Math In FM

7/29/2019 Math In FM

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Chapter 5Mathematics 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

Amortization


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Simple Interest

Simple interest is most often used for loans of shorter

duration.

The money borrowed in a loan is called theprincipal.

The number of dollars received by the borrower is the

present value.

In a simple interest loan, the principal and present value

are the same. The interest rate 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 borrowed and

not paid on interest already earned.

Section 5.1


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Simple Interest

I=Prtwhere P= principal (amount borrowed)

r= interest rate per year (in decimal form)

t= time in years.

The simple interest of a loan can be calculated using the

following formula.


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Example

An individual borrows $300 for 6 months at 1% simple

interest per month. How much interest is paid?

SOLUTIONRemember to check that rand tare consistent in time

units. Here, it will be months.

I=Prt= 300 0.01 6 = $18


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Another Example

Jane borrowed $950 for 15 months. The interest was

$83.13. Find the interest rate.

SOLUTION

First, determine the information we have been given.

P= 950

t= 15/12 = 1.25 years

I= 83.13

UsingI=Prt, we need to find r

83.13 950 (1.25)1187.5

83.130.07

1187.5

r

r

r

The interest rate was 7%.


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Future Value

A loan made at simple interest requires that the borrower pay

back the sum borrowed (principal) plus the interest. This total

is called thefuture value, oramountand is equal toP+I.

A =P+I

= P + Prt

= P(1 + rt)

where P= principal or present value

r= annual interest rate (in decimal form)

t= time in years

A = amount or future value


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Total Number of Compounding Periods

Annual Interest rate expressed as a percent

Present Value: What it is worth to you right now

Amount of the payment being made if any.

Future Value: What it will be worth to you

in the future

The number of payments per year

The number of compounds per year.

Is Interest/Payment due at the beginning or

end of the compound period?


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Example

Find the amount (future value) of a $2400 loan for 9 months

at 11% interest.

SOLUTIONWe need to findA which equalsP+I=P+Prt.

Now, determine the information we have been given.9

122400, 0.11, and 0.75 yearsP r t

FindI: I= 2400(0.11)(0.75) = 198Now, findA: A = 2400 + 198 = 2598

A could have also been found

using the formulaA =P(1 + rt)

2400(1 0.11(0.75))2400(1 0.0825)2400(1.0825)2598

A


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Simple Discount

The 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 asimple discount note.

The interest deducted is the discount.

The amount received by the borrower is theproceeds.

The discount rate is the percentage used.

The amount repaid is the maturity value.


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Simple Discount Note

D =Mdt

PR = MD

= M

Mdt= M(1dt)

where M= maturity value (principal)

d= annual discount rate (in decimal form)t= time in years

D = discount

PR = proceeds or amount the borrower receives


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Example

Find the discount and the amount a borrower receives (proceeds)

on a $1500 simple discount loan at 8% discount rate for 1.5

years.


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Example

A bank paid $987,410 for a 90-day $1 million treasury

bill. What was the simple discount rate?


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Example

A bank wants to earn 7.5% simple discount interest on a

90-day $1 million treasury bill. How much should it bid?

SOLUTIONFirst, determine what information is given.

90

3601,000,000, 0.075, and 0.75 yearsM d t

We need to find the proceeds which equalsPR.

90360

1,000,000 1 (0.075)

1,000,000 1 0.01875

1,000,000 0.98125

981,250

PR

The bank should bid $981,250.


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Example

How much should a bank bid on a 30-day $2 million

treasury bill if the bank wants to earn 5.125% on its money.


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HW 5.1

Pg 348-350 1-32, 34-64 even


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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 timeperiod. At the next time period, the bank will pay interest on

the new total, this is called compound interest.

Amount of Annual Compound Interest

WhenPdollars are invested at an annual interest rate randthe interest is compounded annually, the amountA at the end

oftyears is

A =P(1 + r)t


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Example

Suppose $800 is invested at 6%, and it is compounded

annually. What is the amount in the account at the end

of 4 years?

SOLUTION `

First, determine the information that is given.

P= 800, r= 0.06, and t= 4

4800(1.06) 800(1.26248) 1009.98A

There will be $1009.98 in the account after 4 years.


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Amount (Future Value)

The general formula for finding the amount after a specified number

of compound periods is

A =P(1 + i)n

where r= annual interest rate

m = number of times compounded per year

i = r/m = interest rate per periodn = mt, the number of periods, where tis the number of years

A = amount (future value) at the end ofn compound periods

P= principal (present value)


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Example

Suppose $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.

SOLUTION

First, determine what information is given then useA =P(1 + i)n.

P= 800, r= 0.12, and t= 2 years.

2 2800(1 0.12)

a) 1, so

8

0.12/1 0.

00(1.12) 800(1.2544) 1003.5

12, 2;

2A

m i n

4 4800(1 0.06) 800(1.06) 800(1.26248)b) 2, so 0.12/ 2 0.06, 4;

1009.98m

A

i n

8 8800(1 0.03) 800(1.03) 800(1.26677)

c) 4, so 0.12/4 0.03, 8;

1013.42

m

A

i n


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Total Number of Compounding Periods



Amount of the payment being made if any.

Future Value: What it will be worth to you

in the future

The number of payments per year

The number of compounds per year.

Is Interest/Payment due at the beginning or

end of the compound period?


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(compounds per year)(# of years)



0

Future Value: What it will be worth

Compounds per year

Always on End


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Example

Suppose $800 is invested at 12% for 2 years. Find the amount at

the end of 2 years if the interest is compounded (a) annually

1(2) 1 compound every year for two years

12 Yearly interest rate

-800 The amount being invested todayPresent Value

0 No regular payments are being made

What we want to solve forAlpha Enter

1There is 1 compound per year

1

1003.52


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Example


the end of 2 years if the interest is compounded (b) semiannually






2There are 2 compound per year

2

1009.98


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Example


the end of 2 years if the interest is compounded (c) quarterly.







4

1013.42


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1(5) 1 compound every year for five years


-100 We do not know the present value so use $100


If we started with $100 and we want to double it?


1

14.8698355

200


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Vocabulary

Nominal Rate

Compound rate thatreturns the same total

value as a simpleinterest investment.

Effective Rate

simple interest rate

that produces thesame total value ofinvestment per yearas the compoundinterest.


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Effective Rate

The effective rate of an annual interest rate rcompounded m

times per year is the simple interest rate that produces the same

total value of investment per year as the compound interest.

If money is invested at an annual rate rand compounded m

times per year, the effective rate,x, in decimal form is

x = (1 + i)m1 where i = r/m


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Example

The 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 11.017533 10.017533

i

i

ii

i

The annual rate = 4(0.017533) = 0.070133 = 7.013% (rounded).

The annual rate just found is also called the nominal rate.


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HW 5.2

Pg 359-360 1-17 odd, 18-63 every 3rd

Section 5 3


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Ordinary Annuity

An annuity refers to equal payments paid at equal time

intervals.

The time between successive payments is called the

payment period.

The amount of each payment is theperiodic payment.

The interest on an annuity is compound interest.

An ordinary annuity is an annuity with periodic payments

made at the end of each payment period.

Section 5.3


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Future Value (Amount)

Payments are made at the end of each period.

where i = interest rate per period

n = number of periods

R = amount of each periodic paymentA = future value of amount

(1 ) 1n

iA R

i


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Example

How much money will you have when you retire if you save $20

each month from graduation until retirement? Lets assume you

start saving at age 22 until age 65, 43 years, and the interest rate

averages 6.6% annual rate compounded monthly.SOLUTION

The periodic rate = 0.066/12 = 0.0055 since payments are

made monthly. The number of periods is n = 12(43) = 516.

The periodic payments areR = 20. Substitute these valuesinto the formula for future value.516(1 ) 1 (1.0055) 1

20 57,997.300.0055

ni

A Ri

You will accumulate $57,997.30 in 43 years.

How much money will you have when you retire if you save $20 each


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How much money will you have when you retire if you save $20 each

month from graduation until retirement? Lets assume you start saving at

age 22 until age 65, 43 years, and the interest rate averages 6.6% annual

rate compounded monthly.

12(43) 12 compounds every year for 43 years

6.6 Yearly interest rate

0 The amount being invested todayPresent Value-20 You are putting 20 per month in the account


12

There is 12 compound per year12

57997.30


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1(20) 1 compounds every year for 20 years

8.5 Yearly interest rate

0 The amount being invested todayPresent Value-2000 You are putting 2000 per year in the account


1

There is 1 compound per year1

96754.03


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Sinking Funds

A sinking fund refers to a fund that is created when an amount of

money will be needed 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.

Formula for Periodic Payments of a Sinking Fund

where A = value of the annuity after n payments

n = number of payments

i = periodic interest rate

R = amount of each periodic payment

(1 ) 1

n

AiR

i


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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?

SOLUTION

First, determine the information that is given.

10

65,000, 0.07, and 10

65000(0.07) 45504704.54

(1.07) 1 0.9671513

A i n

R

The annual payments are $4704.54.


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HW 5.3

Pg 372-374 1-47 odd

Section 5.4


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Present Value

Section 5.4

The present value of an annuity is the lump sum payment

that yields the same total amount as that obtained through

equal periodic payments made over the same period of time.

The present value of an annuity is

where i = periodic rate

n = number of periods

R = periodic payments

P= present value of the annuity

(1 ) 1 1 (1 )

(1 )

n n

n

i iP R R

i i i


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Example

Find 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.

SOLUTIONUse the values given,R = 2000, i = 0.06/2 = 0.03 and n = 20

in the formula for present value.20 20

20

(1 ) 1 (1.03) 1 1 (1 0.03)2000 2000

(1 ) 0.03(1.03) 0.03

2000(14.8774749) 29754.95

n

n

iP R

i i

The present value of the annuity is $29,754.95. This lump sum

will accumulate the same amount in 10 years as investing $2000

semiannually for 10 years.


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Equal Periodic Payments

The amount needed to provide equal periodic

payments can be found using the formula

or equivalently,

where P= amount needed in the fund

R = amount of periodic payments

i = periodic interest rate


(1 ) 1

(1 )

n

n

iP R

i i

1 (1 ) niP R

i


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Example

Find 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.

SOLUTION

16

0.01Given 1000, 0.025, and 16 quarters,

4

1 (1.025)

1000 0.025

1000(13.0550027) 13,055

R i n

P

A lump sum investment of $13,055 will provide $1000 per

quarter for 4 years.


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Amortization

The amortization of a debt (repayment of a debt) requires no

new formula because the amount borrowed is just the

present value of an annuity.

Amortization of a Loan (Debt Payments)The amount borrowed,P, is related to the periodic payments,R,

by the formula

where i = periodic interest rate


Note: This is the present value formula for an ordinary annuity.

(1 ) 1 1 (1 )

(1 )

n n

n

i iP R R

i i i


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Example

A 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?

SOLUTION

Since the amount borrowed is the present value of the

annuity, we have

395.420.12

0.0112

24

R

i

n

(Monthly rate)

(Number of months)

241 (1.01)

395.42

0.01

395.42(21.2433873)

8400.06

P

The amount borrowed was $8400.


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Balance of an Amortization

The balance aftern 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 borrowed

i = periodic interest raten = number of time periods elapsed

R = monthly payments


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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?

SOLUTION 1212

60,000, 1% per month, 617.17, 24 monthsP i R n 24

24 (1.01) 1Balance 60,000(1.01) 617.17

0.01

60,000(1.26973) 617.17(26.9734649)76,184.08 16,647.21

59,536.87

The part of the loan repaid is the equity which after

2 years is is 60,00059,536.87 = $463.13.

This is the balance of the

loan after two years.


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HW 5.4

Pg 389-391 1-47 Odd

Math In FM

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