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Chapter 5: Finance
Most adults have to deal with the financial topics in this chapter regardless of their job or income. Understanding these topics helps us to make wise decisions in our private lives as well as in any business situations.
Section 5.1: Basic Budgeting
Budgeting is an important step in managing your money and spending habits. To create a budget you need to identify how much money you are spending. Some expenses to keep in mind when creating a budget are rent, car payment, fuel, auto insurance, utilities, groceries, cell phone, personal, gym membership, entertainment, gifts, dining out, medical expenses, etc.
There are several apps out there that can help you budget your money. Just a few examples are Mint, Manilla, and Check. These are all free apps that help you keep track of bills and your accounts. Your bank also keeps track of your spending and what categories each item falls under. Log into your bank account online and look for “Track Spending” or a similar item. Many banks give you a pie chart showing you how much you spent in each category in the last month. You can edit your categories, change the number of months, and sometimes even set a budget goal.
Table 5.1.1: Example of Budget in Excel
When you are creating a monthly budget, many experts say if you want to have control of your money, you should know where every dollar is going. In order to keep track of this, a written budget is essential. Below is one example of a budget in Excel. This was a free template from the “Life After College” blog. There are hundreds of free templates out there so you should find the template that suits you the best – or create your own Excel budget!
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(“Four-Step Budget Template,” n.d.)
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Example 5.2.1: Simple Interest—Using a Table
Sue borrows $2000 at 5% annual simple interest from her bank. How much does she owe after five years?
Table 5.2.1: Simple Interest Using a Table
Year Interest Earned Total Balance Owed 1 $2000.05 = $100 $2000 + $100 = $ 2 $2000.05 = $100 $2100 + $100 = $ 3 $2000.05 = $100 $2200 + $100 = $ 4 $2000.05 = $100 $2300 + $100 = $ 5 $2000*.05 = $100 $2400 + $100 = $ After 5 years, Sue owes $2500.
Example 5.2.2: Simple Interest—Using the Formula
Chad got a student loan for $10,000 at 8% annual simple interest. How much does he owe after one year? How much interest will he pay for that one year?
P = $10,000, r = 0.08, t = 1
F = 10000(1 + 0.08(1)) =$10,
Chad owes $10,800 after one year. He will pay $10800 - $10000 = $800 in interest.
Example 5.2.3: Simple Interest—Finding Time
Ben wants to buy a used car. He has $3000 but wants $3500 to spend. He invests his $3000 into an account earning 6% annual simple interest. How long will he need to leave his money in the account to accumulate the $3500 he wants?
F = $3500, P = $3000, r = 0.
3500 = 3000 1 ( +0.06 t )
(^3500 1) 0. 3000
= + t
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(^3500 1) 0. 3000
− = t
t
Figure 5.2.2: Calculation to Find t on a TI 83/84 Calculator
years
Ben would need to invest his $3000 for about 2.8 years until he would have $ to spend on a used car.
Note: As shown above, wait to round your answer until the very last step so you get the most accurate answer.
Section 5.3: Compound Interest
Most banks, loans, credit cards, etc. charge you compound interest, not simple interest. Compound interest is interest paid both on the original principal and on all interest that has been added to the original principal. Interest on a mortgage or auto loan is compounded monthly. Interest on a savings account can be compounded quarterly (four times a year). Interest on a credit card can be compounded weekly or daily!
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Example 5.3.2: Comparing Simple Interest versus Compound Interest
Let’s compare a savings plan that pays 6% simple interest versus another plan that pays 6% annual interest compounded quarterly. If we deposit $8,000 into each savings account, how much money will we have in each account after three years?
6% Simple Interest : P = $8,000, r = 0.06, t = 3
F = P ( 1 + rt )
F = 8000 1 ( + 0.06 3⋅ )
F = 9440
Thus, we have $9440.00 in the simple interest account after three years.
6% Interest Compounded Quarterly: P = $8,000, r = 0.06, t = 3, n=
r^ n t F P n
⋅ = ^ +
0.06^ 4 3
F
⋅ = ^ +
0.06^12
F = ^ +
Figure 5.3.3: Calculation for F for Example 5.3.
F = 9564.
So, we have $9564.95 in the compounded quarterly account after three years.
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With simple interest we earn $1440.00 on our investment, while with compound interest we earn $1564.95.
Example 5.3.3: Compound Interest—Compounded Monthly
In comparison with Example 5.3.2 consider another account with 6% interest compounded monthly. If we invest $8000 in this account, how much will there be in the account after three years?
P = $8,000, r = 0.06, t = 3, n = 12
r^ n t F P n
⋅ = ^ +
0.06^ 12 3 8000 1 12
F
⋅ = ^ +
0.06^36 8000 1 12
F = ^ +
Figure 5.3.4: Calculation for F for Example 5.3.
F = 9573.
Thus, we will have $9573.44 in the compounded monthly account after three years.
Interest compounded monthly earns you $9573.44 - $9564.95 = $8.49 more than interest compounded quarterly.
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Example 5.3.5: Continuous Compounding Interest
Isabel invested her inheritance of $100,000 into an account earning 5.7% interest compounded continuously for 20 years. What will her balance be after 20 years?
P = $100,000, r = 0.057, t = 20
F = 100000 e^ 0.057 20 ⋅
F = 100000 e^ 1.
Figure 5.3.6: Calculation for F for Example 5.3.
F = 312, 676.
Isabel’s balance will be $312,676.84 after 20 years.
Annual Percentage Yield (APY) : the actual percentage by which a balance increases in one year.
Example 5.3.6: Annual Percentage Yield (APY)
Find the Annual Percentage Yield for an investment account with a. 7.7% interest compounded monthly b. 7.7% interest compounded daily c. 7.7% interest compounded continuously.
To find APY, it is easiest to examine an investment of $1 for one year.
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a. P = $1, r = 0.077, t = 1, n = 12 0.077^ 12 1 1 1 1. 12
F
⋅ = ^ + =
The percentage the $1 was increased was 7.9776%. The APY is 7.9776%.
b. P = $1, r = 0.077, t = 1, n = 365 0.077^ 365 1 1 1 1. 365
F
⋅ = ^ + =
The percentage the $1 was increased was 8.0033%. The APY is 8.0033%.
c. P = $1, r = 0.077, t = 1 F = 1 e 0.0077 1 ⋅ =1.
The percentage the $1 was increased was 8.0042%. The APY is 8.0042%.
Section 5.4: Savings Plans
Sometimes it makes better financial sense to put small amounts of money away over time to purchase a large item instead of taking out a loan with a high interest rate. When looking at depositing money into a savings account on a periodic basis we need to use the savings plan formula.
Savings Plan Formula :
nt r F PMT n r n
+^ −
= ^ ^
where, F = Future value PMT = Periodic payment r = Annual percentage rate (APR) changed to a decimal t = Number of years n = Number of payments made per year
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nt r F PMT n r n
+^ −
= ^ ^
nt
nt nt
r r r F n^ PMT n n r r r n n n
+^ −
• ^ ^ = ^ ^ ^ • ^
+^ −^ +^ −
^ ^ ^
nt
r F n PMT r n
• ^ =
+^ −
^
nt
r PMT F n r n
= • ^
+^ −
^
Example 5.4.2: Savings Plan—Finding Payment
Joe wants to buy a pop-up trailer that costs $9,000. He wants to pay in cash so he wants to make monthly deposits into an account earning 3.2% APR. How much should his monthly payments be to save up the $9,000 in 3 years?
F = $9,000, r = .032, t = 3, n = 12
nt
r PMT F n r n
= • ^
+^ −
^
12 3
PMT ⋅
= • ^
+^ −
^
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Figure 5.4.2: Calculation for PMT for Example 5.4.
PMT = 238.
Joe has to make monthly payments of $238.52 for 3 years to save up the $9,000.
Example 5.4.3: Savings Plan—Finding Time
Sara has $300 a month she can deposit into an account earning 6.8% APR. How long will it take her to save up the $10,000 she needs?
F = $10,000, PMT = $300, r = 0.068, n = 12
Note: We will use the original savings plan formula which solves for the future value, F to solve this problem.
nt r F PMT n r n
+^ −
= ^ ^
0.068^12
^ t (^) + − (^) = ^ ^
12 1.005667 1 10, 000 300
^ t − = (^)
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First, she deposits $1200 quarterly at 7% for 10 years. PMT = $1200, r = 0.07, t = 10, n = 4
r^ nt F PMT n r n
+^ −
= ^ ^
0.07^ 4 10
F
= ^ ^
Figure 5.4.4: Calculation for F for Part One of Example 5.4.
F = 68, 680.
Second, she puts this lump sum plus $300 a month for 5 years at 9%. Think of the lump sum and the new monthly deposits as separate things. The lump sum just sits there earning interest so use the compound interest formula. The monthly payments are a new payment plan, so use the savings plan formula again.
Total = (lump sum + interest) + (new deposits + interest) 12 5 12 5
⋅ ⋅
+^ −
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Figure 5.4.5: Calculation for the Lump Sum Plus Interest
Figure 5.4.6: Calculation for the New Deposits Plus Interest
Total = 107,532.48 + 22, 627.24 =130,159.
She will have $130,159.72 when she reaches age 65.
Section 5.5: Loans
It is a good idea to try to save up money to buy large items or find 0% interest deals so you are not paying interest. However, this is not always possible, especially when buying a house or car. That is when it is important to understand how much interest you will be charged on your loan.
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Figure 5.5.1: Calculation of P for Example 5.5.
P = 1087.
The price Ed paid for the iPad was $1,087.83. That’s a lot more that $500!
Also, the total amount he paid over the course of the loan was $30 × 12 × 4 = $1440. Therefore, the total amount of interest he paid over the course of the loan was $1440 - $1087.83 = $352.17.
For some problems, you will have to find the payment instead of the present value. In that case, it is helpful to just solve the loan payment formula for PMT. Since PMT is multiplied by a fraction, to solve for PMT , you can just multiply both sides of the formula by that fraction. You should just think of the loan payment formula in two different forms, one solving for present value, P , and one solving for payment, PMT.
r^ nt P PMT n r n
− ^ +
= ^ ^
nt
nt nt
r r r P n^ PMT n n r r r n n n
−
− −
−^ +
• ^ ^ = ^ ^ ^ • ^
−^ +^ −^ +
^ ^ ^
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nt
r PMT P n r n
−
= • ^
−^ +
^
Example 5.5.2: Loan Formula—Finding Payment
Jack goes to a car dealer to buy a new car for $18,000 at 2% APR with a five-year loan. The dealer quotes him a monthly payment of $425. What should the monthly payment on this loan be?
P = $18,000, r = 0.02, n = 12, t = 5
nt
r PMT P n r n
−
= • ^
−^ +
^
12 5
PMT − ⋅
= • ^
−^ +
^
Figure 5.5.2: Calculation for PMT for Example 5.5.
PMT = 315.
Jack should have a monthly payment of $315.50, not $425.
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