Toward Quantitative Literacy: Interesting Problems in Finance

Toward Quantitative Literacy: Interesting Problems in FinanceJim Ham

http://www.delta.edu/jaham 2008 AMATYC Conference

Washington, D.C.Saturday, November 22, 2008

•Pick up a graphing calculator.

•Complete page 1 of the handout by filling in the blanks.




1.1. 51 %51 % of workers age 55 and up have saved less than $50,000 for of workers age 55 and up have saved less than $50,000 for retirement (not including the value of a primary residence).retirement (not including the value of a primary residence).

2.2. The average household has a net worth of just The average household has a net worth of just $264,000$264,000 at retirement, at retirement, not including home equity.not including home equity.

3.3. The savings rate for all of 2006 was The savings rate for all of 2006 was -1%-1%..4.4. 50%50% of all retirement plan participants who change jobs fail to roll over of all retirement plan participants who change jobs fail to roll over

their accounts.their accounts.5.5. People in the 18 to 24 age bracket spend nearly People in the 18 to 24 age bracket spend nearly 30%30% of their monthly of their monthly

income just on debt repayment.income just on debt repayment.6.6. Only about 1 in Only about 1 in 2020 American households owes $8,000 or more on credit American households owes $8,000 or more on credit

cards.cards.7.7. 55%55% of Americans owe nothing on their credit cards. of Americans owe nothing on their credit cards.8.8. More than More than 1%1% of all U.S. households were in some phase of the of all U.S. households were in some phase of the

foreclosure process last year.foreclosure process last year.9.9. More than More than 75%75% of undergraduates began the 2004 school year with of undergraduates began the 2004 school year with

credit cards.credit cards.10.10. 24%24% of undergraduates use their credit cards for tuition. The average of undergraduates use their credit cards for tuition. The average

student credit card balance is student credit card balance is $2,347$2,347..11.11. In 2005, students in a survey believed when they got older that they In 2005, students in a survey believed when they got older that they

would earn an average salary of would earn an average salary of $145,000$145,000. In 2005, adults with a . In 2005, adults with a bachelor's degree earned an average of bachelor's degree earned an average of $54,689$54,689..

12.12. 26%26% of teens and young adults say their parents taught them how to of teens and young adults say their parents taught them how to manage money.manage money.

13.13. 15%15% of high school graduates nationally have taken a course covering of high school graduates nationally have taken a course covering personal finance content.personal finance content.

14.14. The average 30-year fixed mortgage rate is approximately The average 30-year fixed mortgage rate is approximately 6%6%..15.15. Subprime mortgages charge an interest rate lower than the prime Subprime mortgages charge an interest rate lower than the prime

lending rate in the first couple of years, but the rate goes up rather lending rate in the first couple of years, but the rate goes up rather dramatically after a period of two to three years. dramatically after a period of two to three years. FalseFalse..

16.16. Each time you open a store credit card, Each time you open a store credit card, 2020 points are taken off of your points are taken off of your credit score.credit score.

17.17. In the first quarter of 2008, home values decreased by In the first quarter of 2008, home values decreased by 7.7%7.7%..

•51 % of workers age 55 and up have saved less than $50,000 for retirement (not including the value of a primary residence).

•The average household has a net worth of just $264,000 at retirement, not including home equity.

•The savings rate for all of 2006 was -1%.

•Only 11% of workers under 35 years of age indicate they are participating in their company's 401(k). (American Institute of Certified Public Accountants) 50% of all retirement plan participants who change jobs fail to roll over their accounts.

•62% of college graduates will have a student loan debt averaging $27,236 (Student Monitor)

•People in the 18 to 24 age bracket spend nearly 30% of their monthly income just on debt repayment.

•The average household credit card debt is about $8000.

•Only about 1 in 20 American households owes $8,000 or more on credit cards.

•55% of Americans owe nothing on their credit cards.

•26% of teens and young adults say their parents taught them how to manage money.

•In most cases, economics and personal financial literacy programs are elective classes so “only 15% of Americans graduate from high school having learned anything about money at all.”

•Research shows that individuals that have taken personal finance education course have a higher savings rate, higher net worth and make larger contributions to their 401k. (The Department of the Treasury)

Students do not take courses in Students do not take courses in personal financepersonal finance Saving & InvestingSaving & Investing Debt ManagementDebt Management BudgetingBudgeting

Here’s where we come in ….Here’s where we come in …. Integrate financial applications in our Integrate financial applications in our

classesclasses

The problems in your handoutThe problems in your handout A 3-week unit in “Finite Mathematics”A 3-week unit in “Finite Mathematics” Proposed for a Liberal Arts Math Proposed for a Liberal Arts Math

CourseCourse Individual problems used in lower level Individual problems used in lower level

coursescourses Many, but not all, use the TVM Solver Many, but not all, use the TVM Solver

of the graphing calculator.of the graphing calculator.

The Compound Interest Formula The Compound Interest Formula

What mathematical skills do What mathematical skills do students employ when using this students employ when using this formula to solve problems?formula to solve problems?

mt

m

rPA

1

The TVM (Time Value of Money) SolverThe TVM (Time Value of Money) Solver

NN = = Number of Payment Periods (Number of Payment Periods (N = mt)N = mt)

I%I% = = Annual Interest Rate Annual Interest Rate Do not convert the APR to a decimal. If APR = 9.5%, then I% = 9.5.Do not convert the APR to a decimal. If APR = 9.5%, then I% = 9.5.

PVPV = = Present ValuePresent ValueAmount of a loan or beginning lump sum investment.Amount of a loan or beginning lump sum investment.Enter Enter Cash InflowsCash Inflows as positive values and as positive values and Cash OutflowsCash Outflows as negative values. as negative values.

PMT =PMT = Periodic Payment (or Deposit)Periodic Payment (or Deposit)The payment is usually a The payment is usually a Cash Outflow, and hence, aCash Outflow, and hence, a negative value. negative value.

FVFV = = Future ValueFuture Value

P/YP/Y = = Payments Periods per YearPayments Periods per Year

C/Y =C/Y = Compounding Periods per YearCompounding Periods per YearC/Y is automatically set to match P/Y. If C/Y is different from P/Y, enter P/Y first, C/Y is automatically set to match P/Y. If C/Y is different from P/Y, enter P/Y first, then C/Y.then C/Y.

The TVM (Time Value of Money) SolverThe TVM (Time Value of Money) Solver

PMT: END BEGIN (When the regular payments are PMT: END BEGIN (When the regular payments are made: at the BEGINing of the period or at the END)made: at the BEGINing of the period or at the END)PMT: END is used for an ordinary annuity, where payments occur at the end of PMT: END is used for an ordinary annuity, where payments occur at the end of each payment period. Most loans are in this category. PMT: BEGIN is used for each payment period. Most loans are in this category. PMT: BEGIN is used for an annuity due, where payments occur at the beginning of each payment an annuity due, where payments occur at the beginning of each payment period. Most leases are in this category.period. Most leases are in this category.

To get to the TMV Solver on the TI-84, enter APPS, To get to the TMV Solver on the TI-84, enter APPS, Finance, TVM Solver.Finance, TVM Solver.

While the cursor is blinking on the value to be While the cursor is blinking on the value to be calculated, enter ALPHA ENTER (SOLVE).calculated, enter ALPHA ENTER (SOLVE).

Problem 1:

Lump Sum Investment: When Bud Uronner was born, his grandfather made an initial deposit of $3,000 into an account for his college education. Assuming an interest rate of 6% compounded quarterly, how much will the account be worth in 18 years?

mA (r = .06; P = 3000; t =

18)

1 $8,563.02

2 $8,694.83

4 $8,763.47

12 $8,810.30

52 $8,828.54

365 $8,833.25

rA (m = 4; P = 3000; t =

18)

.01 $3,590.85

.05 $7,337.76

.08 $12,483.42

.09 $14,889.50

.13 $30,005.83

.20 $100,635.40

Problem 2:

Rule of 72: Orson Buggy wants his $5,000 investment to double in 6 years. What annual interest rate must he earn? Assume interest is compounded annually.

t r r * t

2 41.4214 82.8427

3 25.9921 77.9763

4 18.9207 75.6828

6 12.2462 73.4772

8 9.0508 72.4062

9 8.0060 72.0538

12 5.9463 71.3557

18 3.9259 70.6666

24 2.9302 70.3254

36 1.9441 69.9863

Problem 4:

Future Value Annuity: How long will it take Dot Snice to accumulate $1,000,000 if she invests $3,000 per year at an annual interest rate of 8%? Assume interest is compounded annually.

mrmr

PMTFV

mt11

rt (m = 1; R = 3000; S =

1000000)

.01 147.37

.05 58.86

.08 43.14

.09 39.85

.13 31.02

.20 23.12

Problem 7:

Present Value Annuity – Monthly Payment: Megan Model borrows $25,000 at 7.53% compounded monthly. If she wishes to pay off the loan after 15 years, how much would the monthly payment be?

mrmr

PMTPV

mt11

tR (m = 12; r = .065; S =

135000)

10 $1,532.90

15 $1,175.99

16 $1,132.75

20 $1,006.52

25 $911.53

30 $853.29

Problem 6a: Invest Early and OftenMuch has been written about the importance of investing early and often. Two friends saved for retirement over a 40-year period in two different ways. Johnny on the Spot invested $4,000 per year at 8% annual interest for the first twenty years, then invested nothing over the last 20 years. During the last 20 years, his investments accumulated interest at 9% annual interest. Johnny Come Lately invested nothing for the first twenty years, but then invested $10,000 per year over the next 20 years at 9% annual interest.

• How much did Johnny on the Spot invest over the 40-year period?

• How much did Johnny Come Lately invest over the 40-year period?

• How much did Johnny on the Spot accumulate over the 40-year period?

• How much did Johnny Come Lately accumulate over the 40-year period?

• Who was the wiser investor and why?

Problem 6a:•How much did Johnny on the Spot invest over the 40-year period? $80,000•How much did Johnny Come Lately invest over the 40-year period? $200,000•How much did Johnny on the Spot accumulate over the 40-year period? $1,025,875.40; $183,047.86 after the first 20 years.

•How much did Johnny Come Lately accumulate over the 40-year period? $511,601.20•Who was the wiser investor and why? Johnny on the Spot. He invested less than half the money, yet earned about twice as much or a half million dollars more.

N= 20I%= 8PV= 0PMT= -4000FV= 183047.8572P/Y= 1C/Y= 1PMT: END BEGIN

N= 20I%= 9PV= -183047.86PMT= 0FV= 1025875.398P/Y= 1C/Y= 1PMT: END BEGIN

Problem 8fProblem 8f Suppose Bob and Barb bought their home Suppose Bob and Barb bought their home

10 years ago and made monthly payments 10 years ago and made monthly payments as scheduled. They plan to move in two as scheduled. They plan to move in two years. They could refinance for 7.25% years. They could refinance for 7.25% right now on a new 20-year mortgage, but right now on a new 20-year mortgage, but closing costs would be $1800. Should closing costs would be $1800. Should they refinance? Assume that they will roll they refinance? Assume that they will roll over the closing costs into the new over the closing costs into the new mortgage.mortgage.

Calculate Original Monthly Payment

Calculate the Present Value

Calculate the New Monthly Payment

N= 360I%= 8.5PV= 182300PMT= -1401.7292FV= 0P/Y= 12C/Y= 12PMT: END BEGIN

N= 240I%= 8.5PV= 161522.5251PMT= -1401.73FV= 0P/Y= 12C/Y= 12PMT: END BEGIN

N= 240I%= 7.25PV= 163322.53PMT= -1290.86FV= 0P/Y= 12C/Y= 12PMT: END BEGIN

Current Mortgage (8.5%)

New Mortgage (7.25%)

Present Value

$161,522.53 $163,322.53

Monthly Payment?

$1401.73 $1,290.86

Savings per month?

$110.87

Number of months to recoup the closing costs?

16.235 months

Yes, they should refinance. They will save $110.87 per month for about 7.8 months before they move.

Problem 8hProblem 8h

Bob and Barb Noxious took out an Bob and Barb Noxious took out an $182,300 loan at 8.5% interest for 30 $182,300 loan at 8.5% interest for 30 years for the purchase of a new house. years for the purchase of a new house. The loan requires monthly mortgage The loan requires monthly mortgage payments. If, on the original loan, they payments. If, on the original loan, they paid an additional $100 per month, how paid an additional $100 per month, how long would it take to pay off the loan? long would it take to pay off the loan?

N= 278.4190833I%= 8.5PV= 182300PMT= -1501.73FV= 0P/Y= 12C/Y= 12PMT: END BEGIN

Additional $100:

t = 278.4190833 months

t = 23.2 years

Years reduced = 6.8

Problem 8h:

Problem 10aProblem 10a

Paige is offered two options when Paige is offered two options when purchasing a new $17,000 car. Option 1 purchasing a new $17,000 car. Option 1 offers 6.75% financing for 4 years and $2500 offers 6.75% financing for 4 years and $2500 “cash back.” Option 2 offers 4.75% financing “cash back.” Option 2 offers 4.75% financing for 5 years with no cash back. The financing for 5 years with no cash back. The financing requires monthly payments. Find the requires monthly payments. Find the monthly payment for each financing option. monthly payment for each financing option. Assume that the cash back in Option 1 will Assume that the cash back in Option 1 will be used to reduce the amount of the original be used to reduce the amount of the original loan. If Paige’s goal is to pay the minimum loan. If Paige’s goal is to pay the minimum amount for financing over the life of the amount for financing over the life of the loan, which option should she choose? loan, which option should she choose? Explain why using specific numbers Explain why using specific numbers

Option 1: Interest = 48($345.54) – $14,500 = $2,085.92

Option 2: Interest = 60($318.87) – $17,000 = $2,132.20

Option 1 (the cash back option) is best since less interest is paid and the loan is paid off sooner.

Option 1 Option 2



Problem 11bProblem 11b

Repeat the calculations of 11a to determine Repeat the calculations of 11a to determine which car has the lowest cost to own, A Chevy which car has the lowest cost to own, A Chevy Cavalier or a Toyota Camry. The Chevy Cavalier Cavalier or a Toyota Camry. The Chevy Cavalier costs $21011 and has a residual of 26.3% after costs $21011 and has a residual of 26.3% after 3 years, and a Toyota Camry costs $29650 and 3 years, and a Toyota Camry costs $29650 and has a residual of 63% after 3 years? Assume an has a residual of 63% after 3 years? Assume an annual interest rate of 8%, and that you will sell annual interest rate of 8%, and that you will sell the vehicle at its residual value after three the vehicle at its residual value after three years years

Chevy Cavalier Toyota Camry

List Price $17,510 $29,650

8% 3-year loan pymt

$548.70 $929.12

Total Payments $19,753.19 $33,448.44

Residual 26.3% 63%

Residual Value $4,605.13 $18,679.50

Total Cost $15,148.06 $14,768.94

Cost per Month $420.78 $410.25

Problem 12a:Problem 12a:

Dell has advertised a Dimension E521 Dell has advertised a Dimension E521 computer for $1149 ($1218 after tax) or $35 computer for $1149 ($1218 after tax) or $35 per month. You are in need of a new computer per month. You are in need of a new computer and this model seems to satisfy all of your and this model seems to satisfy all of your needs. Suppose that you pay only the needs. Suppose that you pay only the minimum due of $35 (at 19.99% APR) each minimum due of $35 (at 19.99% APR) each month on your new computer.month on your new computer.

How long will it take you to pay off the How long will it take you to pay off the computer? How much will you have paid on computer? How much will you have paid on the $1218 balance when the computer is finally the $1218 balance when the computer is finally paid off? paid off?

N= 52.46684889

I%= 19.99PV= 1218PMT= -35FV= 0P/Y= 12C/Y= 12PMT: END

BEGIN

About 52.5 months or 4 years, 4.5 months; $1,837.50

Problem 13a:Problem 13a:

In a $131,250 subprime loan with a 3/27 In a $131,250 subprime loan with a 3/27 ARM, the initial interest rate is 8% and it ARM, the initial interest rate is 8% and it will remain 8% for three years; at the end will remain 8% for three years; at the end of the first three years it will increase to of the first three years it will increase to 12%. Complete the table below. For both 12%. Complete the table below. For both a regular and interest-only loan, by how a regular and interest-only loan, by how much will the monthly payment increase much will the monthly payment increase after the third year when the rate after the third year when the rate increases?increases?

Regular 3/27 ARM

Interest-only 3/27 ARM

Monthly Payments

(first 3 years)

$963.07 Varies: $875.00 down to $851.94

Interest Paid over 1st 3

years

$31,100.72 $31,100.72

Principal Paid over 1st 3

years

$3,569.80 $0

Present Value of Loan after 3

years

$127,680.20 $131,250.00

Monthly Payment after

3rd year

$1,329.72 $1,366.90

Increase in payment

$366.65 $403.83

Problem 5:

Earn 32% Rate of Return the Easy Way. Many employers offer a 401K or 403B plan (tax sheltered annuity or TSA) that allows employees to invest for retirement. The beauty of the plan is that employees who invest $15,000 in a year, will pay federal taxes on $15,000 less in income – a tremendous tax savings. If we assume that the tax saved equals the rate of return on an investment, calculate the return on investment for the two employees below .

Salary $80,000 $80,000

Investment in TSA

$15,000 $0

Taxable Income $65,000 $80,000

Fed Tax Paid $12,902 $17,102

State Tax Paid (4%)

$2,600 $3,200

Tax Savings: $4,800

Rate of return: 32%

Problem 9Problem 9

Suppose that you are going to finance the Suppose that you are going to finance the purchase of a new $21,000 car. There are purchase of a new $21,000 car. There are three financing options available to you: three financing options available to you: 1.9% financing for 3 years, 3.9% financing 1.9% financing for 3 years, 3.9% financing for 4 years, or 5.9% financing for 5 years. for 4 years, or 5.9% financing for 5 years. Compare the financing costs for each of the Compare the financing costs for each of the three loans. Which would be best for you three loans. Which would be best for you and why? and why?

$21,000 Car Loan

Loan Term 3 Years (1.9%)

4 Years (3.9%)

5 Years (5.9%)

Monthly Payments

$600.58 $473.22 $405.01

Total Number of Payments

36 48 60

Total payout during the term

$21,620.80

$22,714.61

$24,300.78

Cost to Finance - Interest

$620.80$1,714.6

1$3,300.7

8




Thank You!

Economy & Finance

Toward Quantitative Literacy: Interesting Problems in Finance