# Mathematics of Finance Revision

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### Welcome :

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