I =

Math 166 Chapter F : Finance TAMU – Spring 2016

F.1 Simple Interest : If a sum of money P, called the Principal or the Present Value, is invested for t years, at an annual simple interest of r % , then the interest I at the end of the year is

I =

The future value F at the end of the year is F =

Note :

1. Simple interest depends on Principal only, not the interest.

2. The interest earned is constant for every period. So, if the interest earned is I for a year, then the

interest every month would be I / 12.

3 . t is in years. So, if the question says the interest is earned for 6 months, we will take t = 0.5 years.

Example 1.

Suppose you invest $5000 at an interest rate of 5%. i)What will be the interest earned in a) 1 year b) 1 month c) 3 months d) 5 th month ii) What will be the value of your money after a) 6 months b) 1 year c) 3 years

Example 2 . a) If I want $1500 at the end of 5 years, at what interest rate should I invest $1000?

b) If I use the interest rate in part (a) with the same present value, how long will it take the money to

increase to $3000?

1


F.2 Compound Interest

Compound Interest : Suppose a principal P earns interest at a nominal rate of r (expressed as a decimal) , compounded annually, and the interest is compounded m times a year. Then, the future value

F is the amount in the account after t

years,

where n = mt is the number of time periods .

Calculator :

1) Compound Interest

APPS → 1: FINANCE → 1: TVM Solver →

N = Total number of compounding periods during the entire time period

I% = Annual Interest rate as a percent

PV = Present Value (Initial amount)

PMT = Recurring payments

FV = Future value of the amount

P/Y = Number of payments per year

C/Y = Number of compounding per year

** C/Y and P/Y will be equal.

** END should be highlighted.

** Outflow of money is Negative; Inflow of money is Positive.

**Once you input all the values, scroll to the unknown quantity and press SOLVE (ALPHA ANTER).

Example 3.

Justin borrowed $5000 from a friend at an annual interest of 4%, for 3 years. What will

Justin pay back if the interest is compounded

Annually Semi-Anually Quarterly Monthly Weekly

N

I%

PV

PMT

FV

P/Y

C/Y b) What will Justin pay back if the interest is compounded continuously?

Daily

2


Example 4. Repeat Example 1 if the the money is borrowed at a semi-annual interest rate of 4%.

Annually Semi-Anually Quarterly Monthly Weekly

N

I%

PV

PMT

FV

P/Y

C/Y b) What will Justin pay back if the interest is compounded continuously?

Daily

Example 5. Your parents invest a sum of money in your account at an interest rate of 6.2%. The money will be worth $1,500,000 at the end of 25 years. How much did they invest, if the interest is compounded annually?

N

I%

PV

PMT

FV

P/Y

C/Y

3


Example 6. At what interest should you invest $10,000 in a hedge fund so that the money doubles in 5 year, when it is compounded quarterly?

N

I%

PV

PMT

FV

P/Y

C/Y

Example 7. How many years will it take for $1275 to increase to $5000, if it is invested at an interest rate of 8%, compounded monthly?

N

I%

PV

PMT

FV

P/Y

C/Y

4


Effective Yield : Suppose a sum of money is invested at a nominal rate of r and is compounded m times a year. The effective yield ( or effective rate) is the simple interest rate that would produce the same accumulated amount in 1 year as the nominal rate compounded m times a year.

This serves as a common basis for comparing different interest rates, compounded with different frequencies.

Calculator :

APPS → 1: FINANCE → C: Eff → Enter r, m

where,

r : rate of interest m : number of compounding per year

Example 8. Bank A advertises a nominal rate of 6.1% compounded semi-annually. Bank B advertises a nominal rate of 6% compounded daily. a) What are the effective yields?

b) In which bank should you deposit your money?

c) From which bank should you borrow money?

5

6

N

I%

PV

PMT

FV

P/Y

C/Y


F.3 Annuities and Sinking Funds

Annuity: Sequence of equal payments made at made at equal time periods.

Ordinary Annuity: Payments are made at the END of time periods of compounding.

Term: of an annuity is the time from beginning of the first period to the end of the last period.

Future Value of an Annuity : is the TOTAL AMOUNT (including the interest) in the account at the

end of the term.

e.g. Monthly insurance payments , Regular deposits, Monthly rent payments

Sinking Fund : An account that is established for accumulating funds to meet a future need.

Example 9. How much money should Ted deposit in the account today, to be able to withdraw $5000 at the end of every 6 months, for the next 10 years, if the account earns an interest of 5.8% .

N

I%

PV

PMT

FV

P/Y

C/Y

Example 10. Paula deposited $5000 into here bank account at the beginning of this month, and will deposit $150 per month, into the same account, at the end of each month (including the current month), for the next 6 years. If the bank pays an interest of 3%, compounded monthly, how much money will

Paula have in her account at the end of 6 years?


Example 11.

I deposit $500 at the end of every month in my bank account, at an interest rate of 3%. a) How much money will I have in my account after 5 years, if the interest is compounded monthly?

N

I%

PV

PMT

FV

P/Y

C/Y b) What is the total interest earned in these 5 years?

c) What will be the interest earned in the 4 th month of the second year?

Example 12. Rob needs to save $20,000 to put a down payment on a house at the end of 6 years. If the interest rate for his savings account is 8% , and the interest is compounded monthly, how much should he deposit at the end of each month into the savings account to meet his goal?

N

I%

PV

PMT

FV

P/Y

C/Y

7


Example 13. Chris borrowed $50,000 from a bank for 7 years, at an interest rate of 8%, compounded monthly. He makes payments of $750 every quarter. What will be his outstanding balance after the 60 payment?

th

Example 14.

Ted and Rob need to set up a sinking fund to start a business in 10 years. They will need to save up $100,000. If the deposits are made per quarter, and the annual interest rate is 8% a) How much per quarter should be deposited into the account compounded quarterly to meet their goal?

N

I%

PV

PMT

FV

P/Y

C/Y b) What will be the interest earned by the sinking fund during the first quarter of the 5 th year?

c) What is their outstanding balance after the 28 th payment?

8

N

I%

PV

PMT

FV

P/Y

C/Y


F.4 Amortization

Example 15.

I bought a car for a certain price, at a monthly interest rate of 1.99%, for 60 months. If I made a down payment of $2000, and I make monthly payments of $390.00, then a) What is the cash price of my car?

b) What is the total interest I would have paid on my car at the end of 60 months?

c) What is my outstanding balance after 2 years?

d) What is my equity after 2 years?

9


Example 16.

You want to sell a piece of land and agree to grant a loan of $2,500 to the buyer, with an annual interest of 9.701%, compounded annually, for 3 years. He makes equal payments at the end of each year. Make an Amortization Schedule for the 3 years.

N

I%

PV

PMT

FV

P/Y

C/Y

0

1

End of

Period

2

3

Payment Made Interest Owed Payment toward

Principal

Outstanding

Principal

Equity

10


Example 17.

Paula purchased a house for $150,000. She made an initial down payment of $ 20,000 and secured a mortgage for the rest of the balance. The loan was was taken at an interest rate of 7%. If the loan is to be amortized over 30 years, then a) Find the monthly payments made by Paula?

b) What is the outstanding balance after 12 years?

c) What will be her equity after 12 years?

d) What is her equity after 23 years?

11

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