This chapter is part of Business Precalculus © David Lippman 2016.

This content is remixed from Math in Society © Lippman 2013.

This material is licensed under a Creative Commons CC-BY-SA license.

Chapter 6: Finance

Section 6.1: Simple and Compound Interest …………………………………………………………… 221

Section 6.2: Annuities …………………………………………………………………………………………. 230

Section 6.3: Payout Annuities ………………………………………………………………………………. 235

Section 6.4: Loans ………………………………………………………………………………………………. 239

Section 6.5: Multistage Finance Problems ……………………………………………………………… 246

Section 6.1: Simple and Compound Interest

Discussing interest starts with the principal, or amount your account starts with. This could

be a starting investment, or the starting amount of a loan. Interest, in its most simple form, is

calculated as a percent of the principal. For example, if you borrowed $100 from a friend

and agree to repay it with 5% interest, then the amount of interest you would pay would just

be 5% of 100: $100(0.05) = $5. The total amount you would repay would be $105, the

original principal plus the interest.

Simple One-time Interest

(1 )

I Pr

A P I P Pr P r

I is the interest

A is the end amount: principal plus interest

P is the principal (starting amount)

r is the interest rate in decimal form. Example: 5% = 0.05

Example 1

A friend asks to borrow $300 and agrees to repay it in 30 days with 3% interest. How much

interest will you earn?

P = $300 the principal

r = 0.03 3% rate

I = $300(0.03) = $9. You will earn $9 interest.

One-time simple interest is only common for extremely short-term loans. For longer term

loans, it is common for interest to be paid on a daily, monthly, quarterly, or annual basis. In

that case, interest would be earned regularly. For example, bonds are essentially a loan made

to the bond issuer (a company or government) by you, the bond holder. In return for the

loan, the issuer agrees to pay interest, often annually. Bonds have a maturity date, at which

time the issuer pays back the original bond value.

222 Chapter 6 Finance

Example 2

Suppose your city is building a new park, and issues bonds to raise the money to build it.

You obtain a $1,000 bond that pays 5% interest annually that matures in 5 years. How

much interest will you earn?

Each year, you would earn 5% interest: $1000(0.05) = $50 in interest. So over the course

of five years, you would earn a total of $250 in interest. When the bond matures, you

would receive back the $1,000 you originally paid, leaving you with a total of $1,250.

We can generalize this idea of simple interest over time.

Simple Interest over Time

(1 )

I Prt

A P I P Prt P rt

I is the interest

A is the end amount: principal plus interest

P is the principal (starting amount)

r is the interest rate in decimal form

t is time

The units of measurement (years, months, etc.) for the time should match the time period

for the interest rate.

APR – Annual Percentage Rate

Interest rates are usually given as an annual percentage rate (APR) – the total interest that

will be paid in the year. If the interest is paid in smaller time increments, the APR will be

divided up.

For example, a 6% APR paid monthly would be divided into twelve 0.5% payments.

A 4% annual rate paid quarterly would be divided into four 1% payments.

Example 3

Treasury Notes (T-notes) are bonds issued by the federal government to cover its expenses.

Suppose you obtain a $1,000 T-note with a 4% annual rate, paid semi-annually, with a

maturity in 4 years. How much interest will you earn?

Since interest is being paid semi-annually (twice a year), the 4% interest will be divided

into two 2% payments.

P = $1000 the principal

r = 0.02 2% rate per half-year

t = 8 4 years = 8 half-years

I = $1000(0.02)(8) = $160. You will earn $160 interest total over the four years.

6.1 Simple and Compound Interest 223

Try it Now

- A loan company charges $30 interest for a one month loan of $500. Find the annual

interest rate they are charging.

Compound Interest

With simple interest, we were assuming that we pocketed the interest when we received it.

In a standard bank account, any interest we earn is automatically added to our balance, and

we earn interest on that interest in future years. This reinvestment of interest is called

compounding. We looked at this situation earlier, in the chapter on exponential growth.

Compound Interest

kt

k

A P r

1

A is the balance in the account after t years.

P is the starting balance of the account (also called initial deposit, or principal)

r is the annual interest rate in decimal form

k is the number of compounding periods in one year.

If the compounding is done annually (once a year), k = 1.

If the compounding is done quarterly, k = 4.

If the compounding is done monthly, k = 12.

If the compounding is done daily, k = 365.

The most important thing to remember about using this formula is that it assumes that we put

money in the account once and let it sit there earning interest.

Example 4

A certificate of deposit (CD) is a savings instrument that many banks offer. It usually gives

a higher interest rate, but you cannot access your investment for a specified length of time.

Suppose you deposit $3000 in a CD paying 6% interest, compounded monthly. How much

will you have in the account after 20 years?

In this example,

P = $3000 the initial deposit

r = 0.06 6% annual rate

k = 12 12 months in 1 year

t = 20 since we’re looking for how much we’ll have after 20 years

So

20 12 3000 1 0.06 $9930.61

12

A

(round your answer to the nearest penny)

224 Chapter 6 Finance

Let us compare the amount of money earned from compounding against the amount you

would earn from simple interest

Years Simple Interest

($15 per month)

6% compounded

monthly = 0.5%

each month.

5 $3900 $4046.55

10 $4800 $5458.19

15 $5700 $7362.28

20 $6600 $9930.61

25 $7500 $13394.91

30 $8400 $18067.73

35 $9300 $24370.65

As you can see, over a long period of time, compounding makes a large difference in the

account balance. You may recognize this as the difference between linear growth and

exponential growth.

Evaluating exponents on the calculator

When we need to calculate something like 53 it is easy enough to just multiply 5·5·5=125.

But when we need to calculate something like 1.005240 , it would be very tedious to

calculate this by multiplying 1.005 by itself 240 times! So to make things easier, we can

harness the power of our calculators.

Most scientific calculators have a button for exponents. It is typically either labeled like:

^ , yx , or xy .

To evaluate 1.005240 we’d type 1.005 ^ 240, or 1.005 yx 240. Try it out – you should get

something around 3.3102044758.

Example 5

You know that you will need $40,000 for your child’s education in 18 years. If your

account earns 4% compounded quarterly, how much would you need to deposit now to

reach your goal?

In this example,

We’re looking for P.

r = 0.04 4%

k = 4 4 quarters in 1 year

t = 18 Since we know the balance in 18 years

A = $40,000 The amount we have in 18 years

In this case, we’re going to have to set up the equation, and solve for P.

0

5000

10000

15000

20000

25000

0 5 10 15 20 25 30 35

Years

Account Balance ($)

6.1 Simple and Compound Interest 225

18 4 40000 1 0.04

4

40000 (2.0471)

40000 $19539.84

2.0471

P

P

P

So you would need to deposit $19,539.84 now to have $40,000 in 18 years.

Rounding

It is important to be very careful about rounding when calculating things with exponents.

In general, you want to keep as many decimals during calculations as you can. Be sure to

keep at least 3 significant digits (numbers after any leading zeros). Rounding 0.00012345

to 0.000123 will usually give you a “close enough” answer, but keeping more digits is

always better. If your calculator allows it, do all your calculations without rounding in the

calculator and only round the final answer.

Example 6

To see why not over-rounding is so important, suppose you were investing $1000 at 5%

interest compounded monthly for 30 years.

P = $1000 the initial deposit

r = 0.05 5%

k = 12 12 months in 1 year

t = 30 since we’re looking for the amount after 30 years

If we first compute r/k, we find 0.05/12 = 0.00416666666667

Here is the effect of rounding this to different values:

If you’re working in a bank, of course you wouldn’t round at all. For our purposes, the

answer we got by rounding to 0.00417, three significant digits, is close enough – $5 off of

$4500 isn’t too bad. Certainly keeping that fourth decimal place wouldn’t have hurt.

r/k rounded to:

Gives Ato be: Error

0.004 $4208.59 $259.15

0.0042 $4521.45 $53.71

0.00417 $4473.09 $5.35

0.004167 $4468.28 $0.54

0.0041667 $4467.80 $0.06

no rounding $4467.74

226 Chapter 6 Finance

Using your calculator

In many cases, you can avoid rounding completely by how you enter things in your

calculator. For example, in the example above, we needed to calculate

12 30

12

1000 1 0.05

A

We can quickly calculate 12×30 = 360, giving

360

12

1000 1 0.05

A .

Now we can use the calculator.

The previous steps were assuming you have a “one operation at a time” calculator; a more

advanced calculator will often allow you to type in the entire expression to be evaluated. If

you have a calculator like this, you will probably just need to enter:

1000 × ( 1 + 0.05 ÷ 12 ) yx 360 = .

Or

1000 × ( 1 + 0.05 ÷ 12 ) ^ 360 = .

Try it Now - If $70,000 are invested at 7% compounded monthly for 25 years, find the end balance.

Because of compounding throughout the year, with compound interest the actual increase in

a year is more than the annual percentage rate. If $1,000 were invested at 10%, the table

below shows the value after 1 year at different compounding frequencies:

Frequency Value after 1 year

Annually $1100

Semiannually $1102.50

Quarterly $1103.81

Monthly $1104.71

Daily $1105.16

Type this Calculator shows

0.05 ÷ 12 = . 0.00416666666667

- 1 = . 1.00416666666667

yx 360 = . 4.46774431400613

× 1000 = . 4467.74431400613

6.1 Simple and Compound Interest 227

If we were to compute the actual percentage increase for the daily compounding, there was

an increase of $105.16 from an original amount of $1,000, for a percentage increase of

0.10516

1000

105.16 = 10.516% increase. This quantity is called the annual percentage yield

(APY).

Notice that given any starting amount, the amount after 1 year would be

k

k

A P r

1 . To find the total change, we would subtract the original amount, then to find

the percentage change we would divide that by the original amount:

1 1

1

k

k

k

r

P

P

k

P r

Annual Percentage Yield

The annual percentage yield is the actual percent a quantity increases in one year. It can

be calculated as

1 1

k

k

APY r

Notice this is equivalent to finding the value of $1 after 1 year, and subtracting the original

dollar.

Example 7

Bank A offers an account paying 1.2% compounded quarterly. Bank B offers an account

paying 1.1% compounded monthly. Which is offering a better rate?

We can compare these rates using the annual percentage yield – the actual percent increase

in a year.

Bank A: 1 0.012054

4

1 0.012

4

APY = 1.2054%

Bank B: 1 0.011056

12

1 0.011

12

APY

#

1.1056%

Bank B’s monthly compounding is not enough to catch up with Bank A’s better APR.

Bank A offers a better rate.

228 Chapter 6 Finance

Example 8

If you invest $2000 at 6% compounded monthly, how long will it take the account to

double in value?

This is a compound interest problem, since we are depositing money once and allowing it to

grow. In this problem,

P = $2000 the initial deposit

r = 0.06 6% annual rate

k = 12 12 months in 1 year

So our general equation is

12 2000 1 0.06

12

t

A

. We also know that we want our ending

amount to be double of $2000, which is $4000, so we’re looking for t so that A = 4000. To

solve this, we set our equation for A equal to 4000.

12 4000 2000 1 0.06

12

t

Divide both sides by 2000

12 2 1.005 t To solve for the exponent, take the log of both sides

12 log 2 log 1.005 t Use the exponent property of logs on the right side

log 2 12t log 1.005 Now we can divide both sides by 12log(1.005)

log 2

12log 1.005

t Approximating this to a decimal

t = 11.581

It will take about 11.581 years for the account to double in value. Note that your answer

may come out slightly differently if you had evaluated the logs to decimals and rounded

during your calculations, but your answer should be close. For example if you rounded

log(2) to 0.301 and log(1.005) to 0.00217, then your final answer would have been about

11.577 years.

Important Topics of this Section

APR

Simple interest

Compound interest

Compounding frequency

APY

Evaluating on a calculator

6.1 Simple and Compound Interest 229

Try it Now Answers

1.

I = $30 of interest

P = $500 principal

r = unknown

t = 1 month

Using I = Prt, we get 30 = 500·r·1. Solving, we get r = 0.06, or 6%. Since the time was

monthly, this is the monthly interest. The annual rate would be 12 times this: 72% interest.

2.

12(25) 70,000 1 0.07 400,779.27

12

A

6.2 Annuities 230

This chapter is part of Business Precalculus © David Lippman 2016.

This content is remixed from Math in Society © Lippman 2013.

This material is licensed under a Creative Commons CC-BY-SA license.

Section 6.2: Annuities

For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we

save for the future by depositing a smaller amount of money from each paycheck into the

bank. This idea is called a savings annuity. Most retirement plans like 401k plans or IRA

plans are examples of savings annuities.

Suppose we will deposit $100 each month into an account paying 6% interest. How much

will we have after a year? We assume that the account is compounded with the same

frequency as we make deposits unless stated otherwise. In this example:

r = 0.06 (6%)

k = 12 (12 compounds/deposits per year)

d = $100 (our deposit per month)

t = 1 year

With ordinary annuities we assume the payment is made at the end of the period. The $100

we deposit at the end of the first month will earn interest for 11 months and at the end of the

year will be worth

11

100 1 0.06 100(1.005)11

12

A

.

The $100 deposited at the end of the second month will have 10 months to grow, and will be

worth A 100(1.005)10 at the end of the year. This pattern continues down to the last

deposit, which has no time to compound, and will be worth A = 100.

In total, we will have accumulated:

A 100(1.005)11 100(1.005)10 100(1.005)2 100(1.005)1 100

This equation leaves a lot to be desired, though – it doesn’t make calculating the ending

balance any easier! To simplify things, multiply both sides of the equation by 1.005:

1.005A 1.005100(1.005)11 100(1.005)10 100(1.005)2 100(1.005) 100

Distributing on the right side of the equation gives

1.005A 100(1.005)12 100(1.005)11 100(1.005)3 100(1.005)2 100(1.005)

Now we’ll line this up with like terms from our original equation, and subtract each side

12 11

11

1.005 100 1.005 100 1.005 100(1.005)

100 1.005 100(1.005) 100

A

A

Almost all the terms cancel on the right hand side when we subtract, leaving

12 1.005A A 100 1.005 100

Now we solve this equation for A.

6.2 Annuities 231

12

12

0.005 100 1.005 1

100 1.005 1

0.005

A

A

Recall 0.005 was r/k, 100 was the deposit d, and 12 was the number of months, kt.

Generalizing this result, we get the saving annuity formula.

Annuity Formula

k

r

k

d r

A

kt

1 1

A is the balance in the account after t years.

d is the regular deposit (the amount you deposit each year, each month, etc.)

r is the annual interest rate in decimal form.

k is the number of compounding periods in one year.

If the compounding frequency is not explicitly stated, assume there are the same number of

compounds in a year as there are deposits made in a year.

For example, if the compounding frequency isn’t stated:

If you make your deposits every month, use monthly compounding, k = 12.

If you make your deposits every year, use yearly compounding, k = 1.

If you make your deposits every quarter, use quarterly compounding, k = 4.

Etc.

When do you use this

Annuities assume that you put money in the account on a regular schedule (every month,

year, quarter, etc.) and let it sit there earning interest.

Compound interest assumes that you put money in the account once and let it sit there

earning interest.

Compound interest: One deposit

Annuity: Many deposits.

232 Chapter 6 Finance

Example 1

A traditional individual retirement account (IRA) is a special type of retirement account in

which the money you invest is exempt from income taxes until you withdraw it. If you

deposit $100 each month into an IRA earning 6% interest, how much will you have in the

account after 20 years?

In this example,

d = $100 the monthly deposit

r = 0.06 6% annual rate

k = 12 since we’re doing monthly deposits, we’ll compound monthly

t = 20 we want the amount after 20 years

Putting this into the equation:

20(12)

240

100 1 0.06 1

12

0.06

12

100 1.005 1

0.005

100 3.310 1

0.005

100 2.310

$46200

0.005

A

A

A

A

The account will grow to $46,200 after 20 years.

Notice that you deposited into the account a total of $24,000 ($100 a month for 240

months). The difference between what you end up with and how much you put in is the

interest earned. In this case it is $46,200 – $24,000 = $22,200.

Example 2

You want to have $200,000 in your account when you retire in 30 years. Your retirement

account earns 8% interest. How much do you need to deposit each month to meet your

retirement goal?

In this example,

We’re looking for d.

r = 0.08 8% annual rate

k = 12 since we’re depositing monthly

t = 30 30 years

A = $200,000 The amount we want to have in 30 years

6.2 Annuities 233

In this case, we’re going to have to set up the equation, and solve for d.

30(12)

360

1 0.08 1

12

200,000

0.08

12

1.00667 1

200,000

0.00667

200,000 (1491.57)

200,000 $134.09

1491.57

d

d

d

d

So you would need to deposit $134.09 each month to have $200,000 in 30 years if your

account earns 8% interest

Try it Now

- A more conservative investment account pays 3% interest. If you deposit $5 a day into

this account, how much will you have after 10 years? How much is from interest?

Example 3

If you invest $100 each month into an account earning 3% compounded monthly, how long

will it take the account to grow to $10,000?

This is a savings annuity problem since we are making regular deposits into the account.

d = $100 the monthly deposit

r = 0.03 3% annual rate

k = 12 since we’re doing monthly deposits, we’ll compound monthly

We don’t know t, but we want A to be $10,000.

Putting this into the equation:

12 100 1 0.03 1

12

10,000

0.03

12

t

Simplifying the fractions a bit

12 100 1.0025 1

10,000

0.0025

t

234 Chapter 6 Finance

We want to isolate the exponential term, 1.002512t, so multiply both sides by 0.0025

12 25 100 1.0025 1 t Divide both sides by 100

12 0.25 1.0025 1 t Add 1 to both sides

12 1.25 1.0025 t Now take the log of both sides

12 log 1.25 log 1.0025 t Use the exponent property of logs

log 1.25 12t log 1.0025 Divide by 12log(1.0025)

log 1.25

12log 1.0025

t Approximating to a decimal

t = 7.447 years

It will take about 7.447 years to grow the account to $10,000.

Important Topics of this Section

Finding the future value of an annuity

Finding deposits needed to fund an annuity

Try it Now Answers

1.

d = $5 the daily deposit

r = 0.03 3% annual rate

k = 365 since we’re doing daily deposits, we’ll compound daily

t = 10 we want the amount after 10 years

365

0.03

1

365

5 1 0.03

365 10

A $21,282.07

We would have deposited a total of $5·365·10 = $18,250, so $3,032.07 is from interest