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Using Recursion
to Explore Financial Matters


 
Part
1: Determining the Best Pay Plan with Paper and Pencil
(1) Resolve the following dilemma, using paper and pencil: Suppose you are able to do a certain job that few people can do. An employer wants to hire you to do this job 20 times, and offers you a choice of compensate plans. Plan 1 pays you $10,000 each time you do the job, and Plan 2 offers you 1 cent the first time you do the job, 2 cents the second time, 4 cents the third time, 8 cents the fourth time, and so on. How do you want to be paid? (2) Explain
how you resolved this dilemma. (3) Which
plan would you choose if you were hired to do the job 30 times? (4) Generate
algebraic expressions for the amount paid for the nth time doing
the job, and for the cumulative amount paid for doing the job n times. Part
2: Exploring the Payment Options with Lists (5) Use
your graphing calculator to generate five lists of data – one listing
the number of times the job was performed, two listing the amounts paid
for each time it was performed (one for each payment plan), and two listing
the cumulative amounts paid through each time the job was performed.
(6) Examine
your lists, and compare your observations with your results from Part
1. (7) On the same screen, generate graphs for the two cumulative payment lists. Comment on how these graphs help one explore the situation, and on how they illustrate the underlying mathematics. (See Task 11 below). (8) Compare the method used in Part 1 with that used in Part 2. Part
3: Exploring the Payment Options with Recursion (9)
Write two sets of recursive
equations to model the two pay plans. (Note: recursive equations can be
written for both the amount paid for the nth time the job was done, and
for the cumulative amount paid through the nth time the job was done.)
(10)
Enter your equations into your graphing calculator and generate tables
to help you explore this dilemma in the short run and long run. Examine
the values in these tables and comment on your observations. Discuss the
short and long run for this dilemma. (11)
Graph these table values, and comment on the shapes of the graphs. Compare
and contrast the tabular representation and the graphical representation
of this data. (12)
Discuss features of the mathematical functions inherent in this situation. (13)
Discuss real world lessons that can be learned from exploring this dilemma. (14)
Compare the list method used in Part
2 with the recursive method used in Part
3. Part 4: Adjusting the Consumer Price Index The
Consumer Price Index (CPI) is an index used to gauge inflation. Many economists
think the current CPI overestimates inflation by 1.1% because of a number
of biases, and many believe it should be adjusted downward to account
for these biases. But, adjusting the CPI downward is controversial because
it is used to determine cost of living increases for social security and
other expenditures (e.g. food stamps and school lunches). The newspapers
reported recently that the cost of living increase for social security
recipients would be 2.1 %. Also reported was that the average monthly
social security check this year is $749, and that the maximum check is
$1,326. (15)
Write a general recursive equation to determine the amount of money a
retiree would receive from Social Security next year, assuming a cost
of living increase of i% over this year’s amount. (16)
Use this equation to generate a recursive equation for the total amount of money a retiree would receive from Social Security
over the next n years, if each year there was a cost of living increase
of i% each year.
(17)
Use this equation and the data given above to generate recursive tables
on your calculator to determine the total amount of money a retiree receiving
the average payment today would receive over the next 20 years
assuming: (i) a constant 2.1% increase each year, and (ii) a CPI downwardadjusted
constant 1% increase each year by using the . (18)
Determine the total amount of money a retiree receiving the maximum
payment today would receive over the next 20 years assuming: (i) a constant
2.1% increase each year, and (ii) a CPI downwardadjusted constant 1%
increase each year. (19)
Estimate the total amount of money such an adjustment would save Social
Security over the next 20 years, given that there are approximately 45
million people receiving social security today. Part 5: Saving for Retirement with a 403b Plan (20)
The 403b regulation allows teachers and other employees of nonprofit
organizations save for retirement on a taxdeferred basis. Money contributed
to a 403b account through payroll deduction is contributed pretax and
is not taxed until money is withdrawn at retirement or at age 591/2. Use
recursion to calculate the amount of money a new teacher would accumulate
after 20 years of contribution into a 403b plan if the teacher contributes
$100 per month to a money market
account paying 6% per year, compounded monthly. (21)
Calculate the amount of money accumulated if the teacher can contribute
an additional $10 per month
each year.
Extension
1: Write a Program Write
a calculator program that can determine the total amount accumulated after
M months, with interest rate I, initial contribution A and increment B.
Extension
2: Create a Spreadsheet Create
a spreadsheet that calculates and displays the new contribution each month,
the interest received each month, and the total accumulation, for any
given monthly contribution amount, interest rate, and increment. Part 6: Using Recursion to Analyze Amortization (22)
Suppose that you took out a $100,000 mortgage at an annual interest rate
of 8%. The mortgage broker tells you that your monthly payment will be
$733.76 if you pay off the mortgage in 30 years, or 360 months. Derive
a recursive formula to determine your balance after each mortgage payment
in terms of the balance after the previous payment. Use this formula to
generate a table of values of your balance after each payment. Verify
that a payment of $733.76 per month will pay off the loan after 360 payments.
Generate a graph of the balances after each payment. (23)
Comment on the connections you see between the algebraic, numerical, and
graphical representations of this situation. (24)
Use the above recursive formula to determine how many months it would
take to pay off the mortgage if you made monthly payments of $600, $1000,
$1500. Generate graphs for the balances after each payment, and compare
the graphs and number of months to pay off the loan.
(25) Make several interpretive statements about the relationship between the term of mortgage and the payment amount. Extension:
Credit Card Debt
(from
Charles Lowery, Cedar Bridge Academy, Bermuda) Credit
cards usually have high interest rates. If your credit card has an interest
rate of 19.8% and you have a balance of $5000 and a minimum monthly payment
of $20, how long will it take you to pay off the loan? What if you paid $60 a month? What if you paid $82.50 per month?
Compare the number of months required to pay off the loan with payments
of $150 and $300. Discuss your results.


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