10 The Mathematics of Money - Huntsville, TXjga001/Math 1332 chapter 10 slides.pdf · Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7e: 10.1 - 14

Excursions in Modern Mathematics, 7e: 10.1 - 1 Copyright © 2010 Pearson Education, Inc.

10 The Mathematics of Money

10.1 Percentages

10.2 Simple Interest

10.3 Compound Interest

10.4 Geometric Sequences

10.5 Deferred Annuities: Planned Savings

for the Future

10.6 Installment Loans: The Cost of

Financing the Present


• People don’t like dealing with fractions.

• The most likely culprit for “fraction phobia” is the difficulty of dealing with fractions with different denominators.

• One way to get around this difficulty is to express fractions using a common, standard denominator, and in modern life the commonly used standard is the denominator 100.

Fractions


A “fraction” with denominator 100 can be

interpreted as a percentage, and the

percentage symbol (%) is used to indicate

the presence of the hidden denominator

100. Thus,

Percentages

x%x

100


Percentages are useful for many reasons.

• give us a common yardstick to compare

different ratios and proportions;

• provide a useful way of dealing with fees,

taxes, and tips;

• help us better understand how things

increase or decrease relative to some

given baseline.

Percentages


decimals can be converted to percentages

through multiplication by 100

Examples:

(a) 1.325 = 132.5%

(b) 0.005 = 0.5%

Convert Decimals to Percents


percentages can be converted to decimals

through division by 100

Examples:

1.100% = 1.00

2.7 1/2 % = 7.5%=0.075

Convert Percents to Decimals


Fractions can be converted to percents by

first converting to decimals then multiplying

by 100.

Examples:

Convert Fractions to Percent

%7575.0434

3 1.

%4.71714.0757

5 2.


Fractions can be converted to percents by

solving a proportion problem for x:

Example:

Convert Fractions to Percent

75 1004

3x

x

100

x

b

a

%754

3 :so


Suppose that in your English Lit class you

scored 19 out of 25 on the quiz, 49.2 out of 60

on the midterm, and 124.8 out of 150 on the

final exam. Which one was your best score?

Example 10.1 Comparing Test Scores


• The numbers 19, 49.2, and 124.8 are

called raw scores.

• We can compare raw scores if we express

each score as a percentage of the total

number of points possible.



• Quiz score = 19/25=0.76=76%

• Midterm score = 49.2/60= 0.82 = 82%

• Final Exam = 124.8/150= 0.832 = 83.2%



• A is P percent of B means:

• Or that:

Solving Percent Problems

BP

A100

100

P

B

A


• Problem 3 on page 392


Example


If you start with a quantity Q and

increase that quantity by x%, you end up

with the quantity

PERCENT INCREASE

1001

100

xQ

xQQy


To find the percent increase, from

solve for x:

PERCENT INCREASE

100quantity original

quantity original of increase x

Q

Qy

yQ to



Example


If you start with a quantity Q and

decrease that quantity by x%, you end

up with the quantity

PERCENT DECREASE

1001

100

xQ

xQQy


To find the percent decrease, from

solve for x:

PERCENT DECREASE

100quantity original

quantity original of decrease x

Q

yQ

yQ to


Percentage decreases are often used

incorrectly, mostly intentionally and in an

effort to exaggerate or mislead.

The misuse is usually framed by the claim

that if an x% increase changes A to B, then

an x% decrease changes B to A.

Not true!

Misleading Use of Percent Changes


With great fanfare, the police chief of

Happyville reports that crime decreased by

200% in one year. He came up with this

number based on reported crimes in

Happyville going down from 450 one year to

150 the next year. Since an increase from

150 to 450 is a 200% increase (true), a

decrease from 450 to 150 must surely be a

200% decrease, right? Wrong.

Example 10.5 The Bogus 200%

Decrease


300/450 = 0.666 . . . ≈ 66.67%.

Example 10.5 The Bogus 200%

Decrease

450

300

450

150450

quantity original

quantity original of decrease



10.1 Percentages





for the Future




• Money has a present value and a future value

• if you invest $P today (the present value) for a promise of getting $A at some future date (the future amount), you expect A to be more than P.

Present Value and Future Value


• If you are getting a present value of P today from someone else (either in cash or in goods), you expect to have to pay a future amount of A back at some time in the future.

• If we are given the present value P, how do we find the future amount A (and vice versa)?

Present Value and Future Value


• The answer depends on several variables, the most important of which is the interest rate.

• Interest is the return the lender or investor expects as a reward for the use of his or her money, and

• The standard way to describe an interest rate is as a yearly rate commonly called the annual percentage rate (APR).

Interest Rate


• In simple interest, only the original

money invested or borrowed (called the

principal) generates interest over time.

• This is in contrast to compound

interest, where the principal generates

interest, then the principal plus the

interest generate more interest, and so

on.

Simple Interest vs. Compound Interest


• P = principal ($)

• r = annual percentage rate (given as %,

but convert to a decimal number)

• t = time (years)

• I = interest

Simple Interest Formula

trPtrPI


Your parents purchased a $1000 savings bond that pays 5% annual simple interest. What is interest earned on the bond after 18 years?

Savings Bonds


• P = $1000

• r = 5%=0.05

• t = 18 year

• I = interest

Example (cont.)

900$1805.01000$trPI


The future value A of P dollars invested

under simple interest for t years at an

APR of r is given by

(where r denotes the APR written as a

decimal).

Future Value Formula

IPtrPPA


On the day you were born your parents purchased a $1000 savings bond that pays 5% annual simple interest. What is the value of the bond on your 18th birthday?

Example 10.7 Savings Bonds



Future value on your 18th birthday

1900$900$1000$IPA


What is the value of the bond on any given birthday?




■ Future value of the bond when you become t years old

t

tA

50$1000$

05.0 1000$1000$





Example


Generally speaking, credit cards charge

exceptionally high interest rates, but you

only have to pay interest if you don’t pay

your monthly balance in full. Thus, a credit

card is a two-edged sword: if you make

minimum payments or carry a balance from

one month to the next, you will be paying a

lot of interest; if you pay your balance in full,

you pay no interest.

Credit Cards


In the latter case you got a free, short-term

loan from the credit card company. When

used wisely, a credit card gives you a rare

opportunity–you get to use someone else’s

money for free. When used unwisely and

carelessly, a credit card is a financial trap.

Credit Cards


Imagine that you recently got a new credit

card. Like most people, you did not pay much

attention to the terms of use or to the APR,

which with this card is a whopping 24%. To

make matters worse, you went out and spent

a little more than you should have the first

month, and when your first statement comes

in you are surprised to find out that your new

balance is $876.

Example 10.9 Credit Card Use: The

Good, the Bad and the Ugly


Like with most credit cards, you have a little

time from the time you got the statement to

the payment due date (this grace period is

usually around 20 days). You can pay a

minimum payment of $20, the full balance of

$876, or any other amount in between. Let’s

consider these three different scenarios

separately.




■ Option 1: Pay the full balance of $876

before the payment due date. This one is

easy. You owe no interest and you got free

use of the credit card company’s money for

a short period of time. When your next

monthly bill comes, the only charges will be

for your new purchases.




■ Option 2: Pay the minimum payment of

$20. When your next monthly bill comes,

you have a new balance of $1165

consisting of:

1. The previous balance of $856. (The $876

you previously owed minus your payment

of $20.)




2. The charges for your new purchases. Let’s

say, for the sake of argument, that you

were a little more careful with your card

and your new purchases for this period

were $288.




3. The finance charge of $21 calculated as

follows:

(i) Periodic rate = 0.02

(ii) Balance subject to finance charge

= $1050

(iii) Finance charge = (0.02)$1050 = $21

You might wonder, together with millions of

other credit card users, where these

numbers come from. Let’s take them one

at a time.




(i) The periodic rate is obtained by dividing

the annual percentage rate (APR) by the

number of billing periods. Almost all credit

cards use monthly billing periods, so the

periodic rate on a credit card is the APR

divided by 12. Your credit card has an

APR of 24%, thus yielding a periodic rate

of 2% = 0.02.




(ii) The balance subject to finance charge

(an official credit card term) is obtained by

taking the average of the daily balances

over the previous billing period. Since this

balance includes your new purchases, all

of a sudden you are paying interest on all

your purchases and lost your grace

period! In your case, the balance subject

to finance charge came to $1050.




(iii) The finance charge is obtained by

multiplying the periodic rate times the

balance subject to finance charge. In this

case, (0.02)$1050 = $21.

■ Option 3: You make a payment that is

more than the minimum payment but less

than the full payment.




Let’s say for the sake of argument that you make a payment of $400. When your next monthly bill comes, you have a new balance of $777.64. As in option 2, this new balance consists of:



1. The previous balance, in this case $476

(the $876 you previously owed minus the

$400 payment you made)

2. The new purchases of $288


3. The finance charges, obtained once again by multiplying the periodic rate (2% = 0.02) times the balance subject to finance charges, which in this case came out to $682.



Thus, your finance charges turn out to be

(0.02)$682 = $13.64, less than under option

2 but still a pretty hefty finance charge.


1. Make sure you understand the terms of

your credit card agreement.

Know the APR (which can range widely

from less than 10% to 24% or even

more), know the length of your grace

period, and try to understand as much of

the fine print as you can.

Two Important Lessons


2. Make a real effort to pay your credit card balance in full each month. This practice will help you avoid finance charges and keep you from getting yourself into a financial hole. If you can’t make your credit card payments in full each month, you are living beyond your means and you may consider putting your credit card away until your balance is paid.

Two Important Lessons



10.1 Percentages





for the Future




• Rewrite the previous formula for future

value of simple interest

Simple Interest Future Value (Shortcut)

rtPtrPPA 1


On the day you were born, your Uncle Nick deposited $5000 in your name in a trust fund that pays a 6% APR. One of the provisions of the trust fund was that you couldn’t touch the money until you turned 18. How much money would there be in the trust when you turn 18 if the future value is due to simple interest?

Example 10.10 Your Trust Fund Found!



040,10$

)08.2(5000$

1806.015000$A


• Under simple interest the gains on an investment are constant–only the principal generates interest.

• Under compound interest, not only does the original principal generate interest, so does the previously accumulated interest.

Compound Interest


• All other things being equal, money invested under compound interest grows a lot faster than money invested under simple interest, and this difference gets magnified over time.

Compound Interest


On the day you were born, your Uncle Nick deposited $5000 in your name in a trust fund that pays a 6% APR. One of the provisions of the trust fund was that you couldn’t touch the money until you turned 18. How much money would there be in the trust when you turn 18? Assume the interest is compounded each year.



Here is an abbreviated timeline of the money

in your trust fund, starting with the day you

were born:


■ Day you were born: Uncle Nick deposits

$5000 in trust fund.

■ First birthday: 6% interest is added to the

account.

Balance in account is $5000(1.06).


■ Second birthday: 6% interest is added to the previous balance (in red). Balance in account is

$5000(1.06) (1.06)= $5000(1.06)2


previous (year one) balance


■ Third birthday: 6% interest is added to the previous balance (in red). Balance in account is

$5000(1.06)2 (1.06)= $5000(1.06)3


previous (year two) balance



The exponent of (1.06) in the right-hand expression goes up by 1 on each birthday and in fact matches the birthday.


Thus,

■ Eighteenth birthday: The balance in the

account is $5000(1.06)18

Use a calculator and do the computation:

$5000(1.06)18 = $14,271.70

(rounded to the nearest penny)



How much money would there be in the trust

fund if I left the money in for retirement and

waited until I turned 60?



■ 60th birthday: The future value of the

account is

$5000(1.06)60 = $164,938.45

which is an amazing return for a $5000

investment (if you are willing to wait, of

course)!



This figure plots the growth of the money in

the account for the first 18 years.



This figure plots the growth of the money in

the account for 60 years.



The future value A of P dollars

compounded annually for t years at an

APR of r (given as %, but written in

decimal form) is given by

A = P(1 + r)t

ANNUAL COMPOUNDING FORMULA



Example


You have $875 in savings that you want to

invest. Your goal is to have $2000 saved in 7

1/2 years. (You want to send your mom on a

cruise on her 50th birthday.) The credit union

around the corner offers a certificate of

deposit (CD) with an APR of 6 3/4%

compounded annually. What is the future

value of your $875 in 7 1/2 years?

Example 10.11 Saving for a Cruise


To answer the first question, we just apply the

annual compounding formula with P = $875,

r = 0.0675, and t = 7 (with annual

compounding, fractions of a year don’t count)

and get

A = $875(1.0675)7 = $1382.24




Unfortunately, this is quite a bit short of the

$2000 you want to have saved. To determine

how much principal to start with to reach a

future value target of A = $2000 in 7 years at

6.75% annual interest, we solve for P in terms

of A in the annual compounding formula. In

this case substituting $2000 for A gives

$2000 = P(1.0675)7



$2000 = P(1.0675)7

and solving for P gives

This is quite a bit more than the $875 you

have right now, so this option is not viable.


P$2000

1.06757

$1266.06


Let’s now return to our story from Example

10.11: You have $875 saved up and a 7 1/2 -

year window in which to invest your money.

As discussed in Example 10.11, the 6.75%

APR compounded annually gives a future

value of only $1382.24 – far short of your goal

of $2000.

Example 10.12 Saving for a Cruise:

Part 2


You find another bank that is advertising a

6.75% APR that is compounded monthly (i.e.,

the interest is computed and added to the

principal at the end of each month). Unlike

the case of annual compounding, you get

interest for that extra half a year at the end.


Part 2


To do the computation we will have to use a

variation of the annual compounding formula.

The key observation is that since the interest

is compounded 12 times a year, the monthly

interest rate is 6.75% ÷ 12 = 0.5625%

(0.005625 when written in decimal form).

■ Original deposit: $875.


Part 2


■ Month 1: 0.5625% interest is added to the

account. The balance in the account is now

$875 (1.005625).


previous balance. The balance in the

account is now $875(1.005625)2.


previous balance. The balance in the

account is now $875(1.005625)3.


Part 2


■ Month 12: At the end of the first year the

balance in the account is

$875(1.005625)12 = $935.92

After 7 1/2 years = 90 months,

■ Month 90: The balance in the account is

$875(1.005625)90 = $1449.62


Part 2


You find a bank that pays a 6.75% APR that

is compounded daily. Compute the future

value of $875 in 7 1/2 years. The analysis is

the same as in Example 10.12, except now

the interest is compounded 365 times a year

(never mind leap years–they don’t count in

banking), and the numbers are not as nice.


Part 3


First, we divide the APR of 6.75% by 365.

This gives a daily interest rate of

6.75% ÷ 365 ≈ 0.01849315% = 0.0001849315

Next, we compute the number of days in the 7

1/2 year life of the investment

365 7.5 = 2737.5

Since parts of days don’t count, we round

down to 2737. Thus,

$875(1.0001849315)2737 = $1451.47


Part 3


Let’s summarize the results of Examples

10.11, 10.12, and 10.13. Each example

represents a scenario in which the present

value is P = $875, the APR is 6.75% (r =

0.0675), and the length of the investment is

t = 7 1/2 years. The difference is the

frequency of compounding during the year.

Differences: Compounding Frequency


■ Annual compounding (Example 10.11):

Future value is A = $1382.24.

■ Monthly compounding (Example 10.12):


■ Daily compounding (Example 10.13):




A reasonable conclusion from these

numbers is that increasing the frequency of

compounding (hourly, every minute, every

second, every nanosecond) is not going to

increase the ending balance by very much.

The explanation for this surprising law of

diminishing returns will be given shortly.



The future value of P dollars in t years at

an APR of r (convert percent to decimal)

compounded n times a year is

GENERAL COMPOUNDING FORMULA

nt

n

rPA 1


r/n represents the periodic interest rate expressed as a decimal, and the exponent

n • t represents the total number of compounding periods over the life of the investment.

Terminology


• Problem 39(a) on page 394

Example


• One of the remarkable properties of the

general compounding formula is that

even as n (the frequency of

compounding) grows without limit, the

future value A approaches a limiting

value L.

Continuous Compounding


• This limiting value L represents the future

value of an investment under

continuous compounding (i.e., the

compounding occurs over infinitely short

time intervals) and is given by the

following continuous compounding

formula.

Continuous Compounding


The future value A of P dollars

compounded continuously for t

years at an APR of r (converted to

decimal form) is

Calculator has “e” key.

CONTINUOUS COMPOUNDING

FORMULA

rtPeA

84590462.71828182e


You finally found a bank that offers an APR of

6.75% compounded continuously. Using the

continuous compounding formula and a

calculator, you find that the future value of

your $875 in 7 1/2 years is

A = $875(e7.5 0.0675)

= $875(e0.50625)

= $1451.68


Part 4


The most disappointing thing is that when you

compare this future value with the future

value under daily compounding (Example

10.13), the difference is 21¢.


Part 4


• The annual percentage yield (APY) of

an investment (sometimes called the

effective rate) is the percentage of profit

that the investment generates in a one-

year period.

Annual Percentage Yield


Suppose that you invest $835.25. At the end

of a year your money grows to $932.80. (The

details of how your money grew to $932.80

are irrelevant for the purposes of our

computation.) Here is how you compute the

APY:

Example 10.15 Computing an APY

APY

$932.80 $835.25

$835.250.1168 11.68%


In general, if you start with P dollars at the

beginning of the year and your investment

grows to A dollars by the end of the year,

This is the annual percentage increase of

your investment.


P

PAAPY


• The APY can be used to easily compare

different interest rates that involve

different compounding periods.



Which of the following three investments is

better: (a) 6.7% APR compounded

continuously, (b) 6.75% APR compounded

monthly, or (c) 6.8% APR compounded

quarterly?

Example 10.16 Comparing Investments

Through APY


• To compare these investments we will

compute their APYs.

• The question is independent of the

principal P and the length of the

investment t.

• We use the future value formulas with a

convenient value for P. Let’s use P=$1.


Through APY


(a) The future value of $1 in 1 year at 6.7%

interest compounded continuously is

given by e0.067 ≈ 1.06930.

The APY in this case is 6.93%.


Through APY

06930.01

106930.1APY


(b) The future value of $1 in 1 year at 6.75%

interest compounded monthly is

(1 + 0.0675/12)12 ≈ 1.00562512 ≈ 1.06963



Through APY


(c) The future value of $1 in 1 year at 6.8%

interest compounded quarterly is

(1 + 0.068/4)4 ≈ 1.0174 ≈ 1.06975



Through APY


Although they are all quite close, we can now

see that (c) is the best choice, (b) is the

second-best choice, and (a) is the worst

choice. Although the differences between the

three investments may appear insignificant

when we look at the effect over one year,

these differences become quite significant

when we invest over longer periods.


Through APY



10.1 Percentages





for the Future




• A geometric sequence starts with an

initial term P and from then on every term

in the sequence is obtained by

multiplying the preceding term by the

same constant c:

The second term equals the first term

times c, the third term equals the second

term times c, and so on.

• The number c is called the common

ratio of the geometric sequence.

Geometric Sequence


5, 10, 20, 40, 80, . . .

• The above is a geometric sequence with

initial term 5 and common ratio c = 2

• Notice that since the initial term and the

common ratio are both positive, every term

of the sequence will be positive.

Example 10.17 Some Simple

Geometric Sequences


5, 10, 20, 40, 80, . . .

• Also notice that the sequence is an

increasing sequence: Every term is bigger

than the preceding term. This will happen

every time the common ratio c is bigger

than 1.


Geometric Sequences



initial term 27 and common ratio

• Notice that this is a decreasing sequence,

a consequence of the common ratio being

between 0 and 1.


Geometric Sequences

c

1

3.

,...9

1,

3

1,1,3,9,27



initial term 27 and common ratio

• Notice that this sequence alternates

between positive and negative terms, a

consequence of the common ratio being a

negative number.


Geometric Sequences

c

1

3.

,...9

1,

3

1,1,3,9,27


A generic geometric sequence with initial

term P and common ratio c can be written in

the form P, cP, c2P, c3P, c4P, . . .

We will use a common letter–in this case, G

for geometric–to label the terms of a generic

geometric sequence, with subscripts

conveniently chosen to start at 0. In other

words,

G0 = P, G1 = cP, G2 = c2P, G3 = c3P, …

Generic Geometric Sequence


GN = cGN–1 ; G0 = P (recursive formula)

GN = CNP (explicit formula)

GEOMETRIC SEQUENCE


Consider the geometric sequence with initial

term P = 5000 and common ratio c = 1.06.

The first few terms of this sequence are

G0 = 5000,

G1 = (1.06)5000 = 5300,

G2 = (1.06)25000 = 5618,

G3 = (1.06)35000 = 5955.08

Example 10.18 A Familiar Geometric

Sequence


If we put dollar signs in front of these numbers, we get the principal and the balances over the first three years on an investment with a principal of $5000 and with an APR of 6% compounded annually. These numbers might look familiar to you–they come from Uncle Nick’s trust fund example (Example 10.10). In fact, the Nth term of the above geometric sequence (rounded to two decimal places) will give the balance in the trust fund on your Nth birthday.

Example 10.18 A Familiar Geometric

Sequence


Principal of P and periodic interest rate r/n,

the balances in the account at the end of

each compounding period are the terms of a

geometric sequence with initial term P and

common ratio (1 + r/n)

P, P(1 + r/n), P(1 + r/n)2, P(1 + r/n)3, . . .

Compound Interest


Thanks to improved vaccines and good public

health policy, the number of reported cases of

the gamma virus has been dropping by 70%

a year since 2008, when there were 1 million

reported cases of the virus. If the present rate

continues, how many reported cases of the

virus can we predict by the year 2014?

Example 10.19 Eradicating the

Gamma Virus


• Because the number of reported cases of

the gamma virus decreases by 70% each

year, the number of reported cases is 30%

of what it was the preceding year.

• This gives a formula for the number of

viruses after N years (denoted ) if we

know the number of viruses in the previous

year (denoted )


Gamma Virus

11 30.070.01 NNN GGG

1NG

NG


• Therefore, we can model this number by a

geometric sequence with common ratio

c = 0.30


Gamma Virus


• We will start the count in 2008 with the

G0 = 1,000,000 reported cases.

• In 2009 the numbers will drop to

G1 = 300,000 reported cases,

• in 2010 the numbers will drop further to

G2 = 90,000 reported cases,

and so on.


Gamma Virus


• By the year 2014 we will be in the sixth

iteration of this process, and thus the

number of reported cases of the gamma

virus will be

G6 =(0.30)6 1,000,000 = 729


Gamma Virus


Useful formula–the geometric sum formula–

that allows us to add a large number of

terms in a geometric sequence without

having to add the terms one by one.

Geometric Sum Formula


THE GEOMETRIC SUM FORMULA

1

1

12

c

cP

PcPccPP

N

N


• Problem 58(c) on page 395

Example



10.1 Percentages




10.5 Deferred Annuities: Planned

Savings for the Future




A fixed annuity is a sequence of equal payments made or received over regular (monthly, quarterly, annually) time intervals.

Fixed Annuity


Examples of annuities:

1.You may be making regular deposits to save for a vacation, a wedding, or college, or you may be making regular payments on a car loan or a home mortgage.

2.You could also be at the receiving end of an annuity, getting regular payments from an inheritance, a college trust fund set up on your behalf, or a lottery jackpot.

Fixed Annuity


• When payments are made so as to produce a lump-sum payout at a later date (e.g., making regular payments into a college trust fund), we call the annuity a deferred annuity;

• When a lump sum is paid to generate a series of regular payments later (e.g., a car loan), we call the annuity an installment loan.

Deferred Annuity vs. Installment Loan


• In a deferred annuity the pain (in the form of payments) comes first and the reward (a lump-sum payout) comes in the future,

• In an installment loan the reward (car, boat, house) comes in the present and the pain (payments again) is stretched out into the future.

• In this section we will discuss deferred annuities. In the next section we will take a look at installment loans.

Deferred Annuity vs. Installment Loan


Given the cost of college, parents often set up

college trust funds for their children by setting

aside a little money each month over the

years. A college trust fund is a form of forced

savings toward a specific goal, and it is

generally agreed to be a very good use of a

parent’s money–it spreads out the pain of

college costs over time, generates significant

interest income, and has valuable tax

benefits.

Example 10.21 Setting Up a College

Trust Fund


A mother decides to set up a college trust

fund for her new-born child. Her plan is to

have $100 withdrawn from her paycheck

each month for the next 18 years and

deposited in a savings account that pays 6%

annual interest compounded monthly. What is

the future value of this trust fund in 18 years?


Trust Fund


What makes this example different from

Uncle Nick’s trust fund example (Example

10.10) is that money is being added to the

account in regular installments of $100 per

month.


Trust Fund


Each $100 monthly installment has a different

“lifespan”: The first $100 compounds for 216

months (12 times a year for 18 years), the

second $100 compounds for only 215

months, the third $100 compounds for only

214 months, and so on.


Trust Fund


Thus, the future value of each $100

installment is different. To compute the future

value of the trust fund we will have to

compute the future value of each of the 216

installments separately and add. Sounds like

a tall order, but the geometric sum formula

will help us out.


Trust Fund


• Each installment is for a fixed amount ($100) and that the periodic interest rate is always the same:

6% ÷ 12 = 0.5% = 0.005

• When we use the general compounding formula, each future value looks the same except for the compounding exponent:


Trust Fund


The future value of P dollars in t years at

an APR of r =0.06 compounded n=12

times a year is


t

nt

n

rPA 12)005.1(100$1

Note: 12t represents total number of compounding

periods (months)


• Future value of the first installment ($100 compounded for 216 months):

• Future value of the second installment ($100 compounded for 215 months):


Trust Fund

216

1 )005.1(100$A

215

2 )005.1(100$A


• Future value of the third installment ($100 compounded for 214 months):

…

• Future value of the last installment ($100 compounded for one month):


Trust Fund

214

3 )005.1(100$A

1

216 )005.1(100$A


• The future value F of this trust fund at the

end of 18 years is the sum of all the above

future values.

• If we write the sum in reverse

chronological order (starting with the last

installment and ending with the first), we

get the following: (next slide)


Trust Fund



Trust Fund

216215

21

12215216

)005.1(100$)005.1(100$

)005.1(100$)005.1(100$

AAAAF



Trust Fund

215214

1

)005.1(50.100$)005.1(50.100$

)005.1(50.100$50.100$


The sum for F is a geometric sum with

common ratio c = 1.005 and a total of N = 216

terms. Applying the geometric sum formula

from chapter 10.4 gives:


Trust Fund

F $100.501.005

216

1

1.005 1$38,929


Let’s determine how much interest she

earned. 216 installments of $100 gives a

total of:

Then, total interest earned is:


Trust Fund

600,21$100$216

17,329$600,21$929,38$


The future value F of a fixed deferred

annuity consisting of nt payments of $P

having a periodic interest of r/n (written

in decimal form) is

where L denotes the future value of the

last payment.

FIXED DEFERRED

ANNUITY FORMULA

nr

nrLF

nt

/

1)/1(


• The value of L in the previous formula is either of the following:

– If the payments are made at the start of each compounding period, then the last payment generates interest (as in example 10.21) and

– If the payments are made at the end of each compounding period, then the last payment generates no interest and

Fixed Deferred Annuity Formula

nrPL /1

PL


• The value of L in the previous formula is either of the following:

– If the payments are made at the start of each compounding period, then the last payment generates interest (as in example 10.21) and

– If the payments are made at the end of each compounding period, then the last payment generates no interest and


nrPL /1

PL



Example


If we have a target for the future value F (that is, we know what F should be) and we know the number of compounding periods and the APR, we can use the formula to find the payments P needed to achieve F.



In Example 10.21 we saw that an 18-year

annuity of $100 monthly payments at an APR

of 6% compounded monthly is $38,929. For

the same APR and the same number of

years, how much should the monthly

payments be if our goal is an annuity with a

future value of $50,000?


Trust Fund: Part 2



Trust Fund: Part 2

005.0

1)005.1($50,000

/

1)/1(

216

L

nr

nrLF

nt

387.3532

005.0

21.93676597

L

L



Trust Fund: Part 2

Solving for L gives

Because payments were made at the

beginning of each period:

$129.08

387.3532000,50$L

)005.1(/1 PnrPL


Recall now that L is the future value of the

last payment, and since the payments are

made at the beginning of each month,

Thus, since L=$129.08


Trust Fund: Part 2

P

$129.08

1.005$128.44

)005.1(/1 PnrPL



10.1 Percentages





for the Future




An installment loan (also know as a fixed

immediate annuity) is a series of equal

payments made at equal time intervals for

the purpose of paying off a lump sum of

money received up front. Typical installment

loans are the purchase of a car on credit or

a mortgage on a home.

Installment Loan


The most important distinction between an

installment loan and a fixed deferred annuity

is that an installment loan has a present

value that we compute by adding the

present value of each payment, whereas a

fixed deferred annuity has a future value

that we compute by adding the future value

of each payment.

Installment Loan vs. Deferred Annuity


You’ve just landed a really good job and

decide to buy the car of your dreams, a

brand-new red Mustang. You negotiate a

good price ($23,995 including taxes and

license fees). You have $5000 saved for a

down payment, and you can get a car loan

from the dealer for 60 months at 6.48%

annual interest. If you take out the loan from

the dealer for the balance of $18,995, what

would the monthly payments be?

Example 10.24 Financing That Red

Mustang


Every time you make a future payment on an

installment loan, that payment has a present

value, and the sum of the present values of

all the payments equals the present value of

the loan–in this case, the $18,995 that you

are financing.


Mustang


You will be paying interest on the $18,995

loan in 60 equal installments.

Every time you make a payment, that

payment has a present value based on the

time at which the payment is made.

The sum of the present values of all the

payments equals the present value of the

loan which is $18,995.


Mustang


Although each monthly loan payment of A

has a different present value, each of

these present values can be computed

using the general compounding formula.


Mustang


The future value A of P dollars in t years

at an APR of r =0.0648 compounded

n=12 times a year is


t

nt

Pn

rPA 12)0054.1(1

Note: 12t represents total number of compounding

periods (months)



Mustang

■ Solving this formula for P gives:

tAP 12)0054.1/(



Mustang

■ The first payment of A is at t=1/12 year

and has present value

Interest you pay on this amount is accrued

over one compounding period.

1

1 )0054.1/(AP



Mustang

■ The second payment of A is at t=2/12 year



over two compounding periods.

2

2 )0054.1/(AP



Mustang

■ The third payment of A is at t=3/12 year



over three compounding periods.

3

3 )0054.1/(AP



Mustang

■ If we continue this, the last payment of A is

at t=5 years and has present value


over sixty compounding periods.

60

60 )0054.1/(AP



Mustang

■ The total present value of our loan is

$18,995

60

21

6021

)0054.1/(

)0054.1/()0054.1/(

995,18$

A

AA

PPP



Mustang

59

1

0054.1

1

0054.1

0054.1

1

0054.10054.1

A

AA


The sum is a geometric sum with common ratio

and a total of N = 60 terms. Applying the geometric sum formula from chapter 10.4 gives:


Mustang

0054.1

1c



Mustang

10054.1

1

10054.1

1

0054.1995,18$

60

A



Mustang

51.14282][

0054.1

51.41899

51.418990054.1

0.00537

0.27612

0054.1995,18$

A

A

A

A


This gives:


Mustang

41.371$14282.51995,18$A


Let’s determine how much interest you paid.

60 installments of $371.41 gives a total of:

Then, total interest paid is:


Trust Fund

22,284.60$41.371$60

60.2893,$995,18$60.284,22$


If an installment loan of P dollars is paid

off in nt payments of A dollars at a

periodic interest of r/n (written in decimal

form), then

where

AMORTIZATION FORMULA

q

qAqP

nt 1

nrq

/1

1


– If the payments are due at the end of each compounding period, then the present value of the first payment Aq as in the formula.

– If the payments are due at the beginning of each compounding period, then the present value of the first payment is A which replaces Aq in the formula.

Amortization Formula


During certain times of the year automobile

dealers offer incentives in the form of cash

rebates or reduced financing costs (including

0% APR), and often the buyer can choose

between those two options. Given a choice

between a cash rebate or cheap financing, a

savvy buyer should be able to figure out

which of the two is better, and we are now in

a position to do that.


Mustang: Part 2


We will consider the same situation we

discussed in Example 10.24. You have

negotiated a price of $23,995 (including taxes

and license fees) for a brand-new red

Mustang, and you have $5000 for a down

payment. The big break for you is that this

dealer is offering two great incentives: a

choice between a cash rebate of $2000 or a

0% APR for 60 months.


Mustang: Part 2


If you choose the cash rebate, you will have a

balance of $16,995 that you will have to

finance at the dealer’s standard interest rate

of 6.48%. If you choose the free financing,

you will have a 0% APR for 60 months on a

balance of $18,995. Which is a better deal?

To answer this question we will compare the

monthly payments under both options.


Mustang: Part 2


Option 1: Take the $2000 rebate. Here the

present value is $P = $16,995, amortized

over 60 months at 6.48% APR. The periodic

rate is p = 0.0054

Applying the amortization formula we get


Mustang: Part 2

F

1.0054

1

1.0054

60

1

1

1.00541

$16,995


Option1 (continued)

Solving the above equation for F gives the

monthly payment under the rebate option:

F = $332.37



Mustang: Part 2


Option 2: Take the 0% APR. Here the present

value is P = $18,995, amortized over 60

months at 0% APR. There is no need for any

formulas here: With no financing costs, your

monthly payment amortized over 60 months

is


Mustang: Part 2

F

18,995

60$316.59



Clearly, in this particular situation the 0%

APR option is a lot better than the $2000

rebate option (you are saving approximately

$16 a month, which over 60 months is a

decent piece of change). Of course, each

situation is different, and it can also be true

that the rebate offer is better than the cheap

financing option.


Mustang: Part 2



Example

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10 The Mathematics of Money - Huntsville, TXjga001/Math 1332 chapter 10 slides.pdf · Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics, 7e: 10.1 - 14