MAT 110, Chapter 4

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Math for Liberal ArtsMAT 110: Chapter 4 NotesManaging MoneyDavid J. Gisch

Taking Control of Your Finances

Controlling Your Finances• Know your bank balance.▫ Never bounce a check or have a debit card rejected.

• Know what you spend.▫ Keep track of debit and credit card spending.

• Don’t buy on impulse.▫ Think first; buy only if the purchase makes sense.

• Make a budget, and don’t overspend it.

▫ Many people do not do this. Number one mistake!!!!!

A Four-Step budget1. List all monthly income, including a prorated

amount for any income not received monthly (such as once-a-year payments).

2. List all monthly expenses, including a prorated amount for expenses that don’t recur monthly.

3. Subtract total expenses from total income to determine your net monthly cash flow.

4. Make adjustments as needed.

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Net Cash FlowNet cash flow is the amount you have left at the end of a period.

Computing your net cash flow is vital when budgeting your money!• If you have a positive net cash flow, then great. Maybe

consider saving or investing more.• If your net cash flow is negative you need to cut expenses

or try to increase your income, the latter of which is usually hard.

For most people, the most difficult part is making sure you do not leave anything out.

Prorating• Prorating just means that you turn all amounts into a

certain unit of measure. For budgeting, that is often the amount per month.

Example: Dave orders a Venti-Iced-No Whip-Peppermint-Mocha and a reduced fat* turkey-bacon sandwich at Starbucks about once a week. It is his Friday treat after a hard week of disseminating knowledge to America's future. About how much does he spend a year if the cost is $8.50 for the two items?

*The no-whip and reduced fat turkey bacon makes it ok, right?

College ExpensesExample: You pay $3500 for tuition, $750 in student fees, and $500 for textbooks each semester. How should you handle these expenses in computing your monthly budget? Assume you attend 2 semesters a year.

Net Cash FlowExample : The following shows income and expenses for an individual. Find the monthly net cash flow. Assume income is after taxes and that 52 1 4 1 .

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• Find a way to make your budget allow for savings;understand how savings work and how to chooseappropriate savings plans. (We learn about savings in Section 4C)

• Understand the basic mathematics of loans. (We do this in section 4D)

• Understand how taxes are computed and howthey can affect your financial decisions. (This is in section 4E, which we are skipping)

• Understand how the federal budget affects futurepersonal finances. (I listen to NPR a lot to stay informed on all U.S. and world issues)

Base Financial Goals on Solid Understanding

The Power of Compounding

I have the power!

If compounding interest were a cartoon character.

Definitions & Formulas• The principal in financial formulas is the balance upon

which interest is paid.

• Simple interest is interest paid only on the original principal, and not on any interest added at later dates.

• Compound interest is interest paid on both the original principal and on all interest that has been added to the original principal.

Simple Interest

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A = Amount after t yearsI = Interest EarnedP = Principal or Initial Amountr = rate of interest (percent as a

decimal)t = time in years

Definitions & Formulas

• Note that compound interest can accrue multiple times a year ( value).▫ Annually, 1▫ Semi-Annually, 2▫ Quarterly, 4▫ Monthly, 12▫ Daily, 365.

Compound Interest

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A = Amount after t yearsP = Principal or Initial Amountr = rate of interest (percent as a decimal)n = number of times compounded per year.

t = time in years

These compound interest formulas should be

thought of as Lump-Sum (one-time payment)

scenarios.

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Simple and Compound InterestExample : Compare the growth in a $100 investment for 5 years at 10% simple interest per year and at 10% interest compounded annually.

Simple and Compound Interest

Simple Interest* Compound Interest*End

of Year

Interest Paid Old Balance + New Balance

1 1000 .10 1 100 1000 100 1100

2 1000 .10 1 100 1100 100 1200

3 1000 .10 1 100 1200 100 1300

4 1000 .10 1 100 1300 100 1400

5 1000 .10 1 100 1400 100 1500

End of

Year

Interest Paid Old Balance + New Balance

1 1000 .10 1 100 1000 100 1100

2 1100 .10 1 110 1100 110 1210

3 1210 .10 1 121 1210 121 1331

4 1331 .10 1 133.10 1331 133.101464.10

5 1464.10 .10 1146.41

1464.10 146.411610.51

Example : Compare the growth in a $1000 investment for 5 years at 10% simple interest per year and at 10% interest compounded annually.

*Remember, simple interest is only on the original amount, being $100.

*Since it is compounding, the interest is calculated using the newest balance.

You ended with $110.51 more! It would be even greater if it was compounded multiple times a year.

Simple and Compound InterestExample : You invest $500 in an account with an APR of 5%.

(a) Calculate the future amount after 8 years using simple interest.

(b) Calculate the future amount after 8 years using quarterly compounding interest.

1 500 1 .05 8

1 500 1.054

∗

Simple and Compound InterestExample : You invest $200 in an account with an APR of 8% when you are 20.

(a) Calculate the future amount when you are 65 using simple interest.

(b) Calculate the future amount when you are 65 using monthly compounding interest.

1

200 1 .08 45

1 200 1.0812

∗

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Annual Percentage Yield• APY is not the actual rate. It is the simple interest rate

you would need to achieve the same amount due to compounding interest over a year.▫ WHAT????▫ Remember earlier, we had the $1000 over 5 years comparison. Compound Interest: $110 for a year. So to achieve $110 with simple interest we would have needed a

rate of 11%.• Basically, APY allows you to compare loans with

different terms.▫ Which is better? A loan with 5% interest compounded monthly, or A loan with 6% interest compounded semi-annually

• Note: APY>APR

APY=5.12%

APY=6.09%

US Bank Example

Savings Plans and Investments

Common Savings Plans• Bank Savings Plans, CD’s, Bonds▫ Guaranteed but extremely low interest rate.

• IRA, 401(k)▫ Take money out of your check which lowers your taxable income

for that current year.▫ It is taxed when you withdraw it.▫ Companies often have matching plans (e.g. match up to 3% of

your salary). ▫ Limits on amount you can contribute in a year.

• ROTH IRA▫ After your current income is taxed you then invest it.▫ It is not taxed when you withdraw it at retirement.▫ Limits on amount you can contribute in a year.

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Mutual Fund• Both types of IRA’s pool your money in with everyone

else who invest in the “general fund” and it is used to buy several stocks. Therefore it is less risky than buying individual stocks on your own.

Stocks• Very risky as your money is all in on one company.• You make money by selling your stock for more than you

purchased it or when the company pays out dividends to its stock holders.▫ When a corporation earns a profit or surplus, that money can be

put to two uses: it can either be re-invested in the business (called retained earnings), or it can be distributed to shareholders (called a dividend)

• A capital gains tax (CGT) is a tax on capital gains, profit from selling stocks, bonds, or from dividends.

What’s a Dividend What is an Annuity• You purchase this “investment” from a company and the

company pays out a stream of payments to the you at a later point in time. ▫ Annuities are primarily used as a means of securing a steady

(think guaranteed) cash flow for an individual during their retirement years.

• SPDIA▫ Single Premium Deferred Income Annuity.▫ Say you buy an annuity for $100,000 now and when you retire in

10 years you start getting $400 a month until you die.• SPIA▫ Single Premium Income Annuity▫ Say you buy an annuity for $100,000 at retirement and

immediately you start getting $300 a month until you die.

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Why Annuities?• It is Guaranteed.• Lots of options.▫ Life▫ Cash refund.▫ Period certain▫ Etc

Annuities (Savings Plan Formula)Annuities (Savings Plan Formula)

APMT 1 r

n 1rn

PMTA r

n

1 rn 1

A = Amount after t years.PMT = Payments Amountr = rate of interest (percent as a

decimal)n = number of times compounded

per year.t = time in years

• An annuity is any series of equal, regular payments.• The future value, A, of an annuity is the accumulated

amount at some future date.

AnnuitiesExample : You decide to dedicate some money every month into an annuity. You find you can manage $100 a month on which you earn, on average, 8% interest compounded monthly. If you started when you were 20, how much would you have at age 60?

APMT 1 r

n 1rn

100 1 .0812

∗1

.0812

$349,100.78

*If you don’t type it in as one “chunk” carry out your decimal at least 6-8 places!!!!

2327.3386.0067 $347,363.97

2327.338554.006667 $349,083.33

2327.338554.00666667 $349,100.61

Advantages of Annuities (from )• Unlike other tax-deferred retirement accounts such as 401(k)s

and IRAs, there is no annual contribution limit for an annuity. That allows you to put away more money for retirement, and is particularly useful for those that are closest to retirement age and need to catch up.

• All the money you invest compounds year after year without any tax bill from Uncle Sam. That ability to keep every dollar invested working for you can be a big advantage over taxable investments.

• When you cash out, you can choose to take a lump-sum payment from your annuity, but many retirees prefer to set up guaranteed payments for a specific length of time or the rest of your life, providing a steady stream of income.

• The annuity serves as a complement to other retirement income sources, such as Social Security and pension plans.

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SavingsExample : You have a child and want to save $100,000 for their college expenses. If you will invest for 17 years at an average rate of 6.5%, how much should your monthly payments be?

PMTA r

n

1 rn 1

PMT100,000 .065

12

1 .06512

∗1

$269.45

Total and Annual ReturnThe total return is the relative change in the investment.

The annual return is the annual percentage yield (APY) that would give the same overall growth.

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Stocks/InvestmentsExample : Suppose that you decided to invest in some real estate property in the year 2004. The amount of your original investment is $27,500. In the year 2013 you decide to sell and receive $43,400 for the property.What is your total return?

Stocks/InvestmentsExample : You buy 12 shares of Google stock at $540 a share. You sell it 3 years later at $574 a share. What is your total and annual return?

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Financial Planning Tips• Handout• The contents of this handout are considered class

reading and the contents are therefore eligible for test questions.

Loan Payments, Credit Cards, and Mortgages

Credit Score• Your credit score greatly effects your interest rate on

loans. (↓ ↑

Loan Basics• The principal is the amount of money owed at any

particular time.

• An installment loan (or amortized loan) is a loan that is paid off with equal regular payments.

• An amortization schedule is a table of principal and interest payments over the life of a loan.

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Understanding Amortization• You make the same

payment each month.• The amount of interest

you pay that month is calculated using simple interest▫ , here is

because time is measured in years for simple interest but it is the interest for 1 month.

▫ P is the current balance of the loan.

▫ After interest is taken out of your payment, whatever is left goes toward your balance.

For example.Say you owe $5000 on a loan charging 5% interest and your monthly payment is $125

5000 .05 112

$20.83

$125 $20.83 $104.17

$5000 $104.17 $4895.83

Amortization Table

Example : What is your balance after Month 4?

Note: They use 0.0075 as it is equal to .09 ⁄ 0.0075.

Calculating Loan Payments

• This formula works for any type of installment loan where you pay off what you borrowed with equal payments over time.

• On www.gischsource.com I have a link to an online amortization calculator.

• You can also get amortization calculators for free as a cell phone app.

Amortization (Loan Payment Formula)

1 1

Credit CardsCredit cards differ from installment loans in that you are not required to pay off your balance in any set period of time.• A minimum monthly payment is required.• Monthly payment generally covers all the interest but

very little principal.• It takes a very long time to pay off a credit card loan if

only the minimum payments are made.

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Credit CardsExample : Suppose you have a balance of $3,200 on your credit card which has an APR of 23% (or 1.92% per month).

(a) If the minimum monthly payment is ⁄ of the previous months balance, what is your minimum payment amount?

(b) If you made no more charges, what would your payment be if you wanted to pay off your credit card in 2 years?

Loan PaymentsExample: Suppose you have $15,000 in student loans upon graduation. The interest rate is 5.25% and the term of the loan is 20 years.

(a) What are your monthly payments?

(b) How much will you pay over the lifetime of the loan?

(c) How much did you pay in interest?

1 1

15,000 .052512

1 1 .052512

∗ $101.08

Mortgages• The longer the loan, the more money in interest you pay.▫ For example, $175,000 for the house PLUS $163,000 in interest

on a 25 year loan!!!!

Mortgages• A home mortgage is an installment loan designed

specifically to finance a home.▫ The mortgage is the amount AFTER your down payment.

• The down payment is the amount of money you must pay up front in order to be given a mortgage or other loan.

• Closing costs are fees you must pay in order to be given the loan. These include direct costs, or fees charged as points, where each point is 1% of the loan amount.▫ For example, if your mortgage is $100,000 and the closing cost is

3 points, it is $100,000 .03 $3,000

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MortgagesExample: You want to buy a house and your mortgage would be $125,000 at a rate of 4.75% for 30 years. Your closing costs are $1200 plus 2 points.

(a) What are your monthly payments?

(b) How much did you pay at closing?

(c) How much will you pay over the lifetime of the loan?

(d) How much did you pay in interest and fees?

1 1

125,000 .047512

1 1 .047512

∗ $652.06

A Closer Look at Amortization• The bank gets most of their money in interest at the

beginning of the loan.• In this example the payment is $734 over 30 years at 8%

interest.

A Closer Look at Amortization• The same $734 payment show in an amortization table.

Taxes

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Taxes• You will not have homework nor be tested over taxes.• We are taxed using what is called a progressive tax

bracket. It is NOT a flat percentage.

Tax Bracket Married Filing Jointly Single

10% Bracket $0 – $17,400 $0 – $8,700

15% Bracket $17,400 – $70,700 $8,700 – $35,350

25% Bracket $70,700 – $142,700 $35,350 – $85,650

28% Bracket $142,700 – $217,450 $85,650 – $178,650

33% Bracket $217,450 – $388,350 $178,650 – $388,350

35% Bracket Over $388,350 Over $388,350

$75,000

Taxes• Lets say you (and your spouse together) make $75,000.

People often would say, “Oh, I pay 25% in taxes.” This not TRUE!!!!!▫ You get an exemption (approximately $5000 for each family

member) subtracted from your income. So let’s say this is for a husband and wife with 1 child, that would give us

$75,000 3 5,000 $60,000▫ Then you subtract any legal deductions determined by the IRS tax

code. Deductions can include: Interest paid on home mortgage, interest paid on student loans, contributions to 401(k), medical deferment account, work related expenses, etc. You subtract this or a minimum of $12,000 (for a couple), whichever is higher. So now we have

$60,000 12,000 $48,000orless

Taxes• So you actually only pay taxes on $48,000, or possibly less if you

had a lot of deductions.▫ Looking back at the chart you then assume you pay 15% of

$48,000, but you don’t.▫ You pay 10% on $17,400. After you take out $17,400 you get

$48,000 $17,400 $30,600▫ You then pay 15% on the remaining $30,600.▫ So you pay the following in taxes.

17,400 .10 30,600 .15 $6,330• This is effectively paying 8.44% of your income in taxes. That is a

huge difference in paying 25% as originally claimed.633075000 0.0844 8.44%

Tax Bracket Married Filing Jointly Single

10% Bracket $0 – $17,400 $0 – $8,700

15% Bracket $17,400 – $70,700 $8,700 – $35,350

Taxes

• Or think of it this way.

$48,000

$17,400

$30,600

You pay 10% on this portion.

$1,740

You pay 15% on this portion.

$4,590

Together that is $6,330.

Amount left of your $75,000 after

exemptions and deductions.

$6,330 is only 8.84% of $75,000.

Tax Bracket Married Filing Jointly Single

10% Bracket $0 – $17,400 $0 – $8,700

15% Bracket $17,400 – $70,700 $8,700 – $35,350

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Business and Investments• Business get taxed too but it is a little different. However,

they have tons of deduction possibilities. This allowed Bank of America to pay $0.00 in taxes in 2010.

• Also, income made on investments such as stocks, some retirement accounts, etc are taxed with a flat 15% tax rate known as the capital gains tax.

• Some states have taxes, like Iowa, but whatever you pay in state tax can be deducted from your federal taxes.