Finance

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

1. 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
       
 
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
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
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
2. 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
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
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 content is remixed from Math in Society © Lippman 2013.
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 1.005100(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

1. 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
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