the identical period receipt/payment

Amortization Example :

Amortization Example You took a loan of $1000 to buy a computer and agree to pay back this amount in a year. The bank charges interests at 10% compounded monthly. How much is your monthly deposits?
Formula:
P: Present Value (loan at the beginning), R: Periodic amount (Payment), r is nominal bank rate per year, m is number of compounding periods in a year, t is period of finance in years n=mt & i=r/m.
P = 1000, r=0.10, t = 1, m=12 n=12, i=0.10/12
Monthly deposit (payment) R = $87.92

Sinking Fund Example :

Sinking Fund Example You need $1000 to buy a computer at the end of 1 year and save an amount monthly. The bank pays interests at 10% compounded monthly. How much is your monthly deposit?
Formula:
P: Present Value (loan at the beginning), R: Periodic amount (Payment), r is nominal bank rate per year, m is number of compounding periods in a year, t is period of finance in years n=mt & i=r/m.
S = 1000, r=0.10, t = 1, m=12 n=12, i=0.10/12
Monthly Deposit (payment) R = $79.58

Revision :

Revision (-) are payments like loan payments, deposits in a savings account
(+) are receipts like loan value you receive at the beginning of the loan period or the amount you withdraw at the end of savings period
Present Value (single):
Future Value (single):
Loan Amortization; identical periodic repayment, R=
Sinking Fund; identical periodic payment, R=

End of Mathematics of Finance Revision :

End of Mathematics of Finance Revision You should now attempt more Mathematics of Finance questions from:
Basic Text (Book), Applied Mathematics for the Managerial, Life and Social Sciences, 4th Edition, ST Tan, Thomson Brooks/Cole, 2007
Past Examination Paper

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