Chapter 5 Mathematics of Finance

Section 5.1 Simple Interest Section 5.2 Compound Interest Section 5.3 Annuities and Sinking Funds Section 5.4 Present Value of an Annuity and Amor

tization

Simple Interest Simple interest is most often used for __________of

_____________ duration. The money borrowed in a loan is called the ___________. The number of dollars received by the borrower is the

________________________. In a simple interest loan, the ___________ and present value are

the same. The ______________ is the fee for a simple interest loan and

usually is expressed as a percent of the principal. Simple interest is paid on the principal _____________ and not

paid on interest already earned.

Section 5.1

Simple InterestThe simple interest of a loan can be calculated using the following formula.

ExampleAn individual borrows $300 for 6 months at 1% simple interest per month. How much interest is paid?

Another ExampleJane borrowed $950 for 15 months. The interest was $83.13. Find the interest rate.

Future ValueA loan made at simple interest requires that the borrower pay back the sum borrowed (principal) plus the interest. This total is called the future value, or amount and is equal to P + I.

ExampleFind the amount (future value) of a $2400 loan for 9 months at 11% interest.

Simple DiscountThe simple discount loan differs from the simple interest loan in that the interest is deducted from the principal and the borrower receives less than the principal.

This type of loan is referred to as a simple ____________ note.

The interest deducted is the _________________.

The amount received by the borrower is the ______________.

The discount _______________is the percentage used.

The amount repaid is the ___________________________.

Simple Discount Note

D = PR = = =

ExampleFind the discount and the amount a borrower receives (proceeds) on a $1500 simple discount loan at 8% discount rate for 1.5 years.

ExampleA bank paid $987,410 for a 90-day $1 million treasury bill. What was the simple discount rate?

ExampleA bank wants to earn 7.5% simple discount interest on a 90-day $1 million treasury bill. How much should it bid?

ExampleHow much should a bank bid on a 30-day $2 million treasury bill if the bank wants to earn 5.125% on its money.

HW 5.1

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Compound Interest

Section 5.2

Suppose you deposit money into a savings account, the bank will typically pay you interest for the use of your money at a specific period of time, say every three months. The interest is usually credited to your savings account at each time period. At the next time period, the bank will pay interest on the new total, this is called _________________________.

Amount of Annual Compound InterestWhen P dollars are invested at an annual interest rate r and the interest is compounded annually, the amount A at the end of t years is

__________________

ExampleSuppose $800 is invested at 6%, and it is compounded annually. What is the amount in the account at the end of 4 years?

Amount (Future Value)The general formula for finding the amount after a specified number of compound periods is

ExampleSuppose $800 is invested at 12% for 2 years. Find the amount at the end of 2 years if the interest is compounded (a) annually, (b) semiannually, and (c) quarterly.

ExampleSuppose $800 is invested at 12% for 2 years. Find the amount at the end of 2 years if the interest is compounded (a) annually

ExampleSuppose $800 is invested at 12% for 2 years. Find the amount at the end of 2 years if the interest is compounded (b) semiannually

ExampleSuppose $800 is invested at 12% for 2 years. Find the amount at the end of 2 years if the interest is compounded (c) quarterly.

Vocabulary Nominal Rate•

Effective Rate

Effective RateThe effective rate of an annual interest rate r compounded m times per year is the simple interest rate that produces the same total value of investment per year as the compound interest.

ExampleThe Mattson Brothers Investment Firm advertises Certificates of Deposit paying a 7.2% effective rate. Find the annual interest rate, compounded quarterly, that gives the effective rate.

SOLUTIONIf we let i = quarterly rate, then

4

4

4

0.072 (1 ) 11.072 (1 )1.072 1

1.017533 10.017533

ii

ii

i

The annual rate = 4(0.017533) = 0.070133 = 7.013% (rounded). The annual rate just found is also called the nominal rate.

HW 5.2

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Ordinary Annuity

An annuity refers to equal __________ paid at equal ________ intervals.

The time between successive payments is called the ________________________________.

The amount of each payment is the _________________________________.

The interest on an annuity is _________________ interest.

An _______________________ is an annuity with periodic payments made at the end of each payment period.

Section 5.3

Future Value (Amount)

Payments are made at the end of each period.

where i =n =R =A =

ExampleHow much money will you have when you retire if you save $20 each month from graduation until retirement? Let’s assume you start saving at age 22 until age 65, 43 years, and the interest rate averages 6.6% annual rate compounded monthly.

How much money will you have when you retire if you save $20 each month from graduation until retirement? Let’s assume you start saving at age 22 until age 65, 43 years, and the interest rate averages 6.6% annual rate compounded monthly.

Sinking FundsA sinking fund refers to a fund that is created when an amount of money will be __________ at some future date. For example, a family may need a new car in 3 years, or a company may expect to replace a piece of equipment in the future.

ExampleDarden Publishing Company plans to replace a piece of equipment at an expected cost of $65,000 in 10 years. The company establishes a sinking fund with annual payments. The fund draws 7% interest, compounded annually. What are the periodic payments?

Darden Publishing Company plans to replace a piece of equipment at an expected cost of $65,000 in 10 years. The company establishes a sinking fund with annual payments. The fund draws 7% interest, compounded annually. What are the periodic payments?

You figure you will need $30,000 to use as a down payment on a house you want to buy in 5 years. How much should you put in your bank account every month to accumulate $30,000 in 5 years if your bank account pays 5% annual interest compounded monthly?

A couple wants to start a college fund for their new born child. They figure they will need $120,000 in 18 years to pay for the child’s education in college. If they set up an annuity that pays 6.5% compounded quarterly, what is the amount of the quarterly payment they will need to achieve this goal?

HW 5.3

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Present Value

Section 5.4

The present value of an annuity is the ______________ payment that yields the same total amount as that obtained through equal _____________ payments made over the same period of time.

ExampleFind the present value of an annuity with periodic payments of $2000, semiannually, for a period of 10 years at an interest rate of 6% compounded semiannually.

Equal Periodic PaymentsThe amount needed to provide equal periodic payments can be found using the formula

or equivalently,

where P = amount needed in the fundR = amount of periodic paymentsi = periodic interest raten = number of payments

ExampleFind the present value of an annuity (lump sum investment) that will pay $1000 per quarter for 4 years. The annual interest rate is 10%, compounded quarterly.

Two Ways to Save Money Future Value of an

Annuity

Present Value of an Annuity

Two Ways to Save Money You decide to save

for retirement by making regular payments of $50 to an annuity that pays 5% compounded monthly. How much money will you have in 20 years?

Find the lump sum payment that will yield the same amount after 20 years.

Present Value of an Annuity The __________ required to pay out

regular payments over time.

Determine the amount required to collect 60 monthly payments of $2000 at 5% compounded monthly.

Determine the amount required to collect 60 monthly payments of $2000 at 5% compounded monthly.

AmortizationThe amortization of a debt (___________________) requires no new formula because the amount borrowed is just the present value of an annuity.

George want to buy a car for $32,595. He decides to put nothing down on it a want to finance it for 5 years. The car company offers him an interest rate of 5.95%. What are George’s monthly payments and how much does the car actually cost him.

ExampleA student obtained a 24-month loan on a car. The monthly payments are $395.42 and are based on a 12% interest rate. What was the amount borrowed?

Balance of an AmortizationThe balance after n periods is the amount of compound interest minus the amount of an annuity. Mathematically we can find the balance using the formula

(1 ) 1Balance (1 )

nn i

P i Ri

where P = the amount borrowedi = periodic interest raten = number of time periods elapsedR = monthly payments

Bal(# of Payments) ---- Be careful the solver menu must be filled in correctly

ExampleA family borrowed $60,000 to buy a house. The loan was for 30 years at 12% interest rate. The monthly payments were $617.17. What is the balance of their loan after 2 years?

HW 5.4

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